Figure 1. Simulation studies used to derive the appropriate penalty term for .

Each panel plots the difference in log likelihood () normalized by the logarithm of the sample size (number of characters), between best fitting GA models with and rates (), against the number of sites in the alignment. For simulations with a single rate class we plotted , top right. Figures for multiple rate simulations (2–5 rates) show as black dots (left column); and as blue dots (right column). Values to the right of row report simulated rates for each class. The left column is a reflection of power, whereas the right column – of the degree of over-fitting. For the case where a single rate was simulated, the degree of over-fitting is the rate of false positives. The desired behavior for is achieved when the model with rate classes is preferred to models with , and rate classes. For a modified BIC criterion with , the former happens if (more definitively with increasing sample size), and the latter if (regardless of sample size).