Power and Effective Study Size Based on Approximations to the Expected Likelihood Ratio Test in Heritability Studies

Determining statistical power and precision of heritability estimates can be difficult particularly in complex pedigrees. Previous work focused on the power to detect heritability by using the expectation of the likelihood ratio test (ELRT). This work is extended to the more general setting involving arbitrary null hypotheses. We derive two Taylor series expansion which approximate the ELRT in terms of one or two relatedness summary parameters, which are simple functions of the eigenvalues (variance) or log-eigenvalues (mean and variance) of twice the kinship matrix. We evaluate the accuracy of these approximations compared to the exact ELRT, and derive formulae for expected confidence interval width which are function of only the relatedness summary parameters. Further, we demonstrate that an effective number of unrelated sibpairs can be calculated from the relatedness summary parameters for any complex pedigree. The result of these approximations provides an easily-calculated summary of the relatedness within a study. For the purposes of planning or comparing studies, these summary parameters provide information about how the relatedness within large complex pedigrees may affect statistical inference. POWER APPROXIMATIONS FOR HERITABILITY ANALYSIS 3