logistic_regression_example.Rmd

---
title: "logistic regression demo"
author: "Clay Ford"
date: "6/9/2021"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Generate data

Let's generate fake data for two groups A and B. If you're in group A, chance of hemetoma is 30%. If you're in group B, chance of hematoma is 50%.

```{r}
grp <- rep(c("A", "B"), each = 50)
set.seed(1)
hematoma <- rbinom(n = 100, size = 1, 
                   prob = rep(c(0.3, 0.5), each = 50))
```

## Examine data

Create a cross tabulation. Hematoma = 1 means the subject experienced hematoma.

```{r}
table(grp, hematoma)
```

Get proportions. It appears group B has a higher chance of hematoma (0.42 vs 0.32).

```{r}
prop.table(table(grp, hematoma), margin = 1)
```

Calculating odds ratio can be done "by hand"

```{r}
# save table of proportions
tab <- prop.table(table(grp, hematoma), margin = 1)
grpA_odds <- tab[1,2]/(1 - tab[1,2])
grpB_odds <- tab[2,2]/(1 - tab[2,2])

# odds ratio
grpB_odds/grpA_odds
```

Appears group B odds for hematoma are about 54% higher than the odds of hematoma for group A.

## Logistic regression for CI and p-value

We can use logistic regression to get odds ratio, confidence interval, and the p-value testing null if odds ratio is 1. To do this we use the `glm` function with `family = binomial`.

```{r}
m <- glm(hematoma ~ grp, family = binomial)
```

Odds ratio is same as we got by hand. Need to exponentiate the coefficient. Some statistics programs return this automatically.

```{r}
exp(coef(m)["grpB"])
```

Confidence interval on odds ratio:

```{r}
exp(confint(m))["grpB",]
```

The p-value:

```{r}
coef(summary(m))["grpB","Pr(>|z|)"]
```

It appears the supposition of no difference in the odds cannot be rejected based on such a large p-value. We generated the data and know there is a real difference, but our analysis of the sample missed it. This is a Type II error: failed to reject null when there was indeed an effect. In real life we don't know if we make these errors.