We revisit the standard conditional likelihood for stratum-matched case-control studies and consider three alternatives that may be more appropriate for family-matched gene association studies: First, the ascertainment-corrected prospective likelihood, that is, Pr(D|G,A); second, the retrospective likelihood, Pr(G|D); and third, the ascertainment-corrected joint likelihood, Pr(D,G|A). We show that these likelihoods provide unbiased estimators of genetic relative risk parameters, as well as population allele frequencies and baseline risks. The parameter estimates based on the retrospective likelihood remain unbiased even when the ascertainment scheme cannot be modeled, as long as ascertainment only depends on families' phenotypes.
Despite the need to estimate additional parameters, the prospective, retrospective, and joint likelihoods can lead to considerable gains in efficiency relative to the standard conditional likelihood. This is true if baseline risks and allele frequencies can be assumed to be homogeneous.
In the presence of heterogeneity, however, the parameter estimates assuming homogeneity can be seriously biased. We discuss the extent of this problem and present a mixed models approach for providing consistent parameter estimates when baseline risks and allele frequencies are heterogeneous. We also compare the efficiency of the mixed-model prospective, retrospective, and joint likelihoods to the efficiency of standard conditional likelihood.