A Model for the Analysis of the Distribution of Qualitative Characters in Sibships

Abstract

A general model for the analysis of the distribution of qualitative characters in sibships is proposed. Beginning with the treatment of potential sibship size as a random variable, a method is formulated for studying the joint distribution of realized size, number of affected children and number of independent ascertainments under sibship-composition-dependent termination rules for 2 types of ascertainment. Alternative assumptions on the conditional distributions of the 3 variables are discussed. Representing sibship size as a geometric and as a Poisson variable, and the number affected and the number of independent ascertainments of a given sibship as binomials, maximum likelihood estimates of the parameters are obtained for 2 ascertainment procedures for the case of no selective termination of reproduction. The results are a direct generalization of Fisher s proband and sib methods where sibship size is treated as a constant and/or where "a posteriori" size distributions are utilized. Under the same model, maximum likelihood estimates of the parameters are obtained for the case of termination of reproduction at the birth of the first affected children.