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The first class, which we call counting methods, involves estimating the number of nonsynonymous and synonymous changes that have occurred at each codon throughout the evolutionary history of the sample. This approach was first proposed by Suzuki and Gojobori (1999) and involves reconstructing the ancestral sequences, for example, using parsimony (Suzuki and Gojobori 1999) or likelihood-based methods (Nielsen 2002; Nielsen and Huelsenbeck 2002; Suzuki 2004); the latter can take into account the uncertainty in the ancestral reconstructions. These methods are attractive as they are computationally fast and hence can be applied to large data sets and do not involve making any assumptions regarding the distribution of rates across sites. However, counting methods may lack power, especially for data sets comprising a small number of sequences or low divergence, as the power of the test is limited by the total number of inferred substitutions at a site. In addition, counting the number of changes between ancestral states may underestimate the true number of substitutions, and hence, the number of changes inferred using this approach may not accurately reflect the rate at which a site is evolving. The second class of methods, originally described by Nielsen and Yang (1998), involves fitting a distribution of substitution rates across sites and then inferring the rate at which individual sites evolve. When this site-by-site inference is based on the maximum likelihood estimates of the rate parameters, this inference is known as empirical Bayes (Nielsen and Yang 1998; Yang et al. 2000), whereas when rate class assignments are based on …