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RPF and PHH jointly ran the simulations and wrote the paper.

The authors have declared that no competing interests exist.

Many phylogenetic comparative methods that are currently widely used in the scientific literature assume a Brownian motion model for trait evolution, but the suitability of that model is rarely tested, and a number of important factors might affect whether this model is appropriate or not. For instance, we might expect evolutionary change in adaptive radiations to be driven by the availability of ecological niches. Such evolution has been shown to produce patterns of change that are different from those modelled by the Brownian process. We applied two tests for the assumption of Brownian motion that generally have high power to reject data generated under non-Brownian niche-filling models for the evolution of traits in adaptive radiations. As a case study, we used these tests to explore the evolution of feeding adaptations in two radiations of warblers. In one case, the patterns revealed do not accord with Brownian motion but show characteristics expected under certain niche-filling models.

Two diagnostic tests reveal when traditional phylogenetic comparative methods relying on a Brownian motion model of trait evolution are not appropriate, and these can help identify underlying ecological mechanisms of adaptation.

Phylogenetic comparative methods are used extensively to explore patterns of correlated evolution of traits or to correct for statistical nonindependence amongst species that arises as a consequence of common ancestry [

Many comparative methods rely on the Brownian motion model of trait evolution [

Niche-filling models are likely to be of particular significance in the evolution of adaptive radiations. In adaptive radiations, speciation and trait evolution may be closely linked if ecological specialisation drives or promotes genetic divergence between populations [

Because of such ecologically constrained trait evolution, statistical comparative methods that assume constant rates of evolution (trait change) through time (e.g., as in Brownian motion) may be severely compromised [

The Brownian model makes a number of simple assumptions [

We tested for patterns of evolution that are inconsistent with Brownian motion. Here we present two simple tests that are generally effective in detecting non-Brownian trait evolution that is consistent with certain niche-filling modes of evolution. We applied these tests to simulated and real data, highlighting that deviations from Brownian models may be indicative of underlying ecological mechanisms.

The rationale of niche-filling models is that species occupy discrete niches, which determine the values of traits required to exploit them. In the models considered to date, niches remain constant through time. For instance, the morphology of beaks of different species within a group of birds may be selected to deal with just one of an array of seed types. If a given species of bird has evolved to feed on a particular seed that does not change in size or shape, then beak morphology remains constant through evolutionary time. Evolution only proceeds when a new niche becomes available. Thus, if a new form of seed appears that is not being exploited, and if there is selection for one of the bird species to feed on this new seed, then there may follow a short period of morphological evolution and perhaps eventual speciation in order to exploit the new resource.

In the simulations described below, niches are characterised by a unique optimum of either one or two characters, modelled as a normal distribution in the case of one character, or a bivariate-normal distribution with correlation coefficient

The model we analyse was originally suggested by Price [

This model has a number of characteristics. First, the correlation between traits is different from the correlation between trait changes, because niches arise within the niche space independently of the location of other occupied niches; different models vary the rules by which species invade the niches [

The conventional model of trait evolution is the Brownian (motion) model, the derivation of which is outlined in some detail elsewhere [

From the point of view of modelling trait evolution, the key features of the Brownian model are as follows: (i) traits evolve constantly, i.e., they do not become fixed at an optimum value for each species; (ii) the evolution of all species is independent, such that potentially several species may adopt similar trait values; (iii) the values of traits are effectively unbounded (i.e., the variance in trait values grows linearly with time of evolution).

We assume three contrasting modes of speciation for both niche-filling and Brownian models. In the niche-filling simulations, the rate of speciation is the rate at which new niches appear and are invaded, whilst in the Brownian model, the rate of speciation is the rate at which lineages split. The first model of speciation assumes that the instantaneous probability of a speciation event occurring is proportional to the number of species present. This corresponds to the familiar Yule process and generates an increasing net rate of speciation, with the intervals between the origin of new species declining through time. The second model assumes that the instantaneous probability of a speciation event is constant irrespective of the number of species present. In the third model, we assume that the instantaneous probability of a speciation event declines with the number of species present.

These three models correspond to different underlying processes that may accompany the invasion of an empty niche space. The Yule model assumes that all species are equally likely to speciate at any given time. On the other hand, a constant rate of speciation would occur when niches become available at a constant rate through time and are invaded by new species as soon as they appear. The declining rate of speciation may be thought of as modelling the initially rapid speciation that accompanies ecological diversification in the early stages of an adaptive radiation, which subsequently slows down as niches are filled and new niches become rarer.

In a true adaptive radiation not all of these scenarios are equally as likely [

The starting point is the method of contrasts [

First, beginning with a pair of adjacent tips (species _{i}_{j}_{ij}_{i}_{i}_{j}_{i}_{j}

Second, we assign

Third, to account for the statistical uncertainty involved in estimating _{k}_{k}

Fourth, the two tips are removed from the tree, leaving

The final node (i.e., the root) will have a zero contrast, by definition, but has a variance, which is the error in the ancestral state at the root, accumulated throughout the tree. Contrasts are calculated for each trait under study and are then correlated to statistically test for correlated evolution between characters.

The usual test of the adequacy of the Brownian model is to test for a significant correlation between the standardised contrasts and their expected standard deviations [

As a test of whether contrasts vary in a manner consistent with niche-filling models, here we correlated phylogenetic contrasts with the heights of the nodes at which they are generated [

In this section, we outline a simple randomisation test for whether data are consistent with Brownian motion. As noted above, the size of a standardised contrast should be independent of its position on the phylogeny, i.e., small and large values should be equally as likely to occur on any part of the tree. This is in contrast with the niche models, where the amount of change in phenotype between a parent and daughter species depends on the position of the parent species in niche space as well as the number of species present at that time. Therefore a further test of the adequacy of the Brownian model that we propose is to use the observed values of the standardised contrasts in order to generate random datasets and to test whether these have similar statistical properties to the data: under the Brownian model, randomly generated data should not differ significantly from the original data.

The procedure runs as follows: (i) using the tree and original data, the trait variance (and covariance if several traits are being studied) is calculated; (ii) standardised contrasts are estimated; (iii) the standardised contrasts are randomised on the tree (see below), with the trait variance and covariance estimated for each pseudoreplicate; and (iv) the distribution of the variances of the pseudoreplicates is examined to determine whether the distribution of the randomised data encompasses the observed values, and the probability of the data is determined. Specifically, the randomisation is then performed in the following way.

First, the trait value (_{0}) is set at the root node.

Second, of the _{i}

Third, this contrast is used to reconstruct character states. The two branches (_{j}_{k}, respectively. By definition, if evolution is Brownian, then
_{j} and _{k}_{k}_{k}:

The second and third steps are then repeated until all nodes have been assigned randomised values, including the tips. The key point about this analysis is that under the Brownian model, any of the values of the

We used 1,000 randomised datasets to estimate empirical confidence intervals for the trait variance (and covariance in the case of two traits) measured at the tips for each dataset studied. Under the Brownian model, the randomised distribution should include the observed data, whereas under non-Brownian models, the observed data may yield values different from the randomised values.

We report below the proportion of simulated trees for which the randomisation test rejects Brownian motion. When data were simulated according to Brownian motion, we found that this proportion of trees for which the tests rejected the Brownian model was the expected 5%. For simplicity of presentation, therefore, in the results below, the rejection rate of the Brownian model is represented by the dotted line at the level of

The tests are applied to the variance in traits that have evolved singly. The graphs show the proportion of significant results obtained from the test applied, which, in the case of the Brownian model, is the type I error rate of the test and is represented by the dashed line. The data points show the power of the tests to reject the Brownian model when data were generated according to the niche-filling model. The mode of speciation was varied as follows: black circle, decelerating net rate of speciation; gray circle, constant net rate of speciation; white circle, increasing net rate of speciation. (A) Standard diagnostic (correlation of absolute values of standardised contrasts with expected standard deviations). (B) Correlation of absolute values of standardised contrasts with node heights. (C) Randomisation test. In (C), the dashed line is the randomisation test, applied without using branch length information.

Correlation of node heights with absolute values of contrasts is a more reliable indicator of deviation from the Brownian model for single traits evolved under the niche-filling model (

The topology of phylogenies generated under the niche-filling model should be unaffected by the choice of model for speciation: it is only the distribution of branch lengths that is affected by this component of the model. Consequently, we examined the power of the randomisation test ignoring branch lengths information (i.e., with all branch lengths set to one). The result of doing this is shown by the dashed line in

We analysed the performance of the node height correlation and randomisation test for traits with a moderate (^{2} = 0.5) and strong correlation (^{2} = 0.9). Under speciation according to the Yule process, the performance of the node-height correlation was depressed for moderately correlated traits (

The tests are applied to data on the correlated evolution of two traits. The graphs show the proportion of significant results obtained from the test applied, which, in the case of the Brownian model, is the type I error rate of the test and is represented by the dashed line. The data points show the power of the tests to reject the Brownian model when data were generated according to the niche-filling model. The mode of speciation was varied as follows: black circles, decelerating net rate of speciation; gray circles, constant net rate of speciation; white circles, increasing net rate of speciation. (A and B) node height test; (C and D) randomisation test applied to data on trait variances. (E and F) randomisation test applied to data on trait co-variances. The strength of association between traits was set at ^{2} = 0.5 (A, C, and E) or ^{2} = 0.9 (B, D, and F) . In (C) and (F), the dashed line is the randomisation test, applied without using branch length information.

In the case of two characters, we used the randomisation approach to generate sampling distributions for both the variance in each trait as well as the covariance between traits. The power of the randomisation test to detect non-Brownian trait evolution through analyses of trait variance is affected by the model of speciation, especially in the case of the Yule process, again because the test is confounded under this model of speciation and trait evolution (

In general, it would be expected that the randomisation test would perform best when analysing traits that evolve closely together. In terms of statistical analyses (as opposed to distinguishing models of evolution), this is unlikely to be an issue, because it has been shown that comparative methods tend to be most compromised by the niche-filling model when the correlation between traits is high [

As in the case of single traits, by ignoring branch lengths, the compromise in the power of tests resulting from confounding effects of the branch-length distribution resulting from different models of speciation can be overcome. As shown in

In summary, the power of the two tests to detect non-Brownian evolution depends on the model of speciation as well as the correlation between traits. In both cases, by assuming a model of speciation that produced a distribution of branch lengths that exactly matched the pattern of accumulation of variance in traits, it was possible to confound the test and hence decrease power. However, by ignoring branch lengths and applying the randomisation approach, it is possible to deal with this problem.

Here we discuss two case studies from classic adaptive radiations. In both cases, independent morphological and phylogenetic information are available. The aim here is to show how the tests we have described may be applied to deduce the existence of non-Brownian modes of trait evolution.

We used the tests to analyse trait evolution in Old World Leaf warblers (

The histograms show the distribution of randomised values of variance in (A) body size and (B) prey size, as well as (C) randomised values of the covariance between body size and prey size, using the randomisation procedure described in the text. The observed values are indicated by arrows and in each case, lie significantly to the right of the randomised values indicating deviation from the Brownian model. (D) The relationship between contrast values (absolute) and node heights. The symbols show correlations for contrasts in body size (black circles;

We analysed data from a similar radiation of

The histograms show the distribution of randomised values of variance in (A) beak size and (B) prey size, as well as (C) randomised values of the covariance between beak size and prey size, using the randomisation procedure described in the text. The observed values are indicated by arrows and in each case lie near the centre of the distribution, revealing no evidence of deviation from the Brownian model. (D) The relationship between contrast values (absolute) and node heights. The symbols show correlations for contrasts in beak size (black circles;

It was previously shown that the model proposed by Price [

A range of parametric approaches exist for diagnosing phylogenetic dependence in comparative data [

The niche-filling model yields punctuated and time-varying rates of evolution; thus, appropriate diagnostics may yield important evidence of the existence of non-Brownian evolution. Harmon et al. [

Many adaptive radiations contain few species (e.g., Darwin's finches, 14 species;

The Brownian model of trait evolution is not incompatible with traits evolving in an adaptive manner nor subject to adaptive constraints. The key assumption is that trait change is random and the likelihood of change is unaffected by the state of a trait or by other species. This does not preclude the possibility that the Brownian model can model adaptive change in cases where species are evolving independently of each other. An important novel feature of niche-filling models is that as niche-space becomes filled, the amount of possible future change in traits tends to become less. It is this element of the evolution of traits that simple variants of the Brownian models are currently unable to capture.

It is important to emphasise that the analysis presented here is diagnostic analysis; further work is required on developing techniques for inferring deviations from Brownian motion under a range of ecological processes. Apart from niche-filling models [

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