A parametric copula model for analysis of familial binary data

Modeling the joint distribution of a binary trait (disease) within families is a tedious challenge, owing to the lack of a general statistical model w ith desirable properties such as the multivariate Gaussian model for a quan titative trait. Models have been proposed that either assume the existence of an underlying liability variable, the reality of which cannot be checked , or provide estimates of aggregation parameters that are dependent on the ordering of family members and on family size. We describe how a class of c opula models for the analysis of exchangeable categorical data can be incor porated into a familial framework. In this class of models, the joint distr ibution of binary outcomes is characterized by a function of the given marg inals. This function, referred to as a "copula," depends on an aggregation parameter that is weakly dependent on the marginal distributions. We propos e to decompose a nuclear family into two sets of equicorrelated data (paren ts and offspring), each of which is characterized by an aggregation paramet er (alpha(FM) and alpha(ss) respectively). The marginal probabilities are m odeled through a logistic representation. The advantage of this model is th at it provides estimates of the aggregation parameters that are independent of family size and does not require any arbitrary ordering of sibs. It can be incorporated easily into segregation or combined segregation-linkage an alysis and does not require extensive computer time. As an illustration, we applied this model to a combined segregation-linkage analysis of levels of plasma angiotensin I-converting enzyme (ACE) dichotomized into two classes according to the median. The conclusions of this analysis were very simila r to those we had reported in an earlier familial analysis of quantitative ACE levels.