Multiplex relative risk and estimation of the number of loci underlying an inherited disease

Knowledge of the number of causative loci is necessary to estimate the power of mapping studies of complex diseases. In the present article, we reexamine a theory developed by Risch and its implications for estimating the number L of causative loci affecting a complex inherited disease. We first show that methods based on Risch's analysis can produce estimates of L that are inconsistent with the observed population prevalence of the disease. We demonstrate this point by showing that the maximum-likelihood estimate for L produced by the method of Farrall and Holder for cleft lip/cleft palate data is not consistent with the prevalence under the multiplicative model. We show how to incorporate disease prevalence and develop a maximum-likelihood method for estimating L that uses the entire distribution of numbers of affected individuals in families containing an affected individual. This method avoids the potential inconsistencies of the Risch method and has greater precision. We apply our method to data on cleft lip/cleft palate and schizophrenia.