Taken together, the above results confirm that approximation by EMMAX is appreciably more accurate than GRAMMAR, even in cases, such as the WTCCC data, where the sample structure is subtle. The comparisons also demonstrate that the accuracy of the EMMAX approximation can vary from case to case. Consequently, the potential gain in power from doing exact vs approximate tests will also vary among datasets. For the HMDP data, the potential gain in power from the exact calculations appears considerable, and this is confirmed by simulations (Supplementary Fig. 1). For the WTCCC Crohn's disease data the power gain is negligible, and as noted in ref1 only a small gain in power is generally expected at SNPs with small effect size. Of course, one nice feature of being able to do the exact tests is that it obviates the need to consider which approximations work best under what circumstances, or to consider ways in which the approximations could be improved. We also note that the computational tricks employed here also apply to other settings, including the combined “variable selection plus random effects”
