Question

Another Test for Cocaine Addiction Exercise 7.42 on page 532 describes an experiment on helping cocaine addicts break the cocaine addiction, in which cocaine addicts were randomized to take desipramine, lithium, or a placebo. The results (relapse or no relapse after six weeks) are summarized in Table \(7.38 .\) (a) In Exercise 7.42, we calculate a \(\chi^{2}\) statistic of 10.5 and use a \(\chi^{2}\) distribution to calculate a p-value of 0.005 using these data, but we also could have used a randomization distribution. How would you use cards to generate a randomization sample? What would you write on the cards, how many cards would there be of each type, and what would you do with the cards? (b) If you generated 1000 randomization samples according to your procedure from part (a) and calculated the \(\chi^{2}\) statistic for each, approximately how many of these statistics do you expect would be greater than or equal to the \(\chi^{2}\) statistic of 10.5 found using the original sample? $$ \begin{array}{l|cc|c} \hline & \text { Relapse } & \text { No Relapse } & \text { Total } \\ \hline \text { Desipramine } & 10 & 14 & 24 \\ \text { Lithium } & 18 & 6 & 24 \\ \text { Placebo } & 20 & 4 & 24 \\ \hline \text { Total } & 48 & 24 & 72 \end{array} $$

Short answer

The randomization procedure involves making 72 cards, with 'Relapse' written on 48 of them and 'No Relapse' on 24. After shuffling these cards, they are divided into three groups (desipramine, lithium, and placebo) while maintaining the same size of groups (24 each) to simulate random assignment of patients to treatments and possibility of outcomes. By repeating this process 1000 times, it's possible to compute chi-square statistics of these randomization samples and compare them with the chi-square statistics from the original sample (10.5) . The estimated number of chi-squared statistics out of 1000 that would be equal to or more than 10.5 from the randomization samples can be found by counting the instances where this condition is met in these 1000 simulated samples.

Step-by-step solution

Simulate The Randomization Sample Using Cards

Write 'Relapse' on 48 cards and 'No Relapse' on 24 cards to reflect the total number of instances for each outcome very well (48 relapse and 24 no relapse). Randomly draw without replacement from this deck of 72 cards to simulate a random assignment of patients to the three treatments. Fill a new 3x2 table with counts from this simulated sample, keeping the marginal totals fixed. Repeat this process many times to represent different possible outcomes of the experiment.

Generate Chi-Square Statistics For Randomization Samples

For each simulated sample, calculate a chi-squared statistic. The chi-squared statistic measures the discrepancy between the observed data and what would be expected if there was no effect of the treatments. It takes into account the actual and expected frequencies for each category in the table.

Predict The Number of Chi-Square Statistics ≥ 10.5

After generating 1000 chi-square statistics from the randomization samples, estimate the proportion that are at least as extreme as the observed chi-square statistic from the original sample of 10.5. This proportion is a simulated p-value, the probability of observing a chi-square statistic as extreme as 10.5 under the null hypothesis which states that there is no difference among the treatments.

Key concepts

Chi-square Statistic

The chi-square statistic is a number that tells us how much the observed data differs from what we would expect to see by chance alone. It is used in hypothesis testing, especially for categorical data, and is pivotal in determining whether there's a significant association between variables. For instance, in the exercise regarding the cocaine addiction study, we calculated a chi-square statistic of 10.5.

To calculate the chi-square statistic, you take each observed frequency, subtract the expected frequency, square the result, and then divide by the expected frequency. In formula terms, this is represented as: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) denotes the observed frequencies, and \(E_i\) denotes the expected frequencies.

This statistic helps us understand whether the discrepancy we see in the data is due to random chance or if there may be an underlying relationship. In other words, if the chi-square statistic is large (and therefore the p-value is small), we have evidence to suggest that the treatment had a noticeable effect on the outcome.

Randomization Distribution

Randomization distribution is a concept in statistics where we create a distribution of possible outcomes by simulating the random assignment of treatments. This is particularly useful in experiments where you want to know if the results you got could have happened just by chance. By using a deck of cards to represent the different possible outcomes, such as 'Relapse' and 'No Relapse' in our exercise, and then drawing them at random to simulate different treatment assignments, you're essentially creating many alternate realities of your study.

To build a randomization distribution, you'd repeat this drawing many times - in our case, 1000 times - and calculate a statistic, such as the chi-square statistic, for each one. This creates a set of statistics that represent what could happen if the null hypothesis were true. Analyzing this distribution aids in understanding the likelihood of observing a statistic as extreme as the one calculated from your actual data.

Expected Frequencies

Expected frequencies are predictions of how often something will happen, based on the assumption that there are no differences or effects (i.e., under the null hypothesis). When comparing groups, like the treatment groups for cocaine addiction, we calculate what counts we would expect in each group if the treatment had no real effect.

To find these expected frequencies, you would typically use the margins of a contingency table - the totals of rows and columns - because we assume each treatment has the same effect. For instance, if all treatments are equally effective, the relapse and no relapse counts should be proportionally similar across treatments. Therefore, we can estimate expected counts by multiplying the marginal totals and then dividing by the grand total. Mathematically, this can be expressed for a cell in the table as: \[ E_i = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}} \] These expected frequencies are then used in calculating the chi-square statistic.

Null Hypothesis

The null hypothesis is a key concept in hypothesis testing. It's the default assumption that there is no effect or no difference in the context of your research. For the cocaine addiction study, the null hypothesis would be that none of the treatments (desipramine, lithium, or placebo) makes a difference in relapse rates.

The null hypothesis serves as a baseline against which you measure your actual findings. If your findings deviate significantly from what would be expected under the null hypothesis, you could have enough evidence to reject the null and consider your alternative hypothesis - that there is an effect or difference - to be likely. In the chi-square test, you determine if the observed frequencies (what actually happened) are statistically different from the expected frequencies (what you predict under the null hypothesis).