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

Abstract

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.

Figure 1

Allowed values of sib relative risk λ_S. The curves correspond to the overall disease prevalence K, as shown. Only combinations of λ_S and L under the curve are possible. The curve is given by equation (B7) and is valid only for the case of no dominance (h=1/2).

Figure 2

Effect of varying L. a, Effect of number of loci L on disease-allele frequency p. The curves correspond to values of the dominance coefficient h, as shown, and to K=0.01, δ_L=1, and π_L=0.2. b, Effect of L on sib relative risk. The curves were generated with π_L values of 0, 0.02, 0.05, 0.1, 0.15, and 0.2, ordered from top to bottom, as shown. For all plots, δ_L=1, and h=0.5.

Figure 3

Aunt-nephew relative risk as a function of L for a parent-offspring relative risk of 10. This curve was generated by equation (5) with λ_R=10 and C_R/C_S=1/2.

Figure 4

Effect of number of disease loci on multiplex-disease probabilities. a, Probability that two of three sibs of an affected individual are also affected. b, Probability that three of three sibs of an affected individual are also affected. Both curves were generated with K=0.01, h=0.5, δ_L=1, and π_L=0.

Figure 5

Likelihood profiles for CLCP data. a, Likelihood profile for L. b, Contour plot of the likelihood versus L and h. The likelihood value shown is the maximum across the parameters not shown. The values shown on the contour lines are the likelihood of parameter combinations on that contour relative to the maximum-likelihood value. The likelihood for the multiplex-relative-risk method was generated using equation (9) and data from Carter et al. (1982) on offspring and sibs of probands. The log(0.1) curve shows the cutoff for the support interval (which includes all L values that are at least one-tenth as likely).

Figure 6

Likelihood contours for CLCP data. a, Likelihood of the CLCP data versus L and δ_L. b, Likelihood versus L and π_L. For each L, δ_L ranged from slightly above K^1/L (the lowest possible value) to 1, in increments of . The Y-axis units gives these increments. Thus, the value of the Y-axis units in panel a changes with the X-axis variable L. The same holds for panel b: π_L ranged from 0 to K^1/L in increments of K^1/L/10, and the Y-axis units have the same value.