The HMDP GWAS represents a situation where approximation methods such as EMMAX or GRAMMAR may yield inaccurate test statistics. In particular, because individuals in the data set are closely related, and the strongly associated SNPs contribute to a significant proportion of phenotypic variation in HDL-C13, using estimates of variance components or fitted residuals from the null model for testing may be expected to yield conservative p values, leading to a potential loss of power. Our empirical comparison (Figure 1c) confirms this: in this case, approximation by EMMAX leads to systematic and appreciable underestimation of the most significant p values (almost two orders of magnitude), while approximation by GRAMMAR leads to dramatic underestimation of all p values. Indeed, in contrast to the exact p values, no p values generated by EMMAX are significant at the conventional 0.05 level after Bonferroni correction, and no p values generated by GRAMMAR are significant even before Bonferroni correction. The fact that the exact p values for the most significant results are substantially more significant than the approximate p values from EMMAX suggests that, in this type of setting, the exact p values may produce a more powerful test; simulation results confirm this (Supplementary Fig. 1).
