Fisher’s Exact Test

Introduction

This lab will help us explore and hopefully better understand the motivation and calculations involved with Fisher’s Exact Test. We’ll begin with a very quick overview of some functions that will be useful in working through the questions to follow. For this, refrain from using any external R packages (i.e., epitools).

Generating matrices

Matrices, by default, are generated (and stored) in column order, meaning that the indicies for a matrix start in the top left and work their way down, moving on to the next column once it finishes:

matrix(1:9, nrow = 3)

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

The byrow option allows us to specify row order instead:

matrix(1:9, nrow = 3, byrow = TRUE)

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

Choose Function

For binomial coefficients:

## 5 choose 2
choose(5, 2)

## [1] 10

Fisher Exact Test

See ?fisher.test, in particular the argument for alternative

Lab

Below is a 2x2 contingency table tabulating the results of a small sample study in which participants were given either Vitamin C tablets or a placebo and recorded whether or not they got sick in the month of February.

Vitamin C	Sick
	Yes	No
Placebo	3	2
Drug	1	3

Use the textbook to answer the following questions:

Question 0: List the names of the people with whom you are working on this lab. Together, determine why it would be inappropriate to use a \(\chi^2\) approximation to assess independence for this table?

Question 1: For a 2x2 table, identify the distribution for the cell counts under independence for each of the following situations:

• That only the total number of participants, \(n\), is fixed
• That only the row margins \(n_{i+}\) are fixed
• That both row and column margins are fixed

In addition, identify how many degrees of freedom each table has under each scenario (e.g., when only \(n\) is fixed, how many cells are allowed to freely vary before the remaining are fixed?)

Question 2: Assuming that the row and column totals are fixed, what are all of the possible values that \(n_{11}\) can take for the table provided? On a separate piece of paper (or in your RMD document), write down what all of these tables will look like.

Question 3: For 2x2 tables, assumptions of independence correspond with an odds ratio of \(H_0: \theta = 1\). Write out the formula for \(\hat{\theta}\) in terms of \(n_{ij}\).

• What happens to \(\hat{\theta}\) when we increase \(n_{11}\), leaving the margins fixed? When we decrease it? Does this make sense in the context of the problem?
• In the context of our problem, state what the alternative hypothesis is for \(H_{A}: \theta > 1\), i.e., what does \(\theta > 1\) imply for the data we have been given? State this in terms of groups and effects.
• Does \(n_{11}\) need to get larger or smaller to offer more evidence for \(H_A: \theta > 1\)?

Question 4: Under \(H_0\), find the probability \(P(n_{11} = 3)\) (It may be useful to create a function for the PMF).

Question 5: The \(p\) value is defined as the probability of observing data as extreme or more so under the null hypothesis. Find directly (that is, without using fisher.test) the p-value associated with the table above for \(H_A: \theta > 1\). Once you have found it, confirm its correctness by comparing it against the results from fisher.test using the correct alternative hypothesis. Interpret the \(p\)-value in the context of the problem.

Question 6: Find the p-value directly again, this time for \(H_A: \theta \not= 1\). Explain how you found it and how this was different than in Question 5. Confirm its correctness with fisher.test. Interpret the \(p\)-value in the context of the problem.

Question 7: For this study, explain whether \(H_A: \theta > 1\) or \(H_A: \theta \not= 1\) is the more appropriate hypothesis.