Estimation of interclass correlations in familial data

SUMMARY The maximum likelihood estimates of the parent-child interclass correlation and other parameters from familial data when the families have unequal numbers of offspring is given. Also given are two alternative sets of estimates which are easier to compute than the maximum likelihood estimates. Using these estimates, a test for the interclass correlation to be zero requires only the ordinary Student's t-distribution. Similarly other tests use only known distributions. In the analysis of familial data the degree of mother-sib resemblance is measured by the interclass correlation, the correlation between two different types of family member. Several estimates of this correlation have been proposed in the literature; see Rosner, Donner & Hennekens (1977) for details and their Monte Carlo comparisons of mean squared errors. In particular, they gave maximum likelihood estimates when the sib sizes are equal. However, when the sib sizes are not equal, Rosner (1979) proposed an algorithm for finding the maximum likelihood estimates which involves iterative maximization of an implicit function of two parameters. This algorithm is difficult to implement and may not even converge for some sets of data. Recently Mak & Ng (1981) used the linear model approach of Kempthorne & Tandan (1953) to obtain the maximum likelihood estimates. This reduces the problem to iterative maximization of a function of one parameter. However, nothing is known about the convergence of the procedure. Another iterative method of finding the maximum likelihood estimates has been given by Smith (1980). In this paper, an alternative approach is given which requires solving only one equation. Also, two alternative sets of estimates are given. These estimates are easier to compute than the maximum likelihood estimates. Using these estimates, a test for the interclass correlation to be zero is given; it requires only the ordinary Student's t distribution. Similarly, other tests use only known results.