On the Statistical Interpretation of Site-Specific Variables in Phylogeny-Based Substitution Models

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ABSTRACT Phylogeny-based modeling of heterogeneity across the positions of multiple-sequence alignments has generally been approached from two main perspectives. The first treats site specificities as random variables drawn from a statistical law, and the likelihood function takes the form of an integral over this law. The second assigns distinct variables to each position, and, in a maximum-likelihood context, adjusts these variables, along with global parameters, to optimize a joint likelihood function. Here, it is emphasized that while the first approach directly enjoys the statistical guaranties of traditional likelihood theory, the latter does not, and should be approached with particular caution when the site-specific variables are high dimensional. Using a phylogeny-based mutation-selection framework, it is shown that the difference in interpretation of site-specific variables explains the incongruities in recent studies regarding distributions of selection coefficients.

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MODELING the heterogeneity of evolutionary regimes across the different positions of genes is of great interest in evolutionary genetics. Among the phylogeny-based approaches taken are two main formulations. The first approach is generalized as follows: considering the ith alignment column, written as Di, one defines a (potentially multivariate) random variable, denoted here as xi. Then, given a set of global parameters u, the likelihood for site i takes the form of an integral over a chosen statistical law V(xi) and is written as .... The random variable is said to be integrated away. Moreover, the global (multidimensional) parameter u may include elements controlling the form of the statistical law V. Assuming independence between the sites of the alignment, the overall likelihood is a product across all site likelihoods, written explicitly as ...

The most well-known instance of the random-variable approach is the gamma-distributed rates across sites model proposed by Yang (1993). In this model, the random variable is the rate at a given position-which acts as a branchlength multiplier-and the statistical law governing it is a gamma distribution of mean 1, and of variance 1/a. In a maximum-likelihood framework, the shape parameter a is included as an element of u, and this overall hypothesis vector is adjusted to u, which maximizes the likelihood. It should be noted that in practice, integrating over the statistical law governing a random variable can be difficult, and in the case of the gamma-distributed rates model most implementations rely on either a discretization method [which reduces the integral to a weighted sum (Yang 1994, 1996)] or on MCMC sampling in a Bayesian framework (e.g., Mateiu and Rannala 2006). In the latter context, the sampling system straightforwardly enables the evaluation of posterior distributions of random variables, but analogous calculations are also possible in a maximum-likelihood context, through empirical Bayes methods (see, e.g., Anisimova 2012). Other examples of phylogeny-based random variable approaches are plentiful and include models for heterogeneous nonsynonymous rates (Yang et al. 2000), for heterogeneous nonsynonymous and synonymous rates (Kosakovsky Pond and Muse 2005), and for heterogeneous amino acid profiles (Lartillot and Philippe 2004).

A second line of work has taken what might be called an extensive parameterization approach, within which each sitespecific variable xi is itself treated as part of the parameters of the model, with the likelihood function optimized being p(D|u, x) so as to obtain estimatesu and x (e.g., Bruno 1996; Halpern and Bruno 1998; Kosakovsky Pond and Frost 2005; Massingham and Goldman 2005; Delport et al. 2008; dos Reis et al. 2009; Holder et al. 2008; Tamuri et al. 2009, 2012; Murrell et al. 2012). These articles continue to drive several lines of research. For instance, the seminal article by Halpern and Bruno (1998) has greatly stimulated developments in population-genetics-based substitution modeling (see Thorne et al. …