Random trees and comparative method: A cautionary tale.
As a potential solution to this problem, Losos (1994) introduced the application of random trees into the analysis of comparative data. He argued that random trees can be used to determine whether the statistical conclusions from a given comparative study are sensitive to the universe of possible phylogenies for the taxon in question. According to this approach, for example, it would be safe to conclude that the results derived from a nonphylogenetic analysis are significant if every conceivable phylogeny for the taxonomic group also yielded significant results.
Martins (1996) recently expanded and further developed Losos's (1994) proposal into an explicit statistical method to conduct comparative studies when the phylogeny of a group of organisms is unknown. This method consists of using computer simulation techniques to generate a random subset of all possible trees with branch lengths in units of expected variance of character change. Interspecific data are then analyzed by a phylogenetically based comparative method to calculate the evolutionary statistic of interest, for example, the evolutionary correlation or regression coefficient, for each of the randomly generated trees. The mean of the resulting distribution is then used as an estimate of the true evolutionary statistic for these data. The confidence limits for this evolutionary statistic can be calculated and are analytically designed to make statistical conclusions based on this method more conservative by incorporating an added element of uncertainty due to unknown phylogenetic information. Therefore, Martins's (1996) method provides an estimate of the evolutionary correlation or regression coefficient and a conservative procedure for hypothesis testing.
Prior to the widespread application of this method to comparative data, however, several questions should be addressed. First, what do the evolutionary parameters derived from random trees really estimate, and how do they compare to those parameters estimated from nonphylogenetic analyses? How do both of these sets of evolutionary parameters compare to those estimated from real phylogenies? Second, how conservative is this method? Will it simply decrease the probability of committing Type I errors, which is the potential problem inherent in most nonphylogenetic analyses, or will this method be so conservative as to mask any interesting evolutionary patterns?
The objective of the present study is to address these questions using real data from the literature. I will compare the parameter estimates and statistical conclusions obtained from traditional cross-species (nonphylogenetic) analyses to those obtained from Felsenstein's (1985) independent contrasts technique when applied with a random subset of all possible trees. The results of these two types of analyses will then be compared to Felsenstein's (1985) independent contrasts technique when applied with real phylogenetic information.
As a basis for these comparisons, I use interspecific data drawn from a recent comparative study evaluating the validity and generality of Rensch's rule across 21 independent animal taxa (Abouheif and Fairbairn 1997 and references therein). Rensch's rule states that sexual size dimorphism (SSD) increases with body size in taxa where males are the larger sex and decreases with body size where females are larger (Rensch 1960; Fairbairn 1997). Abouheif and Fairbairn (1997) used the independent contrasts technique to quantify allometric trends within each of the 21 animal taxa. They found significant allometry for SSD in 38% of the taxa and significant allometry consistent with Rensch's rule in 33%. A meta-analysis of these results revealed that Rensch's rule is general and highly significant in 20 of the 21 taxonomic groups examined.
Felsenstein's (1985) independent contrasts technique applied with real phylogenetic information as well as the cross-species analyses for the 21 animal taxa are described in detail in Abouheif (1995) and Abouheif and Fairbairn (1997). To apply Martins's (1996) method to the same 21 datasets, I used the COMPARE package (vers. 1.0, 1995) by E. P. Martins. I generated all random trees using a Markovian model of speciation (for a general discussion of the strengths and weaknesses of this and other speciation models see Martins 1996). This branching process model assumes that all taxa are equally likely to divide at any moment in time and that each speciation event is independent of all others. I specified the branch lengths for the randomly generated Markovian trees in two different ways. The first employs what Martins (1996) calls a "time only" model. This model assumes a Markovian branching process in which the probabilities of speciation and extinction are constant and assumes that the relative timing of these events depends upon the amount of time that has passed since they last occurred. The second method of specifying branch lengths employs a "coalescent" model. This model also assumes a Markovian branching process, but the time of divergence of a lineage is determined by the total number of other lineages present at that instance in time. Thus, the coalescent model determines the relative timing of speciation and extinction events in a density-dependent fashion.
For each taxonomic group, I randomly generated 1000 "time only" and 1000 "coalescent" trees and estimated an allometric slope and correlation coefficient from independent contrasts for each of these randomly generated trees. I estimated the allometric slope by using Model II major axis regression through the origin, regressing the contrasts of log (female body size) against the contrasts of log (male body size) (Fairbairn and Preziosi 1994), and estimated the correlation coefficient by calculating the coefficient of correspondence (i.e., correlation through the origin) between the contrasts of log (female body size) and the contrasts of log (male body size) (Garland et al. 1992). I then used the mean allometric slope and correlation coefficient derived from the 1000 randomly generated "time only" trees and the 1000 randomly generated "coalescent" trees as estimates of the true evolutionary parameters for each taxa.
Felsenstein's (1985) independent contrasts method applied with real phylogenetic information will from this point on be abbreviated as "FIC-real." The independent contrasts technique applied with a random subset of all possible trees will be abbreviated as "FIC-randomT" when the random trees were generated according to a "time only" branching process model or "FIC-randomC" when the random trees were generated according to a "coalescent" branching process model. The nonphylogenetic (ahistorical) analyses will continue to be referred to as "cross-species" throughout the text. To compare the parameter estimates obtained from FIC-real, FIC-randomT, FIC-randomC, and cross-species, I used regression and correlation analysis.
I used two-tailed probabilities to assess the statistical significance of the correlation coefficient and of the allometric slope (i.e., a slope significantly different from 1.0) within each taxon (Abouheif and Fairbairn 1997). To compare the statistical conclusions of FIC-real, FIC-randomT, and FIC-randomC for each taxonomic group, I examined two methods at a time and tabulated the frequency of the following four possible outcomes: both statistical methods show significant correlations or patterns of allometry, both show no significant correlations or patterns of allometry, and one of the two methods shows significant correlations or patterns of allometry whereas the other does not. Statistical conclusions based on cross-species analyses within these 21 animal taxa have been previously shown to be subject to inflated Type I error rates (Abouheif 1995), and thus this result will be taken as a given because it is has also been generally demonstrated several times in previous simulation studies (Martins and Garland 1991; Diaz-Uriarte and Garland 1996).
Allometric slopes based on FIC-randomT and FIC-randomC are not good predictors of FIC-real slopes ([ILLUSTRATION FOR FIGURE 1A OMITTED], [r.sup.2] = 0.504; lB, [r.sup.2] = 0.472) but are accurate predictors of each other ([ILLUSTRATION FOR FIGURE 1C OMITTED]; [r.sup.2] = 0.982), which is reflected in the re-values of the three scatter plots. Furthermore, there are no significant systematic biases among the allometric slopes of these three methods (i.e., the slope of the Model II major axis regression for each scatter plot is not significantly different from 1.0). The results obtained from comparisons among correlation coefficients are almost identical [ILLUSTRATION FOR FIGURE 2 OMITTED]. FIC-randomT and FIC-randomC correlations are poor predictors of FIC-real correlations ([ILLUSTRATION FOR FIGURE 2A OMITTED], [r.sup.2] = 0.780; 2B, [r.sup.2] = 0.757) but accurate predictors of each other ([ILLUSTRATION FOR FIGURE 2C OMITTED], [r.sup.2] = 0.996). However, FIC-randomT or FIC-randomC correlations are significantly greater in magnitude than are FIC-real correlations.
Comparison of the results of cross-species and independent contrasts analyses indicate that cross-species slopes are poor predictors of FIC-real slopes ([ILLUSTRATION FOR FIGURE 3A OMITTED], [r.sup.2] = 0.421) and are not systematically biased in any direction (b = 1.432, 95% Cl = 0.831-2.810). There is a tight correlation between cross-species and FIC-randomT/FIC-randomC slopes ([ILLUSTRATION FOR FIGURE 3B OMITTED], [r.sup.2] = 0.967; 3C, [r.sup.2] = 0.958). Cross-species correlation coefficients, like allometric slopes, are also poor predictors of FIC-real correlations ([ILLUSTRATION FOR FIGURE 4A OMITTED]; [r.sup.2] = 0.658), and are also tightly correlated with FIC-randomT and FIC-randomC correlations ([ILLUSTRATION FOR FIGURE 4B OMITTED], [r.sup.2] = 0.970; 4C, [r.sup.2] = 0.992). Regression analyses in all three scatter plots, however, indicate that cross-species correlations are significantly greater in magnitude [ILLUSTRATION FOR FIGURES 4A-C OMITTED].
FIC-randomT and FIC-randomC analyses are unable to detect significant patterns of allometry, whereas FIC-real analyses can in 28% (FIC-real alone) and 38% (FIC-real together with the paired t-test) of the taxa examined (Table 1). A common observation consistent with these results is the unusually large degree of inflation of the 95% confidence intervals surrounding FIC-randomT or FIC-randomC slopes (data not shown). For example, waterstriders (one of the 21 independent animal taxa) show clearly significant patterns of allometry when based on cross-species or FIC-real analyses (b = 0.859, 95% CI = 0.805-0.916, power (1 - [Beta]) = 0.998; Abouheif 1995). However, when this taxon is analyzed using FIC-randomT or FIC-randomC, the confidence intervals become distressingly wide (b = 0.946, 95% CI = -12.6155-14.4632) making any significant patterns of allometry no longer detectable. These results suggest that the use of random trees in the analysis of comparative data may sometimes render the statistical conclusions much too conservative, thereby masking the existence of any broad-scale evolutionary patterns.
Conversely, all methods (even cross-species analyses) yielded significant positive correlations for all 21 animal taxa. Such strong agreement among all three methods is most likely caused by the inherently high genetic correlations between male and female size.
Incorporating random trees into the statistical assessment of Rensch's rule has revealed several important observations: (1) the parameter estimates of allometric slopes and correlation [TABULAR DATA FOR TABLE 1 OMITTED] coefficients obtained from a random subset of all possible trees are good predictors of those derived from a traditional cross-species analyses, but are poor predictors of those obtained from real phylogenetic hypotheses; (2) the magnitude of cross-species correlation coefficients are significantly larger than those obtained from the independent contrast technique applied with real and random phylogenies; and (3) the statistical conclusions derived from random trees failed to detect any significant patterns of allometry in any of the 21 independent taxonomic groups analyzed. These observations can now serve to address some of the questions posed at the outset of the paper.
Let us assume for a moment that the true historical relationship among a group of organisms is depicted by a star phylogeny and that phenotypic evolution of the characters in question follows a Brownian motion model of evolutionary change. Under these strict conditions, the parameters obtained from a traditional cross-species analysis, the independent contrasts technique applied with a random subset of all possible trees, and the independent contrast technique applied with real phylogenetic hypotheses (which in this case is a star) will all yield exactly the same estimate of the evolutionary regression or correlation coefficient (Pagel 1993; Purvis and Garland 1993). However, one cannot expect that these strict conditions will apply to real data, and, therefore, any observed differences between the parameter estimates obtained from the three methods above will be due to deviations from the assumption of a star phylogeny or a Brownian motion model of evolutionary change.
How then does one explain the first observation noted above, that is, that the evolutionary parameters derived from random trees are good predictors of those obtained from cross-species analyses and yet are poor predictors of those obtained from real phylogenetic hypotheses? I argue that, under the currently employed branching process models, the mean evolutionary parameter estimate derived from a random subset of all possible phylogenies is in fact equivalent to obtaining a parameter estimate from a star phylogeny. Each randomly generated tree possesses its own unique sequence of historical events, and thus, contains a unique "phylogenetic effect" on the parameters of interest. The mean parameter estimate from a random subset of all possible trees, however, averages out the unique phylogenetic effects contained within each of the random trees, rendering it equivalent to a parameter estimate obtained from a star phylogeny.
Figure 5 clearly demonstrates this using a hypothetical three-taxon tree. For three taxa, there exists only three possible ordered histories. When the independent contrasts technique is applied to each of them to estimate the evolutionary regression or correlation coefficient, the resulting estimates appear to be quite different from one another. However, the mean of these estimates averages out the phylogenetic effects contained within each of the trees and becomes almost identical to the estimates derived from the star phylogeny.
This observation and explanation seems to apply generally to other datasets, regardless of the number of taxa analyzed, branch lengths used, and the particular taxonomic group selected for study. For example, Donoghue and Ackerly (1996) obtained similar results when they performed sensitivity analyses on a 500 taxon tree of seed plants derived from rbcL sequences. They used the independent contrasts technique in conjunction with a computer simulated dataset to assess how estimates of the evolutionary correlation coefficient would change when calculated with all 7670 most-parsimonious trees derived from the phylogenetic analysis of the 500 rbcL sequences. All estimates from the 7670 most-parsimonious trees clustered tightly together. They then performed the same analyses using a random subset of all possible trees and found that the cluster of parameters obtained from random phylogenies centered around the estimate derived from a traditional cross-species analysis, which differed significantly from the cluster obtained from the maximum-parsimony trees. Thus, Martins's (1996) method and cross-species analyses both provide parameter estimates under the assumption of a star phylogeny and both will deviate from estimates obtained from real phylogenetic hypotheses to a degree that depends upon the hierarchical nature of the phylogeny in question.
The second observation, that is, that the correlation coefficients derived from cross-species analyses are significantly greater in magnitude than those derived from the independent contrasts technique applied with real and random phylogenies, may most likely be explained by deviation of the data from a Brownian motion model evolutionary change. This interpretation is consistent with the results of a recent simulation study conducted by Price (1997), who assessed the effect of using an alternative model of evolutionary change (i.e., an adaptive radiation) on the parameter estimates obtained from the independent contrasts technique. He found that, under a non-Brownian motion model of evolutionary change, the correlation coefficients derived from the cross-species analyses were generally greater in magnitude than those of the independent contrasts technique. In this context, comparing the parameter estimates derived from traditional cross-species analyses to those of the independent contrasts technique applied with real and random phylogenies can be biologically informative. Incorporating random trees into the analyses of interspecific data can potentially reveal the degree to which the data deviate from the assumption of a Brownian motion model of evolutionary change, as well as the degree to which the estimated parameters are influenced by the structure of the phylogeny in question (Kirsch and Lapointe 1997; Price 1997).
Martins's (1996) method was analytically designed to be conservative to account for the nonindependence of species data points. However, the statistical assessment of Rensch's rule using random trees clearly shows that this approach may in fact be too conservative. Such a high degree of conservatism is almost or equivalently as undesirable as committing Type I errors (claiming a relationship exists when one, in fact, does not; Sokal and Rohlf 1995). Thus, in the absence of any phylogenetic information, one is forced to make a difficult choice between the elevated risk of committing Type I errors associated with nonphylogenetic analyses and the elevated risk of committing Type II errors associated with using a random subset of all possible trees. Of course, this choice should be made on a case-by-case basis, taking into account the inherent strength of the statistical relationship between the traits under study.
Using Martins's (1996) method alone, for example, would have led one to claim that Rensch's rule did not exist in any of the taxonomic groups analyzed. However, given that phylogenies were available a priori, Abouheif and Fairbairn (1997) found that Rensch's rule is general and highly significant. Therefore, this suggests that the failure to detect evolutionary patterns using random trees should not be interpreted as a "result," but rather as "no result" until a real phylogenetic hypothesis for the taxon in question becomes available. However, if relationships are actually found to be statistically significant with this approach, such as the correlations between male and female size, then the results are likely to be robust to the comparative method used (Losos 1994).
As a final caveat, these results imply that the random trees generated by "time only" and "coalescent" branching process models may not be producing trees that coincide with those commonly found in the literature (for discussion of this issue see Savage 1983; Maddison and Slatkin 1991; Losos 1994; Losos and Adler 1995; Martins 1996), and as a consequence, may introduce unknown biases into Martins's (1996) method. Furthermore, the branch lengths created under these branching process models may violate the assumptions of the independent contrasts technique (Losos 1994). Because Felsenstein's (1985) method assumes that characters have evolved according to a Brownian motion model of evolutionary change, the absolute magnitude of each contrast score is a function of the branch lengths separating any two taxa. Thus, each score must be standardized so that all contrasts are drawn from a distribution with the same mean and variance. However, as Losos (1994) argues, if the branching process models used to generate the random trees are not coinciding with the true evolutionary history of the taxon in question, then the characters will not conform to the expectations of a Brownian motion model of evolutionary change and, as a consequence, the contrast scores will not be adequately standardized. This problem is important, as it can potentially increase the risk of committing both Type I or Type II errors in an independent contrasts analysis depending on the sample of random trees used. These situations may be avoided and corrected statistically by applying the diagnostics and branch length transformations proposed by Garland et al. (1992; e.g., see Losos 1994). It is important to note, however, that applying such procedures will change the way in which random trees are sampled, as well as the underlying evolutionary assumptions used to create them.
To conclude, several interesting questions remain with regard to assessing the potential strengths and weaknesses of incorporating random trees into the analysis of interspecific data. A particularly important issue will be to determine the performance of this approach when applied with other phylogenetically based comparative methods, as well as when it is applied to cases in which partial phylogenetic information for a taxonomic group is available (e.g., see Stamps et al. 1997). Moreover, assessing the effects of random tree generation on the analysis of comparative data clearly warrants future investigation. Detailed studies evaluating and addressing the limitations of phylogenetic comparative methods are becoming increasingly important (Diaz-Uriarte and Garland 1996; Donoghue and Ackerly 1996; Martins 1996; Price 1997) as biologists are realizing the potential of these methods for revealing the patterns and processes underlying general evolutionary phenomena.
I thank D. Fairbairn, D. Adams, F. Rohlf, T. Garland Jr., J. Losos, P. Inchausti, and M. Rosenberg for helpful advice and discussion as well as for providing insightful comments on earlier versions of this manuscript. Reviews by E. Martins, D. Ackerly, and two anonymous reviewers have greatly improved the overall quality of this paper. This work was supported by a graduate fellowship from the Fonds pour la Formation de Chercheurs et l'Aides a ia Recherche (FCAR).
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