Robust asymptotic sampling theory for correlations in pedigrees

Methods to unravel the genetic determinants of non-Mendelian diseases lie at the next frontier of statistical approaches for human genetics. It is generally agreed that, before proceeding with segregation or linkage analysis, the trait under study ought to be shown to exhibit familial correlation. By coding dichotomous traits as binary variables, a single robust approach in the estimation of pedigree correlations, rather than two distinct approaches, can be used to assess the potential heritability of a trait, and, latterly, to examine the mode of inheritance. The asymptotic theory to conduct hypothesis tests and confidence intervals for correlations among different members of nuclear families is well established but is applicable only if the nuclear families are independent. As a further contribution to the literature, we derive the asymptotic sampling distribution of correlations between random variables among arbitrary pairs of members in extended families for the Pearson product-moment estimator with generalized weights. This derivation is done without assuming normality of the traits. The sampling distribution is shown to be asymptotically normal to first order, and hence large-sample hypothesis tests and confidence intervals with estimates of the variances and correlation coefficients are proposed. Discussion concludes with an example and a suggestion for future research.