An Approximation to the Multivariate Normal Integral: Its Application to Multifactorial Qualitative Traits

Abstract

Segregation analysis or pedigree analysis for quasicontinuous multifactorial traits requires probabilities under the multivariate normal distribution. Mendell and Elston (1974) describe a method for approximating recurrence risks which is used to provide an approximation to these probabilities. Three variants of this approximation are investigated and compared to results obtained using a method of Curnow (1972) in the equi-correlational case. These approximations have the advantage that no special covariance or threshold structure is required for their use. Although only genetic applications are given, the approximations are relevant to any problem which requires either the evaluation of the multivariate normal integral or estimation of the parameters of the multivariate normal distribution from polychotomized data. The accuracy of each approximation is assessed over a range of truncation points for an s-variable normal, s = 2, 3, 4, 6, 8, 10, and specific recommendations are made as to which method can be used to achieve the best results. The sources of error in the approximations are discussed, and may serve as guidelines in implementing these procedures for specific applications.