In summary, we have presented an efficient method for computing exact values of standard test statistics in linear mixed models. This method is comparable in speed with approximation methods such as EMMAX while yielding exact test statistics. Using two examples we illustrate our method, and show that the approximation methods can yield inaccurate p values when the sample structure is strong and/or when the marker effect size is large. We also find that the approximation by EMMAX is more accurate than the approximation by GRAMMAR genome-wide (a comparison made possible only by the availability of an efficient exact method).
