Question

In murder trials in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black, in 11 out of 63 cases in which a black killed a white, and in 6 out of 103 cases in which a black killed a black (M. Radelet, Am. Sociol. Rev., 46: 918–927, 1981).a) Exhibit the data as a three-way contingency table

b) Construct the partial tables needed to study the conditional association between defendant’s race and the death penalty verdict. Find and interpret the sample conditional odds ratios.

c) Compute and interpret the sample marginal odds ratio between defendant’s race and the death penalty verdict. Do these data exhibit Simpson’s paradox? Explain. d) Conduct the Cochran–Mantel–Haenszel test of the hypothesis that defendant’s race is independent of the death penalty verdict, interpret.

e) In case that in (d) you reject the hypothesis that defendant’s race is independent of the death penalty verdict, please perform a Breslow-Day test for homogenous association. Interpret.

Answer 1

To answer the question, we need to exhibit the data as a three-way contingency table, construct partial tables to study the conditional association, compute the marginal odds ratio, conduct the Cochran–Mantel–Haenszel test, and perform the Breslow-Day test if necessary.

Step-by-step explanation:

To exhibit the data as a three-way contingency table, we can organize the data according to the four possible combinations of defendant's race and the death penalty verdict. The table will have rows representing defendant's race (white or black), columns representing the death penalty verdict (given or not given), and the cells will contain the corresponding frequencies.

To construct the partial tables needed to study the conditional association between defendant's race and the death penalty verdict, we need to calculate the conditional frequencies. For example, the conditional frequency of a death penalty given, given that the defendant is white can be calculated as 19/(19+151).

To compute and interpret the sample marginal odds ratio between defendant's race and the death penalty verdict, we can calculate the odds ratio for each combination of defendant's race and the death penalty verdict. Simpson's paradox occurs when there is a reversal in the direction of association between two variables when a third variable is included. To determine if the data exhibit Simpson's paradox, we can compare the conditional odds ratios across the different levels of the third variable.

To conduct the Cochran–Mantel–Haenszel test of the hypothesis that defendant's race is independent of the death penalty verdict, we can calculate the test statistic and compare it to the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is an association between defendant's race and the death penalty verdict.

Finally, if the hypothesis is rejected, we can perform a Breslow-Day test for homogenous association to determine if the association between defendant's race and the death penalty verdict is consistent across the different levels of the third variable.