New, effective AIDS vaccines are now being developed using the process of "sieving" (i.e., 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, Gilbert reported the results of VaxGen's preliminary HIV vaccine trial using the 2 × 2 table below. 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 table (saved in the HIV1 file) shows the number of patients who tested positive and negative for the MN strain in the trial follow-up period.

a. Conduct a test to determine whether the vaccine is effective in treating the MN strain of HIV. Use α = .05.
b. Are the assumptions for the test, part a, satisfied? What are the consequences if the assumptions are violated?
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 uses the hypergeometric probability distribution (a discrete probability distribution not covered in Chapter 4). Consider the hypergeometric probability

This represents the probability that 2 out of 7 vaccinated AIDS patients test positive and 22 out of 31 unvaccinated patients test positive (i.e., the probability of the table result given the null hypothesis of independence is true). Compute this probability (called the probability of the contingency table).
d. Refer to part c. Two contingency tables (with the same marginal totals as the original table) that are more contradictory to the null hypothesis of independence than the observed table follow and are saved in the HIV2 and HIV3 files. First, explain why these tables provide more evidence to reject H0 than the original table: then compute the probability of each table using the hypergeometric formula.

GROUP * MNSTRAIN Cross tabulation

Chi-Square Tests

e. The p-value of Fisher's exact test is the probability of observing a result at least as contradictory to the null hypothesis as 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 at the bottom of the SPSS printout on the previous page.) Interpret this value in the context of the vaccine trial.

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MN Strain Patient Group Positive Negative Totals Unvaccinated Vaccinated 31 Totals 24 14 38 2 MN Strain Patient Group Positive Negative Totals Unvaccinated Vaccinated 23 31 6 Totals 24 14 38 MNSTRAIN NEG POS Total GROUP UNVAC Count 31 31.0 11.4 VACC Count5 2.6 14 14.0 Expected Count 19.6 Expected Count Count Expected Count 7.0 38 38.0 Total 24 Asymp. Sig Exact Sig Exact Sig (2-sided)(2-sided) 1-sided) Pearson Chi-Square Continuity Correctione Likelihood Ratio Fisher's Exact Test N of Valid Cases 4.411 2.777 4.289 036 096 038 077 050 38 a. Computed only for a 2x2 table D. 2 cells (50.0%) have expected count less than 5. The minimum expected count is 2.58 MN Strain Patient Group Positive Negative 7 14 Totals Unvaccinated Vaccinated 24 31 Totals 24 38