Familial association of disease and the structure of trivariate distributions

In its usual form, the multifactorial model of disease transmission assumes that the liabilities to disease have a multivariate normal distribution. This paper studies how sensitive to this assumption are the quantitative results from the model. Accordingly, bounds are established for the probability of a child having a disease, given that both parents have it and taking the heritability of the disease to be known. Unfortunately, these bounds turn out to be wide. For example, a probability that is 0.38 under the trivariate normal model may be as low as 0.12 or as high as 0.78 under other trivariate models, even if attention is restricted to those of variables-in-common form. The broader statistical issue of the meaning of trivariate dependence, as distinct from bivariate dependence, is also discussed.