91Ó°ÊÓ

Desipramine vs Placebo in Cocaine Addiction 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 4.9 on page \(267 .\) The comparison of lithium to the placebo is the subject of Example \(4.29 .\) In this exercise, we test the success of desipramine against a placebo using a different statistic than that used in Example \(4.29 .\) 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 than a placebo. (a) What are the null and alternative hypotheses? (b) From Table 4.9 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?

Step by step solution

01

Null and Alternative Hypotheses

For the null hypothesis (H0), it's assumed that the proportion of relapses in the control group, denoted by \( p_{c} \), is the same as the proportion of relapses in the desipramine group, denoted by \( p_{d} \). Therefore, \( p_{d} = p_{c} \). For the alternative hypothesis (Ha), it's assumed that the proportion of relapses is lower in the desipramine group compared to the control group. Therefore, \( p_{d} < p_{c} \).

02

Observed Difference in Relapses

From the given data, 20 out of 24 placebo patients relapsed, and 10 of the 24 desipramine patients relapsed. The observed difference in relapses is calculated as the number of desipramine relapses minus the number of placebo relapses which results in \(10 - 20 = -10 \). This negative value indicates that the desipramine group had fewer relapses.

03

Randomization Distribution Center

If we use the difference in the number of relapses as our sample statistic, the randomization distribution will be centered around 0. This is because, under the null hypothesis, there should be no consistent difference between the placebo and desipramine relapses. Therefore, the differences would occasionally be positive, occasionally negative, and typically cluster around zero.

04

Simulated Sample Using Index Cards

Under the null hypothesis, the treatment assignment will not affect the relapse rate. We can simulate this by writing 'relapse' on 30 index cards (for the 30 total relapses) and 'no relapse' on 18 index cards (for the 18 total patients who didn't relapse). After shuffling, randomly assign 24 index cards to the 'desipramine group' and the remaining 24 to the 'placebo group'. Count the 'relapse' cases in each group and find the difference (desipramine - placebo). Repeat this process many times to build the null distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis in Categorical Data

When conducting a statistical test in scenarios involving categorical data—like the aforementioned exercise with desipramine and placebo in treating cocaine addiction—the first step is to clearly state the null hypothesis (ullHypothesis{}). This hypothesis reflects a standpoint of no change, no effect, or no difference; it is the skeptic's perspective before any evidence is provided. For instance, when comparing relapse rates between a drug and a placebo, the null hypothesis posits that the drug has no effect on relapse rates.

In mathematical terms, if we denote the proportion of patients who relapse in the control group (placebo) as ullHypothesis{} and that in the treatment group (desipramine) as ullHypothesis{}, we state the null hypothesis as ullHypothesis{}. This asserts that any observed difference in relapse rates is due to random chance rather than a true effect of the treatment. To test this hypothesis, researchers calculate an observed statistic based on sample data and determine whether this statistic falls within a range of values likely if the null hypothesis were true. This assessment hinges on the concept of a randomization test, which we will explore in a later section.

Alternative Hypothesis in Categorical Data

The alternative hypothesis (ullHypothesis{}) is the foil to the null hypothesis and postulates the presence of an actual effect, change, or difference in the categorical data under scrutiny. It is an assertion that requires evidence for acceptance. In our example, the researchers hope to demonstrate that desipramine decreases the relapse rate compared to the placebo. The alternative hypothesis is hence formulated as ullHypothesis{}, directly opposite to the null stance, signifying that the treatment has a statistically significant impact on the outcome.

Statistically, the burden of proof lies with the alternative hypothesis; it must 'overcome' the null hypothesis. Researchers conduct tests to see if the data provide sufficient evidence to reject the null in favor of the alternative. The results of these tests, when consistent and repeatable, can ultimately persuade the scientific community that the alternative hypothesis warrants consideration as a possible reality.

Randomization Test

The randomization test is a non-parametric method for assessing the significance of the hypothesis test when the data involves categorical variables. It is based on the assumption that if the null hypothesis is true, all configurations of the data are equally probable because the treatment would have no effect. In the context of our exercise, the randomization test simulates numerous scenarios in which participants are randomly assigned to either the treatment (desipramine) or the control (placebo) groups.

Simulating the Randomization

To simulate one instance of patient group assignment, one might use physical objects like index cards, as described in the solution. With 48 cards representing 48 participants, 'relapse' would be written on the number of cards corresponding to observed relapses and 'no relapse' on the rest. Shuffling and distributing these cards into two groups simulates a random allocation to treatment and control, akin to what might happen under the null hypothesis.

This process is repeated many times, computing the difference in relapse numbers between the groups each time. The collection of these differences forms the randomization distribution—a representation of what we might observe if the null hypothesis is true. If the observed sample statistic (such as the difference in relapse rates between groups) is extreme compared to this distribution, it implies that such an outcome is unlikely under the null hypothesis, leading researchers to consider the alternative hypothesis as more probable.