Let X1, . . . , Xk be independent random variables such that Xi has the binomial distribution with parameters ni and pi . We wish to test the null hypothesis H0: p1 = . . . = pk versus the alternative hypothesis H1 that H0 is false. Assume that the numbers n1, . . . , nk are known constants.
a. Show that the likelihood ratio test procedure is to reject H0 if the following statistic is greater than or equal to some constant c:

b. Describe how you could use simulation techniques to estimate the constant c in order to make the likelihood ratio test have a desired level of significance α0. (Assume that you can simulate as many binomial pseudo-random variables as you wish.)
c. Consider the depression study in Example 2.1.4. Let pi stand for the probability of success (no relapse) for the subjects in group i of Table 2.1 on page 57, where i = 1 means imipramine, i = 2 means lithium, i = 3 means combination, and i = 4 means placebo. Test the null hypothesis that p1 = p2 = p3 = p4 by computing the p-value for the likelihood ratio test.

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