Question

In this exercise, we see that it is possible to use counts instead of proportions in testing a categorical variable. Data 4.7 describes an experiment to investigate the effectiveness of the two drugs desipramine and lithium in the treatment of cocaine addiction. The results of the study are summarized in Table 4.14 on page \(323 .\) The comparison of lithium to the placebo is the subject of Example 4.34 . In this exercise, we test the success of desipramine against a placebo using a different statistic than that used in Example 4.34. Let \(p_{d}\) and \(p_{c}\) be the proportion of patients who relapse in the desipramine group and the control group, respectively. We are testing whether desipramine has a lower relapse rate then a placebo. (a) What are the null and alternative hypotheses? (b) From Table 4.14 we see that 20 of the 24 placebo patients relapsed, while 10 of the 24 desipramine patients relapsed. The observed difference in relapses for our sample is $$\begin{aligned}D &=\text { desipramine relapses }-\text { placebo relapses } \\\&=10-20=-10\end{aligned}$$ If we use this difference in number of relapses as our sample statistic, where will the randomization distribution be centered? Why? (c) If the null hypothesis is true (and desipramine has no effect beyond a placebo), we imagine that the 48 patients have the same relapse behavior regardless of which group they are in. We create the randomization distribution by simulating lots of random assignments of patients to the two groups and computing the difference in number of desipramine minus placebo relapses for each assignment. Describe how you could use index cards to create one simulated sample. How many cards do you need? What will you put on them? What will you do with them?

Short answer

In summary, (a) the null hypothesis is that there's no difference in relapse rates (\(p_d = p_c\)), and the alternative hypothesis is that desipramine has a lower relapse rate (\(p_d < p_c\)). (b) The randomization distribution is centered around 0, demonstrating that if the null hypothesis is true, the observed difference may favor either group. (c) You can simulate the randomization distribution using 48 index cards, labeled with 'Relapse' or 'No Relapse'; randomly assign them to two groups (24 each), then compare the relapse numbers.

Step-by-step solution

Defining Null and Alternative Hypotheses

The null hypothesis will state that there's no difference in relapse rates between the groups (desipramine and placebo), i.e., \(p_d = p_c\). The alternative hypothesis, which is what the study hopes to prove, will state that the relapse rate is lower in the desipramine group, i.e., \(p_d < p_c\).

Identify Difference in Proportions and Randomization Distribution

The difference in proportions is calculated by subtracting the relapses in the placebo group from the relapses in the desipramine group, the result of which is -10. The randomization distribution under the null hypothesis is centered around 0 because there is no difference between the two groups. It implies that any difference observed in the actual experiment is just as likely to favor the placebo as desipramine.

Simulate the Randomization Distribution

Use 48 cards (equal to the total number of patients). On 30 cards write 'Relapse' (total number who relapsed) and on the remaining 18 cards write 'No Relapse'. To create one simulated sample, shuffle the cards and draw 24 for the desipramine group. Count the number of 'Relapse' cards in this group and compare to the number in the remaining (placebo) group. The difference indicates whether there are fewer relapses in the desipramine group. Repeat this process multiple times to simulate the randomness in the sample assignment.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial for hypothesis testing in statistics. These hypotheses are the backbone of making inferences about populations based on sample data. For the task at hand, the null hypothesis (often symbolized as H₀) represents the status quo or the assumption that there is no effect or difference. In the context of the exercise, it posits that the relapse rate for patients on desipramine is the same as the relapse rate for those on the placebo, mathematically defined as \( p_d = p_c \).

The alternative hypothesis (H₁ or H_a), on the other hand, represents what the researcher is trying to demonstrate. In this case, it suggests that desipramine is associated with a lower relapse rate than the placebo, or \( p_d < p_c \). It's important to note that the alternative hypothesis is what researchers hope to find evidence for, in order to reject the null hypothesis.

When conducting a hypothesis test, data are used to determine whether there is sufficient evidence to reject the null hypothesis. If so, the results support the alternative hypothesis. It's akin to a court trial where the null hypothesis is like the presumption of innocence, and the alternative hypothesis is like the accusation. Only strong evidence (data) can lead the 'jury' (researchers) to reject the null hypothesis in favor of the alternative.

Categorical Data Analysis

When handling categorical data, like in this exercise where the treatment outcomes are categorized as 'relapse' or 'no relapse', categorical data analysis techniques come into play. This kind of statistical analysis is used for data that can be sorted into categories, rather than numerical values. It's pertinent in research fields like medicine, social sciences, and economics where outcomes are often non-numeric.

The exercise provided involves a basic form of categorical data analysis: comparing the proportion of patients who relapsed between two groups to figure out the efficacy of a treatment. To present the data more understandably, it's often tabulated in a contingency table or visualized using bar charts or pie charts. This makes it easier to spot patterns and differences that might be significant. However, mere observation isn't enough; statistical tests, like the Chi-square test, are often used to determine if there's a statistically significant difference between the groups.

By making use of the raw counts of relapses, a clearer picture can emerge of how many more people relapsed in one group versus another. In educational contexts, this concrete approach can aid understanding as students can visualize moving 'relapse' cards between groups to see how the proportions change, thereby solidifying their grasp of the concept.

Randomization Distribution

The concept of a randomization distribution is instrumental in understanding how random chance might affect the results of a study. Essentially, it's a hypothetical distribution of a statistic that we would observe if we could repeat a study many times, randomly assigning subjects to groups each time. It's a fundamental component of a permutation test.

With reference to the exercise, when testing the null hypothesis that there's no difference between the effectiveness of desipramine and a placebo, we assume the two treatments are interchangeable. By creating a randomization distribution, we're visualizing all possible outcomes that could occur if subjects were randomly assigned to each treatment group. This distribution helps us to understand where the observed statistic (like the difference in relapse counts) falls within the context of random variation.

To simulate this with index cards, as suggested in the exercise, is an excellent hands-on method for demonstrating randomness and variation. You'd need 48 cards (reflecting the total number of patients) with 'Relapse' or 'No Relapse' written based on the observed data. By shuffling and reassigning these cards, students can physically model the randomization process and visualize the resulting distribution. This can be a powerful teaching tool to drive home the concept of randomness and its impact on data interpretation.