Fisher’s exact test

If we want to see whether there’s an association between two categorical variables and the assumption for the expected frequencies in the contingency table is not fulfilled, an alternative test to the chi-square test can be used.

Fisher came up with a method for computing the exact probability of the chi-square statistic that is accurate when sample sizes are small. This method is called Fisher’s exact test even though it’s not so much a test as a way of computing the exact probability of the chi-square statistic. This procedure is normally used on 2 by 2 contingency tables and with small samples. However, it can be used on larger contingency tables and with large samples, but in this case it becomes computationally intensive.

To perform Fisher’s exact test in R, you can use the fisher.test() function. This test is used to compare the relationship between two categorical variables in a contingency table, where the sample sizes are small and the assumptions for the chi-square test may not be met.

Here is an example code for performing Fisher’s exact test in R:

# Create a contingency table
data <- matrix(c(10, 20, 15, 25), nrow = 2, dimnames = list(Sex = c("Male", "Female"), Smoke = c("Yes", "No")))

# Print the contingency table
data

# Perform Fisher's exact test
fisher.test(data)

In this example, we have a contingency table that shows the relationship between the variables "Sex" and "Smoke". The rows of the table represent the different values of "Sex" (Male and Female) and the columns represent the different values of "Smoke" (Yes and No). The values in the table represent the frequency counts of each combination of "Sex" and "Smoke".

The fisher.test() function produces the following output:

Fisher's Exact Test for Count Data

data:  data
p-value = 0.3466
alternative hypothesis: two.sided

The output includes the p-value of the test, which is 0.3466 in this case. The null hypothesis for Fisher’s exact test is that there is no association between the two categorical variables, while the alternative hypothesis is that there is an association.

In this case, since the p-value is greater than 0.05 (i.e., the commonly used significance level), we fail to reject the null hypothesis. This means that there is no statistically significant association between "Sex" and "Smoke" in our sample.

In summary, Fisher’s exact test is a useful tool for comparing the relationship between two categorical variables in a contingency table. The p-value produced by the test can be used to determine whether there is a statistically significant association between the variables, and the interpretation of the p-value depends on the chosen significance level.