Multivariate probit analysis of binary familial data using stochastic representations

The probit function is an alternative transformation to the logistic function in the analysis of binary data. However, use of the probit function is prohibitively complicated for cases of multivariate or repeated-measure binary responses, as integrations involving the multivariate normal distribution can be difficult to compute. In this paper, we propose an alternative form to stochastically represent random variables in the case of familial binary data that simplifies calculation of the multivariate normal integrals involved in the probit link. We provide examples of these stochastic representations for one- and two-parent families, and compare the performance of this methodology with that of moment estimators by calculating asymptotic relative efficiencies and through a real-life data example. Particular attention is paid to analyzing the properties of regression parameter estimates from these two methods with respect to the feasible ranges of the correlation parameters.