Polytomies and Phylogenetically Independent Contrasts : Examination of the Bounded Degrees of Freedom Approach

—We examined the effect of soft polytomies on the performance (Type I error rate and bias) of Felsenstein’s (1985; Am. Nat. 125:1–15) method of phylogenetically independent contrasts for estimating a bivariate correlation. We specically tested the adequacy of bounding degrees of freedom, as suggested by Purvis and Garland (1993; Syst. Biol. 42:569–575). We simulated bivariate character evolution under Brownian motion (assumed by independent contrasts) and eight other models on ve phylogenetic trees. For non-Brownian motion simulations, the adequacy of branchlength standardization was checked with a simple diagnostic (Garland et al., 1992; Syst. Biol. 41: 18–32), and transformations were applied as indicated. Surprisingly, soft polytomies tended to have negligible effects on Type I error rates when models other than Brownian motion were used. Overall, and irrespective of evolutionary model, degrees of freedom were appropriately bounded for hypothesis testing, and unbiased estimates of the correlation coefcient were obtained. Our results, along with those of previous simulation studies, suggest that independent contrasts can reliably be applied to real data, even with phylogenetic uncertainty. [Comparative method; computer simulation; hypothesis testing; polytomies. ] Many available phylogenetic trees include polytomies. Hard polytomies represent lack of available evidence to resolve branching order and may indicate true simultaneous speciation events. Soft polytomies represent contradictory results from conicting data and lack of information about the true bifurcating pattern of speciation (Maddison, 1989); they are also known as unrecognized phylogeny (Grafen, 1989). When conducting acomparativestudy, biologists will often be faced with phylogenies that contain polytomies. In the presence of hard polytomies, Felsenstein’s (1985) independent contrasts method gives the correct estimates of slopes, correlations, and P values. Hard polytomies are represented as a series of bifurcations with branch lengths of 0, and analyses proceed in the standard way (see Purvis and Garland, 1993). The way in which the branches within a hard polytomy are joined is arbitrary and has no effect on the usual calculations done with phylogenetically independent contrasts (Garland et al., 1992). For soft polytomies, ve methods have been proposed. One of them involves using simulated phylogenies (Losos, 1994). For a set of tip data, we generate a random sample of bifurcating phylogenies consistent with what is known (assumed to be true) about the phylogenetic relationships of that set of species. The correlation coefcient (or other statistic of interest) is computed by independent contrasts for each of these phylogenies. If all phylogenies yield the same result, then we may conclude that the obtained result is reliable, regardless of the true phylogeny. However, if not all of the hypothetical phylogenies yield the same result, then it is necessary toexamine which phylogenies causedivergent results; conclusions will depend on the number and likelihood of the phylogenies that yield contradictory results (see details in Losos, 1994: 121). A similar proposal, butusing completely random trees (Martins, 1996), has been shown to produce estimates that are equivalent to a non-phylogenetic analysis and to yield hypothesis tests that are overly conservative (Abouheif, 1998). The other three methods do not involve use of random trees. Instead, adjustments are made in the number of contrasts com1Author for correspondence. D ow naded rom http/academ ic.p.com /sysbio/articlact/48/3/547/1641528 by gest on 01 M arch 2019 548 SYSTEMATIC BIOLOGY VOL. 48 puted or in the number of degrees of freedom used. Grafen (1989, 1992) and Pagel and Harvey (1989, 1992; Pagel, 1992; Harvey and Pagel, 1991) compute only 1 contrast for each multifurcating node in the polytomous phylogeny. These two methods are arbitrary in the sense that they use information from tip data (the traits under analysis) to indicate how to compute a single contrast for each polytomy. Specically, Pagel and Harvey suggest “resolving” the tree based on phenotypic similarity; Grafen uses the residuals from the regression of y on x to obtain linear contrasts. Hence, for both methods, the way the single contrast is constructed will vary, depending on the tip data. Alternatively, following Felsenstein’s (1985:10) original suggestion, Purvis and Garland (1993) suggested computing m – 1 contrasts for each multifurcating node with m simultaneously radiating branches. This procedure is similar to that used with hard polytomies: the soft polytomy is also represented as a series of bifurcations of length 0, and m – 1 contrasts are computed. (To our knowledge, Konarzewski [1995] is the rst empirical example to follow Felsenstein’s [1985] original suggestion.) For testing hypotheses about bivariate character correlations, Purvis and Garland (1993) suggested bounding the degrees of freedom. If the polytomy were really a hard polytomy, then total degrees of freedom would be N – 2, where N is the total number of species (and N – 1 the total number of independent contrasts); this constitutes the maximum degrees of freedom. The lower bound on degrees of freedom for a polytomous phylogeny is p – 1, where p is the number of real nodes (note that p – 1 = N – 2 – z, where z is the number of branches that are set to 0 to represent the polytomies; i.e., p = N – 1 – z is equal to the number of contrasts that do not proceed from 0-length bifurcations). When using the lower-bound degrees of freedom then, although the total number of contrasts is still N – 1, critical values are established with p – 1 df (instead of N – 2 df). According to Purvis and Garland (1993), their procedure is simple and less arbitrary than the proposals of Grafen (1989, 1992) or Pagel and Harvey (1989, 1992). The method for computing contrasts is the same, regardless of whether polytomies are hard, soft, or rm (rm polytomies are those in which a node is fully dichotomous in reality but resembles a hard polytomy because it is the result of near-simultaneous radiations; see Purvis and Garland, 1993:573) ; the only needed modication is an adjustment to degrees of freedom. Use of the lower bound on the degrees of freedom should protect against inated Type I error rates, while at the same time improving estimation by using a full set of N – 1 contrasts (see Purvis and Garland, 1993:573). However, the performance of Purvis and Garland’s approach has never been examined by computer simulations (Stamps et al. [1997] presented an empirical example comparing this method with that of Losos [1994].) So far, only Grafen (1989) and Purvis et al. (1994) have examined the effects of polytomies on the performance of independent contrasts. The former examined the performance of independent contrasts and the phylogenetic regression (Grafen, 1989) in the presence of unrecognized phylogeny; the latter examined theperformance of Pagel’s (1992) procedure. Here, we use computer simulations to examine whether Purvis and Garland’s (1993) procedure effectively protects against inated Type I error rates and whether it produces unbiased estimates of correlation coefcients. We analyze simultaneously the effects of polytomies and misspecication of the model of bivariate character evolution because deviations from a Brownian motion model can have large adverse effects on the performance of independent contrasts (Martins and Garland, 1991; D ṍ az-Uriarte and Garland, 1996, 1998). We show that the procedure of Purvis and Garland effectively protects against inated Type I error rates and results in unbiased estimates.