Bayesian models for comparative analysis integrating phylogenetic uncertainty

Abstract

BackgroundUncertainty in comparative analyses can come from at least two sources: a) phylogenetic uncertainty in the tree topology or branch lengths, and b) uncertainty due to intraspecific variation in trait values, either due to measurement error or natural individual variation. Most phylogenetic comparative methods do not account for such uncertainties. Not accounting for these sources of uncertainty leads to false perceptions of precision (confidence intervals will be too narrow) and inflated significance in hypothesis testing (e.g. p-values will be too small). Although there is some application-specific software for fitting Bayesian models accounting for phylogenetic error, more general and flexible software is desirable.MethodsWe developed models to directly incorporate phylogenetic uncertainty into a range of analyses that biologists commonly perform, using a Bayesian framework and Markov Chain Monte Carlo analyses.ResultsWe demonstrate applications in linear regression, quantification of phylogenetic signal, and measurement error models. Phylogenetic uncertainty was incorporated by applying a prior distribution for the phylogeny, where this distribution consisted of the posterior tree sets from Bayesian phylogenetic tree estimation programs. The models were analysed using simulated data sets, and applied to a real data set on plant traits, from rainforest plant species in Northern Australia. Analyses were performed using the free and open source software OpenBUGS and JAGS.ConclusionsIncorporating phylogenetic uncertainty through an empirical prior distribution of trees leads to more precise estimation of regression model parameters than using a single consensus tree and enables a more realistic estimation of confidence intervals. In addition, models incorporating measurement errors and/or individual variation, in one or both variables, are easily formulated in the Bayesian framework. We show that BUGS is a useful, flexible general purpose tool for phylogenetic comparative analyses, particularly for modelling in the face of phylogenetic uncertainty and accounting for measurement error or individual variation in explanatory variables. Code for all models is provided in the BUGS model description language.

A simple and general method for simultaneously accounting for phylogenetic and species sampling uncertainty via Rubin’s rules in comparative analysis

“…In contrast, methods ignoring phylogenetic uncertainty performed poorly. These findings are concordant with the previous work by de Villemereuil et al (2012). Our proposed method using Rubin's rules, and with the original degrees of freedom (df), provided the expected coverage of the confidence interval, CI, for the slope with N = 50.…”

“…The required number of trees is far less than 1000 (as in Garamszegi and Mundry 2014), and probably less than 100 (as in de Villemereuil et al 2012). It is likely to be a matter of dozens.…”

“…In order to assess the overall quality of our new method and compare it to existing ones, we performed a simulation study using 100 trees estimated from a real data set, using BEAST (reduced species and tree samples from Wells et al, Submitted; see also de Villemereuil et al, 2012). We simulated data sets in which a variable y was linearly predicted from a variable x, with an intercept of 5 and a slope of 2.…”

“…Losos 1994; Martin 1996; Housworth & Martin 2001;Huelsenbeck et al 2000). Among these methods, probably the best one is to use Bayesian Markov Chain Mote Carlo, MCMC (Huelsenbeck et al 2000;Huelsenbeck and Rannala 2003;de Villemereuil et al 2012); the Bayesian MCMC methods utilize phylogenetic trees sampled from posterior tree set obtained from Bayesian phylogenetic tree estimation programs such as BEAST (Brummond and Rambaut 2007) and MrBayes (Ronquist and Huelsenbeck 2003). .…”

“…4! phylogenetic trees (e.g., see Figure 6 in de Villemereuil et al 2012). Recently, Garamszegi and Mundry (2014) have proposed a frequentist solution, which employs model averaging with Akaike information criterion (AIC) in PGLS incorporating many phylogenetic trees (see also Mahler et al 2010).…”