From 78c55b1913517705c5c9448f8e6026618aabcd86 Mon Sep 17 00:00:00 2001 From: Brad Duthie Date: Tue, 20 Apr 2021 22:26:17 +0100 Subject: [PATCH] Initialise --- Female.JPG | Bin 0 -> 11849 bytes GLMs.Rmd | 565 +++++++++++++++++++++++++++++++++++++ GLMs.pdf | Bin 0 -> 3794840 bytes Het1.csv | 17 ++ Het1_T6A | 44 +++ IMG_0036.JPG | Bin 0 -> 621791 bytes IMG_0506.JPG | Bin 0 -> 224017 bytes Male1.jpg | Bin 0 -> 105404 bytes Male2.jpg | Bin 0 -> 72246 bytes Male2.png | Bin 0 -> 90308 bytes data/heterandrium_data.csv | 407 ++++++++++++++++++++++++++ figtree.jpg | Bin 0 -> 1557796 bytes het1_syc.png | Bin 0 -> 76388 bytes het_heads.csv | 38 +++ wasps.png | Bin 0 -> 525357 bytes 15 files changed, 1071 insertions(+) create mode 100755 Female.JPG create mode 100644 GLMs.Rmd create mode 100644 GLMs.pdf create mode 100644 Het1.csv create mode 100644 Het1_T6A create mode 100755 IMG_0036.JPG create mode 100755 IMG_0506.JPG create mode 100755 Male1.jpg create mode 100644 Male2.jpg create mode 100755 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beamer_presentation: + theme: "default" + colortheme: "default" + fonttheme: "default" + ioslides_presentation: default + slidy_presentation: default +header-includes: +- \usepackage{hyperref} +colorlinks: true +linkcolor: blue +urlcolor: blue +--- + + + + +## Quick review of simple linear regression + +```{r, echo = FALSE} +library(knitr); +library(faraway); +``` + +```{r, echo = FALSE} +include_graphics("figtree.jpg") +``` + + +## Quick review of simple linear regression + +Fig trees in Baja, Mexico are visted by several species of wasps + +```{r, echo = FALSE, out.width="100%"} +include_graphics("wasps.png") +``` + +Wasps use their ovipositors to drill into the side of the enclosed inflorescence (syconia, or colloquially "fruit") + +## Quick review of simple linear regression + +```{r, echo = FALSE} +Het1 <- read.csv("Het1.csv"); +mod1 <- lm(Body_size ~ Volume, data = Het1); +mody <- predict(mod1) +par(mar = c(5, 5, 1, 1)); +plot(x = Het1$Volume, y = Het1$Body_size, pch = 20, cex.lab = 1.5, + cex.axis = 1.25, xlab = expression(paste("Fruit volume (mm"^"3",")")), + ylab = expression(paste("Female wasp body volume (mm"^"3",")")), cex = 1.25); +``` + +## Quick review of simple linear regression + +```{r, echo = FALSE} +Het1 <- read.csv("Het1.csv"); +mod1 <- lm(Body_size ~ Volume, data = Het1); +mody <- predict(mod1) +par(mar = c(5, 5, 1, 1)); +plot(x = Het1$Volume, y = Het1$Body_size, pch = 20, cex.lab = 1.5, + cex.axis = 1.25, xlab = expression(paste("Fruit volume (mm"^"3",")")), + ylab = expression(paste("Female wasp body volume (mm"^"3",")")), cex = 1.25); +points(x = Het1$Volume, y = mody, type = "l", lwd = 3); +text(x = 3800, y = 0.2, + expression(paste(y[i] == beta[0] + beta[1]*x[i] + epsilon[i])), cex = 3); +``` + +## Quick review of simple linear regression + +```{r, echo = FALSE} +Het1 <- read.csv("Het1.csv"); +mod1 <- lm(Body_size ~ Volume, data = Het1); +mody <- predict(mod1) +par(mar = c(5, 5, 1, 1)); +plot(x = Het1$Volume, y = Het1$Body_size, pch = 20, cex.lab = 1.5, + cex.axis = 1.25, xlab = expression(paste("Fruit volume (mm"^"3",")")), + ylab = expression(paste("Female wasp body volume (mm"^"3",")")), cex = 1.25); +points(x = Het1$Volume, y = mody, type = "l", lwd = 3); +text(x = 3800, y = 0.2, + expression(paste(y[i] == beta[0] + beta[1]*x[i] + epsilon[i])), cex = 3); +for(i in 1:dim(Het1)[1]){ + x0 <- Het1$Volume[i]; + x1 <- Het1$Volume[i]; + y0 <- Het1$Body_size[i]; + y1 <- mody[i]; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +``` + + +## Generalising the linear model: error (i.e., residual) structure + +**General** linear models assume normally distributed errors + +Actual errors can violate the asumption of normality in several ways + +```{r, echo = FALSE, fig.height = 5} +x <- rnorm(1000000) +hist(x, breaks = 100, col = "blue", main = "", lwd = 2, + xlab = "Residual around the predictive value"); +``` + +Strong skew, kurtosis, Strict bounds (e.g., values between 0 and 1 as shown earlier, predicted values never below zero as with counts) + +## Common problems of general linear models + +**What if the response ($y$) variable residuals do not fit general linear model assumptions? This can happen under the following conditions:** + +- Residuals ($\epsilon$) do not have a constant variance across $x$ values (heteroscadisticity) +- Residuals ($\epsilon$) are not normally distributed \pause + +**Four situations of interest$^1$:** + +1. Count data +2. Proportion data +3. Binary responses +4. "Time to event" data + +\begin{footnotesize} +\footnotetext[1]{Logan, M. 2011. {\it Biostatistical design and analysis using R: a practical guide}. John Wiley \& Sons.} +\end{footnotesize} + +## A concrete example: fig wasp survival + +\begin{columns} +\begin{column}{0.35\textwidth} + \begin{center} + \begin{itemize} + \item[] \includegraphics{het1_syc.png} \vspace{5 mm} + \item[] \includegraphics[width=70px]{Male2.jpg} \vspace{5 mm} + \item[] \includegraphics{Male1.jpg} + \end{itemize} + \end{center} +\end{column} +\begin{column}{0.65\textwidth} +\begin{itemize} +\setlength\itemsep{1em} +\item Female {\it Heterandrium} wasps can produce two types of males +\item Winged males disperse from their natal fruit to mate +\item Wingless males engage in combat within fruit for access to females \pause +\item {\bf Do bigger wingless males have a higher probability of survival?} +\end{itemize} +\end{column} +\end{columns} + +## Fig wasp survival data + +\Large + +```{r, echo = FALSE} +huse <- read.csv("het_heads.csv"); +huse1 <- huse[,-2]; +rownames(huse1) <- NULL; +huse2 <- as.matrix(huse1); +scrmb <- sample(1:dim(huse2)[1], size = dim(huse2)[1], replace = FALSE); +kable(huse2[scrmb[1:9], ]); +``` + +## Fig wasp survival data + +```{r, echo = FALSE} +lmd <- lm(Survival ~ head_width_mm, data = huse); +xpr <- seq(from = 600, to = 790, by = 1); +ypr <- as.numeric(lmd$coefficients[1]) + as.numeric(lmd$coefficients[2]) * xpr; +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5, + type = "n"); +``` + +## Fig wasp survival data + +```{r, echo = FALSE} +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +``` + + +## Fig wasp survival data + +```{r, echo = FALSE} +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 620, y = 1.1, + expression(paste(y[i] == beta[0] + beta[1]*x[i] + epsilon[i])), cex = 2.5); +points(x = xpr, y = ypr, type = "l", lwd = 3); +``` + +## Fig wasp survival data + +```{r, echo = FALSE} +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 620, y = 1.1, + expression(paste(y[i] == beta[0] + beta[1]*x[i] + epsilon[i])), cex = 2.5); +points(x = xpr, y = ypr, type = "l", lwd = 3); +for(i in 1:dim(huse)[1]){ + x0 <- huse$head_width_mm[i]; + x1 <- huse$head_width_mm[i]; + y0 <- huse$Survival[i] + y1 <- as.numeric(lmd$coefficients[1]) + as.numeric(lmd$coefficients[2])*x1; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +``` + + + +## Using the logit link function + +```{r, echo = FALSE} +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 620, y = 1.1, + expression(paste(y[i] == beta[0] + beta[1]*x[i] + epsilon[i])), cex = 2.5); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "response") +for(i in 1:dim(huse)[1]){ + x0 <- huse$head_width_mm[i]; + x1 <- huse$head_width_mm[i]; + y0 <- huse$Survival[i] + y1 <- prd[i]; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "response") +points(x = newd, y = prdp, type = "l", lwd = 3); +``` + + +## Using the logit link function + +\hrule + +The logit link function linearises the binomial probability function + +$$\ln\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 x$$ + +\hrule \pause + +```{r, echo = FALSE, fig.height = 5} +par(mar = c(5, 5, 1, 1), mfrow = c(1, 2)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, type = "n", + xlab = expression(paste("Wingless male head width (mm): ",beta[1])), + cex = 1.5, ylab = "Wingless male survival: p", cex.lab = 1.5, + cex.axis = 1.5); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "response") +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "response") +points(x = newd, y = prdp, type = "l", lwd = 3); + +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-20, 20), pch = 20, type = "n", + ylab = expression(paste("logit: ln(p/(1-p))")), cex = 1.5, + xlab = expression(paste("Wingless male head width (mm): ",beta[1])), + cex.lab = 1.5, cex.axis = 1.5); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "link") +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "link") +points(x = newd, y = prdp, type = "l", lwd = 3); +``` + + +## Using the logit link function + +\Large + +```{r, echo = FALSE} +kable(huse2[scrmb[1:9], ]); +``` + + +## Generalised linear model in R + +\footnotesize + +```{r} +dat <- read.csv("het_heads.csv"); +ghus <- glm(formula = Survival ~ head_width_mm, + family = binomial(link = "logit"), data = huse); +print(ghus); +``` + + + + +## Generalised linear model in R + +```{r, echo = FALSE} +B0 <- round(as.numeric(ghus$coefficients[1]), digits = 4); +B1 <- round(as.numeric(ghus$coefficients[2]), digits = 4); +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 700, y = 1.2, cex = 2.5, + labels = paste("ln(p/(1-p)) = ", B0, " + ", B1, "x", sep = "")); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "response"); +for(i in 1:dim(huse)[1]){ + x0 <- huse$head_width_mm[i]; + x1 <- huse$head_width_mm[i]; + y0 <- huse$Survival[i] + y1 <- prd[i]; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "response") +points(x = newd, y = prdp, type = "l", lwd = 3); +``` + + +## Generalised linear model in R + +```{r, echo = FALSE} +B0 <- round(as.numeric(ghus$coefficients[1]), digits = 4); +B1 <- round(as.numeric(ghus$coefficients[2]), digits = 4); +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 700, y = 1.2, cex = 2.5, + labels = paste("ln(p/(1-p)) = ", B0, " + ", B1, "x", sep = "")); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "response"); +for(i in 1:dim(huse)[1]){ + x0 <- huse$head_width_mm[i]; + x1 <- huse$head_width_mm[i]; + y0 <- huse$Survival[i] + y1 <- prd[i]; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "response") +points(x = newd, y = prdp, type = "l", lwd = 3); +mdi <- which(abs(prd-0.5) == min(abs(prd-0.5))); +points(x = huse$head_width_mm[mdi], prd[mdi], col = "red", pch = 20, cex = 3); +``` + +## What the link function is doing + +```{r, echo = FALSE} +B0 <- round(as.numeric(ghus$coefficients[1]), digits = 4); +B1 <- round(as.numeric(ghus$coefficients[2]), digits = 4); +par(mar = c(5, 5, 1, 1)); +plot(x = huse$head_width_mm, y = huse$Survival, xlim = c(580, 800), + ylim = c(-0.2, 1.25), pch = 20, + xlab = expression(paste("Wingless male head width (mm)")), cex = 1.5, + ylab = "Wingless male survival", cex.lab = 1.5, cex.axis = 1.5); +text(x = 700, y = 1.2, cex = 2.5, + labels = paste("ln(p/(1-p)) = ", B0, " + ", B1, "x", sep = "")); +ghus <- glm(Survival ~ head_width_mm, family = "binomial", data = huse); +prd <- predict(ghus, type = "response"); +for(i in 1:dim(huse)[1]){ + x0 <- huse$head_width_mm[i]; + x1 <- huse$head_width_mm[i]; + y0 <- huse$Survival[i] + y1 <- prd[i]; + points(x = c(x0, x1), y = c(y0, y1), type = "l", lty = "dotted", lwd = 2); +} +newd <- seq(from = 580, to = 800, by = 0.001); +prdp <- predict(ghus, list(head_width_mm = newd), type = "response"); +points(x = newd, y = prdp, type = "l", lwd = 3); +mdi <- which(abs(prd-0.5) == min(abs(prd-0.5))); +points(x = huse$head_width_mm[mdi], prd[mdi], col = "red", pch = 20, cex = 3); +arrows(x0 = huse$head_width_mm[mdi], x1 = huse$head_width_mm[mdi], + y0 = prd[mdi], y1 = -0.25, length = 0.2, lwd = 2, col = "red"); +arrows(x0 = huse$head_width_mm[mdi], x1 = 572, + y0 = prd[mdi], y1 = prd[mdi], length = 0.2, lwd = 2, col = "red"); +text(x = 725, y = -0.15, cex = 1.75, col = "red", + labels = paste("x = ", huse$head_width_mm[mdi], sep = "")); +text(x = 600, y = prd[mdi] + 0.05, cex = 1.75, col = "red", + labels = paste("p = ", round(prd[mdi], digits = 5), sep = "")); +``` + + +## What the link function is doing + +```{r, echo = FALSE} +B0 <- as.numeric(ghus$coefficients[1]); +B1 <- as.numeric(ghus$coefficients[2]); +XX <- 691.14828; +YY <- B0 + B1*XX; +P1 <- exp(YY); +pp <- P1/(1+P1); +``` + + +$$ln\left(\frac{p}{1-p}\right) = \beta_{0}+\beta_{1}x$$ + +$$ln\left(\frac{p}{1-p}\right) = `r B0` + `r B1` x$$ \pause + +$$ln\left(\frac{`r pp`}{1-`r pp`}\right) = `r B0` + `r B1` (`r XX`)$$ \pause + +$$ln\left(\frac{`r pp`}{1-`r pp`}\right) = `r B0 + B1 * XX`$$ \pause + + +$$ln\left(`r pp/(1-pp)`\right) = `r B0 + B1 * XX`$$ + + +## What the link function is doing + +```{r, echo = FALSE} +B0 <- as.numeric(ghus$coefficients[1]); +B1 <- as.numeric(ghus$coefficients[2]); +XX <- 700; +YY <- B0 + B1*XX; +P1 <- exp(YY); +pp <- P1/(1+P1); +``` + + +$$y = `r B0` + `r B1` (`r XX`)$$ + +$$y = `r B0 + B1 * XX`$$ + +$$ln\left(\frac{p}{1-p}\right) = `r B0 + B1 * XX`$$ + +$$\frac{p}{1-p} = e^{`r B0 + B1 * XX`}$$ + +$$p = \frac{e^{`r B0 + B1 * XX`}}{1 + e^{`r B0 + B1 * XX`}} = `r pp`$$ + + +## Generalising the linear model + +\Large + +**GLMs: Generalised linear models** + +\begin{itemize} +\setlength\itemsep{1em} +\item Not to be confused with {\bf general} linear models (also sometimes called GLMs) +\item have three properties +\begin{enumerate} +\item Error structure (e.g., binomial) +\item Linear predictor (e.g., $-124.79 + 0.1797x$) +\item Link function (e.g., $ln\left(\frac{p}{1-p}\right)$) +\end{enumerate} +\end{itemize} + + +## Generalising the linear model: error (i.e., residual) structure + +**Generalised** linear models are characterised by independent random variables (i.e., $y_1$, $y_2$, ..., $y_n$) with an expected value $E(y_i) = \mu_i$, and a density function (error) **from the exponential family**. + +A density function $f(y_i; \theta_i)$ is in the exponential family if it can be expressed as follows, + +$$f(y_i; \theta_i) = e^{y_i \theta_i + b(\theta_i) + c(y_i)}.$$ + +In the above, $\theta_i$ is a parameter of the family. + +Statistical distributions in the exponential family include the Poisson, binomial, exponential, and gamma (also the normal). + +\begin{footnotesize} +\footnotetext[1]{Rencher, AC, \& Schaalje, GB. 2008. {\it Linear models in statistics}. John Wiley \& Sons, 446-448.} +\footnotetext[2]{Nelder, JA, \& Wedderburn, RW. 1972. Generalized linear models. {\it Journal of the Royal Statistical Society: Series A (General)}, 135:370-384.} +\end{footnotesize} + + +## Generalising the linear model: error (i.e., residual) structure + +\Large + +**There are four common error structures:** + +1. Poisson errors (for count data) +2. Binomial errors (for proportion data) +3. Exponential errors (for time to event) +4. Gamma errors (for data with constant coefficient of variation) + +## Generalising the linear model: linear predictor + +The linear predictor ($\eta$) is the sum of linear effects of 1 or more explanatory variables ($\beta$), + +$$\eta = \beta_{0} + \beta_{1} x_{i1} + ... + \beta_{k} x_{ik}.$$ + +\vspace{10mm} + +GLMs compare a *transformed* value from $\eta$ to observations: + +- The transformation is specified by the link function (see next) +- The fitted value is the predicted value multiplied by the reciprocal of the link function + +## Generalising the linear model: link function + +The link function describes how the expected value of the response variable ($\mu_i$) relates to $\eta$, + +$$g(\mu_i) = \eta.$$ + +- Note that this relates the **mean** of a response variable (i.e., $E(y_i) = \mu_i$) to the linear predictor; it is not transforming individual values of $y_i$. \pause + +- The model prediction is not $E(y_i)$, except in the special case of the *identity link* (i.e., $g(\mu_i) = \mu_i = \eta$); i.e., a general linear model. + +## Linear predictors and link functions + +Common GLMs and associated canonical link-distribution pairs.$^{1}$ + +| **Model** | **Response variable** | **Predictor variable(s)** | **Residual dist.** | **Link** | +|---------------------|-----------------------|---------------------------|---------------------------|--------------------------| +| Linear regression | Continuous | Continuous/ categorical | Gaussian (normal) | Identity $g(\mu) = \mu$ | +| | | | | | +| Logistic regression | Binary | Continuous/ categorical | Binomial | Logit $g(\mu) = \ln\left(\frac{\mu}{1-\mu}\right)$ | +| | | | | | +| Log-linear models | Counts | Categorical | Poisson | Log $g(\mu) = \ln\left(\mu\right)$ | + + +\begin{footnotesize} +\footnotetext[1]{Logan, M. 2011. {\it Biostatistical design and analysis using R: a practical guide}. John Wiley \& Sons.} +\end{footnotesize} + +## Linear predictors and link functions + +Common GLMs and associated canonical link-distribution pairs.$^{1}$ + +| **Model** | **R glm argument** | +|---------------------|------------------------------------------| +| Linear regression | `family = gaussian(link = "identity")` | +| | | +| Logistic regression | `family = binomial(link = "logit")` | +| | | +| Log-linear models | `family = poisson(link = "log")` | + + +```{r, eval = FALSE} +ghus <- glm(formula = Survival~head_width_mm, + family = binomial(link = "logit"), + data = huse); +``` + +\begin{footnotesize} +\footnotetext[1]{Logan, M. 2011. {\it Biostatistical design and analysis using R: a practical guide}. John Wiley \& Sons.} +\end{footnotesize} + + +## Further reading suggestions + +- Logan, M. 2011. *Biostatistical design and analysis using R: a practical guide*. John Wiley & Sons. **(Chapter 17)** +- Crawley, MJ. 2012. *The R book*. John Wiley \& Sons. **(Chapters 13, 14, 16)** +- Generalised Linear Mixed Models: http://glmm.wikidot.com +- Rencher, AC, \& Schaalje, GB. 2008. *Linear models in statistics*. John Wiley \& Sons, 446-448. +- Nelder, JA, \& Wedderburn, RW. 1972. Generalized linear models. *Journal of the Royal Statistical Society: Series A (General)*, 135:370-384. [[PDF]](https://www.jstor.org/stable/pdf/2344614.pdf?casa_token=VHKchYH85SYAAAAA:x-ZXXTC2hPI50vwk7M8cwd-lwT566e-WSPX-MI3tFDrSGmn0rJYXd-DzHxyW0cUgd05Noz9S1Cx0wZB-zq7jJvSSTfOwTK8YR0IDrKzrQVXUbgSBeCHs) +- [Generalized Linear Models undertanding the link function](https://www.r-bloggers.com/2018/10/generalized-linear-models-understanding-the-link-function/). 15 OCT 2018. 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