Multivariate genetic analysis of sex limitation and G x E interaction

Sex-limited expression of genetic or environmental factors occurs in two basic forms. First, the effects of a factor may be larger on one sex than on another, which is known as scalar sex limitation. Second, some factors may have an effect on one sex but not on the other, which is called nonscalar sex limitation. In the classical twin study, scalar sex-limited effects cause same-sex male and same-sex female twin correlations to differ. Nonscalar sex-limited effects would cause the correlations between opposite-sex pairs of relatives to be lower than would be expected from the correlations between relatives of the same sex. One approach to modeling such effects is to allow the genetic correlation between opposite-sex dizygotic twins to be less than one-half; another is to allow the common environment correlation for opposite-sex pairs to be less than unity. Extension of this approach to the multivariate case is not straightforward. Direct extension of the Cholesky decomposition such that each Cholesky factor is allowed to correlate less than one-half in opposite-sex pairs yields a model where the order of the variables can change the goodness-of-fit of the model. It is shown that similar problems exist with a variety of multivariate and longitudinal models, and in a variety of models of genotype x environment interaction. Several solutions to these problems are described.