Likelihoods and TDT for the case-parents design

Association studies using diseased cases and their parents avoid biases due to population stratification, and the transmission/disequilibrium test (TDT) is a popular method of analysis. Sample size and power calculations for the TDT method have been reported, but often for the special situation of multiplicative effects of alleles on the genotype relative risks. Furthermore, some of the proposed calculations ignore the dependence of transmitted alleles from a pair of heterozygous parents when the effects are not multiplicative, which can lead to erroneous sample size calculations. We demonstrate how to calculate sample size and power for the TDT method for general genotype relative risks. As an alternative to the TDT method, we present likelihood methods for a variety of genotype relative risk models. Exact likelihood methods are presented to allow for accurate small-sample analyses. We demonstrate by numerical comparisons: (1) that the TDT method is inefficient for recessive patterns of relative risks, (2) for alleles that are not rare, falsely assuming a multiplicative model can lead to gross underestimation of the required sample size for the TDT statistic, and (3) for common alleles, if the true genotype relative risks have an approximately dominant pattern, then the TDT method can be grossly inefficient compared to likelihood methods. An alternative likelihood ratio statistic, based on two degrees of freedom, tends to be robust for a wide range of genotype relative risk models. Finally, we discuss how to use standard software for conditional logistic regression to accurately assess effects of alleles as well as genotype-environment interaction.