Question

Consider conducting Fisher's exact test with the following fictitious table of data. Let the null hypothesis be that treatment and response are independent, and let the alternative be the directional hypothesis that treatment $\mathrm{B}$ is better than treatment $\mathrm{A}$. List the tables of possible outcomes that more strongly support $H_A$. $$\begin{array}{|llcc|}\hline & & \text{ Treatment }& \\ & & \text{ A }& \text{ B }& \text{ Total }\\ \text{ Outcome }& \text{ Die }& 4& 2& 6\\ & \text{ Live }& 10& 14& 24\\ & \text{ Total }& 14& 16& 30\\ \hline\end{array}$$

Short answer

The tables that strongly support the alternative hypothesis are those with fewer 'Die' outcomes and/or more 'Live' outcomes for treatment B while keeping the marginal totals constant. Examples include situations where treatment B has 1 or 0 'Die' outcomes and 15 or 16 'Live' outcomes, respectively.

Step-by-step solution

Understanding Fisher's Exact Test

Fisher's exact test is a statistical test used to analyze the association between two categorical variables in a contingency table. It's applied when sample sizes are small. In this exercise, we're testing the null hypothesis that treatment and response are independent against the alternative hypothesis that treatment B is better than treatment A, i.e., that it leads to better survival rates.

Identify More Favorable Outcomes for Treatment B

To determine the tables that more strongly support the alternative hypothesis, we look for tables where the proportion of the 'Live' outcome is greater for treatment B than in the original table, while keeping the marginal totals fixed. This involves reducing the number of 'Die' outcomes and/or increasing the number of 'Live' outcomes for treatment B.

List Favorable Outcome Tables

Possible tables more favorable to treatment B are as follows: $$\text{Misplaced hline}$$ And $$\text{Misplaced hline}$$ These tables show an increased survival rate for treatment B compared to treatment A, which better supports the alternative hypothesis. Note that we cannot have more than 6 deaths or 24 lives in total because the marginal totals must remain constant.

Key concepts

Categorical Data Analysis

Categorical data analysis involves the examination and interpretation of data that can be sorted into categories, rather than numerical values. This type of data is frequently represented in a non-ordered form, like gender or blood type, or an ordered form, such as levels of satisfaction. When analyzing categorical data, statisticians use specific statistical tests to draw conclusions, especially when the data is shown in a contingency table.

Fisher's exact test, which we are focusing on in this scenario, is a method used to analyze the association between two categorical variables, particularly useful when dealing with small sample sizes. It helps determine whether there are nonrandom associations between the variables. An important concept in this analysis is to maintain the row and column totals of the contingency table, which leads to the examination of all possible rearrangements that are consistent with these marginal constraints.

Contingency Table

A contingency table is a type of table that displays the frequency distribution of variables. For analytical clarity, the variables are often cross-tabulated, and each cell reflects the count or frequency of occurrences for certain combinations of the categorical variables. In this case, our contingency table is comparing the response to two different treatments across two potential outcomes, 'Die' or 'Live'. When performing a Fisher's exact test, the contingency table is the starting point, presenting the observed frequencies that reflect the original dataset.

One crucial aspect of a contingency table in Fisher's exact test is the fixed marginal totals, representing the sum of the rows and columns. These totals must remain unchanged when assessing various hypothetical rearrangements of the data, as they are related to the fixed design of the study or the total counts in each group, which are not subject to variation during the analysis.

Null Hypothesis

The null hypothesis, typically denoted as $H_0$, is a critical concept for any statistical test. It posits that there is no effect or no difference and that any observed data pattern is due to random chance. In this context, the null hypothesis states that there is no association between the type of treatment and patient outcomes, meaning that treatments A and B do not differ in effectiveness, and any observed difference in the outcomes is purely accidental.

When applying the Fisher's exact test, the null hypothesis is directly tested by comparing the observed data to all potential outcomes that could arise if this hypothesis were true. If the observed data seem unlikely under the null hypothesis—based on a calculated probability—the null may be rejected in favor of the alternative hypothesis, but with a strict control over the Type I error rate, i.e., the chance of incorrectly rejecting the null hypothesis.

Alternative Hypothesis

The alternative hypothesis, represented as $H_A$ or $H_1$, is the hypothesis that researchers wish to support, which states that there is an effect or a difference between groups. In the case of Fisher's exact test, the alternative is frequently directional, as shown in our example. Here, the alternative hypothesis proposes that treatment B is superior to treatment A in terms of patient survival rates.

To support the alternative hypothesis, statistical evidence must show that the observed data are more consistent with $H_A$ than with $H_0$. This is done by calculating the probability of observing a data set as extreme as or more extreme than the original data, assuming the null hypothesis is true. If this probability, known as the p-value, is lower than a designated significance level, the null hypothesis is rejected in favor of the alternative hypothesis, suggesting that treatment B may indeed be better.