The expected frequencies in Table 2.16 show a hypothetical relationship among three variables: Y = response, X = drug treatment, and Z = clinic. Show that X and Table 2.16 Expected frequencies illustrating that conditional independence does not imply marginal independence.

Response Drug Clinic Treatment Success Failure 1 A 18 12 B 12 8 2 A 28 B 8 32 Y are conditionally independent, given Z, but marginally associated. Explain how the marginal XY association can be so different from its conditional association, using the values of the conditional XZ and YZ odds ratios. Explain why it would be misleading to study only the marginal table and conclude that successes are more likely with treatment A than with treatment B.