Independent Contrasts and PGLS Regression Estimators Are Equivalent

Abstract

We prove that the slope parameter of the ordinary least squares regression of phylogenetically independent contrasts (PICs) conducted through the origin is identical to the slope parameter of the method of generalized least squares (GLSs) regression under a Brownian motion model of evolution. This equivalence has several implications: 1. Understanding the structure of the linear model for GLS regression provides insight into when and why phylogeny is important in comparative studies. 2. The limitations of the PIC regression analysis are the same as the limitations of the GLS model. In particular, phylogenetic covariance applies only to the response variable in the regression and the explanatory variable should be regarded as fixed. Calculation of PICs for explanatory variables should be treated as a mathematical idiosyncrasy of the PIC regression algorithm. 3. Since the GLS estimator is the best linear unbiased estimator (BLUE), the slope parameter estimated using PICs is also BLUE. 4. If the slope is estimated using different branch lengths for the explanatory and response variables in the PIC algorithm, the estimator is no longer the BLUE, so this is not recommended. Finally, we discuss whether or not and how to accommodate phylogenetic covariance in regression analyses, particularly in relation to the problem of phylogenetic uncertainty. This discussion is from both frequentist and Bayesian perspectives.

Allometric Scaling of the Optic Tectum in Cartilaginous Fishes

“…Allometric scaling relationships were determined by calculating linear least-squares regression lines and 95% confidence intervals (95% CI) using: (1) the raw species data (i.e. species as independent data points), and (2) standardized independent contrasts to account for phylogenetic relatedness [Garland and Ives, 2000], which are considered equivalent to generalized least-squares (GLS) models [Midford et al, 2005;Blomberg et al, 2012]. In both forms of analysis, optic tectum volume, as determined using method S or method E with above correction factor, along with tegmentum volume, were both scaled against brain volume and mesencephalon volume.…”

Morphological divergence in a continental adaptive radiation: South American ovenbirds of the genusCinclodes

“…Our phylogenetic analyses were based on a reconstruction of the family Furnariidae (Derryberry et al 2011), including estimated branch lengths from that study, pruned to include only Cinclodes species. For the regression of body mass versus the first principal component (PC1), we used ordinary least squares (OLS), phylogenetically generalized least squares (PGLS; Martins and Hansen 1997, Freckleton et al 2002, Blomberg et al 2012, and standardized major-axis (SMA; Warton et al 2006) regression approaches on species' mean values. We investigated the evolution of the 2 axes that accounted for most of the variation in morphology in our datasets, PC1 and PC2, using a 2-step approach (Pagel 1999, Claramunt et al 2012.…”

Identical inferences about correlated evolution arise from ancestral state reconstruction and independent contrasts

“…We agree with [13] that this apparent overconfidence is wholly remedied by reducing the number of degrees of freedom in the calculation of confidence bounds by a factor of two when using ancestral state reconstruction, though this manipulation is not necessary to guarantee the identity of point estimates made by regression estimators under ancestral state reconstruction and independent contrasts. In the light of findings that regression estimators based on independent contrasts are also identical to those based on phylogenetic generalized least squares [16], we conclude that all comparative methods based on the Brownian motion model of evolution yield identical inferences about the parameters of correlated evolution and are conceptually indistinguishable. Our response to claims that ancestral state reconstruction is error-prone is to point out that the ancestral states themselves are merely nuisance parameters of the model formulation.…”

“…The mean squared standardized independent contrast across the internal nodes of a phylogeny is an estimator of this parameter, while the mean squared deviation of reconstructed trait value across the branches of a phylogeny is an estimator of half this parameter [13]. The close association of methods based on Brownian motion is further indicated by the facts that the phylogenetic mean trait value inferred under indendent contrasts is identical to the global maximum likelihood estimate of the root's trait value [14,15], that independent contrasts and phylogenetic generalized least squares models yield identical regression estimators for the slope and gradient of two correlated traits [16], and that regression coefficients of bivariate data estimated under directional and nondirectional approaches are highly correlated [10].…”