General epistatic models of the risk of complex diseases.

The range of possible gene interactions in a multilocus model of a complex inherited disease is studied by exploring genotype-specific risks subject to the constraint that the allele frequencies and marginal risks are known. We quantify the effect of gene interactions by defining the interaction ratio, C(R) = K(R)/K(R)(I), where K(R) is the recurrence risk to relatives with relationship R for the true model and K(R)(I) is the recurrence risk to relatives for a multiplicative model with the same marginal risks. We use a Markov chain Monte Carlo (MCMC) procedure to sample from the space of possible models. We find that the average of C(R) increases with the number of loci for both low frequency (p = 0.03) and higher frequency (p = 0.25) causative alleles. Furthermore, the probability that C(R) > 1 is nearly 1. Similar results are obtained when more weight is given to risk models that are closer to the comparable multiplicative model. These results imply that, in general, gene interactions will result in greater heritability of a complex inherited disease than is expected on the basis of a multiplicative model of interactions and hence may provide a partial explanation for the problem of missing heritability of complex diseases.