To compare the power of our LRTs with the power of previous tests based on hominid or primate genomes, we simulated data sets under a range of parameter values and measured the fraction of cases in which positive selection was predicted ( Figure 6) . These experiments show that power increases substantially when the set of species under consideration is expanded from the two hominid species to the three primates then to all six mammals. With hominid species only, power is poor even when selection is quite strong (e.g., ,20% with a constant v = 2 and ,40% with v = 4), suggesting that a genome-wide scan will tend to identify only the most extreme cases of positive selection. If a rigorous correction for multiple testing is applied, a test based on hominids only has essentially no power, even for fairly long genes under strong selection ( Figure S3 ; see also [5] ). The situation is considerably improved by the addition of the macaque genome, but power remains poor when controlling for multiple testing unless genes are long and selection is strong. When all six mammals are considered, however, power increases substantially. With the full data set, power is reasonably good ($70%) even when genes are short and selection is moderate in strength; it remains good when multiple comparisons are considered ( Figure S3 ). The absolute estimates of power from these experiments depend on the simplifying assumptions used in the simulations (including the unrealistic assumption of constant v among lineages and among sites), and they must be interpreted cautiously. However, estimates of relative power-which will be less sensitive to these simplifying assumptions-indicate a substantial improvement is achieved by the addition of the three non-primate mammals.