---
title: "Fisher Exact Test"
author: "CS 3130 / ECE 3530"
date: "November 28, 2017"
output: html_document
---

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

This is how to do Fisher's exact test. First construct a 2x2 contingency
table (this is a perfect result for the lady drinking tea)
```{r}
(C = matrix(c(4,0,0,4), 2, 2))
```

Now run Fisher's test on the table. The `alternative = "greater"` option means that
we are testing the probability of getting an equal or even better table by
random luck (i.e., one with higher numbers on the diagonal of the table).
```{r}
fisher.test(C, alternative = "greater")
```

Notice this is the same as the probability of getting exactly this table by
random luck, because there is no better table. There are "8 choose 4"
possible guesses, so the probability of the correct guess is:
```{r}
(p = 1 / choose(8, 4))
```

Now try if she misclassifies two cups:
```{r}
(C = matrix(c(3,1,1,3), 2, 2))
fisher.test(C, alternative = "greater")
```

And now 4 misclassified cups:
```{r}
(C = matrix(c(2,2,2,2), 2, 2))
fisher.test(C, alternative = "greater")
```

Notice this is just $1 - p$, where $p$ was the probability above of a perfect answer.
This is because there is exactly one combination that is worse.
```{r}
(C = matrix(c(1,3,3,1), 2, 2))
fisher.test(C, alternative = "greater")
```

Everything wrong! Notice the probability of doing this well or better is 1.
```{r}
(C = matrix(c(0,4,4,0), 2, 2))
fisher.test(C, alternative = "greater")
```

Next lets look at how to use Fisher's Exact Test to revisit the question
about if smoking is linked to heart attacks. (This came up in HW 4, #1.)
Again, we are using the CDC BRFSS Survey data. Slightly different from the HW
numbers, I'm only including the data from people in Utah.

Here our null hypothesis is that there is no relationship between smoking and
heart attacks. The alternate hypothesis is that there is a positive
relationship (that is, smoking increases your chance of a heart attack). The
contingency table consists of counts from the CDC survey respondents:

Variables         | Smoke   |   Don't Smoke
------------------|---------|------------------
Heart Attack      | 203     | 2556
No Heart Attack   | 230     | 7095

```{r}
(C = matrix(c(203, 230, 2556, 7095), 2, 2))
fisher.test(C, alternative = "greater")
```