Fig. 2.

Comparison of REML and PCGC regression. (A) REML yields biased estimates for case–control studies of diseases, whereas PCGC regression yields unbiased estimates. We simulated case–control studies for nine combinations of K (prevalence) and P (proportion of cases among overall samples), and for five values of h² (0.1, 0.3, 0.5, 0.7, and 0.9). For each combination of parameters, we show the average of 10 heritability estimates obtained by applying the REML method of Lee et al. (10) and PCGC regression to our simulated case–control data. REML produced biased estimates, whereas PCGC regression produced unbiased estimates for all scenarios. The bias of REML estimates increases as both the true heritability and overrepresentation of cases increase. To demonstrate the severity of the bias, consider the scenario of a disease with prevalence of 0.1% in a balanced case–control study (values typical for Crohn's disease or MS). When the true heritability is 50%, the estimated heritability would be 30% on average, as indicated by the black dots. (B) Heritability estimates for case–control studies with increasing sample size. Simulated case–control studies are as previously described, with the prevalence of the disease, the proportion of cases, and the heritability fixed at 1%, 30%, and 50%, respectively. The size of simulated studies ranged from 2,000 to 8,000. The bias of heritability estimates from REML increases with study size, whereas those from PCGC regression estimates remain unbiased. (C) Heritability estimation in the presence of fixed effects. We simulated case–control studies with an additional “sex” covariate, which either has no effect on the disease or increases the relative risk (RR) by twofold or fourfold. The prevalence of the disease in the population was 0.5%, the heritability was set to 50%, and the numbers of cases and controls were equal. Applying REML with or without accounting for the additional covariate resulted in underestimation of the heritability. Moreover, inclusion of the covariate as a fixed effect resulted in even lower estimates of heritability when the effect of the covariate on the phenotype was considerable. By contrast, PCGC regression correctly accounted for the presence of the covariate.