Efficacy of an HIV vaccine. New, effective AIDS vaccines are now being developed through t...

Efficacy of an HIV vaccine. New, effective AIDS vaccines are now being developed through the process of “sieving”—that is, sifting out infections with some strains of HIV. Harvard School of Public Health statistician Peter Gilbert demonstrated how to test the efficacy of an HIV vaccine in Chance (Fall 2000). As an example, using the 2 × 2 table shown next, Gilbert reported the results of VaxGen’s preliminary HIV vaccine trial. The vaccine was designed to eliminate a particular strain of the virus called the “MN strain.” The trial consisted of 7 AIDS patients vaccinated with the new drug and 31 AIDS patients who were treated with a placebo (no vaccination). The first table (saved in the HIVVAC1 file) shows the number of patients who tested positive and negative for the MN strain in the trial follow-up period.

Source: Gilbert, P. “Developing an AIDS vaccine by sieving.” Chance, Vol. 13, No. 4, Fall 2000. Reprinted with permission from Chance. Copyright 2000 by the American Statistical Association. All rights reserved.

a. Conduct a test to determine whether the vaccine is effective in treating the MN strain of HIV. Use α = .05.

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b. Are the assumptions for the test you carried out in part a satisfied? What are the consequences if the assumptions are violated?

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c. In the case of a 2 × 2 contingency table, R. A. Fisher (1935) developed a procedure for computing the exact p-value for the test (called Fisher’s exact test ). The method utilizes the hypergeometric probability distribution. Consider the hypergeometric probability

which represents the probability that 2 out of 7 vaccinated AIDS patients test positive and 22 out of 31 unvaccinated patients test positive—that is, the probability of the result shown in the table, given that the null hypothesis of independence is true. Compute this probability (called the probability of the contingency table ).

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SAS output

d. Refer to part c. Two contingency tables (with the same marginal totals as the original table) that are more unsupportive of the null hypothesis of independence than the observed table are shown below (These data are saved in the HIVVAC2 and HIVVAC3 files respectively.). First, explain why these tables provide more evidence to reject H0 than the original table does. Then compute the probability of each table, using the hypergeometric formula.

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e. The p-value of Fisher’s exact test is the probability of observing a result at least as unsupportive of the null hypothesis as is the observed contingency table, given the same marginal totals. Sum the probabilities of parts c and d to obtain the p-value of Fisher’s exact test. (To verify your calculations, check the p-value labeled Left-sided Pr<= F at the bottom of the SAS printout. Interpret this value in the context of the vaccine trial.