The LRT for selection on any branch of the phylogeny is essentially Nielsen and Yang's [31] test of site models 2a versus 1a, and the lineage-and clade-specific LRTs are essentially instances of Yang and Nielsen's [26] test 2 (see also [83, 27] ). However, to reduce the number of parameters estimated per gene, the complete set of 17,489 genes was divided into eight equally sized classes by G+C content in third codon positions. The branch lengths and the transition-transversion rate ratio k were estimated for each class under the null model, and these estimates were subsequently held fixed, in a G+C dependent way, for the LRTs. Instead of a complete set of branch lengths, a single scale parameter m was estimated per gene. Thus, only the parameters m, v 0 ,1 and p 0 for the null model and the additional parameters v 2 .1 and p 1 for the alternative model, were estimated per gene (see [31, 26] ). This parameterization speeds up calculations substantially compared to estimating k and a set of branch length per gene, while its sensitivity, specificity and power to detect positive selection are comparable (Text S1). We developed our own software for likelihood computation and parameter estimation to support this parameterization.