Rcourse_Session2.Rmd

---
title: "Session 2: Modelling/Analysis"
author: "Tanya Flynn"
output: html_document
---

### A note on P values
A p value tests whether your data fits with your null hypothesis, or in other words whether you are likely to get the answer you got by chance alone, it is a statistical tool, not a definitive answer.

   **null hypothesis** - your null hypothesis will always be "there is nothing exciting going on"
   
   **alternate hypothesis** - what you actually think is going on

#### Statistical Significance
Convention makes p < 0.05 the threshold for statistical significance
   - this value means if your null hypothesis is true you would see a result like yours in 5 out of every 100 tests
   - because 5 out of 100 is not often you assume your null hypothesis is not true
      - without replication/validation you cannot say whether you are right in this assumption

For instance, if I have a population of 1000 people, 300 have brown hair, 400 have blue eyes and I want to test if having blue eyes makes you more or less likely to have brown hair (for some reason).

- blue eyes and brown hair being *unrelated* is my **null hypothesis**
- blue eyes and brown hair being *related* is my **alternate hypothesis**

      - I choose one hundred people randomly from the group
      - I would expect to get approximately 12 people with both blue eyes and brown hair
         - This expectation assumes that there is nothing dependent between having blue eyes and brown hair
      - What if in this population most people with blue eyes don't have brown hair?
         - When I pick the people randomly I get less people with both blue eyes and brown hair than I expect, say I get 6
      
      - 12 and 6 are quite different to each other, p = 0.01
         - This p value says that if I did this same test 100 times I would only see a result this extreme by chance once.
         - That means it is unlikely my **null hypothesis** (of blue eyes/brown hair being unrelated) is correct, so I can reject it and *assume* the **alternate hypothesis** is true.
         
#### Clinical Significance
Clinical significance is when something has an effect large enough that it has some kind of relevance to real life.

   - statistical significance does not rely on clinical significance
   - something could have a very different effect to what is expected, but still not have an effect large enough to matter in a clinical setting
   
Be wary of claiming something is of importance solely on the basis of a p value < 0.05

# Lesson 5

Covered in this lesson:

   - chi-squared

   - t tests

   - correlations

### Chi-squared

   - tests whether what you *expect to get* fits with what you *actually got*
   
      - you need either:
         - a 2x2 contingency table
         - or two data columns with factor variables

2x2 CONTINGENCY
```{r}
eyes_hair = data.frame("Brown"=c(6, 24), "Other"=c(34, 36) )
row.names(eyes_hair) = c("Blue", "Other")
print(eyes_hair)
```

```{r}
x = chisq.test(eyes_hair)
print(x)
x$observed
x$expected
```

TWO FACTOR COLUMNS
```{r}
mydata = read.delim(file = "Tanya_Data.txt", header = TRUE, sep = "\t", stringsAsFactors=FALSE)
y = chisq.test(mydata$SNP, mydata$GOUT)
print(y)
y$observed
round(y$expected, 0)
```

### t tests

   - A way to test whether the mean of two variables is different
   
      - this could be between two groups (cases vs controls)
      - or this could be in the same group (before or after)
      
TWO GROUPS   
```{r}
t.test(mydata$URATE ~ mydata$GOUT)
```

SAME GROUP
```{r}
vinha = read.delim(file = "Vinha_Data.txt", header = TRUE, sep = "\t", stringsAsFactors=FALSE)
t.test(y = vinha$UricAcid_B, x = vinha$UricAcid_30, paired=TRUE)
```

&nbsp;

*if you have data from multiple time points of the same sample your t test needs to be paired - this takes out any between sample variations*

&nbsp;

```{r}
t.test(y = vinha$UricAcid_B, x = vinha$UricAcid_30, paired=FALSE)
```

### Correlations

   - testing whether two measures are linearly related to each other, or how similar the values in two variables are for each person/data point
   
   - a correlation coefficient ranges from -1 to 1
      - [-1] - as one variable goes up the other goes down
      - [ 0] - no correlation at all
      - [ 1] - as one variable goes up the other one does too
   
```{r}
cor.test(mydata$STANCESTRY, mydata$GPANCESTRY)
```

# Lesson 6

Covered in this lesson:

   - linear regression

   - logistic regression

   - covariates

   - interactions

### Linear regression

   - tests whether one measure has a linear relationship to another
   - builds an equation representing this relationship
   
      - simple linear regression equation: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**y = B~0~ + B~1~x~1~ + error**   
   
&nbsp;      &nbsp;**y** - is the response variable, what you want to predict

&nbsp;      &nbsp;**B~0~** - the intercept of the line

&nbsp;      &nbsp;**B~1~** - the slope of the line

&nbsp;      &nbsp;**x~1~** - the explanatory variable


```{r}
simple = lm(formula = URATE ~ BMI, data = mydata)
summary(simple)
confint(simple)
```

   - Estimate = a beta value and centres around 0
      - [< 0] - as the explanatory variable goes up the response variable goes down
      - [ 0 ] - no relationship at all
      - [> 0] - as the explanatory variable goes up the response variable does too
      
   - if the confidence interval of the estimate crosses zero there is no significant relationship between your two variables, *check your p value and confidence interval are saying the same thing!!*

   - you can plot this information easily to help yourself see the relationship
```{r}
plot(x = mydata$BMI, y = mydata$URATE, xlab = "BMI (kg/m^2)", ylab = "Serum Urate (mmol/L)")
abline(simple, col = "red" )
```

   - multiple linear regression &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **y = B~0~ + B~1~x~1~ + B~2~x~2~ + ... + B~n~x~n~ + error**
      - when you have more than one explanatory variable
      
```{r}
multiple = lm(formula = URATE ~ BMI + AGE, data = mydata)
summary(multiple)
confint(multiple)
```

   - notice that the estimate for BMI is now slightly different from before: 0.0044121 vs 0.0046408
      - this is because a multiple regression adjusts the answer of every variable by the effect of every other variable
      - sometimes this adjustment can result in a *loss* of significance for a variable
      - sometimes this adjustment can result in a *gain* of significance for a variable

### Logistic regression

   - tests whether a variable influences an outcome
   - works in a similar way to linear regression, but with a categorical variable as the response variable
      
      - simple logistic regression equation: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**log(yes/no) = B~0~ + B~1~x~1~**   
   
&nbsp;      &nbsp;**log(yes/no)** - is the odds of success, where yes and no are the presence or absence of the outcome you are interested in

&nbsp;      &nbsp;**B~0~** - the intercept of the line

&nbsp;      &nbsp;**B~1~** - the slope of the line

&nbsp;      &nbsp;**x~1~** - the explanatory variable

```{r, error=TRUE}
simpleLOG = glm(formula = GOUTAFFSTAT ~ BMI, data = mydata, family = binomial)
```

```{r}
mydata$GOUTAFFSTAT = mydata$GOUTAFFSTAT - 1
```

```{r}
simpleLOG = glm(formula = GOUTAFFSTAT ~ BMI, data = mydata, family = binomial)
summary(simpleLOG)
```

   - the estimates produced by a *glm* are just like those produced by a *lm*
   - because our response variable is the log(yes/no) we need to convert our estimates from the log scale back to the response scale
      - this is how we get an **odds ratio**

```{r}
exp(coef(simpleLOG) )
exp(confint(simpleLOG) )
```

   - exp(Estimate) = the odds ratio and centres around 1
      - [< 1] - as the explanatory variable goes up the chance of "yes" happening goes down: protective response
      - [ 1 ] - no relationship at all
      - [> 1] - as the explanatory variable goes up the chance of "yes" happening goes up: risk response
   - if the confidence interval of the exp(estimate) crosses 1 there is no significant relationship between your two variables, *check your p value and confidence interval are saying the same thing!!*

   - you can also run multiple logistic regression just the same as in a linear model
   
   - you can also run regressions on only a subset of data within the regression command
   
```{r}
Euro_multiple = glm(formula = GOUTAFFSTAT ~ BMI + AGE + SNP, data = mydata, subset = ETHCLASS == "European")
summary(Euro_multiple)
exp(coef(Euro_multiple) )
exp(confint(Euro_multiple) )
```

### Interactions

   - testing whether the effect of one variable is modified by another variable
      - for instance a gene increases gout risk 2-fold in a population.
      - eating pies also increases gout risk by 2-fold.
      - you would assume having the gene and eating pies would increase your risk ~ 4-fold
      - when you test this interaction it turns out eating pies and having the gene decreases your risk 1.5-fold
         - what is going on? I dunno, but pies are delicious.

&nbsp;

   - you can test for interactions by adding an interaction term to your regression model
   
```{r}
interaction = glm(formula = GOUTAFFSTAT ~ BMI*SEX, data = mydata, family = binomial)
summary(interaction)
exp(coef(interaction))
exp(confint(interaction))
```