Estimating blood phenotype probabilities and their products

We consider two methods of estimating phenotype probabilities for a number of standard genetic markers like the ABO, MNSs, and PGM markers. The first method is based on the maximum likelihood estimates of the allele probabilities, and the second (multinomial) method uses the phenotype proportions in the sample. The latter is easy to use, the estimates are always unbiased, and simple formulae for variances are available. The former method, although giving more efficient estimates, requires the assumption of panmixia so that the Hardy-Weinberg law can be used. The two methods are compared theoretically, where possible, or by simulation. Under panmixia, the maximum likelihood estimates can be substantially more efficient than the multinomial estimates. The estimates are also compared in the codominant allele case for nonpanmictic populations. The question of efficiency is of importance when estimating the probability of obtaining a given set of phenotypes, i.e., the product of individual phenotype estimators. This problem is discussed briefly.