GEMMA or EMMA; see below for further discussion.) The main additional contribution of our work here beyond that in Lippert et al is that we provide, and make use of, efficient methods for evaluation of not only the likelihood, but also both its first and second derivatives. This allows us to make use of the Newton--Raphson optimization method, which has better theoretical convergence properties than Brent's algorithm (quadratic, vs super-linear), potentially reducing per-SNP computation time by reducing the number of iterations required for convergence, t. The practical effect of this is expected to depend on the sample size n. Examining the theoretical computational complexity, if p is large (and we assume the simplest case with no additional covariates, so c=1) then the per-SNP complexity of the algorithms is O(nw2 + tn). Thus if n is large then the n2 term will dominate and the number of iterations will have only a small effect of computation time; if n is moderate then the number of iterations may play a more important role. Consistent with this, we found GEMMA to be 12 times faster than the Lippert et al algorithm, implemented in FaST-LMM, for the smaller HMDP dataset (33 minutes vs 6.8
