Figure 1.

The MEDUSA algorithm. 1) The one-rate model, $M_0$, is first fit to the data using maximum likelihood. 2) Every possible two-rate model is then fit to the data, and the models are ranked by their likelihood score (with the best model, $M_1$, at the bottom). 3) Every possible three-rate model that is consistent with the best two-rate model is then fit to the data, and the models are ranked as in the previous step. 4) Step 3 is repeated for increasingly complex models until the most complex model has been evaluated. 5) The AIC score is computed for the best model in each index (i.e., with $0,1,\dots$ rate shifts). 6) The $\Delta$AIC value is calculated for each pair of adjacent models (as the difference in their AIC scores). Each of the $\Delta {\text{AIC}}_i$ comparisons are evaluated in succession, where the more complex model is selected if the computed $\Delta$AIC value exceeds some prespecified threshold, $\Delta {\text{AIC}}_{\text{crit}}$. This process continues until either the most complex model is accepted or the improvement in the AIC score is too small to exceed the $\Delta {\text{AIC}}_{\text{crit}}$ threshold.