We introduce a Liability Threshold Mixed Linear Model (LTMLM) association statistic for ascertained case-control studies that increases power vs. existing mixed model methods, with a well-controlled false-positive rate. Recent work has shown that existing mixed model methods suffer a loss in power under case-control ascertainment, but no solution has been proposed. Here, we solve this problem using a chi-square score statistic computed from posterior mean liabilities (PML) under the liability threshold model. Each individual’s PML is conditional not only on that individual’s case-control status, but also on every individual’s case-control status and on the genetic relationship matrix obtained from the data. The PML are estimated using a multivariate Gibbs sampler, with the liability-scale phenotypic covariance matrix based on the genetic relationship matrix (GRM) and a heritability parameter estimated via Haseman-Elston regression on case-control phenotypes followed by transformation to liability scale. In simulations of unrelated individuals, the LTMLM statistic was correctly calibrated and achieved higher power than existing mixed model methods in all scenarios tested, with the magnitude of the improvement depending on sample size and severity of case-control ascertainment. In a WTCCC2 multiple sclerosis data set with >10,000 samples, LTMLM was correctly calibrated and attained a 4.1% improvement (P=0.007) in chi-square statistics (vs. existing mixed model methods) at 75 known associated SNPs, consistent with simulations. Larger increases in power are expected at larger sample sizes. In conclusion, an increase in power over existing mixed model methods is available for ascertained case-control studies of diseases with low prevalence.
Estimation in a multiplicative mixed model involving a genetic relationship matrix
