From 09da392c5b21d413f3f991a5f93309f62dabc4c6 Mon Sep 17 00:00:00 2001 From: Quarto GHA Workflow Runner Date: Wed, 6 Mar 2024 15:50:11 +0000 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- 01-introduccion.html | 64 +- 02-flujo-basico-2.html | 42 +- 02-flujo-basico.html | 40 +- 03-modelos-genericos.html | 48 +- 05-dags.html | 156 ++- .../figure-html/unnamed-chunk-10-1.png | Bin 30669 -> 30670 bytes .../figure-html/unnamed-chunk-13-1.png | Bin 530766 -> 546561 bytes .../figure-html/unnamed-chunk-14-1.png | Bin 521893 -> 538538 bytes .../figure-html/unnamed-chunk-22-1.png | Bin 19905 -> 19737 bytes .../figure-html/unnamed-chunk-25-1.png | Bin 14740 -> 14472 bytes .../figure-html/unnamed-chunk-28-1.png | Bin 22820 -> 22761 bytes .../figure-html/unnamed-chunk-29-1.png | Bin 26754 -> 26827 bytes .../figure-html/unnamed-chunk-32-1.png | Bin 109738 -> 108607 bytes .../figure-html/unnamed-chunk-33-1.png | Bin 227705 -> 226885 bytes .../figure-html/unnamed-chunk-37-1.png | Bin 27514 -> 27376 bytes .../figure-html/unnamed-chunk-38-1.png | Bin 29033 -> 28884 bytes .../figure-html/unnamed-chunk-41-1.png | Bin 80036 -> 80843 bytes .../figure-html/unnamed-chunk-42-1.png | Bin 187429 -> 185495 bytes .../figure-html/unnamed-chunk-50-1.png | Bin 100433 -> 109683 bytes .../figure-html/unnamed-chunk-52-1.png | Bin 90156 -> 89141 bytes .../figure-html/unnamed-chunk-8-1.png | Bin 28929 -> 29254 bytes 06-calculo-do.html | 92 +- 07-buenos-malos-controles.html | 68 +- 08-mcmc.html | 1035 +++++++++++++++++ .../figure-html/unnamed-chunk-12-1.png | Bin 0 -> 338054 bytes .../figure-html/unnamed-chunk-13-1.png | Bin 0 -> 470526 bytes .../figure-html/unnamed-chunk-3-1.png | Bin 0 -> 46368 bytes .../figure-html/unnamed-chunk-4-1.png | Bin 0 -> 68012 bytes .../figure-html/unnamed-chunk-6-1.png | Bin 0 -> 37180 bytes .../figure-html/unnamed-chunk-7-1.png | Bin 0 -> 75839 bytes .../figure-html/unnamed-chunk-9-1.png | Bin 0 -> 65498 bytes index.html | 32 +- search.json | 48 +- site_libs/bootstrap/bootstrap.min.css | 2 +- site_libs/quarto-nav/quarto-nav.js | 1 + 36 files changed, 1462 insertions(+), 168 deletions(-) create mode 100644 08-mcmc.html create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-12-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-13-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-3-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-4-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-6-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-7-1.png create mode 100644 08-mcmc_files/figure-html/unnamed-chunk-9-1.png diff --git a/.nojekyll b/.nojekyll index 9eb15cc..cab66bf 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -f2bbd923 \ No newline at end of file +74fb27e2 \ No newline at end of file diff --git a/01-introduccion.html b/01-introduccion.html index 0983243..d462897 100644 --- a/01-introduccion.html +++ b/01-introduccion.html @@ -2,7 +2,7 @@ - + @@ -242,6 +242,12 @@ 7  Buenos y malos controles + +
  • +
    + + 8  Markov Chain Monte Carlo +
    • @@ -348,24 +354,24 @@

    Ejemp -B -grandes +A +chicos mejora -B -chicos +A +grandes mejora -B -chicos +A +grandes mejora A grandes -mejora +sin_mejora A @@ -374,13 +380,13 @@

    Ejemp A -chicos -mejora +grandes +sin_mejora B -chicos -mejora +grandes +sin_mejora A @@ -389,7 +395,7 @@

    Ejemp B -chicos +grandes mejora @@ -648,8 +654,8 @@

    Ejemp ", width = 200, height = 50)
    -
    - +
    +

    Es decir, el tamaño de los cálculos es una causa común de tratamiento (T) y resultado (M). Veremos más adelante que la decisión de condicionar a el tipo de cálculos proviene de un análisis relativamente simple de este diagrama causal, independientemente de los métodos que usemos para estimar las proporciones de interés (en este ejemplo, examinar las tablas cruzadas es equivalente a hacer estimaciones de máxima verosimlitud).

    @@ -770,8 +776,8 @@

    Eje ", width = 200, height = 50)
    -
    - +
    +

    Nótese que el análisis más apropiado no está en los datos: en ambos casos la tabla de datos es exactamente la misma. Los supuestos acerca del proceso que genera los datos sin embargo nos lleva a respuestas opuestas.

    @@ -1226,6 +1232,24 @@

    - + @@ -220,6 +220,12 @@ 7  Buenos y malos controles + +
  • +
    + + 8  Markov Chain Monte Carlo +
    • @@ -308,8 +314,8 @@

    ")#, width = 200, height = 50)
    -
    - +
    +

    Donde vemos ahora que el estado real de cada persona de la prueba es desconocido, aunque el resultado de la prueba depende de ese estado, y la cantidad de positivos que observamos es ahora \(N_{obs}\), que depende también de la sensibilidad y especificidad de la prueba.

    @@ -551,8 +557,8 @@

    ")#, width = 200, height = 50)
    -
    - +
    +

    Usando argumentos como los del modelo original, las distribuciones de esp y sens son beta y podemos incorporarlas en la simulación de la posterior. Nuestra nueva función para simular el proceso generativo es:

    @@ -646,7 +652,7 @@ 

    sims <- ajuste$draws(c("theta", "sens", "esp"), format = "df")
 resumen <- ajuste$summary(c("theta"))
    @@ -778,6 +784,24 @@

    - + @@ -222,6 +222,12 @@ 7  Buenos y malos controles + +
  • +
    + + 8  Markov Chain Monte Carlo +
    • @@ -327,8 +333,8 @@

    ", width = 300, height = 100)
    -
    - +
    +

    Que también podríamos simplificar (suponiendo la \(N\) fija y conocida, pues \(N_+\) y \(M\) dan \(N_{-}\)) como:

    @@ -361,8 +367,8 @@

    ", width = 300, height = 100)
    -
    - +
    +

    Y ahora construimos el modelo generativo. Supondremos que la muestra de \(N\) personas se toma de manera aleatoria de la población (una población grande, así que podemos ignorar el efecto de muestreo). Supondremos provisionalmente, además, que la prueba es perfecta, es decir, no hay falsos positivos o negativos.

    @@ -1684,6 +1690,24 @@

    - + @@ -240,6 +240,12 @@ 7  Buenos y malos controles + +
  • +
    + + 8  Markov Chain Monte Carlo +
    • @@ -333,8 +339,8 @@

    ")#, width = 200, height = 50)
    -
    - +
    +

    Nótese que no consideramos \(W\to H\), porque podemos pensar en varias intervenciones que podrían cambiar el peso por no cambian la estatura. Por otro lado, es difícil pensar en alguna intervención que cambie la estatura pero no cambie el peso de una persona. Adicionalmente, hay otros factores desconocidos no observados \(U\) que afectan el peso de cada persona adicionalmente a su estatura.

    @@ -928,8 +934,8 @@

    ", width = 200, height = 50)
    -
    - +
    +

    Omitiendo del diagrama las variables no observadas que también son causas únicamente de \(S\) y \(W, H\):

    @@ -954,8 +960,8 @@

    ", width = 200, height = 50)
    -
    - +
    +

    Si queremos saber cómo influye el sexo en el peso, este diagrama indica que hay dos tipos de preguntas que podemos hacer:

    @@ -1445,8 +1451,8 @@

    ")#, width = 200, height = 50)
    -
    - +
    +

    En este caso, el modelo causal es como sigue: conocemos la distancia \(D\) al hoyo en cada tiro. El éxito (\(Y=1\)) o fracaso (\(Y=0\)) depende de la distancia, junto con la velocidad a la que sale la pelota (muy alto o muy bajo puede dar un tiro fallido), y el ángulo \(\theta\) de salida. Adicionalmente, hay otros factors \(U\) que pueden afectar la probabilidad de éxito. Nótese que no escribiríamos, por ejemplo \(Y \leftarrow D\), porque la distancia no cambia causalmente con el resultado del tiro, aunque es cierto que si intervenimos en la distancia, esperaríamos obtener tasas de éxito diferentes. Igualmente, es necesario poner una flecha de \(V\) a \(D\) y \(V\) a \(Y\).

    @@ -2383,6 +2389,24 @@

    - + @@ -240,6 +240,12 @@ 7  Buenos y malos controles + +
  • +
    + + 8  Markov Chain Monte Carlo +
    • @@ -343,8 +349,8 @@

    ", width = 150, height = 40)
    -
    - +
    +

    Nótese que no describimos exactamente cómo son las funciones que relacionan las variables, sino más bien qué variables son causas directas de qué otras. Por ejemplo, aunque en nuestro ejemplo de arriba \(Y\) puede estar correlacionado con \(Z\), no hay una causa directa a \(Y\), porque cambios en \(Z\) afectan a \(X\), y es el cambio en \(X\) que es causa directa de \(Y\).

    @@ -401,8 +407,8 @@

    ")
    -
    - +
    +

    En este ejemplos no podemos saber \(U1\) y \(U2\), y no nos interesa modelar la física de monedas, manera de lanzarlas, etc. En este ejemplo también no consideraremos qué hace que un día sea soleado o lluvioso (no nos interesa modelar el clima). En este momento, en teoría tenemos ecuaciones determinísticas para todas las variables, y si conocemos todas las variables exógenas \(U1,U2,U3,U4\) podríamos determinar exactamente lo que va a suceder con la ganancia, por ejemplo, o cualquier otra variable del sistema.

    @@ -433,8 +439,8 @@

    ")
    -
    - +
    +
    @@ -511,11 +517,11 @@

    Ejemplo

    # A tibble: 5 × 5
       x d           s1    s2     g
   <dbl> <chr>    <int> <int> <int>
-1 0.234 soleado      1     0     1
-2 0.281 soleado      3     3     3
-3 0.475 soleado      4     3     4
-4 0.944 lluvioso     5     5    10
-5 0.917 soleado      4     5     4
    +1 0.737 soleado 4 5 4 +2 0.713 lluvioso 4 4 8 +3 0.601 lluvioso 2 3 5 +4 0.575 soleado 4 3 4 +5 0.582 soleado 4 3 4
    @@ -577,8 +583,8 @@

    ", width = 200, height = 50)
    -
    - +
    +

    En este caso,

    @@ -627,8 +633,8 @@

    Ejemplo (si 
    cor(sims_confusor |> select(x,y)) |> round(3)
          x     y
-x  1.00 -0.43
-y -0.43  1.00
    +x 1.00 -0.42 +y -0.42 1.00

    Sin embargo, si condicionamos a \(Z\), que puede tomar los valores 0 o 1, vemos que \(X\) y \(Y\) son independientes, o dicho de otra manera, la condicional de \(Y\) dada \(Z\) y \(X\) sólo depende de \(Z\):

    @@ -653,15 +659,15 @@

    Ejemplo (si 
    cor(sims_confusor |> filter(z == 1) |> select(x,y)) |> round(3)
    -
           x      y
-x  1.000 -0.009
-y -0.009  1.000
    +
          x     y
+x 1.000 0.008
+y 0.008 1.000
    cor(sims_confusor |> filter(z == 0) |> select(x,y)) |> round(3)
    -
           x      y
-x  1.000 -0.006
-y -0.006  1.000
    +
      x y
+x 1 0
+y 0 1

    Un ejemplo con variables continuas podría ser como sigue:

    @@ -761,8 +767,8 @@

    E ", width = 200, height = 50)
    -
    - +
    +

    Por la discusión de arriba, es claro que es necesario considerar la edad al casarse si queremos estimar el efecto de tasa de matrimonio en la tasa de divorcio. Es posible que la correlación entre estas dos tasas puede ser explicada solamente por la edad al casarse, y que en realidad al flecha \(M\to D\) sea muy débil o inexistente.

    @@ -913,8 +919,8 @@

    ", width = 200, height = 50)
    -
    - +
    +

    Es decir, borramos todas las flechas que caen en \(M\) (pues la estamos interveniendo al valor que queramos), y luego simulando \(D\).

    @@ -930,7 +936,7 @@

    @@ -982,8 +988,8 @@

    ", width = 150, height = 20)
    -
    - +
    +

    En este caso,

    @@ -1050,15 +1056,15 @@ 

    cor(sims_mediador |> filter(z == 1) |> select(x,y)) |> round(3)
    -
          x     y
-x 1.000 0.004
-y 0.004 1.000
    +
           x      y
+x  1.000 -0.002
+y -0.002  1.000
    cor(sims_mediador |> filter(z == 0) |> select(x,y)) |> round(3)
           x      y
-x  1.000 -0.002
-y -0.002  1.000
    +x 1.000 -0.012 +y -0.012 1.000

    Podemos también hacer un ejemplo continuo:

    @@ -1151,8 +1157,8 @@

    Ejemplo: Burks")
    -
    - +
    +

    Como el NSE es del hogar (una medida general de estatus social), se consideró en principio como una variable pre-tratamiento a la inteligencia de los niños por la que tradicionalmente se controlaba. Burks notó que hacer esto tenía no era apropiado, pues tiene como consecuencia cortar parte del efecto total de la inteligencia sobre el la inteligencia de los hijos. En otras palabras: la inteligencia de los padres hace más probable mejor NSE, y mejor NSE presenta mejores condiciones de desarrollo para sus hijos. Estatificar por esta variable bloquea este efecto.

    @@ -1179,8 +1185,8 @@

    ", width = 200, height = 50)
    -
    - +
    +
    @@ -329,8 +336,8 @@

    ', width = 250, height = 60)
    -
    - +
    +
    @@ -364,8 +371,8 @@

    ")
    -
    - +
    +

    No hay ninguna variable confusora, y una estrategia de estimación es comparar \(PF\) entre los grupos.

    @@ -492,8 +499,8 @@

    ', width = 250, height = 120)
    -
    - +
    +
    @@ -542,8 +549,8 @@

    ', width = 250, height = 120)

    -
    - +
    +

    Hemos añadido un nodo implícito (otros factores que afectan \(Y\) y no tienen relación con otras variables del sistema) para explicar qué es lo que pasa cuando condicionamos a \(Z\): como \(Z\) es un descendiente del colisionador en \(Y\), se activa una ruta no causal entre \(U_y\) y \(T\), y estas dos cantidades aparecen como correlacionadas (es una correlación no causal). Esto en consecuencia modifica la correlación entre \(T\) y \(Y\).

    @@ -622,8 +629,8 @@

    ', width = 250, height = 140)
    -
    - +
    +
    @@ -653,8 +660,8 @@

    }")
    -
    - +
    +

    En la gráfica de arriba, \(T\) indica si la madre es fumadora o no, y \(Y\) la mortalidad. \(Z\) si el bebé nació con bajo peso o no.

    @@ -681,8 +688,8 @@

    }', width = 100, height = 50)
    -
    - +
    +

    En este caso, condicionar a \(Z\) no sesga nuestras estimaciones, pues no activamos ninguna ruta no causal. La dificultad es que típicamente disminuye la precisión de la estimación (usamos un modelo más grande donde no es necesario):

    @@ -728,8 +735,8 @@

    }', width = 100, height = 50)
    -
    - +
    +

    En este caso, tenemos una variable confusora \(U\) que no nos permite estimar sin sesgo el efecto de \(T\) sobre \(Y\). Sin embargo, si condicionamos a \(Z\), la situación puede emperorar (amplificación de sesgo), pues dentro de cada nivel de \(Z\) hay menos variación de \(X\), y eso implica que la covarianza entre \(X\) y \(Y\), en cada nivel de \(Z\), se debe más a la variable confusora.

    @@ -855,6 +862,24 @@

    + + 8  Markov Chain Monte Carlo + diff --git a/08-mcmc.html b/08-mcmc.html new file mode 100644 index 0000000..54d1ac9 --- /dev/null +++ b/08-mcmc.html @@ -0,0 +1,1035 @@ + + + + + + + + + +Métodos analíticos - 8  Markov Chain Monte Carlo + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + +
    + +
    + + +
    + + + +
    + +
    +
    +

    8  Markov Chain Monte Carlo

    +
    + + + +
    + + + + +
    + + + +
    + + +
    +
    +Código +
    library(tidyverse)
+library(kableExtra)
+library(DiagrammeR)
+ggplot2::theme_set(ggplot2::theme_light())
+inv_logit <- \(x) 1 / (1 + exp(-x))
    +
    +
    +

    En esta sección exlicaremos brevemente cómo funcionan paquetes como Stan para producir simulaciones de una posteriores complicadas en dimensión alta.

    +

    En primer lugar, recordemos que si queremos calcular la posterior de un modelo (generalmente para calcular después resúmenes que involucran integrales de esta posterior) tenemos los siguiente enfoques:

    +
      +
    1. Intentar hacer los cálculos analíticamente.
      2. +
    2. Usar una aproximación de rejilla.
      3. +
    3. Simulación por Markov Chain Monte Carlo (MCMC).
      4. +
    +

    Stan utiliza 3, y hay variedad de algoritmos MCMC. Ya discutimos que 1, la aproximación analítica, es en general imposible (a menos fuera de ciertos modelos restringidos). La aproximación 2 excesivamente intensiva, al grado que sólo para modelos muy chicos y con pocos parámetros es posible utilizarla. Existen otros métodos también como aproximaciones cuadráticas que en ciertos casos funcionan, pero son limitados en su aplicación.

    +

    La idea de simulación de variables aleatorias es ahora fundamental en muchas áreas científicas, incluyendo la estadística, y los métodos que la utilizan se llaman métodos de Monte Carlo. Por ejemplo, considera el método bootstrap, pruebas de permutaciones, validación cruzada, y en general simulación para cálculo de resúmenes que son difíciles de calcular directamente (por ejemplo, la mediana de una distribución Gamma, ver Median approximations and bounds aquí).

    +
    +

    8.1 Algoritmos Metropolis-Hastings

    +

    Uno de los primeros algoritmos MCMC fue el de Metropolis-Hastings, que veremos primero en algunos ejemplos. Veremos también por qué ahora tenemos mejores opciones que MH para estimar posteriores de nuestros modelos.

    +
    +

    Ejemplo: dos implementaciones de Metropolis

    +

    Supongamos que queremos simular de una variable aleatoria \(X\) con distribución discreta sobre los valores \(1,2\ldots, k\) con probabilidades \(p(1),p(2),\ldots,p(k)\). Este problema puede resolverse fácilmente de varias maneras, pero utilizaremos un método de Monte Carlo tipo Metropolis. Supongamos que podemos simular de una variable aleatoria \(U\) que es uniforme en \(1,2,\ldots, k\) (con probabilidades iguales a 1/k).

    +

    Lo que podemos hacer es lo que sigue, para \(i=1,\ldots, M\):

    +
      +
    1. Para \(i=1\), comenzamos fijando un valor \(x_1\) en \(1,2,\ldots, k\).
      2. +
    +

    Para cada \(i\),

    +
      +
    1. Simulamos una uniforme en \(1,2,\ldots, k\). Al valor \(u_i\) obtenido le llamamos valor propuesto.
      2. +
    2. Calculamos \(q = p(u_i)/p(x_i)\) (el cociente de la probabilidad del valor propuesto entre la probabilidad del valor actualo).
      3. +
    3. Si \(q > 1\), aceptamos el valor propuesto y ponemos \(x_{i+1} = u_i\).
      4. +
    4. Si \(q < 1\), aceptamos el valor propuesto con probabilidad \(q\) (\(x_{i+1} = u_i\)), y con probabilidad \(1-q\) rechazamos (\(x_{i+1} = x_i\)).
      5. +
    +

    El resultado es una sucesión de valores \(x_1,x_2,\ldots, x_M\). Es posible demostrar que la distribución de estas \(x_i\) converge a la distribución \(p(1),\ldots, p(k)\) si \(M\) es suficientemente grande.

    +

    Este método se llama Metropolis-Hastings. Es un método de Monte Carlo, y como podemos ver, se trata de una cadena de Markov, pues la distribución cada siguiente lugar \(x_{i+1}\), condicionada al valor actual \(x_i\) no depende de valores anteriores de las \(x\).

    +
    +
    #set.seed(451123)
+# definimos estas p
+k <- 40
+p <- exp(-(1:k - k/3)^2 / 10)
+p <- p /sum(p)
+dist_obj <- tibble(x = 1:k, p = p)
+# simulamos
+M <- 200000
+x <- numeric(M)
+x[1] <- 20
+for(i in 1:M){
+  u <- sample(1:k, 1)
+  q <- p[u] / p[x[i]]
+  if(runif(1) < q){
+    x[i+1] <- u
+  } else {
+    x[i+1] <- x[i]  
+  }
+}
    +
    +

    En rojo mostramos las probabilidades objetivo que queremos estimar, y en negro las estimadas con nuestro método de arriba.

    +
    +
    resultados_sim <- tibble(x = x) |> 
+  mutate(n_sim = row_number())
+resumen_sim <- resultados_sim |> count(x) |> 
+  right_join(tibble(x = 1:k, p = p)) |> 
+  mutate(n = ifelse(is.na(n), 0, n)) |> 
+  mutate(p_aprox = n / sum(n))
    +
    +
    Joining with `by = join_by(x)`
    +
    +
    ggplot(dist_obj, aes(x = x)) +
+    geom_point(aes(y = p)) +
+  geom_point(data = resumen_sim, 
+             aes(y = p_aprox), color = "red", size = 3, alpha = 0.5)
    +
    +
    +
    +

    +
    +
    +
    +
    +

    Como vemos, los valores de \(x_1,\ldots, x_M\) se distribuyen aproximadamente como la distribución \(p\) objetivo. Esta es una manera de simular valores de esta distribución discreta \(p\). Podemos ver cómo se ven las simulaciones sucesivas:

    +
    +
    ggplot(resultados_sim |> filter(n_sim < 400), aes(x = n_sim, y = x)) +
+  geom_line() + scale_y_continuous(breaks = 1:20)
    +
    +
    +
    +

    +
    +
    +
    +
    +

    El defecto que tiene este algoritmo es que puede ser relativamente lento, pues vemos que hay periodos largos donde se “atora” en valores de probabilidad relativamente alta. La razón es que en muchos pasos, estamos proponiendo “saltos al vacío” a lugares de probabilidad muy baja, que rara vez se aceptan.

    +

    Podemos hacer más eficiente nuestro algoritmo si le permitimos explorar con mayor facilidad los posibles valores de \(x\). Esto se logra proponiendo saltos locales: si estamos en \(x_i\), entonces proponemos los valores \(x_i - 1\) y \(x_i + 1\) con la misma probabilidad 1/2 (excepto en los extremos donde sólo proponemos \(x_i,x_i+1\) o \(x_i-1,x_i\)).

    +

    Proponemos entonces la suguiente modificación del paso 1 de propuesta:

    +
      +
    1. Para \(i=1\), comenzamos fijando un valor \(x_1\) en \(1,2,\ldots, k\).
      2. +
    +

    Para cada \(i\),

    +
      +
    1. Si \(1< x_i<k\), escogemos al azar un salto a la derecha o al izquierda con igual probabilidad 1/2. En los extremos \(x_i=1\) o \(x_i=k\) escogemos entre \(1,2\) o \(k-1,k\) respectivamente. Al valor \(u_i\) obtenido le llamamos valor propuesto.
      2. +
    2. Calculamos \(q = p(u_i)/p(x_i)\) .
      3. +
    3. Si \(q > 1\), aceptamos el valor propuesto y ponemos \(x_{i+1} = u_i\).
      4. +
    4. Si \(q < 1\), aceptamos el valor propuesto con probabilidad \(q\) (\(x_{i+1} = u_i\)), y con probabilidad \(q\) rechazamos(\(x_{i+1} = x_i\)).
      5. +
    +

    Esto lo escribimos como sigue:

    +
    +
    #set.seed(4511)
+# simulamos
+x <- numeric(M)
+x[1] <- 20
+for(i in 1:M){
+  u <- sample(c(x[i] - 1,  x[i] + 1), 1)
+  if(u == k+1) u <- k
+  if(u == 0) u <- 1
+  q <- p[u] / p[x[i]]
+  if(runif(1) < q){
+    x[i+1] <- u
+  } else {
+    x[i+1] <- x[i]  
+  }
+}
    +
    +

    Obtenemos:

    +
    +
    resultados_sim_2 <- tibble(x = x) |> 
+  mutate(n_sim = row_number())
+resumen_sim_2 <- resultados_sim_2 |> count(x) |> 
+  mutate(p_aprox = n / sum(n))
+ggplot(dist_obj, aes(x = x)) +
+    geom_point(aes(y = p)) +
+  geom_point(data = resumen_sim_2, 
+             aes(y = p_aprox), color = "red", size = 3, alpha = 0.5)
    +
    +
    +
    +

    +
    +
    +
    +
    +

    Y podemos ver cómo evoluciona nuestra cadena de Markov:

    +
    +
    ggplot(resultados_sim_2 |> filter(n_sim < 400), aes(x = n_sim, y = x)) +
+  geom_line() + scale_y_continuous(breaks = 1:10)
    +
    +
    +
    +

    +
    +
    +
    +
    +

    ¿Cómo se comparan estos dos métodos? Podemos ver por ejemplo cómo se comparan las distribuciones aproximadas hasta cierto número de iteraciones con la verdadera distribución objetivo:

    +
    +
    approx_sim <- map_df(seq(200, 30000, by = 200), function(n_sims){
+  resumen_1 <- resultados_sim |> filter(n_sim <= n_sims) |> 
+    count(x) |>
+    mutate(p_aprox = n / sum(n)) |>
+    select(-n) |>
+    right_join(dist_obj, by = "x") |> 
+    mutate(metodo = "MH-1") |> 
+    mutate(n_sims = n_sims) 
+  resumen_2 <- resultados_sim_2 |> filter(n_sim <= n_sims) |>
+    count(x) |>
+    mutate(p_aprox = n / sum(n)) |>
+    select(-n) |>
+    right_join(dist_obj, by = "x") |> 
+    mutate(metodo = "MH-2") |> 
+    mutate(n_sims = n_sims) 
+  bind_rows(resumen_1, resumen_2) |> 
+    mutate(p_aprox = ifelse(is.na(p_aprox), 0, p_aprox))
+})
    +
    +
    +
    approx_sim |> 
+  mutate(dif_abs = abs (p_aprox-p)) |>
+  group_by(metodo, n_sims) |> 
+  summarise(dif_abs = sum(dif_abs)) |> 
+ggplot(aes(n_sims, dif_abs, color = metodo)) +
+  geom_line()
    +
    +
    `summarise()` has grouped output by 'metodo'. You can override using the
+`.groups` argument.
    +
    +
    +
    +
    +

    +
    +
    +
    +
    +

    En este caso, considera qué es lo que sucede en cada uno de estos casos:

    +
      +
    1. El algoritmo que da saltos grandes muchas veces rechaza porque cae en un área de probilidad muy baja.
      2. +
    2. El algoritmo que da saltos chicos puede tardar en explorar regiones de probabilidad relativamente alta con suficiente frecuencia (tarda un moverse de un lugar a otro), por su naturaleza de “caminata aleatoria”. Pero sus tasas de aceptación son más altas.
      3. +
    +

    Este es el primer balance que existe en este algoritmo: tomar pasos grandes y balancear las probabilidades quizá rechazando muy frecuentemente (no es eficiente), o tomar pasos chicos y vagar más tiempo para visitar regiones de alta probabilidad, aunque con menos tasa de rechazo. Dependiendo de la distribución que queremos aproximar podemos inclinarnos más por una o por otra opción.

    +

    ¿Por qué funcionan estos algoritmos? Supongamos que en cada paso, se cumple que (balance detallado): \[{q(x|y)}p(y) = {q(y|x)}{p(x)}\] donde \(q(x|y)\) son las probabilidades de transición de nuestra cadena de Markov propuesta. Esta ecuación dice que si la probabilidad se distribuye como \(p(x)\), entonces al transicionar con \(q\) la probabilidad fluje de manera que se mantiene estática en \(p\), es decir \(p\) es una distribución estacionaria. Esto es fácil de demostrar pues \[\sum_{y} q(x|y)p(y) = \sum_{y} q(y|x)p(x) = p(x) \sum_{y} q(y|x) = p(x).\]

    +

    Bajo otros supuestos adicionales de ergodicidad (aperiodicidad y tiempos de recurrencia finitos, es decir, las transiciones mezclan bien los estados), entonces podemos simular la cadena de Markov por un tiempo suficientemente largo y con esto obtener una muestra de la distribución objetivo \(p\).

    +

    ¿Cómo podemos diseñar entonces las \(q(x|y)\) correspondientes? Comenzamos considerando distribuciones propuesta \(q_0(x|y)\) que no satisfacen la ecuación de balance, y supondremos como en los ejemplos de arriba (verifícalo) que nuestras transiciones tienen probabilidades simétricas \(q_0(y|x) = q_0(x|y)\). Entonces, cuando \(p(y)/p(x) > 1\), queremos transicionar de \(x\) a \(y\) con más frecuencia que de \(y\) a \(x\). Comenzando en \(x\), si la propuesta de \(q_0\) es \(y\), podríamos poner entonces que el sistema transicione con probabilidad 1 a \(y\). Sin embargo, si empezamos en \(y\) y la propuesta es \(x\), el sistema sólo transiciona de \(y\) a \(x\) con probabilidad \(p(x)/p(y)\).

    +

    De esta manera, obtenemos que bajo \(q(y|x)\), \(x\) transiciona a \(y\) con probabilidad \(\min\{1, p(y)/p(x)\}\). Entonces, el cociente \(\frac{q(y|x)}{q(x|y)}\) es igual a \(\frac{p(y)}{p(x)}\) si \(p(y)<p(x)\), y es igual a \(1/\frac{p(x)}{p(y)} = \frac{p(y)}{p(x)}\) si \(p(y)>p(x)\).

    +
    +
    +
    + +
    +
    +Idea básica de MCMC +
    +
    +
    +

    En MCMC, buscamos un cadena de Markov que, en el largo plazo, visite cada posible parámetro proporcionalmente a la probabildad posterior de cada parámetro.

    +
    +
    +
    +
    +

    Ejemplo: Metropolis bivariado

    +

    Supongamos ahora que quisiéramos simular de una normal multivariada con media en c(2,3) y matriz de covarianza \(\Sigma\), que supondremos es tal que la desviación estándar de cada variable es 1 y la correlación es 0.8. La matriz \(\Sigma\) tiene 1 en la diagonal y 0.8 fuera de la diagonal.

    +

    La distribución objetivo \(p\) está dada entonces (módulo una constante de proporcionalidad):

    +
    +
    Sigma <- matrix(c(1, 0.8, 0.8, 1), nrow = 2)
+Sigma_inv <- solve(Sigma)
+m <- c(2, 3)
+log_p <- function(z){
+  - 0.5 * (t(z-m) %*% Sigma_inv %*% (z-m))
+}
    +
    +

    Nótese que como Metropolis hace cocientes de probabilidades, sólo es necesario conocer la densidad objetivo módulo una constante de proporcionalidad.

    +

    Una algoritmo de Metropolis podría ser el siguiente:

    +
    +
    # simulamos
+M <- 50000
+simular_normal <- function(M, m, Sigma, delta_x, delta_y){
+  z <- matrix(nrow = M, ncol = 2)
+  z[1, ] <- c(2.5, 3.2)
+  colnames(z) <- c("x", "y")
+  rechazo <- 0
+  for(i in 1:(M-1)){
+    x_prop <- rnorm(1, z[i, 1], delta_x)
+    y_prop <- rnorm(1, z[i, 2], delta_y)
+    z_prop <- c(x_prop, y_prop)
+    q <- exp(log_p(z_prop) - log_p(z[i, ]))
+    if(runif(1) < q){
+      z[i + 1, ] <- z_prop
+    } else {
+      rechazo <- rechazo + 1
+      z[i + 1, ] <- z[i, ]
+    } 
+  }
+  print(rechazo / M)
+  z_tbl <- as_tibble(z) |> 
+    mutate(n_sim = row_number())
+  z_tbl
+}
+z_tbl <- simular_normal(M, m, Sigma, 1, 1)
    +
    +
    [1] 0.60038
    +
    +
    +

    Vemos que tenemos una tasa alta de rechazos. ¿Por qué? Veamos cómo se ven las simulaciones después de 1000 iteraciones:

    +
    +
    # estas las usamos para graficar
+sims_normal <- mvtnorm::rmvnorm(10000, mean = m, sigma = Sigma)
+colnames(sims_normal) <- c("x", "y")
+sims_normal <- as_tibble(sims_normal)
+graf_tbl <- map_df(seq(10, 500, 20), function(i){
+  z_tbl |> filter(n_sim <= i) |> mutate(num_sims = i)
+})
+ggplot(graf_tbl, aes(x, y)) + 
+  stat_ellipse(data = sims_normal, aes(x, y), 
+               level = c( 0.9), type = "norm", colour = "salmon") +
+  stat_ellipse(data = sims_normal, aes(x, y), 
+               level = c( 0.5), type = "norm", colour = "salmon") +
+  geom_point(alpha = 0.1) +
+  facet_wrap(~num_sims) + theme_minimal()
    +
    +
    +
    +

    +
    +
    +
    +
    +

    Observaciones:

    +
      +
    • Los puntos que tienen intensidad alta son puntos donde hubo varios rechazos. Esto es porque las propuestas a veces caen en elipses de baja probabilidad (en la gráfica mostramos una elipse de 50% probabilidad y otra de 95%).
      • +
    • Esto se debe a que los saltos en cada dirección son de desviación estándar 1, y esto fácilmente nos lleva a una zona de alta probabilidad a una de baja probabilidad.
      • +
    • Sin embargo, a largo plazo, vemos cómo la cadena está visitando las regiones de alta probabilidad con aparentemente la frecuencia correcta.
      • +
    +

    Podemos entonces proponer saltos más chicos, por ejemplo:

    +
    +
    z_tbl <- simular_normal(M, m, Sigma, 0.2, 0.2)
    +
    +
    [1] 0.1568
    +
    +
    graf_tbl <- map_df(seq(10, 500, 20), function(i){
+  z_tbl |> filter(n_sim <= i) |> mutate(num_sims = i)
+})
+ggplot(graf_tbl, aes(x, y)) + 
+  stat_ellipse(data = sims_normal, aes(x, y), 
+               level = c( 0.9), type = "norm", colour = "salmon") +
+  stat_ellipse(data = sims_normal, aes(x, y), 
+               level = c( 0.5), type = "norm", colour = "salmon") +
+  geom_point(alpha = 0.1) +
+  facet_wrap(~num_sims) + theme_minimal()
    +
    +
    +
    +

    +
    +
    +
    +
    +

    Observaciones:

    +
      +
    • En este caso tenemos una tasa de aceptación más alta.
      • +
    • Sin embargo, la cadena parece “vagar” en la regiones de probabilidad alta, y tiene dificultades para explorar correctamente estas regiones.
      • +
    • Sin embargo, a largo plazo, vemos cómo la cadena está visitando las regiones de alta probabilidad con aparentemente la frecuencia correcta.
      • +
    +
    +
    +
    + +
    +
    +Metropolis-Hastings +
    +
    +
    +

    En el algoritmo de Metropolis Hastings hay una tensión natural en tamaño de salto y tasa de aceptación. Si el tamaño de los saltos es muy grande, la tasa de aceptación es baja y esto producen ineficiencias. Si el tamaño de los saltos es muy chico, la tasa de aceptación es más alta, pero esto también es ineficiente.

    +
    +
    +

    Existen métodos que pueden superar este problema, como son muestreo de Gibbs y Monte Carlo Hamiltoniano. El primero no lo discutiremos, pues requiere poder simular fácilmente de cada parámetro dados los otros, y esto no siempre es posible. Veremos más el segundo, donde usaremos información del gradiente de la distribución objetivo para proponer exploración más eficiente.

    +
    +
    +
    +

    8.2 Monte Carlo Hamiltoniano

    + + +
    + +
    + + +
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    @@ -368,6 +374,24 @@

    Software: // clear code selection e.clearSelection(); }); + var localhostRegex = new RegExp(/^(?:http|https):\/\/localhost\:?[0-9]*\//); + var mailtoRegex = new RegExp(/^mailto:/); + var filterRegex = new RegExp('/' + window.location.host + '/'); + var isInternal = (href) => { + return filterRegex.test(href) || localhostRegex.test(href) || mailtoRegex.test(href); + } + // Inspect non-navigation links and adorn them if external + var links = window.document.querySelectorAll('a[href]:not(.nav-link):not(.navbar-brand):not(.toc-action):not(.sidebar-link):not(.sidebar-item-toggle):not(.pagination-link):not(.no-external):not([aria-hidden]):not(.dropdown-item):not(.quarto-navigation-tool)'); + for (var i=0; iSoftware: try { href = new URL(href).hash; } catch {} const id = href.replace(/^#\/?/, ""); const note = window.document.getElementById(id); - return note.innerHTML; + if (note) { + return note.innerHTML; + } else { + return ""; + } }); } const xrefs = window.document.querySelectorAll('a.quarto-xref'); diff --git a/search.json b/search.json index 858c28e..4871ecb 100644 --- a/search.json +++ b/search.json @@ -24,7 +24,7 @@ "href": "01-introduccion.html#diagramas-causales", "title": "1  Introducción", "section": "", - "text": "Causas y mecanismos\n\n\n\nLas razones de cómo hacemos análisis estadístico (que procedimiento o algoritmo seleccionamos, por ejemplo) en un problema dado no están en los datos observados, las causas de los datos.\n\n\n\n\n\nEjemplo (cálculos renales)\nEste es un estudio real acerca de tratamientos para cálculos renales (Julious y Mullee (1994)). Pacientes se asignaron de una forma no controlada a dos tipos de tratamientos para reducir cálculos renales. Para cada paciente, conocemos el tipo de ćalculos que tenía (grandes o chicos) y si el tratamiento tuvo éxito o no.\nLa tabla original tiene 700 renglones (cada renglón es un paciente)\n\ncalculos <- read_csv(\"../datos/kidney_stone_data.csv\")\nnames(calculos) <- c(\"tratamiento\", \"tamaño\", \"éxito\")\ncalculos <- calculos |> \n mutate(tamaño = ifelse(tamaño == \"large\", \"grandes\", \"chicos\")) |> \n mutate(resultado = ifelse(éxito == 1, \"mejora\", \"sin_mejora\")) |> \n select(tratamiento, tamaño, resultado)\nnrow(calculos)\n\n[1] 700\n\n\ny se ve como sigue (muestreamos algunos renglones):\n\ncalculos |> \n sample_n(10) |> kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nresultado\n\n\n\n\nB\ngrandes\nmejora\n\n\nB\nchicos\nmejora\n\n\nB\nchicos\nmejora\n\n\nA\ngrandes\nmejora\n\n\nA\ngrandes\nmejora\n\n\nA\nchicos\nmejora\n\n\nB\nchicos\nmejora\n\n\nA\ngrandes\nmejora\n\n\nB\nchicos\nmejora\n\n\nB\nchicos\nmejora\n\n\n\n\n\n\n\n\nAunque estos datos contienen información de 700 pacientes, los datos pueden resumirse sin pérdida de información contando como sigue:\n\ncalculos_agregada <- calculos |> \n group_by(tratamiento, tamaño, resultado) |> \n count()\ncalculos_agregada |> kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nresultado\nn\n\n\n\n\nA\nchicos\nmejora\n81\n\n\nA\nchicos\nsin_mejora\n6\n\n\nA\ngrandes\nmejora\n192\n\n\nA\ngrandes\nsin_mejora\n71\n\n\nB\nchicos\nmejora\n234\n\n\nB\nchicos\nsin_mejora\n36\n\n\nB\ngrandes\nmejora\n55\n\n\nB\ngrandes\nsin_mejora\n25\n\n\n\n\n\n\n\n\nComo en este caso nos interesa principalmente la tasa de éxito de cada tratamiento, podemos mejorar mostrando como sigue:\n\ncalculos_agregada |> pivot_wider(names_from = resultado, values_from = n) |> \n mutate(total = mejora + sin_mejora) |> \n mutate(prop_mejora = round(mejora / total, 2)) |> \n select(tratamiento, tamaño, total, prop_mejora) |> \n arrange(tamaño) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\ntotal\nprop_mejora\n\n\n\n\nA\nchicos\n87\n0.93\n\n\nB\nchicos\n270\n0.87\n\n\nA\ngrandes\n263\n0.73\n\n\nB\ngrandes\n80\n0.69\n\n\n\n\n\n\n\n\nEsta tabla descriptiva es una reescritura de los datos, y no hemos resumido nada todavía. Pero es apropiada para empezar a contestar la pregunta:\n\n¿Qué indican estos datos acerca de qué tratamiento es mejor? ¿Acerca del tamaño de cálculos grandes o chicos?\n\nSupongamos que otro analista decide comparar los pacientes que recibieron cada tratamiento, ignorando la variable de tamaño:\n\ncalculos |> group_by(tratamiento) |> \n summarise(prop_mejora = mean(resultado == \"mejora\") |> round(2)) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\nprop_mejora\n\n\n\n\nA\n0.78\n\n\nB\n0.83\n\n\n\n\n\n\n\n\ny parece ser que el tratamiento \\(B\\) es mejor que el \\(A\\). Esta es una paradoja (un ejemplo de la paradoja de Simpson) . Si un médico no sabe que tipo de cálculos tiene el paciente, ¿entonces debería recetar \\(B\\)? ¿Si sabe debería recetar \\(A\\)? Esta discusión parece no tener mucho sentido.\nPodemos investigar por qué está pasando esto considerando la siguiente tabla, que solo examina cómo se asignó el tratamiento dependiendo del tipo de cálculos de cada paciente:\n\ncalculos |> group_by(tratamiento, tamaño) |> count() |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nn\n\n\n\n\nA\nchicos\n87\n\n\nA\ngrandes\n263\n\n\nB\nchicos\n270\n\n\nB\ngrandes\n80\n\n\n\n\n\n\n\n\nNuestra hipótesis aquí es que la decisión de qué tratamiento usar depende del tamaño de los cálculos. En este caso, hay una decisión pues A es una cirugía y B es un procedimiento menos invasivo, y se prefiere utilizar el tratamiento \\(A\\) para cálculos grandes, y \\(B\\) para cálculos chicos. Esto quiere decir que en la tabla total el tratamiento \\(A\\) está en desventaja porque se usa en casos más difíciles, pero el tratamiento \\(A\\) parece ser en general mejor. La razón es probablemente un proceso de optimización de recursos y riesgo que hacen los doctores.\n\nEn este caso, una mejor respuesta a la pregunta de qué tratamiento es mejor es la que presenta los datos desagregados.\nLa tabla desagregada de asignación del tratamiento nos informa acerca de cómo se está distribuyendo el tratamiento en los pacientes.\n\n\n\n\n\n\n\nNota\n\n\n\nLos resúmenes descriptivos acompañados de hipótesis causales acerca del proceso generador de datos, nos guía hacia descripciones interpretables de los datos.\n\n\nLas explicaciones no son tan simples y, otra vez, interviene el comportamiento de doctores, tratamientos, y distintos tipos de padecimientos.\nPodemos codificar la información causal con un diagrama:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n T \n M \n C\n edge [minlen = 3]\n T -> M\n C -> T\n C -> M\n{ rank = same; M; T }\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEs decir, el tamaño de los cálculos es una causa común de tratamiento (T) y resultado (M). Veremos más adelante que la decisión de condicionar a el tipo de cálculos proviene de un análisis relativamente simple de este diagrama causal, independientemente de los métodos que usemos para estimar las proporciones de interés (en este ejemplo, examinar las tablas cruzadas es equivalente a hacer estimaciones de máxima verosimlitud).\n\n\nEjemplo (cálculos renales 2)\nContrastemos el ejemplo anterior usando exactamente la misma tabla de datos, pero con el supuesto de un proceso generador diferente. En este caso, los tratamientos son para mejorar alguna enfermedad del corazón. Sabemos que parte del efecto de este tratamiento ocurre gracias a una baja en presión arterial de los pacientes, así que después de administrar el tratamiento, se toma la presión arterial de los pacientes. Ahora tenemos la tabla agregada y desagregada como sigue:\n\ncorazon <- calculos |> \n select(tratamiento, presión = tamaño, resultado) |> \n mutate(presión = ifelse(presión == \"grandes\", \"alta\", \"baja\"))\ncorazon_agregada <- corazon |> \n group_by(tratamiento, presión, resultado) |> \n count()\ncorazon_agregada |> pivot_wider(names_from = resultado, values_from = n) |> \n mutate(total = mejora + sin_mejora) |> \n mutate(prop_mejora = round(mejora / total, 2)) |> \n select(tratamiento, presión, total, prop_mejora) |> \n arrange(presión) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\npresión\ntotal\nprop_mejora\n\n\n\n\nA\nalta\n263\n0.73\n\n\nB\nalta\n80\n0.69\n\n\nA\nbaja\n87\n0.93\n\n\nB\nbaja\n270\n0.87\n\n\n\n\n\n\n\n\n\ncorazon |> group_by(tratamiento) |> \n summarise(prop_mejora = mean(resultado == \"mejora\") |> round(2)) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\nprop_mejora\n\n\n\n\nA\n0.78\n\n\nB\n0.83\n\n\n\n\n\n\n\n\n¿Cuál creemos que es el mejor tratamiento en este caso? ¿Deberíamos usar la tabla agregada o la desagregada por presión?\n\nEn este caso, la tabla agregada es más apropiada (B es mejor tratamiento).\nLa razón es que presión en este caso es una consecuencia de tomar el tratamiento, y como las tablas muestran, B es más exitoso en bajar la presión de los pacientes.\nSi sólo comparamos dentro de los grupos de presión baja o de presión alta, ignoramos lo más importante del tratamiento en la probabilidad de mejorar.\n\nNuestros supuestos causales podemos mostrarlos con el siguiente diagrama:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n P\n T \n M \n edge [minlen = 3]\n T -> P\n P -> M\n T -> M\n{ rank = same; M; T}\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nNótese que el análisis más apropiado no está en los datos: en ambos casos la tabla de datos es exactamente la misma. Los supuestos acerca del proceso que genera los datos sin embargo nos lleva a respuestas opuestas.", + "text": "Causas y mecanismos\n\n\n\nLas razones de cómo hacemos análisis estadístico (que procedimiento o algoritmo seleccionamos, por ejemplo) en un problema dado no están en los datos observados, las causas de los datos.\n\n\n\n\n\nEjemplo (cálculos renales)\nEste es un estudio real acerca de tratamientos para cálculos renales (Julious y Mullee (1994)). Pacientes se asignaron de una forma no controlada a dos tipos de tratamientos para reducir cálculos renales. Para cada paciente, conocemos el tipo de ćalculos que tenía (grandes o chicos) y si el tratamiento tuvo éxito o no.\nLa tabla original tiene 700 renglones (cada renglón es un paciente)\n\ncalculos <- read_csv(\"../datos/kidney_stone_data.csv\")\nnames(calculos) <- c(\"tratamiento\", \"tamaño\", \"éxito\")\ncalculos <- calculos |> \n mutate(tamaño = ifelse(tamaño == \"large\", \"grandes\", \"chicos\")) |> \n mutate(resultado = ifelse(éxito == 1, \"mejora\", \"sin_mejora\")) |> \n select(tratamiento, tamaño, resultado)\nnrow(calculos)\n\n[1] 700\n\n\ny se ve como sigue (muestreamos algunos renglones):\n\ncalculos |> \n sample_n(10) |> kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nresultado\n\n\n\n\nA\nchicos\nmejora\n\n\nA\ngrandes\nmejora\n\n\nA\ngrandes\nmejora\n\n\nA\ngrandes\nsin_mejora\n\n\nA\ngrandes\nmejora\n\n\nA\ngrandes\nsin_mejora\n\n\nB\ngrandes\nsin_mejora\n\n\nA\ngrandes\nmejora\n\n\nB\ngrandes\nmejora\n\n\nB\nchicos\nmejora\n\n\n\n\n\n\n\n\nAunque estos datos contienen información de 700 pacientes, los datos pueden resumirse sin pérdida de información contando como sigue:\n\ncalculos_agregada <- calculos |> \n group_by(tratamiento, tamaño, resultado) |> \n count()\ncalculos_agregada |> kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nresultado\nn\n\n\n\n\nA\nchicos\nmejora\n81\n\n\nA\nchicos\nsin_mejora\n6\n\n\nA\ngrandes\nmejora\n192\n\n\nA\ngrandes\nsin_mejora\n71\n\n\nB\nchicos\nmejora\n234\n\n\nB\nchicos\nsin_mejora\n36\n\n\nB\ngrandes\nmejora\n55\n\n\nB\ngrandes\nsin_mejora\n25\n\n\n\n\n\n\n\n\nComo en este caso nos interesa principalmente la tasa de éxito de cada tratamiento, podemos mejorar mostrando como sigue:\n\ncalculos_agregada |> pivot_wider(names_from = resultado, values_from = n) |> \n mutate(total = mejora + sin_mejora) |> \n mutate(prop_mejora = round(mejora / total, 2)) |> \n select(tratamiento, tamaño, total, prop_mejora) |> \n arrange(tamaño) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\ntotal\nprop_mejora\n\n\n\n\nA\nchicos\n87\n0.93\n\n\nB\nchicos\n270\n0.87\n\n\nA\ngrandes\n263\n0.73\n\n\nB\ngrandes\n80\n0.69\n\n\n\n\n\n\n\n\nEsta tabla descriptiva es una reescritura de los datos, y no hemos resumido nada todavía. Pero es apropiada para empezar a contestar la pregunta:\n\n¿Qué indican estos datos acerca de qué tratamiento es mejor? ¿Acerca del tamaño de cálculos grandes o chicos?\n\nSupongamos que otro analista decide comparar los pacientes que recibieron cada tratamiento, ignorando la variable de tamaño:\n\ncalculos |> group_by(tratamiento) |> \n summarise(prop_mejora = mean(resultado == \"mejora\") |> round(2)) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\nprop_mejora\n\n\n\n\nA\n0.78\n\n\nB\n0.83\n\n\n\n\n\n\n\n\ny parece ser que el tratamiento \\(B\\) es mejor que el \\(A\\). Esta es una paradoja (un ejemplo de la paradoja de Simpson) . Si un médico no sabe que tipo de cálculos tiene el paciente, ¿entonces debería recetar \\(B\\)? ¿Si sabe debería recetar \\(A\\)? Esta discusión parece no tener mucho sentido.\nPodemos investigar por qué está pasando esto considerando la siguiente tabla, que solo examina cómo se asignó el tratamiento dependiendo del tipo de cálculos de cada paciente:\n\ncalculos |> group_by(tratamiento, tamaño) |> count() |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\ntamaño\nn\n\n\n\n\nA\nchicos\n87\n\n\nA\ngrandes\n263\n\n\nB\nchicos\n270\n\n\nB\ngrandes\n80\n\n\n\n\n\n\n\n\nNuestra hipótesis aquí es que la decisión de qué tratamiento usar depende del tamaño de los cálculos. En este caso, hay una decisión pues A es una cirugía y B es un procedimiento menos invasivo, y se prefiere utilizar el tratamiento \\(A\\) para cálculos grandes, y \\(B\\) para cálculos chicos. Esto quiere decir que en la tabla total el tratamiento \\(A\\) está en desventaja porque se usa en casos más difíciles, pero el tratamiento \\(A\\) parece ser en general mejor. La razón es probablemente un proceso de optimización de recursos y riesgo que hacen los doctores.\n\nEn este caso, una mejor respuesta a la pregunta de qué tratamiento es mejor es la que presenta los datos desagregados.\nLa tabla desagregada de asignación del tratamiento nos informa acerca de cómo se está distribuyendo el tratamiento en los pacientes.\n\n\n\n\n\n\n\nNota\n\n\n\nLos resúmenes descriptivos acompañados de hipótesis causales acerca del proceso generador de datos, nos guía hacia descripciones interpretables de los datos.\n\n\nLas explicaciones no son tan simples y, otra vez, interviene el comportamiento de doctores, tratamientos, y distintos tipos de padecimientos.\nPodemos codificar la información causal con un diagrama:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n T \n M \n C\n edge [minlen = 3]\n T -> M\n C -> T\n C -> M\n{ rank = same; M; T }\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEs decir, el tamaño de los cálculos es una causa común de tratamiento (T) y resultado (M). Veremos más adelante que la decisión de condicionar a el tipo de cálculos proviene de un análisis relativamente simple de este diagrama causal, independientemente de los métodos que usemos para estimar las proporciones de interés (en este ejemplo, examinar las tablas cruzadas es equivalente a hacer estimaciones de máxima verosimlitud).\n\n\nEjemplo (cálculos renales 2)\nContrastemos el ejemplo anterior usando exactamente la misma tabla de datos, pero con el supuesto de un proceso generador diferente. En este caso, los tratamientos son para mejorar alguna enfermedad del corazón. Sabemos que parte del efecto de este tratamiento ocurre gracias a una baja en presión arterial de los pacientes, así que después de administrar el tratamiento, se toma la presión arterial de los pacientes. Ahora tenemos la tabla agregada y desagregada como sigue:\n\ncorazon <- calculos |> \n select(tratamiento, presión = tamaño, resultado) |> \n mutate(presión = ifelse(presión == \"grandes\", \"alta\", \"baja\"))\ncorazon_agregada <- corazon |> \n group_by(tratamiento, presión, resultado) |> \n count()\ncorazon_agregada |> pivot_wider(names_from = resultado, values_from = n) |> \n mutate(total = mejora + sin_mejora) |> \n mutate(prop_mejora = round(mejora / total, 2)) |> \n select(tratamiento, presión, total, prop_mejora) |> \n arrange(presión) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\npresión\ntotal\nprop_mejora\n\n\n\n\nA\nalta\n263\n0.73\n\n\nB\nalta\n80\n0.69\n\n\nA\nbaja\n87\n0.93\n\n\nB\nbaja\n270\n0.87\n\n\n\n\n\n\n\n\n\ncorazon |> group_by(tratamiento) |> \n summarise(prop_mejora = mean(resultado == \"mejora\") |> round(2)) |> \n kable() |> \n kable_paper(full_width = FALSE)\n\n\n\n\n\ntratamiento\nprop_mejora\n\n\n\n\nA\n0.78\n\n\nB\n0.83\n\n\n\n\n\n\n\n\n¿Cuál creemos que es el mejor tratamiento en este caso? ¿Deberíamos usar la tabla agregada o la desagregada por presión?\n\nEn este caso, la tabla agregada es más apropiada (B es mejor tratamiento).\nLa razón es que presión en este caso es una consecuencia de tomar el tratamiento, y como las tablas muestran, B es más exitoso en bajar la presión de los pacientes.\nSi sólo comparamos dentro de los grupos de presión baja o de presión alta, ignoramos lo más importante del tratamiento en la probabilidad de mejorar.\n\nNuestros supuestos causales podemos mostrarlos con el siguiente diagrama:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n P\n T \n M \n edge [minlen = 3]\n T -> P\n P -> M\n T -> M\n{ rank = same; M; T}\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nNótese que el análisis más apropiado no está en los datos: en ambos casos la tabla de datos es exactamente la misma. Los supuestos acerca del proceso que genera los datos sin embargo nos lleva a respuestas opuestas.", "crumbs": [ "1  Introducción" ] @@ -184,7 +184,7 @@ "href": "02-flujo-basico-2.html#prevalencia-con-datos-de-referencia", "title": "3  Flujo de trabajo básico: refinando el modelo", "section": "3.2 Prevalencia con datos de referencia", - "text": "3.2 Prevalencia con datos de referencia\nAhora haremos un paso adicional: los valores de sensibilidad y especificidad generalmente no son conocidos con certeza, sino que son estimados a partir de una muestra de “estándar de oro”. En esta prueba particular, el kit identificó correctamente como positivos a 103 de 122 personas infectadas, e identificó correctamente como negativos a 399 de 401 personas no infectadas. Consideraremos 122 y 401 como tamaños de muestra fijos y conocidos (las personas fueron extraídas de otra población).\nDenotamos como \\(Ref\\) a los datos de referencia de “estándar de oro”.\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.3, rankdir = LR]\n node [shape=circle]\n theta\n esp\n sens\n Npos [label = <N<SUB>+</SUB>>]\n node [shape=plaintext]\n Nobs [label = <N<SUB>obs</SUB>>]\n # Nneg [label = <N<SUB>-</SUB>>]\n edge [minlen = 3]\n theta -> Npos\n #p -> Nneg\n N -> Npos\n Npos -> Nobs\n #N -> Nneg\n esp -> Nobs\n sens -> Nobs\n #esp -> Nneg\n #sens -> Nneg\n esp -> Ref\n sens -> Ref\n{ rank = same; theta; N }\n#{ rank = same; Npos; Nneg}\n{ rank = max; sens; esp}\n}\n\")#, width = 200, height = 50)\n\n\n\n\n\n\nUsando argumentos como los del modelo original, las distribuciones de esp y sens son beta y podemos incorporarlas en la simulación de la posterior. Nuestra nueva función para simular el proceso generativo es:\n\nsim_pos_neg <- function(p = 0.01, N = 20, pos_gold = c(103,122), neg_gold = c(399,401)) {\n # Simular especificidad y sensibilidad\n sens <- rbeta(1, pos_gold[1] + 1, pos_gold[2] - pos_gold[1] + 1)\n esp <- rbeta(1, neg_gold[1] + 1, neg_gold[2] - neg_gold[1] + 1)\n # verdaderos positivos que capturamos en la muestra\n Pos_verdadero <- rbinom(N, 1, p)\n Neg_verdadero <- 1 - Pos_verdadero\n # positivos observados en la muestra: si es positivo, calculamos\n # la probabilidad de que realmente sea positivo\n sim_tbl <- tibble(Pos_verdadero, Neg_verdadero) |> \n mutate(Pos = rbinom(N, 1, Pos_verdadero * sens + Neg_verdadero * (1-esp))) |> \n mutate(Neg = 1 - Pos)\n # Observaciones\n sim_tbl |> select(Pos, Neg)\n}\n\nConsiderando que tenemos tres parámetros, en este punto decidimos no hacer la aproximación de rejilla. Es posible hacer otro tipo de aproximaciones (por ejemplo cuadráticas), pero en lugar de esto veremos cómo lo haríamos con Stan. Más adelante discutiremos los algoritmos que Stan utiliza para simular de la posterior de modelos muy generales. Por el momento, notamos que está basado en un algoritmo de simulación MCMC (Markov Chain Montecarlo), que es el estándar para modelos que no son muy simples. Este ejemplo es para ilustrar cómo resolveríamos el problema más general, no es necesario que en este punto entiendas cómo funciona o los detalles de la implementación.\n\nlibrary(cmdstanr)\nmod_sc <- cmdstan_model(\"./src/sclara.stan\")\nprint(mod_sc)\n\ndata {\n int<lower=0> N;\n int<lower=0> n;\n int<lower=0> kit_pos;\n int<lower=0> n_kit_pos;\n int<lower=0> kit_neg;\n int<lower=0> n_kit_neg;\n}\n\nparameters {\n real<lower=0, upper=1> theta; //seroprevalencia\n real<lower=0, upper=1> sens; //sensibilidad\n real<lower=0, upper=1> esp; //especificidad\n}\n\ntransformed parameters {\n real<lower=0, upper=1> prob_pos;\n\n prob_pos = theta * sens + (1 - theta) * (1 - esp);\n\n}\nmodel {\n // modelo de número de positivos\n n ~ binomial(N, prob_pos);\n // modelos para resultados del kit\n kit_pos ~ binomial(n_kit_pos, sens);\n kit_neg ~ binomial(n_kit_neg, esp);\n // iniciales para cantidades no medidas\n theta ~ beta(1.0, 10.0);\n sens ~ beta(2.0, 1.0);\n esp ~ beta(2.0, 1.0);\n}\n\n\n\nn <- 50\nN <- 3300\ndatos_lista <- list(N = 3300, n = 50,\n kit_pos = 103, n_kit_pos = 122,\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.7 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 theta 0.0104 0.00243 0.0174\n\n\nY podemos graficar la posterior de la seroprevalencia:\n\nggplot(sims, aes(x = theta)) + \n geom_histogram()\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY vemos que los datos son consistentes con el dato reportado por los autores (alrededor de 1.2%), pero que no podemos excluir valores de prevalencia muy bajos (por abajo de 0.3% por ejemplo). Por otro lado, también son consistentes valores muy altos de seroprevalencia, de manera que este estudio resultó ser poco informativo de la IFR del COVID.\nPodemos hacer diagnósticos adicionales acerca de la razón de esta variabilidad alta, si graficamos la relación entre especificidad de la prueba y estimación de prevalencia:\n\nggplot(sims, aes(x = esp, y = theta)) + geom_point() +\n xlab(\"Especificidad del kit\") + ylab(\"Prevalencia\") + geom_smooth()\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\n\n\n\n\n\n\n\nLa asociación entre estas dos cantidades es interesante porque conceptualmente (y desde punto de vista del modelo), no hay relación entre estas dos variables: su asociación aparece porque son causas que compiten para explicar una observación.\nNótese que dada la prevalencia baja, la especificidad del kit es un factor importante para explicar la prevalencia observada, pero si no pensamos con cuidado podríamos concluir que los falsos positivos no deberían ser problema por que la especificidad para ser muy buena.\nY notamos que aún con una muestra relativamente grande, el rango de \\(\\theta\\) es considerable: va desde valores cercanos a 0 hasta valores alrededor de 0.025-0.03.", + "text": "3.2 Prevalencia con datos de referencia\nAhora haremos un paso adicional: los valores de sensibilidad y especificidad generalmente no son conocidos con certeza, sino que son estimados a partir de una muestra de “estándar de oro”. En esta prueba particular, el kit identificó correctamente como positivos a 103 de 122 personas infectadas, e identificó correctamente como negativos a 399 de 401 personas no infectadas. Consideraremos 122 y 401 como tamaños de muestra fijos y conocidos (las personas fueron extraídas de otra población).\nDenotamos como \\(Ref\\) a los datos de referencia de “estándar de oro”.\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.3, rankdir = LR]\n node [shape=circle]\n theta\n esp\n sens\n Npos [label = <N<SUB>+</SUB>>]\n node [shape=plaintext]\n Nobs [label = <N<SUB>obs</SUB>>]\n # Nneg [label = <N<SUB>-</SUB>>]\n edge [minlen = 3]\n theta -> Npos\n #p -> Nneg\n N -> Npos\n Npos -> Nobs\n #N -> Nneg\n esp -> Nobs\n sens -> Nobs\n #esp -> Nneg\n #sens -> Nneg\n esp -> Ref\n sens -> Ref\n{ rank = same; theta; N }\n#{ rank = same; Npos; Nneg}\n{ rank = max; sens; esp}\n}\n\")#, width = 200, height = 50)\n\n\n\n\n\n\nUsando argumentos como los del modelo original, las distribuciones de esp y sens son beta y podemos incorporarlas en la simulación de la posterior. Nuestra nueva función para simular el proceso generativo es:\n\nsim_pos_neg <- function(p = 0.01, N = 20, pos_gold = c(103,122), neg_gold = c(399,401)) {\n # Simular especificidad y sensibilidad\n sens <- rbeta(1, pos_gold[1] + 1, pos_gold[2] - pos_gold[1] + 1)\n esp <- rbeta(1, neg_gold[1] + 1, neg_gold[2] - neg_gold[1] + 1)\n # verdaderos positivos que capturamos en la muestra\n Pos_verdadero <- rbinom(N, 1, p)\n Neg_verdadero <- 1 - Pos_verdadero\n # positivos observados en la muestra: si es positivo, calculamos\n # la probabilidad de que realmente sea positivo\n sim_tbl <- tibble(Pos_verdadero, Neg_verdadero) |> \n mutate(Pos = rbinom(N, 1, Pos_verdadero * sens + Neg_verdadero * (1-esp))) |> \n mutate(Neg = 1 - Pos)\n # Observaciones\n sim_tbl |> select(Pos, Neg)\n}\n\nConsiderando que tenemos tres parámetros, en este punto decidimos no hacer la aproximación de rejilla. Es posible hacer otro tipo de aproximaciones (por ejemplo cuadráticas), pero en lugar de esto veremos cómo lo haríamos con Stan. Más adelante discutiremos los algoritmos que Stan utiliza para simular de la posterior de modelos muy generales. Por el momento, notamos que está basado en un algoritmo de simulación MCMC (Markov Chain Montecarlo), que es el estándar para modelos que no son muy simples. Este ejemplo es para ilustrar cómo resolveríamos el problema más general, no es necesario que en este punto entiendas cómo funciona o los detalles de la implementación.\n\nlibrary(cmdstanr)\nmod_sc <- cmdstan_model(\"./src/sclara.stan\")\nprint(mod_sc)\n\ndata {\n int<lower=0> N;\n int<lower=0> n;\n int<lower=0> kit_pos;\n int<lower=0> n_kit_pos;\n int<lower=0> kit_neg;\n int<lower=0> n_kit_neg;\n}\n\nparameters {\n real<lower=0, upper=1> theta; //seroprevalencia\n real<lower=0, upper=1> sens; //sensibilidad\n real<lower=0, upper=1> esp; //especificidad\n}\n\ntransformed parameters {\n real<lower=0, upper=1> prob_pos;\n\n prob_pos = theta * sens + (1 - theta) * (1 - esp);\n\n}\nmodel {\n // modelo de número de positivos\n n ~ binomial(N, prob_pos);\n // modelos para resultados del kit\n kit_pos ~ binomial(n_kit_pos, sens);\n kit_neg ~ binomial(n_kit_neg, esp);\n // iniciales para cantidades no medidas\n theta ~ beta(1.0, 10.0);\n sens ~ beta(2.0, 1.0);\n esp ~ beta(2.0, 1.0);\n}\n\n\n\nn <- 50\nN <- 3300\ndatos_lista <- list(N = 3300, n = 50,\n kit_pos = 103, n_kit_pos = 122,\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.6 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 theta 0.0104 0.00243 0.0174\n\n\nY podemos graficar la posterior de la seroprevalencia:\n\nggplot(sims, aes(x = theta)) + \n geom_histogram()\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY vemos que los datos son consistentes con el dato reportado por los autores (alrededor de 1.2%), pero que no podemos excluir valores de prevalencia muy bajos (por abajo de 0.3% por ejemplo). Por otro lado, también son consistentes valores muy altos de seroprevalencia, de manera que este estudio resultó ser poco informativo de la IFR del COVID.\nPodemos hacer diagnósticos adicionales acerca de la razón de esta variabilidad alta, si graficamos la relación entre especificidad de la prueba y estimación de prevalencia:\n\nggplot(sims, aes(x = esp, y = theta)) + geom_point() +\n xlab(\"Especificidad del kit\") + ylab(\"Prevalencia\") + geom_smooth()\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\n\n\n\n\n\n\n\nLa asociación entre estas dos cantidades es interesante porque conceptualmente (y desde punto de vista del modelo), no hay relación entre estas dos variables: su asociación aparece porque son causas que compiten para explicar una observación.\nNótese que dada la prevalencia baja, la especificidad del kit es un factor importante para explicar la prevalencia observada, pero si no pensamos con cuidado podríamos concluir que los falsos positivos no deberían ser problema por que la especificidad para ser muy buena.\nY notamos que aún con una muestra relativamente grande, el rango de \\(\\theta\\) es considerable: va desde valores cercanos a 0 hasta valores alrededor de 0.025-0.03.", "crumbs": [ "3  Flujo de trabajo básico: refinando el modelo" ] @@ -314,7 +314,7 @@ "href": "05-dags.html#regla-del-producto-y-simulación", "title": "5  Modelos gráficos y causalidad", "section": "5.3 Regla del producto y simulación", - "text": "5.3 Regla del producto y simulación\nEl orden del modelo gráfico también nos indica cómo simular las variables de la gráfica. Como cada modelo gráfico nos da una factorización de la conjunta, podemos utlizar esta para simular datos una vez que conocemos o estimamos las relaciones de dependencia directa. Empezamos con las variables exógenas (que no tienen padres) y vamos simulando hacia adelante.\n\nEjemplo\nEn nuestro ejemplo simulamos primero \\(X\\) y \\(D\\). A partir de \\(X\\) podemos simular \\(X_1\\) y \\(S_2\\), y a partir de \\(D\\), junto con \\(S_1\\) y \\(S_2\\), podemos simular \\(G\\). En nuestro ejemplo tendríamos\n\nsimular_juego <- function(N){\n x <- runif(N)\n d <- sample(c(\"lluvioso\",\"soleado\"), N, replace = TRUE, prob = c(0.3,0.7))\n s1 <- rbinom(N, 5, x)\n s2 <- rbinom(N, 5, x)\n g <- ifelse(d==\"lluvioso\", s1+s2, s1)\n tibble(x, d, s1, s2, g)\n}\nsimular_juego(5)\n\n# A tibble: 5 × 5\n x d s1 s2 g\n <dbl> <chr> <int> <int> <int>\n1 0.234 soleado 1 0 1\n2 0.281 soleado 3 3 3\n3 0.475 soleado 4 3 4\n4 0.944 lluvioso 5 5 10\n5 0.917 soleado 4 5 4", + "text": "5.3 Regla del producto y simulación\nEl orden del modelo gráfico también nos indica cómo simular las variables de la gráfica. Como cada modelo gráfico nos da una factorización de la conjunta, podemos utlizar esta para simular datos una vez que conocemos o estimamos las relaciones de dependencia directa. Empezamos con las variables exógenas (que no tienen padres) y vamos simulando hacia adelante.\n\nEjemplo\nEn nuestro ejemplo simulamos primero \\(X\\) y \\(D\\). A partir de \\(X\\) podemos simular \\(X_1\\) y \\(S_2\\), y a partir de \\(D\\), junto con \\(S_1\\) y \\(S_2\\), podemos simular \\(G\\). En nuestro ejemplo tendríamos\n\nsimular_juego <- function(N){\n x <- runif(N)\n d <- sample(c(\"lluvioso\",\"soleado\"), N, replace = TRUE, prob = c(0.3,0.7))\n s1 <- rbinom(N, 5, x)\n s2 <- rbinom(N, 5, x)\n g <- ifelse(d==\"lluvioso\", s1+s2, s1)\n tibble(x, d, s1, s2, g)\n}\nsimular_juego(5)\n\n# A tibble: 5 × 5\n x d s1 s2 g\n <dbl> <chr> <int> <int> <int>\n1 0.737 soleado 4 5 4\n2 0.713 lluvioso 4 4 8\n3 0.601 lluvioso 2 3 5\n4 0.575 soleado 4 3 4\n5 0.582 soleado 4 3 4", "crumbs": [ "5  Modelos gráficos y causalidad" ] @@ -334,7 +334,7 @@ "href": "05-dags.html#bifurcaciones-o-causa-común", "title": "5  Modelos gráficos y causalidad", "section": "5.5 Bifurcaciones o causa común", - "text": "5.5 Bifurcaciones o causa común\nEn el siguiente ejemplo, llamamos a \\(Z\\) una causa que es común a \\(X\\) y \\(Y\\).\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n Z -> X\n Z -> Y\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEn este caso,\n\n\\(X\\) y \\(Y\\) tienen asociación\nSi condicionamos (o estratificamos) con \\(Z\\), entonces \\(X\\) y \\(Y\\) son condicionalmente independientes.\n\nEste tipo de estructura también se llama bifurcación, o decimos más tradicionalmente que \\(Z\\) es un confusor en esta gráfica. Variación en \\(Z\\) produce variación conjunta de \\(X\\) y \\(Y\\).\nPor ejemplo, podríamos encontrar que el uso de aspirina \\(X\\) está asociado a una mortalidad más alta \\(Y\\). Una causa común es enfermedad grave que produce dolor (\\(Z\\)). Sin embargo, si condicionamos a personas sanas, veríamos que no hay relación entre uso de aspirina y mortalidad, igualmente veríamos que entre las personas enfermas el uso de aspirina no les ayuda a vivir más tiempo.\nEn este caso, tenemos:\n\\[p(x, y, z) = p(z)p(x|z)p(y|z)\\] Y como el lado izquierdo es igual (en general) a \\(p(x,y|z)p(z)\\), obtenemos la independiencia condicional de \\(X\\) y \\(Y\\) dado \\(Z\\).\n\nEjemplo (simulación)\n\nrbern <- function(n, prob){\n rbinom(n, 1, prob = prob)\n} \nsimular_confusor <- function(n = 10){\n z <- rbern(n, p = 0.5) |> as.numeric()\n x <- rbern(n, p = z * 0.3 + (1 - z) * 0.8)\n y <- rbinom(n, 4, z * 0.9 + (1 - z) * 0.3)\n tibble(x, z, y)\n}\nsims_confusor <- simular_confusor(50000)\n\n\\(X\\) y \\(Y\\) están asociadas\n\nsims_confusor |> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\n\nLo cual lo vemos también si calculamos la correlación:\n\ncor(sims_confusor |> select(x,y)) |> round(3)\n\n x y\nx 1.00 -0.43\ny -0.43 1.00\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1, vemos que \\(X\\) y \\(Y\\) son independientes, o dicho de otra manera, la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) sólo depende de \\(Z\\):\n\nsims_confusor |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nUna consecuencia es por ejemplo que la correlación debe ser cero:\n\ncor(sims_confusor |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.009\ny -0.009 1.000\n\ncor(sims_confusor |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.006\ny -0.006 1.000\n\n\nUn ejemplo con variables continuas podría ser como sigue:\n\nsimular_bifurcacion <- function(n = 10){\n z <- rbern(n, p = 0.5)\n x <- rnorm(n, 100 + 20 * z, 15)\n y <- rnorm(n, 100 + 30 * z, 20)\n tibble(x, z, y)\n}\nsims_bifurcacion <- simular_bifurcacion(5000)\n\n\\(X\\) y \\(Y\\) son dependientes (por ejemplo si vemos la media condicional de \\(Y\\) dado \\(X\\):\n\nggplot(sims_bifurcacion, aes(x = x, y = y, colour = z)) + \n geom_point(alpha = 0.2) +\n geom_smooth(span = 1, se = FALSE)\n\n\n\n\n\n\n\n\nSi condicionamos a \\(Z\\), no hay dependencia entre \\(X\\) y \\(Y\\)\n\nggplot(sims_bifurcacion, aes(x = x, y = y, colour = z, group = z)) + \n geom_point(alpha = 0.2) +\n geom_smooth(span = 2)\n\n\n\n\n\n\n\n\n\n\nEjemplo: matrimonio y divorcio\nEn este ejemplo de McElreath (2020), se muestra que regiones de Estados Unidos con tasas más altas de matrimonio también tienen tasas más altas de divorcio.\n\ndata(WaffleDivorce)\nWaffleDivorce |> \n ggplot(aes(x = Marriage, y = Divorce)) +\n geom_point() +\n geom_smooth(method = \"lm\")\n\n`geom_smooth()` using formula = 'y ~ x'\n\n\n\n\n\n\n\n\n\nAunque esta es una correlación clara, lo que nos interesa en este caso el efecto causal \\(M\\to D\\). Es importante notar que hay considerable variabilidad de la edad promedio al casarse a lo largo de los estados:\n\nWaffleDivorce |> \n ggplot(aes(sample = MedianAgeMarriage)) +\n geom_qq() +\n geom_qq_line()\n\n\n\n\n\n\n\n\nPara el modelo causal, tenemos que considerar las siguientes afirmaciones que no son muy difíciles de justificar:\n\nLa edad promedio al casarse de cada estado es un factor que influye en la tasa de divorcio (menor edad a casarse implica mayores tasas de divorcio, pues las parejas tienen más tiempo para divorciarse, porque la gente cambia más cuando es joven).\nAdicionalmente, si la gente tiende a casarse más joven, en cualquier momento hay más gente con probabilidad de casarse, por lo que esperaríamos que la edad al casarse también influye en la tasa de matrimonio.\n\nEsto implica que tenemos que considerar una causa común de la edad al casarse en nuestro diagrama causal:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n M\n D\n Edad\n edge [minlen = 3]\n Edad -> M\n Edad -> D\n M -> D\n{rank=same; M; D;}\n\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nPor la discusión de arriba, es claro que es necesario considerar la edad al casarse si queremos estimar el efecto de tasa de matrimonio en la tasa de divorcio. Es posible que la correlación entre estas dos tasas puede ser explicada solamente por la edad al casarse, y que en realidad al flecha \\(M\\to D\\) sea muy débil o inexistente.\nYa que tenemos este modelo causal básico, tendríamos que proponer un proceso generador, proponer un modelo estadístico, y probar nuestra estimación. Este paso nos lo saltaremos (ver sección anterior), aunque sigue siendo necesario.\nPor el momento recordemos que si condicionamos (se dice también estratificar) por edad al casarse, y no vemos relación condicional entre las dos tasas, la relación que vimos en los datos es factible que haya aparecido por la causa común que induce correlación. Una manera en que estratificamos o condicionamos a una variable continua en un modelo lineal, como sigue:\n\\[D_i\\sim N(\\mu_i, \\sigma)\\] donde \\[\\mu_i = \\alpha + \\beta_M M_i + \\beta_E Edad_i\\] ¿De qué manera estamos estratificando por edad en este ejemplo? Obsérvese que para cada Edad que fijemos, la relación entre \\(M\\) y \\(D\\) es:\n\\[\\mu_i = (\\alpha + \\beta_E Edad) + \\beta_M M_i \\] Cada valor de \\(E\\) produce una relación diferente entre \\(M\\) y \\(D\\) (en este caso particular, una recta diferente con distinta altura).\nAhora tenemos que poner iniciales para terminar nuestro modelo estadístico. En este punto poner iniciales informadas para estos coeficientes puede ser complicado (depende de cuánta demografía sabemos). Podemos usar un enfoque más simple, considerando las variables estandarizadas. De esta forma podemos poner iniciales más estándar. Utilizaremos\n\nescalar <- function(x){\n (x - mean(x))/sd(x)\n}\nWaffleDivorce <- WaffleDivorce |> \n mutate(Marriage_est = escalar(Marriage), \n Divorce_est = escalar(Divorce), \n MedianAgeMarriage_est = escalar(MedianAgeMarriage))\ndatos_lista <- list(\n N = nrow(WaffleDivorce),\n d_est = WaffleDivorce$Divorce_est, \n m_est = WaffleDivorce$Marriage_est, \n edad_est = WaffleDivorce$MedianAgeMarriage_est)\n\n\nmod_mat_div <- cmdstan_model(\"./src/matrimonio-divorcio-1.stan\")\nprint(mod_mat_div)\n\ndata {\n int<lower=0> N;\n vector[N] d_est;\n vector[N] m_est;\n vector[N] edad_est;\n}\n\nparameters {\n real alpha;\n real beta_M;\n real beta_E;\n real <lower=0> sigma;\n}\n\ntransformed parameters {\n vector[N] w_media;\n // determinístico dado parámetros\n w_media = alpha + beta_M * m_est + beta_E * edad_est;\n}\n\nmodel {\n // partes no determinísticas\n d_est ~ normal(w_media, sigma);\n alpha ~ normal(0, 1);\n beta_M ~ normal(0, 0.5);\n beta_E ~ normal(0, 0.5);\n sigma ~ normal(0, 1);\n}\n\ngenerated quantities {\n real dif;\n {\n //simulamos 50 estados\n int M = 50;\n array[M] real dif_sim;\n for(i in 1:M){\n real edad_sim_est = normal_rng(0, 1);\n // fijamos el valor de M en 0 y 1 para el modelo con do(M)\n real M_sim_0 = normal_rng(alpha * beta_M * 0 + beta_E * edad_sim_est, sigma);\n real M_sim_1 = normal_rng(alpha * beta_M * 1 + beta_E * edad_sim_est, sigma);\n dif_sim[i] = M_sim_1 - M_sim_0;\n }\n dif = mean(dif_sim);\n }\n\n}\n\n\n\nsims_mod <- mod_mat_div$sample(data = datos_lista, \n chains = 4, \n init = 0.1, step_size = 0.1,\n iter_warmup = 1000, \n iter_sampling = 1000,\n refresh = 0)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 finished in 0.1 seconds.\nChain 2 finished in 0.1 seconds.\nChain 3 finished in 0.1 seconds.\nChain 4 finished in 0.1 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.1 seconds.\nTotal execution time: 0.6 seconds.\n\n\n\nresumen <- sims_mod$summary(c(\"alpha\", \"beta_M\", \"beta_E\", \"sigma\"))\n\n\nresumen |> \n ggplot(aes(x = variable, y = mean, ymin = q5, ymax = q95)) +\n geom_hline(yintercept = 0, color = \"red\") +\n geom_point() +\n geom_linerange() +\n coord_flip()\n\n\n\n\n\n\n\n\nY el resultado que obtenemos es que no observamos un efecto considerable de las tasas de matrimonio en las tasas de divorcio, una vez que estratificamos por la causa común de edad de matrimonio. Este ejemplo es simple y podemos ver el efecto causal directo en un sólo coeficiente \\(\\beta_M\\), pero de todas formas haremos contrastes como hicimos en la parte anterior.\n\n\n5.5.1 Simulando intervenciones\nLa manera más directa de definir efecto causal, bajo nuestros supuestos causales, es a través de intervenciones (imaginarias o reales).\n\n\n\n\n\n\nNota\n\n\n\nEntendemos saber una causa como poder predecir correctamente las consecuencias de una intervención en el sistema generador de datos.\n\n\nEn nuestro caso, el diagrama de arriba muestra nuestro modelo causal. Si nosotros alteramos este proceso causal, interviniendo en la tasa de matrimonio, la distribución de matrimonio ya no depende de la Edad (pues está bajo nuestro control). Esto quiere decir que ahora consideramos el siguiente diagrama, en donde la nueva dependendencia del divorcio del matrimonio la escribiremos como \\(p(D|do(M))\\):\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n M\n D\n Edad\n edge [minlen = 3]\n Edad -> D\n M -> D\n{rank=same; M; D;}\n\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEs decir, borramos todas las flechas que caen en \\(M\\) (pues la estamos interveniendo al valor que queramos), y luego simulando \\(D\\).\nEn nuestro ejemplo (ve el código de Stan de arriba, la parte de generated quantities) simularemos los 50 estados bajo dos intervenciones: todos tienen la tasa promedio de matrimonio vs. los 50 estados con tasa de matrimonio un error estándar por encima de la tasa promedio. Repetimos esta comparación sobre todas las simulaciones de la posterior:\n\nsims_tbl <- sims_mod$draws(format = \"df\") |> \n select(dif) \nsims_tbl |> summarize(\n q5 = quantile(dif, 0.05),\n q95 = quantile(dif, 0.95)\n)\n\n# A tibble: 1 × 2\n q5 q95\n <dbl> <dbl>\n1 -0.278 0.278\n\n\n\nggplot(sims_tbl, aes(x = dif)) +\n geom_histogram(bins = 50) +\n geom_vline(xintercept = 0, color = \"red\")\n\n\n\n\n\n\n\n\nEn este caso, vemos que el resultado de la intervención no tienen una tendencia clara hacia incrementar o disminuir la tasa de divorcio, aunque existe variabilidad por la incertidumbre que tenemos acerca de las relaciones modeladas.\n\n\n\n\n\n\nTip\n\n\n\nLa relación que vimos entre matrimonio y divorcio en nuestro ejemplo es probablemente producida por la causa común Edad, y no necesariamente es causal.\n\n\nFinalmente, antes de terminar sería apropiado hacer chequeos predictivos posteriores, pero por el momento los omitiremos para avanzar en los otros tipos de estructuras básicas en los DAGs.", + "text": "5.5 Bifurcaciones o causa común\nEn el siguiente ejemplo, llamamos a \\(Z\\) una causa que es común a \\(X\\) y \\(Y\\).\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n Z -> X\n Z -> Y\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEn este caso,\n\n\\(X\\) y \\(Y\\) tienen asociación\nSi condicionamos (o estratificamos) con \\(Z\\), entonces \\(X\\) y \\(Y\\) son condicionalmente independientes.\n\nEste tipo de estructura también se llama bifurcación, o decimos más tradicionalmente que \\(Z\\) es un confusor en esta gráfica. Variación en \\(Z\\) produce variación conjunta de \\(X\\) y \\(Y\\).\nPor ejemplo, podríamos encontrar que el uso de aspirina \\(X\\) está asociado a una mortalidad más alta \\(Y\\). Una causa común es enfermedad grave que produce dolor (\\(Z\\)). Sin embargo, si condicionamos a personas sanas, veríamos que no hay relación entre uso de aspirina y mortalidad, igualmente veríamos que entre las personas enfermas el uso de aspirina no les ayuda a vivir más tiempo.\nEn este caso, tenemos:\n\\[p(x, y, z) = p(z)p(x|z)p(y|z)\\] Y como el lado izquierdo es igual (en general) a \\(p(x,y|z)p(z)\\), obtenemos la independiencia condicional de \\(X\\) y \\(Y\\) dado \\(Z\\).\n\nEjemplo (simulación)\n\nrbern <- function(n, prob){\n rbinom(n, 1, prob = prob)\n} \nsimular_confusor <- function(n = 10){\n z <- rbern(n, p = 0.5) |> as.numeric()\n x <- rbern(n, p = z * 0.3 + (1 - z) * 0.8)\n y <- rbinom(n, 4, z * 0.9 + (1 - z) * 0.3)\n tibble(x, z, y)\n}\nsims_confusor <- simular_confusor(50000)\n\n\\(X\\) y \\(Y\\) están asociadas\n\nsims_confusor |> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\n\nLo cual lo vemos también si calculamos la correlación:\n\ncor(sims_confusor |> select(x,y)) |> round(3)\n\n x y\nx 1.00 -0.42\ny -0.42 1.00\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1, vemos que \\(X\\) y \\(Y\\) son independientes, o dicho de otra manera, la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) sólo depende de \\(Z\\):\n\nsims_confusor |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nUna consecuencia es por ejemplo que la correlación debe ser cero:\n\ncor(sims_confusor |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 0.008\ny 0.008 1.000\n\ncor(sims_confusor |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1 0\ny 0 1\n\n\nUn ejemplo con variables continuas podría ser como sigue:\n\nsimular_bifurcacion <- function(n = 10){\n z <- rbern(n, p = 0.5)\n x <- rnorm(n, 100 + 20 * z, 15)\n y <- rnorm(n, 100 + 30 * z, 20)\n tibble(x, z, y)\n}\nsims_bifurcacion <- simular_bifurcacion(5000)\n\n\\(X\\) y \\(Y\\) son dependientes (por ejemplo si vemos la media condicional de \\(Y\\) dado \\(X\\):\n\nggplot(sims_bifurcacion, aes(x = x, y = y, colour = z)) + \n geom_point(alpha = 0.2) +\n geom_smooth(span = 1, se = FALSE)\n\n\n\n\n\n\n\n\nSi condicionamos a \\(Z\\), no hay dependencia entre \\(X\\) y \\(Y\\)\n\nggplot(sims_bifurcacion, aes(x = x, y = y, colour = z, group = z)) + \n geom_point(alpha = 0.2) +\n geom_smooth(span = 2)\n\n\n\n\n\n\n\n\n\n\nEjemplo: matrimonio y divorcio\nEn este ejemplo de McElreath (2020), se muestra que regiones de Estados Unidos con tasas más altas de matrimonio también tienen tasas más altas de divorcio.\n\ndata(WaffleDivorce)\nWaffleDivorce |> \n ggplot(aes(x = Marriage, y = Divorce)) +\n geom_point() +\n geom_smooth(method = \"lm\")\n\n`geom_smooth()` using formula = 'y ~ x'\n\n\n\n\n\n\n\n\n\nAunque esta es una correlación clara, lo que nos interesa en este caso el efecto causal \\(M\\to D\\). Es importante notar que hay considerable variabilidad de la edad promedio al casarse a lo largo de los estados:\n\nWaffleDivorce |> \n ggplot(aes(sample = MedianAgeMarriage)) +\n geom_qq() +\n geom_qq_line()\n\n\n\n\n\n\n\n\nPara el modelo causal, tenemos que considerar las siguientes afirmaciones que no son muy difíciles de justificar:\n\nLa edad promedio al casarse de cada estado es un factor que influye en la tasa de divorcio (menor edad a casarse implica mayores tasas de divorcio, pues las parejas tienen más tiempo para divorciarse, porque la gente cambia más cuando es joven).\nAdicionalmente, si la gente tiende a casarse más joven, en cualquier momento hay más gente con probabilidad de casarse, por lo que esperaríamos que la edad al casarse también influye en la tasa de matrimonio.\n\nEsto implica que tenemos que considerar una causa común de la edad al casarse en nuestro diagrama causal:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n M\n D\n Edad\n edge [minlen = 3]\n Edad -> M\n Edad -> D\n M -> D\n{rank=same; M; D;}\n\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nPor la discusión de arriba, es claro que es necesario considerar la edad al casarse si queremos estimar el efecto de tasa de matrimonio en la tasa de divorcio. Es posible que la correlación entre estas dos tasas puede ser explicada solamente por la edad al casarse, y que en realidad al flecha \\(M\\to D\\) sea muy débil o inexistente.\nYa que tenemos este modelo causal básico, tendríamos que proponer un proceso generador, proponer un modelo estadístico, y probar nuestra estimación. Este paso nos lo saltaremos (ver sección anterior), aunque sigue siendo necesario.\nPor el momento recordemos que si condicionamos (se dice también estratificar) por edad al casarse, y no vemos relación condicional entre las dos tasas, la relación que vimos en los datos es factible que haya aparecido por la causa común que induce correlación. Una manera en que estratificamos o condicionamos a una variable continua en un modelo lineal, como sigue:\n\\[D_i\\sim N(\\mu_i, \\sigma)\\] donde \\[\\mu_i = \\alpha + \\beta_M M_i + \\beta_E Edad_i\\] ¿De qué manera estamos estratificando por edad en este ejemplo? Obsérvese que para cada Edad que fijemos, la relación entre \\(M\\) y \\(D\\) es:\n\\[\\mu_i = (\\alpha + \\beta_E Edad) + \\beta_M M_i \\] Cada valor de \\(E\\) produce una relación diferente entre \\(M\\) y \\(D\\) (en este caso particular, una recta diferente con distinta altura).\nAhora tenemos que poner iniciales para terminar nuestro modelo estadístico. En este punto poner iniciales informadas para estos coeficientes puede ser complicado (depende de cuánta demografía sabemos). Podemos usar un enfoque más simple, considerando las variables estandarizadas. De esta forma podemos poner iniciales más estándar. Utilizaremos\n\nescalar <- function(x){\n (x - mean(x))/sd(x)\n}\nWaffleDivorce <- WaffleDivorce |> \n mutate(Marriage_est = escalar(Marriage), \n Divorce_est = escalar(Divorce), \n MedianAgeMarriage_est = escalar(MedianAgeMarriage))\ndatos_lista <- list(\n N = nrow(WaffleDivorce),\n d_est = WaffleDivorce$Divorce_est, \n m_est = WaffleDivorce$Marriage_est, \n edad_est = WaffleDivorce$MedianAgeMarriage_est)\n\n\nmod_mat_div <- cmdstan_model(\"./src/matrimonio-divorcio-1.stan\")\nprint(mod_mat_div)\n\ndata {\n int<lower=0> N;\n vector[N] d_est;\n vector[N] m_est;\n vector[N] edad_est;\n}\n\nparameters {\n real alpha;\n real beta_M;\n real beta_E;\n real <lower=0> sigma;\n}\n\ntransformed parameters {\n vector[N] w_media;\n // determinístico dado parámetros\n w_media = alpha + beta_M * m_est + beta_E * edad_est;\n}\n\nmodel {\n // partes no determinísticas\n d_est ~ normal(w_media, sigma);\n alpha ~ normal(0, 1);\n beta_M ~ normal(0, 0.5);\n beta_E ~ normal(0, 0.5);\n sigma ~ normal(0, 1);\n}\n\ngenerated quantities {\n real dif;\n {\n //simulamos 50 estados\n int M = 50;\n array[M] real dif_sim;\n for(i in 1:M){\n real edad_sim_est = normal_rng(0, 1);\n // fijamos el valor de M en 0 y 1 para el modelo con do(M)\n real M_sim_0 = normal_rng(alpha * beta_M * 0 + beta_E * edad_sim_est, sigma);\n real M_sim_1 = normal_rng(alpha * beta_M * 1 + beta_E * edad_sim_est, sigma);\n dif_sim[i] = M_sim_1 - M_sim_0;\n }\n dif = mean(dif_sim);\n }\n\n}\n\n\n\nsims_mod <- mod_mat_div$sample(data = datos_lista, \n chains = 4, \n init = 0.1, step_size = 0.1,\n iter_warmup = 1000, \n iter_sampling = 1000,\n refresh = 0)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 finished in 0.1 seconds.\nChain 2 finished in 0.1 seconds.\nChain 3 finished in 0.1 seconds.\nChain 4 finished in 0.1 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.1 seconds.\nTotal execution time: 0.6 seconds.\n\n\n\nresumen <- sims_mod$summary(c(\"alpha\", \"beta_M\", \"beta_E\", \"sigma\"))\n\n\nresumen |> \n ggplot(aes(x = variable, y = mean, ymin = q5, ymax = q95)) +\n geom_hline(yintercept = 0, color = \"red\") +\n geom_point() +\n geom_linerange() +\n coord_flip()\n\n\n\n\n\n\n\n\nY el resultado que obtenemos es que no observamos un efecto considerable de las tasas de matrimonio en las tasas de divorcio, una vez que estratificamos por la causa común de edad de matrimonio. Este ejemplo es simple y podemos ver el efecto causal directo en un sólo coeficiente \\(\\beta_M\\), pero de todas formas haremos contrastes como hicimos en la parte anterior.\n\n\n5.5.1 Simulando intervenciones\nLa manera más directa de definir efecto causal, bajo nuestros supuestos causales, es a través de intervenciones (imaginarias o reales).\n\n\n\n\n\n\nNota\n\n\n\nEntendemos saber una causa como poder predecir correctamente las consecuencias de una intervención en el sistema generador de datos.\n\n\nEn nuestro caso, el diagrama de arriba muestra nuestro modelo causal. Si nosotros alteramos este proceso causal, interviniendo en la tasa de matrimonio, la distribución de matrimonio ya no depende de la Edad (pues está bajo nuestro control). Esto quiere decir que ahora consideramos el siguiente diagrama, en donde la nueva dependendencia del divorcio del matrimonio la escribiremos como \\(p(D|do(M))\\):\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n M\n D\n Edad\n edge [minlen = 3]\n Edad -> D\n M -> D\n{rank=same; M; D;}\n\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEs decir, borramos todas las flechas que caen en \\(M\\) (pues la estamos interveniendo al valor que queramos), y luego simulando \\(D\\).\nEn nuestro ejemplo (ve el código de Stan de arriba, la parte de generated quantities) simularemos los 50 estados bajo dos intervenciones: todos tienen la tasa promedio de matrimonio vs. los 50 estados con tasa de matrimonio un error estándar por encima de la tasa promedio. Repetimos esta comparación sobre todas las simulaciones de la posterior:\n\nsims_tbl <- sims_mod$draws(format = \"df\") |> \n select(dif) \nsims_tbl |> summarize(\n q5 = quantile(dif, 0.05),\n q95 = quantile(dif, 0.95)\n)\n\n# A tibble: 1 × 2\n q5 q95\n <dbl> <dbl>\n1 -0.280 0.274\n\n\n\nggplot(sims_tbl, aes(x = dif)) +\n geom_histogram(bins = 50) +\n geom_vline(xintercept = 0, color = \"red\")\n\n\n\n\n\n\n\n\nEn este caso, vemos que el resultado de la intervención no tienen una tendencia clara hacia incrementar o disminuir la tasa de divorcio, aunque existe variabilidad por la incertidumbre que tenemos acerca de las relaciones modeladas.\n\n\n\n\n\n\nTip\n\n\n\nLa relación que vimos entre matrimonio y divorcio en nuestro ejemplo es probablemente producida por la causa común Edad, y no necesariamente es causal.\n\n\nFinalmente, antes de terminar sería apropiado hacer chequeos predictivos posteriores, pero por el momento los omitiremos para avanzar en los otros tipos de estructuras básicas en los DAGs.", "crumbs": [ "5  Modelos gráficos y causalidad" ] @@ -344,7 +344,7 @@ "href": "05-dags.html#cadenas-o-mediación", "title": "5  Modelos gráficos y causalidad", "section": "5.6 Cadenas o mediación", - "text": "5.6 Cadenas o mediación\nEn este caso tenemos:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir=LR]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n X -> Z\n Z -> Y\n}\n\", width = 150, height = 20)\n\n\n\n\n\n\nEn este caso,\n\nExiste asociación entre \\(X\\) y \\(Y\\), pero no existe relación directa entre ellas. Decimos que \\(Z\\) es un mediador del efecto de \\(X\\) sobre \\(Y\\).\nSi condicionamos a un valor de \\(Z\\), \\(X\\) y \\(Y\\) son condicionalmente independientes.\n\nPodemos pensar en \\(Z\\) como un mediador del efecto de \\(X\\) sobre \\(Y\\). Si no permitimos que \\(Z\\) varíe, entonces la información de \\(X\\) no fluye a \\(Y\\).\nPor ejemplo, si \\(X\\) tomar o no una medicina para el dolor de cabeza, \\(Z\\) es dolor de cabeza y \\(Y\\) es bienestar general, \\(X\\) y \\(Y\\) están relacionadas. Sin embargo, si condicionamos a un valor fijo de dolor de cabeza, no hay relación entre tomar la medicina y bienestar general.\nEn términos de factorización, podemos checar la independencia condicional: como \\(p(x,y,z) = p(x)p(z|x)p(y|z)\\), entonces\n\\[p(x, y | z) = p(x,y,z) / p(z) = (p(x)(z|x)) (p(y|z) / p(z))\\] y vemos que el lado izquierdo se factoriza en una parte que sólo involucra a \\(x\\) y \\(z\\) y otro factor que sólo tiene a \\(y\\) y \\(z\\): no hay términos que incluyan conjuntamente a \\(x\\), \\(y\\) y \\(z\\). Podemos de cualquier forma continuar notando\n\\[p(x)p(z|x)/p(z) = p(x,z)/p(z) = p(x | z)\\] de modo que\n\\[p(x, y | z) = p(x|z) p(y|z) \\]\nY mostramos un ejemplo simulado:\n\nrbern <- function(n, prob){\n rbinom(n, 1, prob = prob)\n} \nsimular_mediador <- function(n = 10){\n x <- rbern(n, p = 0.5) |> as.numeric()\n z <- rbern(n, p = x * 0.8 + (1 - x) * 0.3)\n y <- rbinom(n, 2, z * 0.7 + (1 - z) * 0.5)\n tibble(x, z, y)\n}\nsims_mediador <- simular_mediador(50000)\n\n\\(X\\) y \\(Y\\) son dependientes:\n\nsims_mediador |> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1:\n\nsims_mediador |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nY vemos que la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) sólo depende de \\(Z\\). Una consecuencia es por ejemplo que la correlación debe ser cero:\n\ncor(sims_mediador |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 0.004\ny 0.004 1.000\n\ncor(sims_mediador |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.002\ny -0.002 1.000\n\n\nPodemos también hacer un ejemplo continuo:\n\nsimular_mediador <- function(n = 10){\n x <- rnorm(n, 100, 10)\n prob <- 1 / (1 + exp(-(x - 100)/5))\n z <- rbern(n, p = prob)\n y <- rnorm(n, 100 + 30 * z, 15)\n tibble(x, z, y)\n}\nsims_mediador <- simular_mediador(2000)\n\n\\(X\\) y \\(Y\\) son dependientes (por ejemplo si vemos la media condicional de \\(Y\\) dado \\(X\\):\n\nggplot(sims_mediador, aes(x = x, y = y, colour = z)) + geom_point() +\n geom_smooth(span = 1, se = FALSE)\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\nWarning: The following aesthetics were dropped during statistical transformation: colour\nℹ This can happen when ggplot fails to infer the correct grouping structure in\n the data.\nℹ Did you forget to specify a `group` aesthetic or to convert a numerical\n variable into a factor?\n\n\n\n\n\n\n\n\n\nSi condicionamos a \\(Z\\), no hay dependencia entre \\(X\\) y \\(Y\\)\n\nggplot(sims_mediador, aes(x = x, y = y, colour = z, group = z)) + \n geom_point() +\n geom_smooth(span = 2)\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\n\n\n\n\n\n\n\nNótese que en este ejemplo sí hay un efecto causal de \\(X\\) sobre \\(Y\\), pero está mediado por otra variable \\(Z\\). Si condicionamos a \\(Z\\), no hay relación entre \\(X\\) y \\(Y\\). El análisis condicionado podría llevarnos a una conclusión errónea de que \\(X\\) no influye sobre \\(Y\\).\n\n\n\n\n\n\nTip\n\n\n\nNota que no existe una diferencia estadística entre una bifurcación y una cadena: en ambos casos, las variables \\(X\\) y \\(Y\\) están correlacionadas, y son independientes una vez que condicionamos o estratificamos por \\(Z\\). Sin embargo, su tratamiento en inferencia causal es muy diferente.\n\n\n\nSesgo post-tratamiento\nEn McElreath (2020) se discute que en algunos estudios experimentales, se estratifica por variables que son consecuencia del tratamiento. Esto induce sesgo post-tratamiento, lo cual puede llevar a equivocaciones en donde parece que el tratamiento no tiene efecto cuando sí lo tiene. Incluso bajo condiciones de experimento (donde el tratamiento es asignado al azar) estratificar por mediadores es una mala idea. Ver más en McElreath (2020), donde por ejemplo cita una fuente que en estudios experimentales de Ciencia Política, casi la mitad de ellos sufre de este tipo de sesgo por estratificación por mediadores.\n\n\nEjemplo: Burks\nEste ejemplo es de Pearl y Mackenzie (2018). En 1926 Burks recolectó datos sobre qué tanto podría esperarse que la inteligencia de padres se hereda a los hijos (medido según una prueba de IQ). Construyó un diagrama parecido al de abajo:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = circle]\n U\n node [shape=plaintext]\n edge [minlen = 3]\n IntPadres -> NSE\n NSE -> IntHijos\n U -> NSE\n U -> IntHijos\n IntPadres -> IntHijos\n{rank = same; U}\n}\n\")\n\n\n\n\n\n\nComo el NSE es del hogar (una medida general de estatus social), se consideró en principio como una variable pre-tratamiento a la inteligencia de los niños por la que tradicionalmente se controlaba. Burks notó que hacer esto tenía no era apropiado, pues tiene como consecuencia cortar parte del efecto total de la inteligencia sobre el la inteligencia de los hijos. En otras palabras: la inteligencia de los padres hace más probable mejor NSE, y mejor NSE presenta mejores condiciones de desarrollo para sus hijos. Estatificar por esta variable bloquea este efecto.\nAdicionalmente, como veremos, condicionar a NSE abre un camino no causal entre Inteligencia de Padres e Hijos.", + "text": "5.6 Cadenas o mediación\nEn este caso tenemos:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir=LR]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n X -> Z\n Z -> Y\n}\n\", width = 150, height = 20)\n\n\n\n\n\n\nEn este caso,\n\nExiste asociación entre \\(X\\) y \\(Y\\), pero no existe relación directa entre ellas. Decimos que \\(Z\\) es un mediador del efecto de \\(X\\) sobre \\(Y\\).\nSi condicionamos a un valor de \\(Z\\), \\(X\\) y \\(Y\\) son condicionalmente independientes.\n\nPodemos pensar en \\(Z\\) como un mediador del efecto de \\(X\\) sobre \\(Y\\). Si no permitimos que \\(Z\\) varíe, entonces la información de \\(X\\) no fluye a \\(Y\\).\nPor ejemplo, si \\(X\\) tomar o no una medicina para el dolor de cabeza, \\(Z\\) es dolor de cabeza y \\(Y\\) es bienestar general, \\(X\\) y \\(Y\\) están relacionadas. Sin embargo, si condicionamos a un valor fijo de dolor de cabeza, no hay relación entre tomar la medicina y bienestar general.\nEn términos de factorización, podemos checar la independencia condicional: como \\(p(x,y,z) = p(x)p(z|x)p(y|z)\\), entonces\n\\[p(x, y | z) = p(x,y,z) / p(z) = (p(x)(z|x)) (p(y|z) / p(z))\\] y vemos que el lado izquierdo se factoriza en una parte que sólo involucra a \\(x\\) y \\(z\\) y otro factor que sólo tiene a \\(y\\) y \\(z\\): no hay términos que incluyan conjuntamente a \\(x\\), \\(y\\) y \\(z\\). Podemos de cualquier forma continuar notando\n\\[p(x)p(z|x)/p(z) = p(x,z)/p(z) = p(x | z)\\] de modo que\n\\[p(x, y | z) = p(x|z) p(y|z) \\]\nY mostramos un ejemplo simulado:\n\nrbern <- function(n, prob){\n rbinom(n, 1, prob = prob)\n} \nsimular_mediador <- function(n = 10){\n x <- rbern(n, p = 0.5) |> as.numeric()\n z <- rbern(n, p = x * 0.8 + (1 - x) * 0.3)\n y <- rbinom(n, 2, z * 0.7 + (1 - z) * 0.5)\n tibble(x, z, y)\n}\nsims_mediador <- simular_mediador(50000)\n\n\\(X\\) y \\(Y\\) son dependientes:\n\nsims_mediador |> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1:\n\nsims_mediador |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nY vemos que la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) sólo depende de \\(Z\\). Una consecuencia es por ejemplo que la correlación debe ser cero:\n\ncor(sims_mediador |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.002\ny -0.002 1.000\n\ncor(sims_mediador |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.012\ny -0.012 1.000\n\n\nPodemos también hacer un ejemplo continuo:\n\nsimular_mediador <- function(n = 10){\n x <- rnorm(n, 100, 10)\n prob <- 1 / (1 + exp(-(x - 100)/5))\n z <- rbern(n, p = prob)\n y <- rnorm(n, 100 + 30 * z, 15)\n tibble(x, z, y)\n}\nsims_mediador <- simular_mediador(2000)\n\n\\(X\\) y \\(Y\\) son dependientes (por ejemplo si vemos la media condicional de \\(Y\\) dado \\(X\\):\n\nggplot(sims_mediador, aes(x = x, y = y, colour = z)) + geom_point() +\n geom_smooth(span = 1, se = FALSE)\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\nWarning: The following aesthetics were dropped during statistical transformation: colour\nℹ This can happen when ggplot fails to infer the correct grouping structure in\n the data.\nℹ Did you forget to specify a `group` aesthetic or to convert a numerical\n variable into a factor?\n\n\n\n\n\n\n\n\n\nSi condicionamos a \\(Z\\), no hay dependencia entre \\(X\\) y \\(Y\\)\n\nggplot(sims_mediador, aes(x = x, y = y, colour = z, group = z)) + \n geom_point() +\n geom_smooth(span = 2)\n\n`geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = \"cs\")'\n\n\n\n\n\n\n\n\n\nNótese que en este ejemplo sí hay un efecto causal de \\(X\\) sobre \\(Y\\), pero está mediado por otra variable \\(Z\\). Si condicionamos a \\(Z\\), no hay relación entre \\(X\\) y \\(Y\\). El análisis condicionado podría llevarnos a una conclusión errónea de que \\(X\\) no influye sobre \\(Y\\).\n\n\n\n\n\n\nTip\n\n\n\nNota que no existe una diferencia estadística entre una bifurcación y una cadena: en ambos casos, las variables \\(X\\) y \\(Y\\) están correlacionadas, y son independientes una vez que condicionamos o estratificamos por \\(Z\\). Sin embargo, su tratamiento en inferencia causal es muy diferente.\n\n\n\nSesgo post-tratamiento\nEn McElreath (2020) se discute que en algunos estudios experimentales, se estratifica por variables que son consecuencia del tratamiento. Esto induce sesgo post-tratamiento, lo cual puede llevar a equivocaciones en donde parece que el tratamiento no tiene efecto cuando sí lo tiene. Incluso bajo condiciones de experimento (donde el tratamiento es asignado al azar) estratificar por mediadores es una mala idea. Ver más en McElreath (2020), donde por ejemplo cita una fuente que en estudios experimentales de Ciencia Política, casi la mitad de ellos sufre de este tipo de sesgo por estratificación por mediadores.\n\n\nEjemplo: Burks\nEste ejemplo es de Pearl y Mackenzie (2018). En 1926 Burks recolectó datos sobre qué tanto podría esperarse que la inteligencia de padres se hereda a los hijos (medido según una prueba de IQ). Construyó un diagrama parecido al de abajo:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = circle]\n U\n node [shape=plaintext]\n edge [minlen = 3]\n IntPadres -> NSE\n NSE -> IntHijos\n U -> NSE\n U -> IntHijos\n IntPadres -> IntHijos\n{rank = same; U}\n}\n\")\n\n\n\n\n\n\nComo el NSE es del hogar (una medida general de estatus social), se consideró en principio como una variable pre-tratamiento a la inteligencia de los niños por la que tradicionalmente se controlaba. Burks notó que hacer esto tenía no era apropiado, pues tiene como consecuencia cortar parte del efecto total de la inteligencia sobre el la inteligencia de los hijos. En otras palabras: la inteligencia de los padres hace más probable mejor NSE, y mejor NSE presenta mejores condiciones de desarrollo para sus hijos. Estatificar por esta variable bloquea este efecto.\nAdicionalmente, como veremos, condicionar a NSE abre un camino no causal entre Inteligencia de Padres e Hijos.", "crumbs": [ "5  Modelos gráficos y causalidad" ] @@ -354,7 +354,7 @@ "href": "05-dags.html#colisionador-o-causas-alternativas", "title": "5  Modelos gráficos y causalidad", "section": "5.7 Colisionador o causas alternativas", - "text": "5.7 Colisionador o causas alternativas\nEn este caso, a \\(Z\\) también le llamamos un colisionador. Este es el caso que puede ser más difícil de entender en un principio. Consiste de la siguiente estructura:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n X -> Z\n Y -> Z\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\n\nEn este caso \\(X\\) y \\(Y\\) son independientes. Tanto \\(X\\) como \\(Y\\) influyen en \\(Z\\).\nSin embargo, si condicionamos a \\(Z\\) entonces \\(X\\) y \\(Y\\) están asociados.\n\nPor ejemplo, si observamos que el pasto está mojado, entonces saber que no llovió implica que probablemente se encendieron los aspersores.\nComo la conjunta se factoriza como:\n\\[p(x,y,z) = p(x)p(y)p(z|x,y)\\] Entonces integrando sobre \\(Z\\):\n\\[p(x,y) = \\int p(x,y,z)dz = p(x)p(y)\\int p(z|x,y)\\, dz\\] pero \\(p(z|x,y)\\) integra uno porque es una densidad, de forma que \\(x\\) y \\(y\\) son independientes.\nMostramos un ejemplo simulado:\n\nsimular_colisionador <- function(n = 10){\n x <- rbern(n, 0.5) \n y <- rbinom(n, 2, 0.7)\n z <- rbern(n, p = 0.1 + 0.7 * x * (y > 1)) \n tibble(x, z, y)\n}\nsims_colisionador <- simular_colisionador(50000)\n\n\\(X\\) y \\(Y\\) son independientes:\n\nsims_colisionador|> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\ncor(sims_colisionador |> select(x,y))\n\n x y\nx 1.000000000 0.008200524\ny 0.008200524 1.000000000\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1:\n\nsims_colisionador |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nY vemos que la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) depende de \\(X\\) y de \\(Z\\).\nLas correlaciones condicionales, por ejemplo, no son cero:\n\nprint(\"Dado Z = 0\")\n\n[1] \"Dado Z = 0\"\n\ncor(sims_colisionador |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.271\ny -0.271 1.000\n\nprint(\"Dado Z = 1\")\n\n[1] \"Dado Z = 1\"\n\ncor(sims_colisionador |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 0.377\ny 0.377 1.000\n\n\nOtro ejemplo con variables continuas:\n\nsimular_colisionador_2 <- function(n = 10){\n x <- rnorm(n, 100, 20) \n y <- rnorm(n, 100, 20)\n z <- rbern(n, p = 0.92 * ((x + y) > 220) + 0.05) \n tibble(x, z, y)\n}\nsims_colisionador <- simular_colisionador_2(1000)\n\n\\(X\\) y \\(Y\\) son independientes:\n\nggplot(sims_colisionador, aes(x = x, y = y)) + geom_point()\n\n\n\n\n\n\n\n\nSin embargo, si condicionamos a un valor de \\(Z\\), \\(X\\) y \\(Y\\) ya no son independientes:\n\nggplot(sims_colisionador, aes(x = x, y = y, group = z, colour = factor(z))) + \n geom_point() + geom_smooth(method = \"lm\", se = FALSE) \n\n`geom_smooth()` using formula = 'y ~ x'\n\n\n\n\n\n\n\n\n\nY vemos que condicional a \\(Z\\), \\(X\\) y \\(Y\\) están correlacionadas, aunque no hay relación causal entre \\(X\\) y \\(Y\\).\n\n5.7.1 Ejemplos de colisionadores\nExisten muchos ejemplos de colisionadores en análisis de datos. Algunos ejemplos se deben a sesgo de selección (puedes dibujar diagramas para cada uno de estos):\n\nPodemos observar correlaciones entre habilidades que en realidad son independientes si observamos muestras de estudiantes seleccionados por un examen de admisión (por ejemplo, para entrar es necesario tener alta habilidad atlética y/o alta habilidad académica).\nEntre los artículos científicos publicados (ver McElreath (2020)), aquellos que son más tomados por las noticias son los menos confiables. Esta correlación puede aparecer aunque no exista relación en proyectos científicos entre confiabilidad e interés de los medios, pues lo que se fondea o publica puede tener dos razones: ser trabajo muy confiable, o ser trabajo que “está de moda” o atrae la atención de los medios.\n\nPero también puede ser consecuencia de condicionar a variables endógenos (que resultan ser colisionadores), y ocurren como parte del procesamiento o construcción de modelos. Un ejemplo interesante de McElreath (2020) es el siguiente:\n\nNos interesa saber si la edad influye en la felicidad o bienestar de las personas.\nAlgún investigador puede pensar que es necesario controlar por sí las personas están casadas o no, por ejemplo, para “quitar” ese efecto o algo así.\nEsto puede ser mala idea si consideramos que un diagrama apropiado puede ser \\(F \\rightarrow Matrim \\leftarrow Edad\\), que se basa en las observaciones de que personas más felices generalmente tienen mayor posibilidad de casarse, y también conforme pasa el tiempo, hay más oportunidades para casarse.\nEsto induce una correlación no causal entre edad y felicidad dentro de los grupos de casados y no casados, y puede llevar a conclusiones incorrectas.", + "text": "5.7 Colisionador o causas alternativas\nEn este caso, a \\(Z\\) también le llamamos un colisionador. Este es el caso que puede ser más difícil de entender en un principio. Consiste de la siguiente estructura:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n edge [minlen = 3]\n X -> Z\n Y -> Z\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\n\nEn este caso \\(X\\) y \\(Y\\) son independientes. Tanto \\(X\\) como \\(Y\\) influyen en \\(Z\\).\nSin embargo, si condicionamos a \\(Z\\) entonces \\(X\\) y \\(Y\\) están asociados.\n\nPor ejemplo, si observamos que el pasto está mojado, entonces saber que no llovió implica que probablemente se encendieron los aspersores.\nComo la conjunta se factoriza como:\n\\[p(x,y,z) = p(x)p(y)p(z|x,y)\\] Entonces integrando sobre \\(Z\\):\n\\[p(x,y) = \\int p(x,y,z)dz = p(x)p(y)\\int p(z|x,y)\\, dz\\] pero \\(p(z|x,y)\\) integra uno porque es una densidad, de forma que \\(x\\) y \\(y\\) son independientes.\nMostramos un ejemplo simulado:\n\nsimular_colisionador <- function(n = 10){\n x <- rbern(n, 0.5) \n y <- rbinom(n, 2, 0.7)\n z <- rbern(n, p = 0.1 + 0.7 * x * (y > 1)) \n tibble(x, z, y)\n}\nsims_colisionador <- simular_colisionador(50000)\n\n\\(X\\) y \\(Y\\) son independientes:\n\nsims_colisionador|> select(x, y) |> \n count(x, y) |> \n group_by(x) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") +\n labs(subtitle = \"Condicional de Y dada X\")\n\n\n\n\n\n\n\ncor(sims_colisionador |> select(x,y))\n\n x y\nx 1.000000000 0.000457974\ny 0.000457974 1.000000000\n\n\nSin embargo, si condicionamos a \\(Z\\), que puede tomar los valores 0 o 1:\n\nsims_colisionador |> \n count(x, y, z) |> \n group_by(x, z) |> \n mutate(p_cond = n / sum(n)) |>\n select(x, y, z, p_cond) |> \nggplot(aes(x = y, y = p_cond, fill = factor(x))) +\n geom_col(position = \"dodge\") + facet_wrap(~ z) +\n labs(subtitle = \"Condicional de Y dada X y Z\")\n\n\n\n\n\n\n\n\nY vemos que la condicional de \\(Y\\) dada \\(Z\\) y \\(X\\) depende de \\(X\\) y de \\(Z\\).\nLas correlaciones condicionales, por ejemplo, no son cero:\n\nprint(\"Dado Z = 0\")\n\n[1] \"Dado Z = 0\"\n\ncor(sims_colisionador |> filter(z == 0) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 -0.278\ny -0.278 1.000\n\nprint(\"Dado Z = 1\")\n\n[1] \"Dado Z = 1\"\n\ncor(sims_colisionador |> filter(z == 1) |> select(x,y)) |> round(3)\n\n x y\nx 1.000 0.368\ny 0.368 1.000\n\n\nOtro ejemplo con variables continuas:\n\nsimular_colisionador_2 <- function(n = 10){\n x <- rnorm(n, 100, 20) \n y <- rnorm(n, 100, 20)\n z <- rbern(n, p = 0.92 * ((x + y) > 220) + 0.05) \n tibble(x, z, y)\n}\nsims_colisionador <- simular_colisionador_2(1000)\n\n\\(X\\) y \\(Y\\) son independientes:\n\nggplot(sims_colisionador, aes(x = x, y = y)) + geom_point()\n\n\n\n\n\n\n\n\nSin embargo, si condicionamos a un valor de \\(Z\\), \\(X\\) y \\(Y\\) ya no son independientes:\n\nggplot(sims_colisionador, aes(x = x, y = y, group = z, colour = factor(z))) + \n geom_point() + geom_smooth(method = \"lm\", se = FALSE) \n\n`geom_smooth()` using formula = 'y ~ x'\n\n\n\n\n\n\n\n\n\nY vemos que condicional a \\(Z\\), \\(X\\) y \\(Y\\) están correlacionadas, aunque no hay relación causal entre \\(X\\) y \\(Y\\).\n\n5.7.1 Ejemplos de colisionadores\nExisten muchos ejemplos de colisionadores en análisis de datos. Algunos ejemplos se deben a sesgo de selección (puedes dibujar diagramas para cada uno de estos):\n\nPodemos observar correlaciones entre habilidades que en realidad son independientes si observamos muestras de estudiantes seleccionados por un examen de admisión (por ejemplo, para entrar es necesario tener alta habilidad atlética y/o alta habilidad académica).\nEntre los artículos científicos publicados (ver McElreath (2020)), aquellos que son más tomados por las noticias son los menos confiables. Esta correlación puede aparecer aunque no exista relación en proyectos científicos entre confiabilidad e interés de los medios, pues lo que se fondea o publica puede tener dos razones: ser trabajo muy confiable, o ser trabajo que “está de moda” o atrae la atención de los medios.\n\nPero también puede ser consecuencia de condicionar a variables endógenos (que resultan ser colisionadores), y ocurren como parte del procesamiento o construcción de modelos. Un ejemplo interesante de McElreath (2020) es el siguiente:\n\nNos interesa saber si la edad influye en la felicidad o bienestar de las personas.\nAlgún investigador puede pensar que es necesario controlar por sí las personas están casadas o no, por ejemplo, para “quitar” ese efecto o algo así.\nEsto puede ser mala idea si consideramos que un diagrama apropiado puede ser \\(F \\rightarrow Matrim \\leftarrow Edad\\), que se basa en las observaciones de que personas más felices generalmente tienen mayor posibilidad de casarse, y también conforme pasa el tiempo, hay más oportunidades para casarse.\nEsto induce una correlación no causal entre edad y felicidad dentro de los grupos de casados y no casados, y puede llevar a conclusiones incorrectas.", "crumbs": [ "5  Modelos gráficos y causalidad" ] @@ -364,7 +364,7 @@ "href": "05-dags.html#razonamiento-de-descendientes", "title": "5  Modelos gráficos y causalidad", "section": "5.8 Razonamiento de descendientes", - "text": "5.8 Razonamiento de descendientes\nCondicionar a un descendiente puede entenderse como “condicionar parcialmente” o “débilmente” a los padres de ese descendiente.\nPor ejemplo, condicionar a un colisionador también produce dependencias condicionales:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n A\n edge [minlen = 3]\n X -> Z\n Y -> Z\n Z -> A\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEn este caso,\n\n\\(X\\) y \\(Y\\) son independientes\n\\(X\\) y \\(Y\\) son dependientes si condicionamos a \\(A\\).\n\nDependiendo de la naturaleza de la asociación entre el colisionador \\(Z\\) y su descendiente \\(A\\), esta dependencia puede ser más fuerte o más débil.\nPor ejemplo, en nuestro ejemplo donde el pasto mojado es un colisionador entre cuánta agua dieron los aspersores y cuánta lluvia cayó, un descendiente del pasto mojado es el estado de las plantas del jardín. Aunque los aspersores trabajan independientemente de la lluvia, si observamos que las plantas se secaron entonces lluvia y aspersores están correlacionados: por ejemplo, si noto que los aspersores están descompuestos, entonces concluimos que no hubo lluvia.\n\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X [label = lluvia]\n Y [label = aspersores]\n Z [label = humedad]\n A [label = plantas]\n edge [minlen = 3]\n X -> Z\n Y -> Z\n Z -> A\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEjemplo\n\nsimular_desc <- function(n = 10){\n x <- rbern(n, 0.5) \n y <- rbinom(n, 2, 0.7)\n z <- rbern(n, p = 0.1 + 0.7 * x * (y > 1)) \n a <- rbern(n, p = 0.5 + 0.5 * z)\n tibble(x, z, y, a)\n}\nsims_colisionador <- simular_desc(50000)\n# No hay correlación\ncor(sims_colisionador$x, sims_colisionador$y)\n\n[1] -0.003243749\n\n\nSin embargo,\n\ncor(sims_colisionador |> filter(a ==0) |> select(x,y))\n\n x y\nx 1.0000000 -0.2822012\ny -0.2822012 1.0000000\n\n\n\ncor(sims_colisionador |> filter(a ==1) |> select(x,y))\n\n x y\nx 1.0000000 0.1097786\ny 0.1097786 1.0000000\n\n\n\n\n5.8.1 Ejemplo: dependencias de colisionador\nVerificamos que en nuestro modelo de Santa Clara, efectivamente nuestro modelo no implica ninguna dependencia no condicional entre sensibilidad de la prueba y prevalencia. Eso debería ser claro de la simulación, pero de todas formas lo checamos\n\nlibrary(cmdstanr)\nmod_sc <- cmdstan_model(\"./src/sclara.stan\")\nprint(mod_sc)\n\ndata {\n int<lower=0> N;\n int<lower=0> n;\n int<lower=0> kit_pos;\n int<lower=0> n_kit_pos;\n int<lower=0> kit_neg;\n int<lower=0> n_kit_neg;\n}\n\nparameters {\n real<lower=0, upper=1> theta; //seroprevalencia\n real<lower=0, upper=1> sens; //sensibilidad\n real<lower=0, upper=1> esp; //especificidad\n}\n\ntransformed parameters {\n real<lower=0, upper=1> prob_pos;\n\n prob_pos = theta * sens + (1 - theta) * (1 - esp);\n\n}\nmodel {\n // modelo de número de positivos\n n ~ binomial(N, prob_pos);\n // modelos para resultados del kit\n kit_pos ~ binomial(n_kit_pos, sens);\n kit_neg ~ binomial(n_kit_neg, esp);\n // iniciales para cantidades no medidas\n theta ~ beta(1.0, 10.0);\n sens ~ beta(2.0, 1.0);\n esp ~ beta(2.0, 1.0);\n}\n\n\nEn este caso, no pondremos información acerca de positivos en la prueba:\n\ndatos_lista <- list(N = 0, n = 0,\n kit_pos = 103, n_kit_pos = 122,\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000, iter_sampling = 400)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 1 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 1 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 2 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 2 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 3 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 3 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 4 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 4 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.5 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nggplot(sims, aes(x = theta, y = sens)) + geom_point() +\n scale_x_sqrt()\n\n\n\n\n\n\n\n\nNo vemos ninguna asocación entre estas dos variables.\nSin embargo, al condicionar al valor de Positivos, creamos una relación que no podemos interpretar como casual. En este caso particular supondremos prácticamente fija la sensibilidad para ver solamente lo que sucede en el colisionador de especificidad y número de positivos (la especificidad en este ejemplo es más crítica):\n\ndatos_lista <- list(N = 3300, n = 50,\n kit_pos = 1030000, n_kit_pos = 1220000, # números grandes para que esté practicamente\n# fija la sensibilidad\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000, iter_sampling = 400)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 1 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 1 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 2 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 2 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 3 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 3 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 4 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 4 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.5 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nggplot(sims, aes(x = theta, y = esp)) + geom_point() \n\n\n\n\n\n\n\n\nY vemos que condiconando al colisionador, obtenemos una relación fuerte entre prevalencia y especificidad de la prueba: necesitaríamos más datos de especificidad para obtener una estimación útil.\n\nLa razón de que la especificidad es más importante en este ejemplo es que la prevalencia es muy baja al momento del estudio, y los falsos positivos pueden introducir más error en la estimación\nTambién repetimos nótese que el análisis correcto de estos datos no se puede hacer con intervalos separados para cada cantidad, sino que debe examinarse la conjunta de estos parámetros.\n\n\nCon estas tres estructuras elementales podemos entender de manera abstracta la existencia o no de asociaciones entre nodos de cualquier gráfica dirigida.", + "text": "5.8 Razonamiento de descendientes\nCondicionar a un descendiente puede entenderse como “condicionar parcialmente” o “débilmente” a los padres de ese descendiente.\nPor ejemplo, condicionar a un colisionador también produce dependencias condicionales:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X\n Y\n Z\n A\n edge [minlen = 3]\n X -> Z\n Y -> Z\n Z -> A\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEn este caso,\n\n\\(X\\) y \\(Y\\) son independientes\n\\(X\\) y \\(Y\\) son dependientes si condicionamos a \\(A\\).\n\nDependiendo de la naturaleza de la asociación entre el colisionador \\(Z\\) y su descendiente \\(A\\), esta dependencia puede ser más fuerte o más débil.\nPor ejemplo, en nuestro ejemplo donde el pasto mojado es un colisionador entre cuánta agua dieron los aspersores y cuánta lluvia cayó, un descendiente del pasto mojado es el estado de las plantas del jardín. Aunque los aspersores trabajan independientemente de la lluvia, si observamos que las plantas se secaron entonces lluvia y aspersores están correlacionados: por ejemplo, si noto que los aspersores están descompuestos, entonces concluimos que no hubo lluvia.\n\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n X [label = lluvia]\n Y [label = aspersores]\n Z [label = humedad]\n A [label = plantas]\n edge [minlen = 3]\n X -> Z\n Y -> Z\n Z -> A\n}\n\", width = 200, height = 50)\n\n\n\n\n\n\nEjemplo\n\nsimular_desc <- function(n = 10){\n x <- rbern(n, 0.5) \n y <- rbinom(n, 2, 0.7)\n z <- rbern(n, p = 0.1 + 0.7 * x * (y > 1)) \n a <- rbern(n, p = 0.5 + 0.5 * z)\n tibble(x, z, y, a)\n}\nsims_colisionador <- simular_desc(50000)\n# No hay correlación\ncor(sims_colisionador$x, sims_colisionador$y)\n\n[1] 0.0003440308\n\n\nSin embargo,\n\ncor(sims_colisionador |> filter(a ==0) |> select(x,y))\n\n x y\nx 1.0000000 -0.2738001\ny -0.2738001 1.0000000\n\n\n\ncor(sims_colisionador |> filter(a ==1) |> select(x,y))\n\n x y\nx 1.0000000 0.1082551\ny 0.1082551 1.0000000\n\n\n\n\n5.8.1 Ejemplo: dependencias de colisionador\nVerificamos que en nuestro modelo de Santa Clara, efectivamente nuestro modelo no implica ninguna dependencia no condicional entre sensibilidad de la prueba y prevalencia. Eso debería ser claro de la simulación, pero de todas formas lo checamos\n\nlibrary(cmdstanr)\nmod_sc <- cmdstan_model(\"./src/sclara.stan\")\nprint(mod_sc)\n\ndata {\n int<lower=0> N;\n int<lower=0> n;\n int<lower=0> kit_pos;\n int<lower=0> n_kit_pos;\n int<lower=0> kit_neg;\n int<lower=0> n_kit_neg;\n}\n\nparameters {\n real<lower=0, upper=1> theta; //seroprevalencia\n real<lower=0, upper=1> sens; //sensibilidad\n real<lower=0, upper=1> esp; //especificidad\n}\n\ntransformed parameters {\n real<lower=0, upper=1> prob_pos;\n\n prob_pos = theta * sens + (1 - theta) * (1 - esp);\n\n}\nmodel {\n // modelo de número de positivos\n n ~ binomial(N, prob_pos);\n // modelos para resultados del kit\n kit_pos ~ binomial(n_kit_pos, sens);\n kit_neg ~ binomial(n_kit_neg, esp);\n // iniciales para cantidades no medidas\n theta ~ beta(1.0, 10.0);\n sens ~ beta(2.0, 1.0);\n esp ~ beta(2.0, 1.0);\n}\n\n\nEn este caso, no pondremos información acerca de positivos en la prueba:\n\ndatos_lista <- list(N = 0, n = 0,\n kit_pos = 103, n_kit_pos = 122,\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000, iter_sampling = 400)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 1 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 1 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 2 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 2 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 3 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 3 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 4 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 4 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.5 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nggplot(sims, aes(x = theta, y = sens)) + geom_point() +\n scale_x_sqrt()\n\n\n\n\n\n\n\n\nNo vemos ninguna asocación entre estas dos variables.\nSin embargo, al condicionar al valor de Positivos, creamos una relación que no podemos interpretar como casual. En este caso particular supondremos prácticamente fija la sensibilidad para ver solamente lo que sucede en el colisionador de especificidad y número de positivos (la especificidad en este ejemplo es más crítica):\n\ndatos_lista <- list(N = 3300, n = 50,\n kit_pos = 1030000, n_kit_pos = 1220000, # números grandes para que esté practicamente\n# fija la sensibilidad\n kit_neg = 399, n_kit_neg = 401)\najuste <- mod_sc$sample(data = datos_lista, refresh = 1000, iter_sampling = 400)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 1 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 1 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 1 finished in 0.0 seconds.\nChain 2 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 2 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 2 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 2 finished in 0.0 seconds.\nChain 3 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 3 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 3 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 3 finished in 0.0 seconds.\nChain 4 Iteration: 1 / 1400 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 1400 [ 71%] (Warmup) \nChain 4 Iteration: 1001 / 1400 [ 71%] (Sampling) \nChain 4 Iteration: 1400 / 1400 [100%] (Sampling) \nChain 4 finished in 0.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 0.0 seconds.\nTotal execution time: 0.5 seconds.\n\nsims <- ajuste$draws(c(\"theta\", \"sens\", \"esp\"), format = \"df\")\nresumen <- ajuste$summary(c(\"theta\"))\n\n\nggplot(sims, aes(x = theta, y = esp)) + geom_point() \n\n\n\n\n\n\n\n\nY vemos que condiconando al colisionador, obtenemos una relación fuerte entre prevalencia y especificidad de la prueba: necesitaríamos más datos de especificidad para obtener una estimación útil.\n\nLa razón de que la especificidad es más importante en este ejemplo es que la prevalencia es muy baja al momento del estudio, y los falsos positivos pueden introducir más error en la estimación\nTambién repetimos nótese que el análisis correcto de estos datos no se puede hacer con intervalos separados para cada cantidad, sino que debe examinarse la conjunta de estos parámetros.\n\n\nCon estas tres estructuras elementales podemos entender de manera abstracta la existencia o no de asociaciones entre nodos de cualquier gráfica dirigida.", "crumbs": [ "5  Modelos gráficos y causalidad" ] @@ -444,7 +444,7 @@ "href": "06-calculo-do.html#bloqueando-puertas-traseras", "title": "6  Identificación y cálculo-do", "section": "6.5 Bloqueando puertas traseras", - "text": "6.5 Bloqueando puertas traseras\nEn las partes anteriores vimos que estratificando por los padres de la variable de tratamiento \\(X\\) podemos construir un estimador del efecto de \\(X\\) sobre otra variable \\(Y\\), pasando de una distribución observacional a una conceptualmente experimental (dado que los supuestos causales sean aproximadamente correctos).\nSin embargo, esta aplicación de la fórmula de ajuste no funciona si existen padres que no fueron observados, y por tanto no podemos estratificar por ellos. El siguiente método (ajuste por “puerta trasera”) nos da una generalización que podemos usar dado ciertos tipos de estructura en nuestro modelo causal (veremos también por ejemplo, que a veces podemos usar menos variables que padres de la variable de interés). Nótese que una vez más, este criterio sólo depende de la gráfica causal \\(G\\) asociada a nuestro modelo, y no los modelos locales que utilizemos para modelar la condicional de cada nodo.\n\n\n\n\n\n\nAjuste de puerta trasera (Pearl)\n\n\n\nSi tenemos dos variables \\(T\\) y \\(Y\\) en una gráfica \\(G\\), un conjunto \\(Z\\) de variables satisface el criterio de puerta trasera relativo a \\(T\\) y \\(Y\\) cuando \\(Z\\) bloquea cualquier camino entre \\(T\\) y \\(Y\\) que tenga una arista que incida en \\(T\\), y ninguna variable de \\(Z\\) es descendiente de \\(T\\).\nEn tal caso, podemos utilizar la fórmula de ajuste, pero en lugar de estratificar por los padres de \\(T\\), estratificamos por las variables en \\(Z\\)\n\n\nLa idea es:\n\nQueremos bloquear todos los caminos no causales entre \\(T\\) y \\(Y\\).\nQueremos no perturbar todos los caminos dirigidos de \\(T\\) a \\(Y\\) (caminos causales).\nNo queremos activar caminos no causales entre \\(T\\) y \\(Y\\) al condicionar.\n\nCumplimos 1 al estratificar por variables que bloquean los caminos que son causas de \\(T\\), pues estos caminos no son causales y distorsionan la relación entre \\(T\\) y \\(Y\\). Al mismo tiempo, no bloqueamos caminos causales porque ningúna variable de \\(Z\\) es descendiente de \\(T\\), de modo que se satisface el criterio 2 (todos los caminos causales comienzan con \\(T\\to\\)). Finalmente, al excluir descendientes de \\(T\\) también implica que no condicionamos a colisionadores del tipo \\(T\\to \\cdots \\to Z_1\\gets Y\\), pues esto activa un camino no causal entre \\(T\\) y \\(Y\\) (se cumple 3).\n\nEjemplo (Pearl)\nConsideramos primero este ejemplo simple, donde queremos evaluar la efectividad de un tratamiento en cierta enfermedad. Los datos que tenemos disponibles son si una persona recibió o no un tratamiento, y si se recuperó o no. No se registró el nivel socioeconómico, pero sabemos que el tratamiento es caro, de forma que fue accedido más por gente de NSE más alto. También que sabemos que para este tipo de tratamiento, el peso de la persona es un factor importante. Nuestros supuestos están en la siguiente gráfica:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir = LR]\n node [shape=plaintext]\n Trata\n Res\n node [shape = circle]\n NSE\n Peso\n U\n edge [minlen = 3]\n NSE -> Peso\n NSE -> Trata\n Trata -> Res\n Peso -> Res\n U -> NSE\n U -> Peso\n}\n\")\n\n\n\n\n\n\nObservamos que no podemos directamente usar la fórmula de ajuste pues NSE no es una variable observada.\nEn esta circunstancia no podríamos identificar el efecto causal, pues existen un caminos abiertos no causales. Quizá el tratamiento no es muy efectivo, y parece ser bueno pues fue aplicado a personas con menor peso que las que no recibieron el tratamiento, a través del efecto de NSE. Sin embargo, supón que tuviéramos disponible la variable Peso:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir = LR]\n node [shape=plaintext]\n Trata\n Res\n Peso\n node [shape = circle]\n NSE\n U\n edge [minlen = 3]\n NSE -> Peso\n NSE -> Trata\n Trata -> Res\n Peso -> Res\n U -> NSE\n U -> Peso\n}\n\")\n\n\n\n\n\n\nEn este caso, todavía no podemos aplicar la fórmula original de ajuste pues no conocemos \\(NSE\\). Sin embargo, podemos bloquear los caminos no causales estratificando por Peso, y entonces podemos usar el criterio de puerta trasera para identificar el efecto del tratamiento, aún cuando no tengamos NSE.\n\n\nEjemplo\nPrimero consideramos un modelo generador:\n\ninv_logit <- function(x) 1 / (1 + exp(-x))\nsimular_bd <- function(n = 10){\n nse <- sample(c(0, 1), n, replace = TRUE)\n peso <- rnorm(n, 70 - 7 * nse, 12 + 2 * nse)\n trata <- rbinom(n, 1, 0.8 * nse + 0.2 * (1 - nse))\n p_trata <- inv_logit(1 * trata - 0.2 * (peso - 70))\n res <- rbinom(n, 1, p_trata)\n tibble(nse, peso, trata, res)\n}\ndatos_bd <- simular_bd(10000)\nhead(datos_bd)\n\n# A tibble: 6 × 4\n nse peso trata res\n <dbl> <dbl> <int> <int>\n1 1 71.9 0 0\n2 0 45.0 0 1\n3 0 73.5 0 0\n4 0 66.1 0 1\n5 1 49.4 1 1\n6 0 69.0 1 1\n\n\nVeamos qué sucede si cruzamos tratamiento con resultado (es una muestra grande y el error de estimación no es importante):\n\ndatos_bd |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 2678 0.533 NA \n2 1 1 3686 0.741 0.208\n\n\nSabemos que esta diferencia en respuesta puede estar confundida por un camino no causal. El verdadero efecto casual podemos calcularlo en nuestras simulaciones como sigue a partir de nuestro modelo (igualmente, usamos una muestra muy grande):\n\nsimular_efecto <- function(n = 10, peso = NULL){\n # cómo es la población\n nse <- sample(c(0, 1), n, replace = TRUE)\n if(is.null(peso)){\n peso <- rnorm(n, 70 - 7 * nse, 12 + 2 * nse)\n }\n # asignar al azar\n trata <- rbinom(n, 1, 0.5)\n p_trata <- inv_logit(1 * trata - 0.2 * (peso - 70))\n res <- rbinom(n, 1, p_trata)\n tibble(nse, peso, trata, res)\n}\nsims_efecto <- simular_efecto(20000)\nresumen <- sims_efecto |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\ndif_real <- resumen$dif[2]\nresumen\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 5929 0.590 NA \n2 1 1 6996 0.703 0.113\n\n\nLa estimación ingenua del cruce simple es mucho más grande que el verdadero efecto.\nPodemos también calcular el efecto para un peso particular:\n\nsims_efecto <- simular_efecto(20000, peso = 70)\nres_70 <- sims_efecto |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\ndif_70 <- res_70$dif[2]\nres_70\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 5002 0.500 NA \n2 1 1 7344 0.735 0.235\n\n\nSuponiendo nuestro diagrama, queremos estimar estratificando por peso. Podríamos usar un sólo modelo logístico, pero pueden ser más simples los cálculos si construimos nuestro modelo en stan. En este caso, podríamos calcular las diferencias para un peso particular, por ejemplo 70 kg (en lugar de modelar estaturas para producir una estimación de diferencia promedio).\nUsaremos una muestra de 2 mil personas:\n\nmod_trata <- cmdstan_model(\"./src/trata-backdoor.stan\")\nprint(mod_trata)\n\ndata {\n int<lower=0> N;\n vector[N] trata;\n array[N] int res;\n vector[N] peso;\n\n}\n\ntransformed data {\n real media_peso;\n\n // centrar\n media_peso = mean(peso);\n}\n\nparameters {\n real gamma_0;\n real gamma_1;\n real gamma_2;\n}\n\ntransformed parameters {\n vector[N] p_logit_res;\n\n p_logit_res = gamma_0 + gamma_1 * trata + gamma_2 * (peso - media_peso);\n\n}\n\nmodel {\n // modelo de resultado\n res ~ bernoulli_logit(p_logit_res);\n gamma_0 ~ normal(0, 2);\n gamma_1 ~ normal(0, 1);\n gamma_2 ~ normal(0, 0.2);\n\n\n}\ngenerated quantities {\n real dif_trata;\n real p_trata;\n real p_no_trata;\n\n real peso_sim = 70;\n {\n array[2000] int res_trata;\n array[2000] int res_no_trata;\n for(k in 1:2000){\n res_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 1 +\n gamma_2 * (peso_sim - media_peso)));\n res_no_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 0 +\n gamma_2 * (peso_sim - media_peso)));\n }\n dif_trata = mean(res_trata) - mean(res_no_trata);\n }\n}\n\n\n\nset.seed(915)\ndatos_bd <- simular_bd(2000)\ndatos_lista <- list(N = nrow(datos_bd),\n trata = datos_bd$trata, res = datos_bd$res,\n peso = datos_bd$peso)\najuste <- mod_trata$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 1.9 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 2.0 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 1.9 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 2.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 2.0 seconds.\nTotal execution time: 8.2 seconds.\n\nsims <- ajuste$draws( format = \"df\")\nresumen <- ajuste$summary(c( \"dif_trata\"))\n\n\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 dif_trata 0.214 0.162 0.268\n\nsims |> select(dif_trata) |> \n ggplot(aes(x = dif_trata)) + geom_histogram() +\n geom_vline(xintercept = dif_70, colour = \"red\")\n\nWarning: Dropping 'draws_df' class as required metadata was removed.\n\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY obtenemos una estimación correcta del efecto en 70 kg. Podríamos también calcular el efecto en distintos pesos (nuestro estimador es una curva), promediar estimando una distribución de pesos modelada, o tomar una distribución fija de pesos para modelar (cada una de estas estrategias tiene propósitos diferentes).\nSi queremos tener un efecto promedio, podemos modelar los pesos. Otra estrategia es promediar sobre los valores observados de la muestra. Nótese que esto ignora una parte de la incertidumbre proveniente de la muestra particular usada.\n\nmod_trata <- cmdstan_model(\"./src/trata-backdoor-promedio.stan\")\nprint(mod_trata)\n\ndata {\n int<lower=0> N;\n vector[N] trata;\n array[N] int res;\n vector[N] peso;\n\n}\n\ntransformed data {\n real media_peso;\n\n // centrar\n media_peso = mean(peso);\n}\n\nparameters {\n real gamma_0;\n real gamma_1;\n real gamma_2;\n}\n\ntransformed parameters {\n vector[N] p_logit_res;\n\n p_logit_res = gamma_0 + gamma_1 * trata + gamma_2 * (peso - media_peso);\n\n}\n\nmodel {\n // modelo de resultado\n res ~ bernoulli_logit(p_logit_res);\n gamma_0 ~ normal(0, 2);\n gamma_1 ~ normal(0, 1);\n gamma_2 ~ normal(0, 0.2);\n\n\n}\ngenerated quantities {\n real dif_trata;\n real p_trata;\n real p_no_trata;\n vector[N] probs;\n\n for(i in 1:N){\n probs[i] = 1.0 / N;\n }\n\n {\n array[2000] int res_trata;\n array[2000] int res_no_trata;\n for(k in 1:2000){\n real peso_sim = peso[categorical_rng(probs)];\n res_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 1 +\n gamma_2 * (peso_sim - media_peso)));\n res_no_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 0 +\n gamma_2 * (peso_sim - media_peso)));\n }\n p_trata = mean(res_trata);\n p_no_trata = mean(res_no_trata);\n }\n dif_trata = p_trata - p_no_trata;\n\n}\n\n\n\ndatos_lista <- list(N = nrow(datos_bd),\n trata = datos_bd$trata, res = datos_bd$res,\n peso = datos_bd$peso)\najuste <- mod_trata$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 10.9 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 11.0 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 10.9 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 10.9 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 10.9 seconds.\nTotal execution time: 44.1 seconds.\n\nsims <- ajuste$draws(c(\"dif_trata\"), format = \"df\")\n\n\nresumen <- ajuste$summary(c( \"dif_trata\"))\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 dif_trata 0.111 0.0805 0.141\n\nsims |> select(dif_trata) |> \n ggplot(aes(x = dif_trata)) + geom_histogram() +\n geom_vline(xintercept = dif_real, colour = \"red\")\n\nWarning: Dropping 'draws_df' class as required metadata was removed.\n\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY recuperamos nuevamente el efecto verdadero que mostramos arriba.", + "text": "6.5 Bloqueando puertas traseras\nEn las partes anteriores vimos que estratificando por los padres de la variable de tratamiento \\(X\\) podemos construir un estimador del efecto de \\(X\\) sobre otra variable \\(Y\\), pasando de una distribución observacional a una conceptualmente experimental (dado que los supuestos causales sean aproximadamente correctos).\nSin embargo, esta aplicación de la fórmula de ajuste no funciona si existen padres que no fueron observados, y por tanto no podemos estratificar por ellos. El siguiente método (ajuste por “puerta trasera”) nos da una generalización que podemos usar dado ciertos tipos de estructura en nuestro modelo causal (veremos también por ejemplo, que a veces podemos usar menos variables que padres de la variable de interés). Nótese que una vez más, este criterio sólo depende de la gráfica causal \\(G\\) asociada a nuestro modelo, y no los modelos locales que utilizemos para modelar la condicional de cada nodo.\n\n\n\n\n\n\nAjuste de puerta trasera (Pearl)\n\n\n\nSi tenemos dos variables \\(T\\) y \\(Y\\) en una gráfica \\(G\\), un conjunto \\(Z\\) de variables satisface el criterio de puerta trasera relativo a \\(T\\) y \\(Y\\) cuando \\(Z\\) bloquea cualquier camino entre \\(T\\) y \\(Y\\) que tenga una arista que incida en \\(T\\), y ninguna variable de \\(Z\\) es descendiente de \\(T\\).\nEn tal caso, podemos utilizar la fórmula de ajuste, pero en lugar de estratificar por los padres de \\(T\\), estratificamos por las variables en \\(Z\\)\n\n\nLa idea es:\n\nQueremos bloquear todos los caminos no causales entre \\(T\\) y \\(Y\\).\nQueremos no perturbar todos los caminos dirigidos de \\(T\\) a \\(Y\\) (caminos causales).\nNo queremos activar caminos no causales entre \\(T\\) y \\(Y\\) al condicionar.\n\nCumplimos 1 al estratificar por variables que bloquean los caminos que son causas de \\(T\\), pues estos caminos no son causales y distorsionan la relación entre \\(T\\) y \\(Y\\). Al mismo tiempo, no bloqueamos caminos causales porque ningúna variable de \\(Z\\) es descendiente de \\(T\\), de modo que se satisface el criterio 2 (todos los caminos causales comienzan con \\(T\\to\\)). Finalmente, al excluir descendientes de \\(T\\) también implica que no condicionamos a colisionadores del tipo \\(T\\to \\cdots \\to Z_1\\gets Y\\), pues esto activa un camino no causal entre \\(T\\) y \\(Y\\) (se cumple 3).\n\nEjemplo (Pearl)\nConsideramos primero este ejemplo simple, donde queremos evaluar la efectividad de un tratamiento en cierta enfermedad. Los datos que tenemos disponibles son si una persona recibió o no un tratamiento, y si se recuperó o no. No se registró el nivel socioeconómico, pero sabemos que el tratamiento es caro, de forma que fue accedido más por gente de NSE más alto. También que sabemos que para este tipo de tratamiento, el peso de la persona es un factor importante. Nuestros supuestos están en la siguiente gráfica:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir = LR]\n node [shape=plaintext]\n Trata\n Res\n node [shape = circle]\n NSE\n Peso\n U\n edge [minlen = 3]\n NSE -> Peso\n NSE -> Trata\n Trata -> Res\n Peso -> Res\n U -> NSE\n U -> Peso\n}\n\")\n\n\n\n\n\n\nObservamos que no podemos directamente usar la fórmula de ajuste pues NSE no es una variable observada.\nEn esta circunstancia no podríamos identificar el efecto causal, pues existen un caminos abiertos no causales. Quizá el tratamiento no es muy efectivo, y parece ser bueno pues fue aplicado a personas con menor peso que las que no recibieron el tratamiento, a través del efecto de NSE. Sin embargo, supón que tuviéramos disponible la variable Peso:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2, rankdir = LR]\n node [shape=plaintext]\n Trata\n Res\n Peso\n node [shape = circle]\n NSE\n U\n edge [minlen = 3]\n NSE -> Peso\n NSE -> Trata\n Trata -> Res\n Peso -> Res\n U -> NSE\n U -> Peso\n}\n\")\n\n\n\n\n\n\nEn este caso, todavía no podemos aplicar la fórmula original de ajuste pues no conocemos \\(NSE\\). Sin embargo, podemos bloquear los caminos no causales estratificando por Peso, y entonces podemos usar el criterio de puerta trasera para identificar el efecto del tratamiento, aún cuando no tengamos NSE.\n\n\nEjemplo\nPrimero consideramos un modelo generador:\n\ninv_logit <- function(x) 1 / (1 + exp(-x))\nsimular_bd <- function(n = 10){\n nse <- sample(c(0, 1), n, replace = TRUE)\n peso <- rnorm(n, 70 - 7 * nse, 12 + 2 * nse)\n trata <- rbinom(n, 1, 0.8 * nse + 0.2 * (1 - nse))\n p_trata <- inv_logit(1 * trata - 0.2 * (peso - 70))\n res <- rbinom(n, 1, p_trata)\n tibble(nse, peso, trata, res)\n}\ndatos_bd <- simular_bd(10000)\nhead(datos_bd)\n\n# A tibble: 6 × 4\n nse peso trata res\n <dbl> <dbl> <int> <int>\n1 1 71.9 0 0\n2 0 45.0 0 1\n3 0 73.5 0 0\n4 0 66.1 0 1\n5 1 49.4 1 1\n6 0 69.0 1 1\n\n\nVeamos qué sucede si cruzamos tratamiento con resultado (es una muestra grande y el error de estimación no es importante):\n\ndatos_bd |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 2678 0.533 NA \n2 1 1 3686 0.741 0.208\n\n\nSabemos que esta diferencia en respuesta puede estar confundida por un camino no causal. El verdadero efecto casual podemos calcularlo en nuestras simulaciones como sigue a partir de nuestro modelo (igualmente, usamos una muestra muy grande):\n\nsimular_efecto <- function(n = 10, peso = NULL){\n # cómo es la población\n nse <- sample(c(0, 1), n, replace = TRUE)\n if(is.null(peso)){\n peso <- rnorm(n, 70 - 7 * nse, 12 + 2 * nse)\n }\n # asignar al azar\n trata <- rbinom(n, 1, 0.5)\n p_trata <- inv_logit(1 * trata - 0.2 * (peso - 70))\n res <- rbinom(n, 1, p_trata)\n tibble(nse, peso, trata, res)\n}\nsims_efecto <- simular_efecto(20000)\nresumen <- sims_efecto |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\ndif_real <- resumen$dif[2]\nresumen\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 5929 0.590 NA \n2 1 1 6996 0.703 0.113\n\n\nLa estimación ingenua del cruce simple es mucho más grande que el verdadero efecto.\nPodemos también calcular el efecto para un peso particular:\n\nsims_efecto <- simular_efecto(20000, peso = 70)\nres_70 <- sims_efecto |> \n count(trata, res) |>\n group_by(trata) |> \n mutate(p = n / sum(n)) |> \n filter(res == 1) |> \n ungroup() |> \n mutate(dif = p - lag(p))\ndif_70 <- res_70$dif[2]\nres_70\n\n# A tibble: 2 × 5\n trata res n p dif\n <int> <int> <int> <dbl> <dbl>\n1 0 1 5002 0.500 NA \n2 1 1 7344 0.735 0.235\n\n\nSuponiendo nuestro diagrama, queremos estimar estratificando por peso. Podríamos usar un sólo modelo logístico, pero pueden ser más simples los cálculos si construimos nuestro modelo en stan. En este caso, podríamos calcular las diferencias para un peso particular, por ejemplo 70 kg (en lugar de modelar estaturas para producir una estimación de diferencia promedio).\nUsaremos una muestra de 2 mil personas:\n\nmod_trata <- cmdstan_model(\"./src/trata-backdoor.stan\")\nprint(mod_trata)\n\ndata {\n int<lower=0> N;\n vector[N] trata;\n array[N] int res;\n vector[N] peso;\n\n}\n\ntransformed data {\n real media_peso;\n\n // centrar\n media_peso = mean(peso);\n}\n\nparameters {\n real gamma_0;\n real gamma_1;\n real gamma_2;\n}\n\ntransformed parameters {\n vector[N] p_logit_res;\n\n p_logit_res = gamma_0 + gamma_1 * trata + gamma_2 * (peso - media_peso);\n\n}\n\nmodel {\n // modelo de resultado\n res ~ bernoulli_logit(p_logit_res);\n gamma_0 ~ normal(0, 2);\n gamma_1 ~ normal(0, 1);\n gamma_2 ~ normal(0, 0.2);\n\n\n}\ngenerated quantities {\n real dif_trata;\n real p_trata;\n real p_no_trata;\n\n real peso_sim = 70;\n {\n array[2000] int res_trata;\n array[2000] int res_no_trata;\n for(k in 1:2000){\n res_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 1 +\n gamma_2 * (peso_sim - media_peso)));\n res_no_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 0 +\n gamma_2 * (peso_sim - media_peso)));\n }\n dif_trata = mean(res_trata) - mean(res_no_trata);\n }\n}\n\n\n\nset.seed(915)\ndatos_bd <- simular_bd(2000)\ndatos_lista <- list(N = nrow(datos_bd),\n trata = datos_bd$trata, res = datos_bd$res,\n peso = datos_bd$peso)\najuste <- mod_trata$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 1.9 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 2.0 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 1.9 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 2.0 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 1.9 seconds.\nTotal execution time: 8.2 seconds.\n\nsims <- ajuste$draws( format = \"df\")\nresumen <- ajuste$summary(c( \"dif_trata\"))\n\n\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 dif_trata 0.214 0.162 0.268\n\nsims |> select(dif_trata) |> \n ggplot(aes(x = dif_trata)) + geom_histogram() +\n geom_vline(xintercept = dif_70, colour = \"red\")\n\nWarning: Dropping 'draws_df' class as required metadata was removed.\n\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY obtenemos una estimación correcta del efecto en 70 kg. Podríamos también calcular el efecto en distintos pesos (nuestro estimador es una curva), promediar estimando una distribución de pesos modelada, o tomar una distribución fija de pesos para modelar (cada una de estas estrategias tiene propósitos diferentes).\nSi queremos tener un efecto promedio, podemos modelar los pesos. Otra estrategia es promediar sobre los valores observados de la muestra. Nótese que esto ignora una parte de la incertidumbre proveniente de la muestra particular usada.\n\nmod_trata <- cmdstan_model(\"./src/trata-backdoor-promedio.stan\")\nprint(mod_trata)\n\ndata {\n int<lower=0> N;\n vector[N] trata;\n array[N] int res;\n vector[N] peso;\n\n}\n\ntransformed data {\n real media_peso;\n\n // centrar\n media_peso = mean(peso);\n}\n\nparameters {\n real gamma_0;\n real gamma_1;\n real gamma_2;\n}\n\ntransformed parameters {\n vector[N] p_logit_res;\n\n p_logit_res = gamma_0 + gamma_1 * trata + gamma_2 * (peso - media_peso);\n\n}\n\nmodel {\n // modelo de resultado\n res ~ bernoulli_logit(p_logit_res);\n gamma_0 ~ normal(0, 2);\n gamma_1 ~ normal(0, 1);\n gamma_2 ~ normal(0, 0.2);\n\n\n}\ngenerated quantities {\n real dif_trata;\n real p_trata;\n real p_no_trata;\n vector[N] probs;\n\n for(i in 1:N){\n probs[i] = 1.0 / N;\n }\n\n {\n array[2000] int res_trata;\n array[2000] int res_no_trata;\n for(k in 1:2000){\n real peso_sim = peso[categorical_rng(probs)];\n res_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 1 +\n gamma_2 * (peso_sim - media_peso)));\n res_no_trata[k] = bernoulli_rng(\n inv_logit(gamma_0 + gamma_1 * 0 +\n gamma_2 * (peso_sim - media_peso)));\n }\n p_trata = mean(res_trata);\n p_no_trata = mean(res_no_trata);\n }\n dif_trata = p_trata - p_no_trata;\n\n}\n\n\n\ndatos_lista <- list(N = nrow(datos_bd),\n trata = datos_bd$trata, res = datos_bd$res,\n peso = datos_bd$peso)\najuste <- mod_trata$sample(data = datos_lista, refresh = 1000)\n\nRunning MCMC with 4 sequential chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 10.9 seconds.\nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 10.9 seconds.\nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 10.9 seconds.\nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 10.9 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 10.9 seconds.\nTotal execution time: 44.1 seconds.\n\nsims <- ajuste$draws(c(\"dif_trata\"), format = \"df\")\n\n\nresumen <- ajuste$summary(c( \"dif_trata\"))\nresumen |> select(variable, mean, q5, q95)\n\n# A tibble: 1 × 4\n variable mean q5 q95\n <chr> <dbl> <dbl> <dbl>\n1 dif_trata 0.111 0.0805 0.141\n\nsims |> select(dif_trata) |> \n ggplot(aes(x = dif_trata)) + geom_histogram() +\n geom_vline(xintercept = dif_real, colour = \"red\")\n\nWarning: Dropping 'draws_df' class as required metadata was removed.\n\n\n`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n\n\n\n\n\n\n\n\n\nY recuperamos nuevamente el efecto verdadero que mostramos arriba.", "crumbs": [ "6  Identificación y cálculo-do" ] @@ -464,7 +464,7 @@ "href": "06-calculo-do.html#el-criterio-de-puerta-delantera", "title": "6  Identificación y cálculo-do", "section": "6.7 El criterio de puerta delantera", - "text": "6.7 El criterio de puerta delantera\nEn algunos casos, puede ser que no sea posible bloquear algún camino no causal con variables observadas. Un ejemplo clásico es el de la discusión acerca de la relación de fumar con cáncer de pulmón. Algunos estadísticos plantearon que los estudios de asociación entre fumar y cáncer de pulmón podrían tener efectos gravemente confundidos, por ejemplo, por aspectos genéticos que hacen a una persona propensa a fumar al mismo tiempo que aumenta su probabilidad de fumar:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n F\n C\n node [shape = circle]\n U\n edge [minlen = 3]\n U -> F\n U -> C\n F -> C\n{rank= same; C; F}\n}\n\")\n\n\n\n\n\n\nEn este caso, el efecto de fumar (\\(F\\)) sobre cáncer (\\(C\\)) no es identificable pues no podemos condicionar a la variable de Genotipo (\\(U\\)). Supongamos que tenemos una medida adicional, que es la cantidad de depósitos de alquitrán den los pulmones de los pacientes. Este es es afectado por \\(F\\), y a su vez, el alquitrán incrementa la probabilidad de cáncer:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n F\n C\n A\n node [shape = circle]\n U\n edge [minlen = 3]\n U -> F\n U -> C\n F -> A\n A -> C\n{rank= same; C; F; A}\n}\n\")\n\n\n\n\n\n\nLa idea es primero estimar el efecto de \\(F\\) sobre \\(A\\), y después estimar el efecto de \\(A\\) sobre \\(C\\). La “composición” de estos dos efectos, dado el diagrama, debe darnos el estimador correcto. Primero consideramos el efecto de \\(F\\) sobre \\(A\\), y tenemos que (regla 2)\n\\[p(a|do(f)) = p(a|f),\\] La igualdad se debe a que una vez que condicionamos a \\(F\\) no hay puertas traseras entre \\(F\\) y \\(A\\) (pues no condicionamos a \\(C\\)). Esta dependencia causal la podemos entonces estimar de los datos.\nEl efecto de \\(A\\) sobre \\(C\\) también es identificable, pues el camino no causal se bloquea cuando condicionamos a \\(A\\), de forma que por la fórmula de ajuste:\n\\[p(c|do(a)) = \\int p(c|a, f') p(f')\\, df'\\]\nAhora encadenamos estas dos ecuaciones:\n\\[p(c|do(f)) = \\int p(c|do(a))p(a|f)\\,da\\]\nque equivale en simulación a: dado un valor de \\(F\\), simulamos \\(A=a\\) con nuestro modelo ajustado con datos naturales. Ahora intervenimos \\(A\\) con el valor a que obtuvimos y simulamos \\(C\\). Sin embargo, para hacer este último paso con datos naturales, necesitamos usar el criterio de puerta trasera como explicamos arriba: simulamos entonces \\(f´\\) de \\(p(f)\\), y después simulamos \\(C\\) en función de \\(a\\) y \\(f´\\) (con una distribución construida a partir de datos).\nRequerimos en este caso construir y estimar la condicional \\(p(c|a, f)\\) basado en los datos.\nEn fórmula, en general, se escribe como:\n\n\n\n\n\n\nCriterio de fuerta delantera (Pearl)\n\n\n\nDecimos que un conjunto de variables \\(A\\) satisface el criterio de puerta delantera en relación a las variables \\(F\\) y \\(C\\) cuando:\n\n\\(A\\) intercepta todos las cadenas dirigidos de \\(F\\) a \\(C\\)\nNo hay ningún camino activo de puerta trasera de \\(F\\) a \\(A\\)\nTodos los caminos de puerta trasera de \\(A\\) a \\(C\\) están bloqueados por \\(F\\).\n\nSi \\(A\\) satisface el criterio de puerta delantera en relación a \\(F\\) y \\(C\\), entonces el efecto causal de \\(F\\) en \\(C\\) es identificable y está dado por la fórmula:\n\\[p(c|do(f)) = \\int \\left [ \\int p(c|a,f´)p(f´)\\,df´ \\right ] p(a|f)\\,da\\]\n\n\nTodas estas cantidades puede estimarse de los datos.\n\nEjemplo: proceso generador\nAntes de aplicar este nuevo procedimiento, describamos el proceso generador que utilizaremos:\n\n# simular distribución natural\nsimular_fd <- function(n = 10, efecto_a = 0.3){\n ## causa común\n u <- rnorm(n, 0, 1);\n # cantidad que fuma\n f <- exp(rnorm(n, 1 + 0.2 * u, 0.1))\n # acumulación de alquitrán\n a <- rnorm(n, 4 * f, 2)\n # probabilidad de cancer\n p_c <- inv_logit(-6 + efecto_a * a + 2 * u)\n c <- rbinom(n, 1, p_c)\n tibble(f, a, c, u)\n}\n# simular datos intervenidos (suponiendo que conocemos todo)\nsim_int_f <- function(n = 100, do_f = 0.3, efecto_a = 0.3){\n a <- rnorm(n, 4 * do_f, 2)\n u <- rnorm(n, 0, 1)\n p_c <- inv_logit(-6 + efecto_a * a + 2 * u)\n c <- rbinom(n, 1, p_c)\n tibble(do_f = do_f, media_c = mean(c))\n}\n\n\nset.seed(4481)\nsims_fd <- simular_fd(5000)\nsims_fd_1 <- simular_fd(10000)\nqplot(sims_fd$f, sims_fd$a)\n\nWarning: `qplot()` was deprecated in ggplot2 3.4.0.\n\n\n\n\n\n\n\n\n\n¿Cómo se ve la relación de fumador con cáncer? En esta gráfica mostramos también el valor de la variable no observada \\(U\\). Nótese que parte de la correlación positiva que existe es debido a esta variable \\(U\\).\n\nggplot(sims_fd, aes(x = f, y = c, colour = u)) + \n geom_jitter() + scale_colour_continuous(type = \"viridis\")\n\n\n\n\n\n\n\n\nAhora veamos cómo se ve el efecto de \\(F\\) sobre \\(C\\) y también cómo se ve el cruce de \\(F\\) y \\(C\\) en los datos naturales:\n\nsims_1 <- map_df(seq(1, 4, 0.5), ~ sim_int_f(100000, .x))\n\nsims_1 |> \n ggplot() + geom_line(aes(x = do_f, y = media_c)) +\n geom_smooth(data = sims_fd_1, aes(x = f, y = c), method = \"loess\", span = 0.3, se = FALSE, colour =\"red\") + xlab(\"Grado de tabaquismo\") +\n xlim(c(1,4))\n\n`geom_smooth()` using formula = 'y ~ x'\n\n\nWarning: Removed 376 rows containing non-finite values (`stat_smooth()`).\n\n\n\n\n\n\n\n\n\nEn efecto causal promedio de fumar, en cada nivel, sobre la incidencia de cáncer de pulmón, suponiendo nuestro proceso generador. Nótese que la relación no es tan fuerte como observamos en los datos naturales (en rojo). Esto se debe a que en los datos naturales, las personas existe una causa común entre no fumar y prevenir cáncer de pulmón.\n\n\nEjemplo: estimación con puerta delantera\nVeamos cómo sería la estimación si tuviéramos datos disponible, y si es que podemos recuperar el efecto correcto dados los datos observados y la técnica de puerta delantera.\nNótese que sólo necesitamos \\(p(c|a, f), p(a|f)\\) y \\(p(f)\\). Estos son modelos estadísticos con el que podemos identificar el efecto que nos interesa. Una vez que los estimemos, podemos usar simulación:\n\nFijamos una \\(f\\).\nSimulamos una \\(a\\) del modelo \\(p(a|f)\\)\nPara calcular \\(\\int p(c|a,f')p(f')\\), tenemos que simular un valor \\(f'\\) de la marginal de \\(p(f)\\), y luego, sustituir junto la \\(a\\) de 1 para simular una \\(c\\) de \\(p(c|a, f')\\).\nConsideramos solamente \\(c\\) y \\(f\\) para resumir el efecto.\n\n\nset.seed(481)\nsims_fd <- simular_fd(2000)\nmod_front_door <- cmdstan_model(\"./src/front-door.stan\")\nprint(mod_front_door)\n\ndata {\n int<lower=0> N;\n int<lower=0> n_f;\n vector[N] f;\n vector[N] a;\n array[N] int<lower=0, upper=1> c;\n array[n_f] real do_f;\n\n}\n\ntransformed data {\n real media_a;\n real media_f;\n\n media_a = mean(a);\n media_f = mean(f);\n}\n\nparameters {\n real<lower=0> alpha;\n real alpha_a;\n real<lower=0> alpha_f;\n real int_a;\n real beta_0;\n real<lower=0> beta_1;\n real<lower=0> beta;\n real<lower=0> a_f;\n real<lower=0> b_f;\n real<lower=0> sigma_a;\n real<lower=0> sigma_f;\n\n}\n\n\n\ntransformed parameters {\n\n\n}\n\nmodel {\n f ~ gamma(a_f, b_f);\n a ~ normal(beta * f, sigma_a);\n c ~ bernoulli_logit(int_a + alpha_a * a + alpha_f * f);\n alpha_a ~ normal(0, 1);\n alpha_f ~ normal(0, 1);\n int_a ~ normal(0, 3);\n sigma_a ~ normal(0, 1);\n sigma_f ~ normal(0, 0.1);\n alpha ~ normal(0, 1);\n beta ~ normal(0, 1);\n beta_0 ~ normal(0, 3);\n beta_1 ~ normal(0, 1);\n\n}\ngenerated quantities {\n array[n_f] real mean_c;\n\n for(i in 1:n_f){\n array[2000] real res_sim;\n for(j in 1:2000){\n real a_sim = normal_rng(beta * (do_f[i]), sigma_a);\n real f_sim = gamma_rng(a_f, b_f);\n res_sim[j] = bernoulli_rng(inv_logit(int_a + alpha_a * a_sim + alpha_f * f_sim));\n }\n mean_c[i] = mean(res_sim);\n }\n\n}\n\n\n\ndo_f <- seq(1, 4, 0.1)\nn_f <- length(do_f)\nsims <- mod_front_door$sample(data = list(N = nrow(sims_fd),\n f = sims_fd$f, a = sims_fd$a,\n c = sims_fd$c, do_f = do_f, n_f = n_f),\n init = 0.01, step_size = 0.01, \n refresh = 1000,\n parallel_chains = 4)\n\nRunning MCMC with 4 parallel chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 42.0 seconds.\nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 43.3 seconds.\nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 44.0 seconds.\nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 finished in 44.1 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 43.3 seconds.\nTotal execution time: 44.2 seconds.\n\n\n\nsims_efecto_tbl <- sims$draws(\"mean_c\", format = \"df\") |> \n pivot_longer(cols = contains(\"mean_c\"), values_to = \"media_c\") |> \n separate(name, c(\"nom\", \"id\"), \n sep = \"[\\\\[\\\\]]\", convert = TRUE, extra = \"drop\") |> \n left_join(tibble(f = do_f) |> \n mutate(id = seq_along(f))) \nresumen_tbl <- sims_efecto_tbl |> \n group_by(id, f) |> \n summarise(media = mean(media_c), \n q5 = quantile(media_c, 0.05),\n q95 = quantile(media_c, 0.95))\n\n\nggplot(resumen_tbl) + \n geom_linerange(aes(x= f, ymax = q95, ymin = q5), colour = \"red\") + \n geom_point(aes(x = f, y = media), colour = \"red\") +\n geom_line(data = sims_1, aes(x = do_f, y = media_c)) +\n xlab(\"Nivel de tabaquismo\") + ylab(\"Prop afectada\")\n\n\n\n\n\n\n\n\nY parece que hemos obtenido una estimación razonable del efecto causal de fumar sobre cáncer. Recordemos también que debemos ser cuidadosos al comparar intervalos que salen del mismo modelo por su nivel de traslape.\nPor ejemplo, si quisiéramos calcular contrastes con el nivel 2 de tabaquismo:\n\nefecto_2 <- sims_efecto_tbl |> filter(f == 2) |> \n select(.draw, efecto_2 = media_c)\ncomp_tbl <- left_join(sims_efecto_tbl, efecto_2) |> \n mutate(dif_2 = media_c - efecto_2)\n\nJoining with `by = join_by(.draw)`\n\ncomp_tbl |> group_by(f) |> \n summarise(media = mean(dif_2), q5 = quantile(dif_2, 0.05),\n q95 = quantile(dif_2, 0.95)) |> \nggplot() + geom_linerange(aes(x= f, ymax = q95, ymin = q5)) + geom_point(aes(x = f, y = media)) +\n xlab(\"Nivel de tabaquismo\") + ylab(\"Prop afectada\")\n\n\n\n\n\n\n\n\nNota: nótese como en este ejemplo hemos evitado incluir en nuestro modelo la variable no observada \\(U\\), gracias al procedimiento de puerta delantera descrito arriba.\nEs posible sin embargo intentar un modelo completo bayesiano, sin necesidad de recordar la fórmula. El procedimiento, que es más difícil de ajustar: considera una variable latente \\(U\\) no observada, y es necesario definir cómo puede ser su relación con sus descendientes. Es necesario más cuidado en definir formas funcionales e iniciales apropiadas para que los muestreadores funcionen apropiadamente.\n\n\n\n\nMcElreath, R. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. A Chapman & Hall libro. CRC Press. https://books.google.com.mx/books?id=Ie2vxQEACAAJ.", + "text": "6.7 El criterio de puerta delantera\nEn algunos casos, puede ser que no sea posible bloquear algún camino no causal con variables observadas. Un ejemplo clásico es el de la discusión acerca de la relación de fumar con cáncer de pulmón. Algunos estadísticos plantearon que los estudios de asociación entre fumar y cáncer de pulmón podrían tener efectos gravemente confundidos, por ejemplo, por aspectos genéticos que hacen a una persona propensa a fumar al mismo tiempo que aumenta su probabilidad de fumar:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n F\n C\n node [shape = circle]\n U\n edge [minlen = 3]\n U -> F\n U -> C\n F -> C\n{rank= same; C; F}\n}\n\")\n\n\n\n\n\n\nEn este caso, el efecto de fumar (\\(F\\)) sobre cáncer (\\(C\\)) no es identificable pues no podemos condicionar a la variable de Genotipo (\\(U\\)). Supongamos que tenemos una medida adicional, que es la cantidad de depósitos de alquitrán den los pulmones de los pacientes. Este es es afectado por \\(F\\), y a su vez, el alquitrán incrementa la probabilidad de cáncer:\n\n\nCódigo\ngrViz(\"\ndigraph {\n graph [ranksep = 0.2]\n node [shape=plaintext]\n F\n C\n A\n node [shape = circle]\n U\n edge [minlen = 3]\n U -> F\n U -> C\n F -> A\n A -> C\n{rank= same; C; F; A}\n}\n\")\n\n\n\n\n\n\nLa idea es primero estimar el efecto de \\(F\\) sobre \\(A\\), y después estimar el efecto de \\(A\\) sobre \\(C\\). La “composición” de estos dos efectos, dado el diagrama, debe darnos el estimador correcto. Primero consideramos el efecto de \\(F\\) sobre \\(A\\), y tenemos que (regla 2)\n\\[p(a|do(f)) = p(a|f),\\] La igualdad se debe a que una vez que condicionamos a \\(F\\) no hay puertas traseras entre \\(F\\) y \\(A\\) (pues no condicionamos a \\(C\\)). Esta dependencia causal la podemos entonces estimar de los datos.\nEl efecto de \\(A\\) sobre \\(C\\) también es identificable, pues el camino no causal se bloquea cuando condicionamos a \\(F\\), de forma que por la fórmula de ajuste:\n\\[p(c|do(a)) = \\int p(c|a, f') p(f')\\, df'\\]\nAhora encadenamos estas dos ecuaciones:\n\\[p(c|do(f)) = \\int p(c|do(a))p(a|f)\\,da\\]\nque equivale en simulación a: dado un valor de \\(F\\), simulamos \\(A\\) con nuestro modelo ajustado con datos naturales. Ahora intervenimos \\(A\\) con el valor \\(a\\) que obtuvimos y simulamos \\(C\\). Sin embargo, para hacer este último paso con datos naturales, necesitamos usar el criterio de puerta trasera como explicamos arriba: simulamos entonces \\(f´\\) de \\(p(f)\\), y después simulamos \\(C\\) en función de \\(a\\) y \\(f´\\) (con una distribución construida a partir de datos).\nRequerimos en este caso construir y estimar la condicional \\(p(c|a, f)\\) basado en los datos.\nEn fórmula, en general, se escribe como:\n\n\n\n\n\n\nCriterio de fuerta delantera (Pearl)\n\n\n\nDecimos que un conjunto de variables \\(A\\) satisface el criterio de puerta delantera en relación a las variables \\(F\\) y \\(C\\) cuando:\n\n\\(A\\) intercepta todos las cadenas dirigidos de \\(F\\) a \\(C\\)\nNo hay ningún camino activo de puerta trasera de \\(F\\) a \\(A\\)\nTodos los caminos de puerta trasera de \\(A\\) a \\(C\\) están bloqueados por \\(F\\).\n\nSi \\(A\\) satisface el criterio de puerta delantera en relación a \\(F\\) y \\(C\\), entonces el efecto causal de \\(F\\) en \\(C\\) es identificable y está dado por la fórmula:\n\\[p(c|do(f)) = \\int \\left [ \\int p(c|a,f´)p(f´)\\,df´ \\right ] p(a|f)\\,da\\]\n\n\nTodas estas cantidades puede estimarse de los datos.\n\nEjemplo: proceso generador\nAntes de aplicar este nuevo procedimiento, describamos el proceso generador que utilizaremos:\n\n# simular distribución natural\nsimular_fd <- function(n = 10, efecto_a = 0.3){\n ## causa común\n u <- rnorm(n, 0, 1);\n # cantidad que fuma\n f <- exp(rnorm(n, 1 + 0.2 * u, 0.1))\n # acumulación de alquitrán\n a <- rnorm(n, 4 * f, 2)\n # probabilidad de cancer\n p_c <- inv_logit(-6 + efecto_a * a + 2 * u)\n c <- rbinom(n, 1, p_c)\n tibble(f, a, c, u)\n}\n# simular datos intervenidos (suponiendo que conocemos todo)\nsim_int_f <- function(n = 100, do_f = 0.3, efecto_a = 0.3){\n a <- rnorm(n, 4 * do_f, 2)\n u <- rnorm(n, 0, 1)\n p_c <- inv_logit(-6 + efecto_a * a + 2 * u)\n c <- rbinom(n, 1, p_c)\n tibble(do_f = do_f, media_c = mean(c))\n}\n\n\nset.seed(4481)\nsims_fd <- simular_fd(5000)\nsims_fd_1 <- simular_fd(10000)\nqplot(sims_fd$f, sims_fd$a)\n\nWarning: `qplot()` was deprecated in ggplot2 3.4.0.\n\n\n\n\n\n\n\n\n\n¿Cómo se ve la relación de fumador con cáncer? En esta gráfica mostramos también el valor de la variable no observada \\(U\\). Nótese que parte de la correlación positiva que existe es debido a esta variable \\(U\\).\n\nggplot(sims_fd, aes(x = f, y = c, colour = u)) + \n geom_jitter() + scale_colour_continuous(type = \"viridis\")\n\n\n\n\n\n\n\n\nAhora veamos cómo se ve el efecto de \\(F\\) sobre \\(C\\) y también cómo se ve el cruce de \\(F\\) y \\(C\\) en los datos naturales:\n\nsims_1 <- map_df(seq(1, 4, 0.5), ~ sim_int_f(100000, .x))\n\nsims_1 |> \n ggplot() + geom_line(aes(x = do_f, y = media_c)) +\n geom_smooth(data = sims_fd_1, aes(x = f, y = c), method = \"loess\", span = 0.3, se = FALSE, colour =\"red\") + xlab(\"Grado de tabaquismo\") +\n xlim(c(1,4))\n\n`geom_smooth()` using formula = 'y ~ x'\n\n\nWarning: Removed 376 rows containing non-finite values (`stat_smooth()`).\n\n\n\n\n\n\n\n\n\nEn efecto causal promedio de fumar, en cada nivel, sobre la incidencia de cáncer de pulmón, suponiendo nuestro proceso generador. Nótese que la relación no es tan fuerte como observamos en los datos naturales (en rojo). Esto se debe a que en los datos naturales, las personas existe una causa común entre no fumar y prevenir cáncer de pulmón.\n\n\nEjemplo: estimación con puerta delantera\nVeamos cómo sería la estimación si tuviéramos datos disponible, y si es que podemos recuperar el efecto correcto dados los datos observados y la técnica de puerta delantera.\nNótese que sólo necesitamos \\(p(c|a, f), p(a|f)\\) y \\(p(f)\\). Estos son modelos estadísticos con el que podemos identificar el efecto que nos interesa. Una vez que los estimemos, podemos usar simulación:\n\nFijamos una \\(f\\).\nSimulamos una \\(a\\) del modelo \\(p(a|f)\\)\nPara calcular \\(\\int p(c|a,f')p(f')\\), tenemos que simular un valor \\(f'\\) de la marginal de \\(p(f)\\), y luego, sustituir junto la \\(a\\) de 1 para simular una \\(c\\) de \\(p(c|a, f')\\).\nConsideramos solamente \\(c\\) y \\(f\\) para resumir el efecto.\n\n\nset.seed(481)\nsims_fd <- simular_fd(2000)\nmod_front_door <- cmdstan_model(\"./src/front-door.stan\")\nprint(mod_front_door)\n\ndata {\n int<lower=0> N;\n int<lower=0> n_f;\n vector[N] f;\n vector[N] a;\n array[N] int<lower=0, upper=1> c;\n array[n_f] real do_f;\n\n}\n\ntransformed data {\n real media_a;\n real media_f;\n\n media_a = mean(a);\n media_f = mean(f);\n}\n\nparameters {\n real<lower=0> alpha;\n real alpha_a;\n real<lower=0> alpha_f;\n real int_a;\n real beta_0;\n real<lower=0> beta_1;\n real<lower=0> beta;\n real<lower=0> a_f;\n real<lower=0> b_f;\n real<lower=0> sigma_a;\n real<lower=0> sigma_f;\n\n}\n\n\n\ntransformed parameters {\n\n\n}\n\nmodel {\n f ~ gamma(a_f, b_f);\n a ~ normal(beta * f, sigma_a);\n c ~ bernoulli_logit(int_a + alpha_a * a + alpha_f * f);\n alpha_a ~ normal(0, 1);\n alpha_f ~ normal(0, 1);\n int_a ~ normal(0, 3);\n sigma_a ~ normal(0, 1);\n sigma_f ~ normal(0, 0.1);\n alpha ~ normal(0, 1);\n beta ~ normal(0, 1);\n beta_0 ~ normal(0, 3);\n beta_1 ~ normal(0, 1);\n\n}\ngenerated quantities {\n array[n_f] real mean_c;\n\n for(i in 1:n_f){\n array[2000] real res_sim;\n for(j in 1:2000){\n real a_sim = normal_rng(beta * (do_f[i]), sigma_a);\n real f_sim = gamma_rng(a_f, b_f);\n res_sim[j] = bernoulli_rng(inv_logit(int_a + alpha_a * a_sim + alpha_f * f_sim));\n }\n mean_c[i] = mean(res_sim);\n }\n\n}\n\n\n\ndo_f <- seq(1, 4, 0.1)\nn_f <- length(do_f)\nsims <- mod_front_door$sample(data = list(N = nrow(sims_fd),\n f = sims_fd$f, a = sims_fd$a,\n c = sims_fd$c, do_f = do_f, n_f = n_f),\n init = 0.01, step_size = 0.01, \n refresh = 1000,\n parallel_chains = 4)\n\nRunning MCMC with 4 parallel chains...\n\nChain 1 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 2 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 3 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1 / 2000 [ 0%] (Warmup) \nChain 4 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 4 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 3 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 1 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 1 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 2 Iteration: 1000 / 2000 [ 50%] (Warmup) \nChain 2 Iteration: 1001 / 2000 [ 50%] (Sampling) \nChain 3 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 3 finished in 42.8 seconds.\nChain 4 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 4 finished in 42.9 seconds.\nChain 1 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 2 Iteration: 2000 / 2000 [100%] (Sampling) \nChain 1 finished in 44.5 seconds.\nChain 2 finished in 44.5 seconds.\n\nAll 4 chains finished successfully.\nMean chain execution time: 43.7 seconds.\nTotal execution time: 44.5 seconds.\n\n\n\nsims_efecto_tbl <- sims$draws(\"mean_c\", format = \"df\") |> \n pivot_longer(cols = contains(\"mean_c\"), values_to = \"media_c\") |> \n separate(name, c(\"nom\", \"id\"), \n sep = \"[\\\\[\\\\]]\", convert = TRUE, extra = \"drop\") |> \n left_join(tibble(f = do_f) |> \n mutate(id = seq_along(f))) \nresumen_tbl <- sims_efecto_tbl |> \n group_by(id, f) |> \n summarise(media = mean(media_c), \n q5 = quantile(media_c, 0.05),\n q95 = quantile(media_c, 0.95))\n\n\nggplot(resumen_tbl) + \n geom_linerange(aes(x= f, ymax = q95, ymin = q5), colour = \"red\") + \n geom_point(aes(x = f, y = media), colour = \"red\") +\n geom_line(data = sims_1, aes(x = do_f, y = media_c)) +\n xlab(\"Nivel de tabaquismo\") + ylab(\"Prop afectada\")\n\n\n\n\n\n\n\n\nY parece que hemos obtenido una estimación razonable del efecto causal de fumar sobre cáncer. Recordemos también que debemos ser cuidadosos al comparar intervalos que salen del mismo modelo por su nivel de traslape.\nPor ejemplo, si quisiéramos calcular contrastes con el nivel 2 de tabaquismo:\n\nefecto_2 <- sims_efecto_tbl |> filter(f == 2) |> \n select(.draw, efecto_2 = media_c)\ncomp_tbl <- left_join(sims_efecto_tbl, efecto_2) |> \n mutate(dif_2 = media_c - efecto_2)\n\nJoining with `by = join_by(.draw)`\n\ncomp_tbl |> group_by(f) |> \n summarise(media = mean(dif_2), q5 = quantile(dif_2, 0.05),\n q95 = quantile(dif_2, 0.95)) |> \nggplot() + geom_linerange(aes(x= f, ymax = q95, ymin = q5)) + geom_point(aes(x = f, y = media)) +\n xlab(\"Nivel de tabaquismo\") + ylab(\"Prop afectada\")\n\n\n\n\n\n\n\n\nNota: nótese como en este ejemplo hemos evitado incluir en nuestro modelo la variable no observada \\(U\\), gracias al procedimiento de puerta delantera descrito arriba.\nEs posible sin embargo intentar un modelo completo bayesiano, sin necesidad de recordar la fórmula. El procedimiento, que es más difícil de ajustar: considera una variable latente \\(U\\) no observada, y es necesario definir cómo puede ser su relación con sus descendientes. Es necesario más cuidado en definir formas funcionales e iniciales apropiadas para que los muestreadores funcionen apropiadamente.\n\n\n\n\nMcElreath, R. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. A Chapman & Hall libro. CRC Press. https://books.google.com.mx/books?id=Ie2vxQEACAAJ.", "crumbs": [ "6  Identificación y cálculo-do" ] @@ -528,5 +528,35 @@ "crumbs": [ "7  Buenos y malos controles" ] + }, + { + "objectID": "08-mcmc.html", + "href": "08-mcmc.html", + "title": "8  Markov Chain Monte Carlo", + "section": "", + "text": "8.1 Algoritmos Metropolis-Hastings\nUno de los primeros algoritmos MCMC fue el de Metropolis-Hastings, que veremos primero en algunos ejemplos. Veremos también por qué ahora tenemos mejores opciones que MH para estimar posteriores de nuestros modelos.", + "crumbs": [ + "8  Markov Chain Monte Carlo" + ] + }, + { + "objectID": "08-mcmc.html#algoritmos-metropolis-hastings", + "href": "08-mcmc.html#algoritmos-metropolis-hastings", + "title": "8  Markov Chain Monte Carlo", + "section": "", + "text": "Ejemplo: dos implementaciones de Metropolis\nSupongamos que queremos simular de una variable aleatoria \\(X\\) con distribución discreta sobre los valores \\(1,2\\ldots, k\\) con probabilidades \\(p(1),p(2),\\ldots,p(k)\\). Este problema puede resolverse fácilmente de varias maneras, pero utilizaremos un método de Monte Carlo tipo Metropolis. Supongamos que podemos simular de una variable aleatoria \\(U\\) que es uniforme en \\(1,2,\\ldots, k\\) (con probabilidades iguales a 1/k).\nLo que podemos hacer es lo que sigue, para \\(i=1,\\ldots, M\\):\n\nPara \\(i=1\\), comenzamos fijando un valor \\(x_1\\) en \\(1,2,\\ldots, k\\).\n\nPara cada \\(i\\),\n\nSimulamos una uniforme en \\(1,2,\\ldots, k\\). Al valor \\(u_i\\) obtenido le llamamos valor propuesto.\nCalculamos \\(q = p(u_i)/p(x_i)\\) (el cociente de la probabilidad del valor propuesto entre la probabilidad del valor actualo).\nSi \\(q > 1\\), aceptamos el valor propuesto y ponemos \\(x_{i+1} = u_i\\).\nSi \\(q < 1\\), aceptamos el valor propuesto con probabilidad \\(q\\) (\\(x_{i+1} = u_i\\)), y con probabilidad \\(1-q\\) rechazamos (\\(x_{i+1} = x_i\\)).\n\nEl resultado es una sucesión de valores \\(x_1,x_2,\\ldots, x_M\\). Es posible demostrar que la distribución de estas \\(x_i\\) converge a la distribución \\(p(1),\\ldots, p(k)\\) si \\(M\\) es suficientemente grande.\nEste método se llama Metropolis-Hastings. Es un método de Monte Carlo, y como podemos ver, se trata de una cadena de Markov, pues la distribución cada siguiente lugar \\(x_{i+1}\\), condicionada al valor actual \\(x_i\\) no depende de valores anteriores de las \\(x\\).\n\n#set.seed(451123)\n# definimos estas p\nk <- 40\np <- exp(-(1:k - k/3)^2 / 10)\np <- p /sum(p)\ndist_obj <- tibble(x = 1:k, p = p)\n# simulamos\nM <- 200000\nx <- numeric(M)\nx[1] <- 20\nfor(i in 1:M){\n u <- sample(1:k, 1)\n q <- p[u] / p[x[i]]\n if(runif(1) < q){\n x[i+1] <- u\n } else {\n x[i+1] <- x[i] \n }\n}\n\nEn rojo mostramos las probabilidades objetivo que queremos estimar, y en negro las estimadas con nuestro método de arriba.\n\nresultados_sim <- tibble(x = x) |> \n mutate(n_sim = row_number())\nresumen_sim <- resultados_sim |> count(x) |> \n right_join(tibble(x = 1:k, p = p)) |> \n mutate(n = ifelse(is.na(n), 0, n)) |> \n mutate(p_aprox = n / sum(n))\n\nJoining with `by = join_by(x)`\n\nggplot(dist_obj, aes(x = x)) +\n geom_point(aes(y = p)) +\n geom_point(data = resumen_sim, \n aes(y = p_aprox), color = \"red\", size = 3, alpha = 0.5) \n\n\n\n\n\n\n\n\nComo vemos, los valores de \\(x_1,\\ldots, x_M\\) se distribuyen aproximadamente como la distribución \\(p\\) objetivo. Esta es una manera de simular valores de esta distribución discreta \\(p\\). Podemos ver cómo se ven las simulaciones sucesivas:\n\nggplot(resultados_sim |> filter(n_sim < 400), aes(x = n_sim, y = x)) +\n geom_line() + scale_y_continuous(breaks = 1:20)\n\n\n\n\n\n\n\n\nEl defecto que tiene este algoritmo es que puede ser relativamente lento, pues vemos que hay periodos largos donde se “atora” en valores de probabilidad relativamente alta. La razón es que en muchos pasos, estamos proponiendo “saltos al vacío” a lugares de probabilidad muy baja, que rara vez se aceptan.\nPodemos hacer más eficiente nuestro algoritmo si le permitimos explorar con mayor facilidad los posibles valores de \\(x\\). Esto se logra proponiendo saltos locales: si estamos en \\(x_i\\), entonces proponemos los valores \\(x_i - 1\\) y \\(x_i + 1\\) con la misma probabilidad 1/2 (excepto en los extremos donde sólo proponemos \\(x_i,x_i+1\\) o \\(x_i-1,x_i\\)).\nProponemos entonces la suguiente modificación del paso 1 de propuesta:\n\nPara \\(i=1\\), comenzamos fijando un valor \\(x_1\\) en \\(1,2,\\ldots, k\\).\n\nPara cada \\(i\\),\n\nSi \\(1< x_i<k\\), escogemos al azar un salto a la derecha o al izquierda con igual probabilidad 1/2. En los extremos \\(x_i=1\\) o \\(x_i=k\\) escogemos entre \\(1,2\\) o \\(k-1,k\\) respectivamente. Al valor \\(u_i\\) obtenido le llamamos valor propuesto.\nCalculamos \\(q = p(u_i)/p(x_i)\\) .\nSi \\(q > 1\\), aceptamos el valor propuesto y ponemos \\(x_{i+1} = u_i\\).\nSi \\(q < 1\\), aceptamos el valor propuesto con probabilidad \\(q\\) (\\(x_{i+1} = u_i\\)), y con probabilidad \\(q\\) rechazamos(\\(x_{i+1} = x_i\\)).\n\nEsto lo escribimos como sigue:\n\n#set.seed(4511)\n# simulamos\nx <- numeric(M)\nx[1] <- 20\nfor(i in 1:M){\n u <- sample(c(x[i] - 1, x[i] + 1), 1)\n if(u == k+1) u <- k\n if(u == 0) u <- 1\n q <- p[u] / p[x[i]]\n if(runif(1) < q){\n x[i+1] <- u\n } else {\n x[i+1] <- x[i] \n }\n}\n\nObtenemos:\n\nresultados_sim_2 <- tibble(x = x) |> \n mutate(n_sim = row_number())\nresumen_sim_2 <- resultados_sim_2 |> count(x) |> \n mutate(p_aprox = n / sum(n))\nggplot(dist_obj, aes(x = x)) +\n geom_point(aes(y = p)) +\n geom_point(data = resumen_sim_2, \n aes(y = p_aprox), color = \"red\", size = 3, alpha = 0.5) \n\n\n\n\n\n\n\n\nY podemos ver cómo evoluciona nuestra cadena de Markov:\n\nggplot(resultados_sim_2 |> filter(n_sim < 400), aes(x = n_sim, y = x)) +\n geom_line() + scale_y_continuous(breaks = 1:10)\n\n\n\n\n\n\n\n\n¿Cómo se comparan estos dos métodos? Podemos ver por ejemplo cómo se comparan las distribuciones aproximadas hasta cierto número de iteraciones con la verdadera distribución objetivo:\n\napprox_sim <- map_df(seq(200, 30000, by = 200), function(n_sims){\n resumen_1 <- resultados_sim |> filter(n_sim <= n_sims) |> \n count(x) |>\n mutate(p_aprox = n / sum(n)) |>\n select(-n) |>\n right_join(dist_obj, by = \"x\") |> \n mutate(metodo = \"MH-1\") |> \n mutate(n_sims = n_sims) \n resumen_2 <- resultados_sim_2 |> filter(n_sim <= n_sims) |>\n count(x) |>\n mutate(p_aprox = n / sum(n)) |>\n select(-n) |>\n right_join(dist_obj, by = \"x\") |> \n mutate(metodo = \"MH-2\") |> \n mutate(n_sims = n_sims) \n bind_rows(resumen_1, resumen_2) |> \n mutate(p_aprox = ifelse(is.na(p_aprox), 0, p_aprox))\n})\n\n\napprox_sim |> \n mutate(dif_abs = abs (p_aprox-p)) |>\n group_by(metodo, n_sims) |> \n summarise(dif_abs = sum(dif_abs)) |> \nggplot(aes(n_sims, dif_abs, color = metodo)) +\n geom_line() \n\n`summarise()` has grouped output by 'metodo'. You can override using the\n`.groups` argument.\n\n\n\n\n\n\n\n\n\nEn este caso, considera qué es lo que sucede en cada uno de estos casos:\n\nEl algoritmo que da saltos grandes muchas veces rechaza porque cae en un área de probilidad muy baja.\nEl algoritmo que da saltos chicos puede tardar en explorar regiones de probabilidad relativamente alta con suficiente frecuencia (tarda un moverse de un lugar a otro), por su naturaleza de “caminata aleatoria”. Pero sus tasas de aceptación son más altas.\n\nEste es el primer balance que existe en este algoritmo: tomar pasos grandes y balancear las probabilidades quizá rechazando muy frecuentemente (no es eficiente), o tomar pasos chicos y vagar más tiempo para visitar regiones de alta probabilidad, aunque con menos tasa de rechazo. Dependiendo de la distribución que queremos aproximar podemos inclinarnos más por una o por otra opción.\n¿Por qué funcionan estos algoritmos? Supongamos que en cada paso, se cumple que (balance detallado): \\[{q(x|y)}p(y) = {q(y|x)}{p(x)}\\] donde \\(q(x|y)\\) son las probabilidades de transición de nuestra cadena de Markov propuesta. Esta ecuación dice que si la probabilidad se distribuye como \\(p(x)\\), entonces al transicionar con \\(q\\) la probabilidad fluje de manera que se mantiene estática en \\(p\\), es decir \\(p\\) es una distribución estacionaria. Esto es fácil de demostrar pues \\[\\sum_{y} q(x|y)p(y) = \\sum_{y} q(y|x)p(x) = p(x) \\sum_{y} q(y|x) = p(x).\\]\nBajo otros supuestos adicionales de ergodicidad (aperiodicidad y tiempos de recurrencia finitos, es decir, las transiciones mezclan bien los estados), entonces podemos simular la cadena de Markov por un tiempo suficientemente largo y con esto obtener una muestra de la distribución objetivo \\(p\\).\n¿Cómo podemos diseñar entonces las \\(q(x|y)\\) correspondientes? Comenzamos considerando distribuciones propuesta \\(q_0(x|y)\\) que no satisfacen la ecuación de balance, y supondremos como en los ejemplos de arriba (verifícalo) que nuestras transiciones tienen probabilidades simétricas \\(q_0(y|x) = q_0(x|y)\\). Entonces, cuando \\(p(y)/p(x) > 1\\), queremos transicionar de \\(x\\) a \\(y\\) con más frecuencia que de \\(y\\) a \\(x\\). Comenzando en \\(x\\), si la propuesta de \\(q_0\\) es \\(y\\), podríamos poner entonces que el sistema transicione con probabilidad 1 a \\(y\\). Sin embargo, si empezamos en \\(y\\) y la propuesta es \\(x\\), el sistema sólo transiciona de \\(y\\) a \\(x\\) con probabilidad \\(p(x)/p(y)\\).\nDe esta manera, obtenemos que bajo \\(q(y|x)\\), \\(x\\) transiciona a \\(y\\) con probabilidad \\(\\min\\{1, p(y)/p(x)\\}\\). Entonces, el cociente \\(\\frac{q(y|x)}{q(x|y)}\\) es igual a \\(\\frac{p(y)}{p(x)}\\) si \\(p(y)<p(x)\\), y es igual a \\(1/\\frac{p(x)}{p(y)} = \\frac{p(y)}{p(x)}\\) si \\(p(y)>p(x)\\).\n\n\n\n\n\n\nIdea básica de MCMC\n\n\n\nEn MCMC, buscamos un cadena de Markov que, en el largo plazo, visite cada posible parámetro proporcionalmente a la probabildad posterior de cada parámetro.\n\n\n\n\nEjemplo: Metropolis bivariado\nSupongamos ahora que quisiéramos simular de una normal multivariada con media en c(2,3) y matriz de covarianza \\(\\Sigma\\), que supondremos es tal que la desviación estándar de cada variable es 1 y la correlación es 0.8. La matriz \\(\\Sigma\\) tiene 1 en la diagonal y 0.8 fuera de la diagonal.\nLa distribución objetivo \\(p\\) está dada entonces (módulo una constante de proporcionalidad):\n\nSigma <- matrix(c(1, 0.8, 0.8, 1), nrow = 2)\nSigma_inv <- solve(Sigma)\nm <- c(2, 3)\nlog_p <- function(z){\n - 0.5 * (t(z-m) %*% Sigma_inv %*% (z-m))\n}\n\nNótese que como Metropolis hace cocientes de probabilidades, sólo es necesario conocer la densidad objetivo módulo una constante de proporcionalidad.\nUna algoritmo de Metropolis podría ser el siguiente:\n\n# simulamos\nM <- 50000\nsimular_normal <- function(M, m, Sigma, delta_x, delta_y){\n z <- matrix(nrow = M, ncol = 2)\n z[1, ] <- c(2.5, 3.2)\n colnames(z) <- c(\"x\", \"y\")\n rechazo <- 0\n for(i in 1:(M-1)){\n x_prop <- rnorm(1, z[i, 1], delta_x)\n y_prop <- rnorm(1, z[i, 2], delta_y)\n z_prop <- c(x_prop, y_prop)\n q <- exp(log_p(z_prop) - log_p(z[i, ]))\n if(runif(1) < q){\n z[i + 1, ] <- z_prop\n } else {\n rechazo <- rechazo + 1\n z[i + 1, ] <- z[i, ]\n } \n }\n print(rechazo / M)\n z_tbl <- as_tibble(z) |> \n mutate(n_sim = row_number())\n z_tbl\n}\nz_tbl <- simular_normal(M, m, Sigma, 1, 1)\n\n[1] 0.60038\n\n\nVemos que tenemos una tasa alta de rechazos. ¿Por qué? Veamos cómo se ven las simulaciones después de 1000 iteraciones:\n\n# estas las usamos para graficar\nsims_normal <- mvtnorm::rmvnorm(10000, mean = m, sigma = Sigma)\ncolnames(sims_normal) <- c(\"x\", \"y\")\nsims_normal <- as_tibble(sims_normal)\ngraf_tbl <- map_df(seq(10, 500, 20), function(i){\n z_tbl |> filter(n_sim <= i) |> mutate(num_sims = i)\n})\nggplot(graf_tbl, aes(x, y)) + \n stat_ellipse(data = sims_normal, aes(x, y), \n level = c( 0.9), type = \"norm\", colour = \"salmon\") +\n stat_ellipse(data = sims_normal, aes(x, y), \n level = c( 0.5), type = \"norm\", colour = \"salmon\") +\n geom_point(alpha = 0.1) +\n facet_wrap(~num_sims) + theme_minimal()\n\n\n\n\n\n\n\n\nObservaciones:\n\nLos puntos que tienen intensidad alta son puntos donde hubo varios rechazos. Esto es porque las propuestas a veces caen en elipses de baja probabilidad (en la gráfica mostramos una elipse de 50% probabilidad y otra de 95%).\nEsto se debe a que los saltos en cada dirección son de desviación estándar 1, y esto fácilmente nos lleva a una zona de alta probabilidad a una de baja probabilidad.\nSin embargo, a largo plazo, vemos cómo la cadena está visitando las regiones de alta probabilidad con aparentemente la frecuencia correcta.\n\nPodemos entonces proponer saltos más chicos, por ejemplo:\n\nz_tbl <- simular_normal(M, m, Sigma, 0.2, 0.2)\n\n[1] 0.1568\n\ngraf_tbl <- map_df(seq(10, 500, 20), function(i){\n z_tbl |> filter(n_sim <= i) |> mutate(num_sims = i)\n})\nggplot(graf_tbl, aes(x, y)) + \n stat_ellipse(data = sims_normal, aes(x, y), \n level = c( 0.9), type = \"norm\", colour = \"salmon\") +\n stat_ellipse(data = sims_normal, aes(x, y), \n level = c( 0.5), type = \"norm\", colour = \"salmon\") +\n geom_point(alpha = 0.1) +\n facet_wrap(~num_sims) + theme_minimal()\n\n\n\n\n\n\n\n\nObservaciones:\n\nEn este caso tenemos una tasa de aceptación más alta.\nSin embargo, la cadena parece “vagar” en la regiones de probabilidad alta, y tiene dificultades para explorar correctamente estas regiones.\nSin embargo, a largo plazo, vemos cómo la cadena está visitando las regiones de alta probabilidad con aparentemente la frecuencia correcta.\n\n\n\n\n\n\n\nMetropolis-Hastings\n\n\n\nEn el algoritmo de Metropolis Hastings hay una tensión natural en tamaño de salto y tasa de aceptación. Si el tamaño de los saltos es muy grande, la tasa de aceptación es baja y esto producen ineficiencias. Si el tamaño de los saltos es muy chico, la tasa de aceptación es más alta, pero esto también es ineficiente.\n\n\nExisten métodos que pueden superar este problema, como son muestreo de Gibbs y Monte Carlo Hamiltoniano. El primero no lo discutiremos, pues requiere poder simular fácilmente de cada parámetro dados los otros, y esto no siempre es posible. Veremos más el segundo, donde usaremos información del gradiente de la distribución objetivo para proponer exploración más eficiente.", + "crumbs": [ + "8  Markov Chain Monte Carlo" + ] + }, + { + "objectID": "08-mcmc.html#monte-carlo-hamiltoniano", + "href": "08-mcmc.html#monte-carlo-hamiltoniano", + "title": "8  Markov Chain Monte Carlo", + "section": "8.2 Monte Carlo Hamiltoniano", + "text": "8.2 Monte Carlo Hamiltoniano", + "crumbs": [ + "8  Markov Chain Monte Carlo" + ] } ] \ No newline at end of file diff --git a/site_libs/bootstrap/bootstrap.min.css b/site_libs/bootstrap/bootstrap.min.css index dc4965a..8c92dbe 100644 --- a/site_libs/bootstrap/bootstrap.min.css +++ b/site_libs/bootstrap/bootstrap.min.css @@ -2,7 +2,7 @@ * Bootstrap v5.3.1 (https://getbootstrap.com/) * Copyright 2011-2023 The Bootstrap Authors * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE) - 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex 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xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.7%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.7%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: 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var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type .accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}.accordion-flush .accordion-item .accordion-button,.accordion-flush .accordion-item .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23f3a4a2'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23f3a4a2'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(34, 34, 34, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(34, 34, 34, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #eb6864;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #fff;--bs-pagination-hover-bg: #eb6864;--bs-pagination-hover-border-color: #eb6864;--bs-pagination-focus-color: #bc5350;--bs-pagination-focus-bg: #eee;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(235, 104, 100, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #eb6864;--bs-pagination-active-border-color: #eb6864;--bs-pagination-disabled-color: rgba(34, 34, 34, 0.75);--bs-pagination-disabled-bg: #eee;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 1px solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #eee;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #eb6864;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #222;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(34, 34, 34, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #222;--bs-list-group-action-active-bg: #eee;--bs-list-group-disabled-color: rgba(34, 34, 34, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #eb6864;--bs-list-group-active-border-color: #eb6864;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;border-radius:var(--bs-list-group-border-radius)}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xxl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(235, 104, 100, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.25rem;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(34, 34, 34, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow);border-radius:var(--bs-toast-border-radius)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width))}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #fff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);border-radius:var(--bs-modal-border-radius);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color);border-top-left-radius:var(--bs-modal-inner-border-radius);border-top-right-radius:var(--bs-modal-inner-border-radius)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;flex-wrap:wrap;-webkit-flex-wrap:wrap;align-items:center;-webkit-align-items:center;justify-content:flex-end;-webkit-justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid 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.value-box-grid>div{align-self:center}.bslib-value-box.showcase-top-right:not([data-full-screen=true]) .value-box-showcase{margin-top:0}@container bslib-value-box (max-width: 300px){.bslib-value-box.showcase-top-right:not([data-full-screen=true]) .value-box-grid .value-box-showcase{padding-left:1rem}}.bslib-value-box.showcase-left-center .value-box-grid{grid-template-columns:var(---bslib-value-box-showcase-w, 30%) auto}.bslib-value-box.showcase-left-center[data-full-screen=true] .value-box-grid{grid-template-columns:var(---bslib-value-box-showcase-w-fs, 1fr) auto}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-showcase{grid-area:left}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-area{grid-area:right}.bslib-value-box.showcase-bottom .value-box-grid{grid-template-columns:1fr;grid-template-rows:1fr var(---bslib-value-box-showcase-h, auto);grid-template-areas:"top" "bottom";overflow:hidden}.bslib-value-box.showcase-bottom .value-box-grid .value-box-showcase{grid-area:bottom;padding:0;margin:0}.bslib-value-box.showcase-bottom .value-box-grid .value-box-area{grid-area:top}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid{grid-template-rows:1fr var(---bslib-value-box-showcase-h-fs, 2fr)}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid .value-box-showcase{padding:1rem}[data-bs-theme=dark] .bslib-value-box{--bslib-value-box-shadow: 0 0.5rem 1rem rgb(0 0 0 / 50%)}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto 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1200px){.bslib-grid.grid.bslib-grid--row-heights--xl{--bslib-grid--row-heights: var(--bslib-grid--row-heights--xl)}}@media(min-width: 1400px){.bslib-grid.grid.bslib-grid--row-heights--xxl{--bslib-grid--row-heights: var(--bslib-grid--row-heights--xxl)}}.bslib-grid>*>.shiny-input-container{width:100%}.bslib-grid-item{grid-column:auto/span 1}@media(max-width: 767.98px){.bslib-grid-item{grid-column:1/-1}}@media(max-width: 575.98px){.bslib-grid{grid-template-columns:1fr !important;height:var(--bslib-grid-height-mobile)}.bslib-grid.grid{height:unset !important;grid-auto-rows:var(--bslib-grid--row-heights--xs, auto)}}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex 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xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.7%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.7%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: 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var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type .accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}.accordion-flush .accordion-item .accordion-button,.accordion-flush .accordion-item .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23f3a4a2'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23f3a4a2'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(34, 34, 34, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(34, 34, 34, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #eb6864;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #fff;--bs-pagination-hover-bg: #eb6864;--bs-pagination-hover-border-color: #eb6864;--bs-pagination-focus-color: #bc5350;--bs-pagination-focus-bg: #eee;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(235, 104, 100, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #eb6864;--bs-pagination-active-border-color: #eb6864;--bs-pagination-disabled-color: rgba(34, 34, 34, 0.75);--bs-pagination-disabled-bg: #eee;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 1px solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #eee;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #eb6864;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #222;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(34, 34, 34, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #222;--bs-list-group-action-active-bg: #eee;--bs-list-group-disabled-color: rgba(34, 34, 34, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #eb6864;--bs-list-group-active-border-color: #eb6864;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;border-radius:var(--bs-list-group-border-radius)}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xxl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(235, 104, 100, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.25rem;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(34, 34, 34, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow);border-radius:var(--bs-toast-border-radius)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width))}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #fff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);border-radius:var(--bs-modal-border-radius);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color);border-top-left-radius:var(--bs-modal-inner-border-radius);border-top-right-radius:var(--bs-modal-inner-border-radius)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;flex-wrap:wrap;-webkit-flex-wrap:wrap;align-items:center;-webkit-align-items:center;justify-content:flex-end;-webkit-justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid 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