Size-Correction and Principal Components for Interspecific Comparative Studies

BRIEF COMMUNICATION doi:10.1111/j.1558-5646.2009.00804.x SIZE-CORRECTION AND PRINCIPAL COMPONENTS FOR INTERSPECIFIC COMPARATIVE STUDIES Liam J. Revell1,2,3 1Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138 2E-mail: [email protected] Received February 25, 2009 Accepted July 21, 2009 Phylogenetic methods for the analysis of species data are widely used in evolutionary studies. However, preliminary data transformations and data reduction procedures (such as a size-correction and principal components analysis, PCA) are often performed without ﬁrst correcting for nonindependence among the observations for species. In the present short comment and attached R and MATLAB code, I provide an overview of statistically correct procedures for phylogenetic size-correction and PCA. I also show that ignoring phylogeny in preliminary transformations can result in signiﬁcantly elevated variance and type I error in our statistical estimators, even if subsequent analysis of the transformed data is performed using phylogenetic methods. This means that ignoring phylogeny during preliminary data transformations can possibly lead to spurious results in phylogenetic statistical analyses of species data. KEY WORDS: Data transformation, evolutionary regression, least squares, linear regression, PCA, phylogenetic comparative methods, principal components analysis. Phylogenetic methods for the statistical analysis of species data independent datapoints—an assumption that is frequently vio- have become widely accepted in recent years (e.g., Cheverud lated by phylogenetic data from species (Harvey and Pagel 1991; et al. 1985; Felsenstein 1985; Grafen 1989; Harvey and Pagel Martins and Hansen 1997; Garland et al. 2005; but see Price 1991; Garland et al. 1992; Martins 1994; Hansen and Martins 1997). 1996; Hansen 1997; Butler et al. 2000; Hansen et al. 2008). A relatively commonly used procedure for phylogenetic This is because biologists are aware that data for species may statistical transformation of interspecific data was provided by be nonindependent due to shared history (Blomberg and Gar- Garland et al. (1992). According to this procedure, we first cal- land 2002; Freckleton et al. 2002; Blomberg et al. 2003; Gar- culate standardized phylogenetically independent contrasts (fol- land et al. 2005; Revell et al. 2008). Although it has become lowing Felsenstein 1985). To size-correct, we then regress the rare to find published studies in which the authors choose to ig- contrasts for each size-correlated variable on the contrasts for nore phylogeny in their primary analysis, it is not uncommon size (conducting regression “through the origin,” i.e., without an to see phylogeny effectively bypassed during preliminary statis- intercept term; Garland et al. 1992) and compute the residuals tical data manipulations, such as size-correction and principal as deviations between the predicted and estimated contrast val- components analysis (PCA). However, both standard linear re- ues. To perform PCA, we first calculate the variance–covariance gression (the most common method of size-correction in evolu- or correlation matrix from our contrasts (again, without intercept tionary ecology) and PCA assume that the sample consists of terms; e.g., Revell et al. 2007), then calculate its eigenvectors and eigenvalues, and, finally, compute PC scores for contrasts in 3Current address: National Evolutionary Synthesis Center, 2024 W. the typical way (e.g., Ackerly and Donoghue 1998; Clobert et al. Main St. A200, Durham, NC 27705 1998). C 2009 The Author(s). Journal compilation C 2009 The Society for the Study of Evolution. 3258 Evolution 63-12: 3258–3268 BRIEF COMMUNICATION Although these methods should yield phylogenetically in- First, my presentation focuses on BM as a model for evo- dependent and size-corrected or rotated evolutionary differences, lution. Various authors have already pointed out that BM is an they have two main disadvantages. The first is that it is difficult inappropriate model under many circumstances (e.g., Felsen- or impossible to use these methods under non-Brownian mo- stein 1988; Hansen 1997; Butler and King 2004; Hansen et al. tion (BM) models for the evolutionary process (although branch- 2008), although it is a suitable model for genetic drift and some length transformations can be used to mimic some other evolu- types of natural selection (e.g., O’Meara et al. 2006; Revell and tionary processes, such as punctuational evolution; e.g., Martins Harmon 2008). In the present comment, I use BM as my model and Garland 1991; Garland et al. 1993; Pagel 1994). The sec- for the evolutionary process—however, as noted below, the meth- ond disadvantage of size-correction and PCA on contrasts is that ods described herein can theoretically be applied equally well the residuals or scores that are returned are residual evolutionary using other evolutionary models (e.g., Pagel 1999; Butler and differences between nodes rather than residual values for species King 2004; Hansen et al. 2008). (i.e., they are associated with internal nodes in the tree rather than Second, I focus on a single, very common method of with the species at the tips of the tree). This latter consideration size-correction in evolutionary studies. This is the method in might not be a problem in some studies (e.g., if the residuals are which we obtain residuals from a least-squares regression of the destined only to be analyzed by a linear regression and merely the size-dependent trait against body size (Gould 1966; Jolicoeur slope and significance of this secondary analysis are of interest). et al. 1984; e.g., Glossip and Losos 1997; Hulsey et al. 2007; However, many evolutionary analyses require the association of Revell et al. 2007). However, size-correction can be performed size-corrected or rotated phenotypic values with tip taxa in the by many methods, only one of which is detailed herein, and tree. Types of analyses that might require transformed values for an extensive literature exists on procedures for size-correction tip species include (but are not limited to): the estimation of phy- (e.g., Humphries et al. 1981; Jolicoeur et al. 1984; Rohlf and logenetic signal (Freckleton et al. 2002; Blomberg et al. 2003); Bookstein 1987; Garc´ıa-Berthou 2001; McCoy et al. 2006). My ancestral state reconstruction (Schluter et al. 1997); the phylo- use of a linear regression for size-correction should not be con- genetic analysis of variance (ANOVA) or analysis of covariance sidered an endorsement of this method. Garc´ıa-Berthou (2001) (ANCOVA) (Garland et al. 1993); and phylogenetic regression in and Freckleton (2002, 2009) both caution against using size- which both slope and intercept (or prediction in the species space) corrected residuals as data for subsequent statistical analyses, and are of concern (Rohlf 2001). instead recommend that size be included as a covariate (Garc´ıa- In the present short comment, I describe statistically cor- Berthou 2001) or as an additional independent variable in a mul- rect procedures for phylogenetic size-correction and PCA. Al- tiple regression (Freckleton 2002). Even when size-correction is though in some cases these procedures are developed elsewhere desired, Rohlf and Bookstein (1987) have pointed out some of the (e.g., Garland and Ives 2000; Rohlf 2001; Blomberg et al. 2003), reasons why a linear regression should not be our size-correction they have not previously been presented in detail and in the ex- procedure of choice, particularly when data for size are collected plicit context of preliminary statistical data transformation. I also with sampling error (as will be true of almost any evolutionary use numerical simulations to examine the consequences of ig- study). In most cases, it is possible to extend the presented phy- noring phylogeny in preliminary transformations prior to a sta- logenetic approach to less commonly used statistical procedures tistical analysis by standard phylogenetic methods, such as in- for size-correction (e.g., shearing; Humphries et al. 1981). dependent contrasts followed by linear regression (Felsenstein Finally, third, although I present methods for phylogenetic 1985). size-correction and phylogenetic PCA, it is important for the The purpose of this comment is almost wholly utilitarian, and reader to note that these procedures provide residuals and scores as such I have appended R (R Development Core Team 2008) and in the original, phylogenetically dependent, species space—not MATLAB (The Mathworks 2006) code for the analyses described in a transformed phylogenetically independent space. This means herein. The inspiration for this comment comes from the frequent that the residuals and scores from these analyses still probably questions posed to me by evolutionary biologists and ecologists need to be analyzed using phylogenetic methods! Why then, one who are interested in phylogenetic methods for size-correction might ask, should we bother to use phylogenetic methods for size- and PCA, but unsure how to proceed. correction or data rotation? I show below, in section Examples, that phylogenetic size-correction and principal components provide estimates of the allometric coefficient and eigenstructure that Methods will have lower variance relative to nonphylogenetic procedures, In the present section, I describe correct procedures for phyloge- thus reducing type I error to its nominal level when residuals and netic size-correction and PCA. Three primary disclaimers

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