Contributions to Zoology, 68 (1) 3-18 (1998)Arne Ø. Mooers; Dolph Schluter: Fitting macroevolutionary models to phylogenies: an example using vertebrate body sizes

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Discussion

Choice of scenarios and models

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The macroevolutionary scenarios presented here are but a sample of those conceivable. DíazUriarte & Garland (1996) simulated comparative data using fifteen different scenarios; not all are tractable. One entire class of theirs which cannot be tested in our framework is a random model with a trend: with only end points, it would be impossible to estimate the rate at which the mean value for a trait changed through time. Another class presented by DíazUriarte & Garland (1996) allowed for random motion within preset boundaries. Boundaries on trait evolution are intuitively appealing (trait evolution must certainly be bounded, certainly on a multiplicative scale) and easy to simulate, but difficult to model analytically: estimating boundaries from data would require arbitrary decisions. In DíazUriarte & Garland’s simulation study, the boundaries were set arbitrarily to maximize differences between this class and others.

A third class of scenario and one which warrants more attention is true punctuated evolution (sensu Eldredge & Gould, 1972): given a model of peripatric speciation, change is restricted to the peripheral isolate, and is uncorrelated with time. Graphically, at every node in the tree, one daughter lineage would be represented with a branch of some length, and the other would have its branch length set to zero. There is no reason that all peripheral isolates should undergo the same expected amount of change (Grafen & Ridley, 1997), but this could serve as a reasonable first approximation. DíazUriarte & Garland (1996) state that this scenario assumes no extinction: this can be relaxed if we assume that extinction is random across lineages, in which case extinction should obscure but not destroy the pattern. This holds for the speciational scenario tested here as well. However, if extinction and speciation rates are positively correlated (Rosenzweig, 1995), then the speciational hypothesis should fare well (Ferris et al., 1979). If there is a bias in future extinction and speciation probabilities between ancestral species and new peripheral isolates (cf. Losos & Adler, 1995, then the speciational hypothesis will be hampered. As with random motion within boundaries, true punctuated evolution is easy to simulate, but difficult to model analytically: it is akin to the free scenario, where the data are used to estimate the branch lengths, but with peculiar and discrete constraints: branches can only be of zero or unit length, and one of each must occur at each node.

Lynch (1990)suggested that change might be rapid following a speciation event, and subsequently slow down, perhaps at an exponential rate. At the limit, this scenario approaches the speciational hypothesis, but could be better fit with the gradual hypothesis and an extra parameter governing the rate at which change slows down after the speciation event. Such a transformation of branch lengths is akin to the power functions advocated by students of the comparative method (Grafen, 1989; Gittleman & Kot, 1990; Garland, 1992; Pagel, 1994) for standardizing phylogenetic independent contrasts. This scenario, and others mentioned above, require the fitting of extra parameters, necessitating loglikelihood ratio tests or MonteCarlo tests (Goldman, 1993) in order to distinguish between them and simpler hypotheses. More work is required to offer guidelines for assessing when data warrant more complex models.

The Brownian motion process is also only one of several possible models of trait evolution. When originally proposed as a model for estimating phylogenies from quantitative character data (Felsenstein, 1981; see Felsenstein, 1988, for a review), recourse was made to genetic drift as the variancegenerating mechanism. For traits under natural selection, Brownian motion was deemed “rather arbitrary” (Felsenstein, 1988: 464). Brownian motion can, however, represent change in traits under selection if the selection pressures are multifarious and constantly changing, or if lineages wander randomly from one regularlyspaced adaptive peak to another, both of which may be reasonable representations over long periods of time. Felsenstein (1985) advocated Brownian motion as a possible model with which to investigate correlated evolution of characters under selection (the main class of characters investigated with the comparative method), and this has become the model of choice in this context. In simulations, the model performs well as an approximation even when the true model is quite different (Martins & Garland, 1991; DíazUriarte & Garland, 1996).

Hansen & Martins (1996; Martins & Hansen, 1997; Martins, 1994; Hansen, 1997; see also Garland et al., 1993; Felsenstein, 1988) have advocated the OrnsteinUhlenbeck process of character change (Lande, 1976), whereby species are randomly perturbed from, and then return towards, some optimum value, with the rate of return increasing with greater perturbance. This model is offered as a theoretical justification for boundaries on trait evolution and so might be seen as a refinement on.Brownian motion. It is more complex than simple random drift, and requires fitting several extra parameters, regardless of the hypothesis tested, which might not be warranted with small data sets. It is hampered by the assumption that all the species in the clade are centered on the same optimum value, which must also be specified or estimated from the data. It also implies that there is no way to recover distantly past events, as evidence of ancient phenotypes will eventually be erased by the pull toward the central point (Felsenstein, 1988). However, the model is grounded in population genetics theory, and links microevolutionary process with macroevolutionary pattern and so may prove to be a valuable and versatile approach. Martins (1994) and Hansen & Martins (1996) discuss ways in which the fit of the OU model can be compared with strict Brownian motion.

Crucially, our formal approach assumes that the rate of change for the character(s) under question is the same throughout the tree. This will not generally be the case (e.g., changes in discrete characters often appear to be clumped in trees (Grafen & Ridley, 1997)) for large, disparate clades, but might be less of a problem for more homogeneous clades like those considered here. This will add noise rather than bias when comparing among the simple hypotheses, however, and the assumption is partially tested by comparison of these simple hypotheses with the free scenario. The test is not perfect, however: where the assumption of equal rates make the simple scenarios underparameterized, assuming a new rate of change for every branch in the free scenario makes it overparameterized.

Our phylogenetic, modelbased approach to macroevolution has antecedants. Ferris et al. (1979) offered a test for comparing the relative importance of speciation events and time on the probability of gene function loss within the catostomid fishes, a discrete character. Their approach was based on the Poisson process (they assumed irreversibility of gene function loss) and they contrasted gradual and speciational models that differed in the number of estimated parameters, necessitating loglikelihood ratio tests. M. Pagel has presented methods for testing the speciational model for discrete (Pagel, 1994) and continuous (Pagel, 1997) characters in a maximum likelihood context. Like Ferris et al., Pagel’s approach compares models with different numbers of parameters and considers relative fits via a loglikelihood ratio test.

Garland (1992) offers a graphical method for investigating the relationship between differences between sister groups in trait values and their expected variances. This approach, presented in the context of correlated evolution, could be used to distinguish between the speciational and gradual models of evolution. Finally, there have been a number of tests of strict punctuated evolution (reviewed by Gould & Eldredge, 1993), using both paleontological and recent data, some of which make explicit use of trees.

Bodysize evolution

Overall, the model which did not incorporate phylogeny (the pitchfork model) offered the best fit to the specieslevel trees. The difference is not significant, but this may suggest that phylogeny may not be an important factor in body size evolution, at least at the species level. Body size may evolve idiosyncratically with respect to time and speciation. It is known that body size can evolve very rapidly (Brown, 1995).

Overall, differences in fits among the models were small (though for the Ursidae, we might accept the speciational model as significantly the best of the lot). The Brownian motion process is inherently noisy, and so each model might accommodate quite a range of data when the trees are small, such as is the case here (most trees have 6 or fewer tips). Under a Brownian motion process, the test may have low disciminatory power for small trees.

To explore this point, we analysed a data set with a wellestablished prior expectation. We took the second axis of a principle component analysis of shape (being mostly sizeindependent beak measures) for members of Geospiza and Zonotrichia (data available on request) and subjected these measures to the same four macroevolutionary models. Previous work (Schluter & Nagel, 1995) has implicated changes in beak morphology with speciation within Geospiza, and so we predicted that the speciational model should best fit the shape data within this clade, with the pitchfork model doing the worst. We had no such expectation for.the sparrows. The results were in accordance with these predictions – the speciational model offered the superior fit in Geospiza, performing 125 times better than the pitchfork scenario. Recall that for body size, the pitchfork model offered the best fit in this clade (Table I). Conversely, for Zonotrichia the pitchfork model offered the best fit, with a 14 fold difference between it and the worst, gradual model. So for Geospiza, body shape does carry a particular phylogenetic signal, consistent with the idea that changes in shape occur in concert with speciation events.

Another factor which might affect our results is tree size. The trees sampled here are so small that single errors (one nonrandom extinction event or one misleading branch length) may have a large effect. This is particularly devastating for the speciational model. Furthermore, if there is phylogenetic signal in bodysize differences among species, the relative success of the pitchfork model for the younger trees may be because those trees are the most misinformative, such that the topology is presenting noise rather than signal. Finally, the gradual model must be viewed with some caution for many of the trees listed here because if the rates of change of the molecules deviate strongly from the molecular clock expectation, then the branch lengths may be poor estimates of elapsed time. Reconstructing the trees without assuming constant rates of evolution did not however improve the fit over the speciational or pitchfork models (unpubl. results).

For both groups of bird families, the speciational model of macroevolution offered a better fit to the bodysize data than did the other models. The phylogeny used has come under some criticism (see Mooers & Cotgreave, 1994), and it is likely that the branch lengths taken from this tree are not accurate. This handicaps the gradual model. However, the difference in relative fit of the pitchfork model between the two clades suggests that there may be differences in the pattern of bodysize evolution in the two groups – incorporating phylogeny causes a much improved fit for the Ciconiiformes versus for the Passeriformes. This is illustrated in Fig. 2, which contrasts the free trees for the two groups. For the Ciconiiformes (Fig. 2A), long branches tend to emanate from long branches (particularly the path leading to the Procellaridae), illustrating that there is a phylogenetic component to the evolution of body size. For the Passeriformes (Fig. 2B), the free tree looks distinctly pitchforklike.

This difference may be explained in several ways. The songbird tree may simply be less accurate, so that the topology is presenting so much noise that a pitchfork phylogeny does well in comparison. In the UPGMA tree of the birds, shorter branches are considered less reliable (Sibley & Ahlquist, 1990; Barraclough et al., 1995). However, the average internode length does not differ greatly between the two clades (mean abovefamily branch lengths: Ciconiiformes=1.3 ΔT50 H units, Passeriformes=1.1 ΔT50 H units; p=0.14 based on a ttest of logarithmically transformed data). A more interesting hypothesis is that changes in body size and diversification are more closely linked in the Ciconiiformes than in the songbirds, such that models which do not incorporate this (the pitchfork model) perform poorly. This has intuitive appeal, as there is more variation in body size among families in the former clade (standard deviation of weights: Ciconiiformes =0.5; Passeriformes=0.3). In addition, the free model offered a relatively better fit to the Ciconiiformes (Table II). Both of these groups are much older than the average of the smaller specieslevel vertebrate trees, where the speciational model performed best for the older clades. This trend, if true, suggests an attenuation in the rate of change through time, and is supported by the work of Lynch (1990).

Fig. 2 also illustrates another use of this explicit modelbased approach. Long branches under the free model are lineages where much change has occurred (either because there has been much elapsed time or high rates of change). A simple regression of the free branch lengths on the actual (time elapsed) branch lengths can help to identify those lineages of the tree where rates of change have been particularly high or low, directly analogous to looking for outliers in plots of standardized contrasts in comparative analyses (Garland, 1992; see also Martins, 1994). The difference is that our method focusses on single branches rather than differences between reconstructed sister groups. These free model trees offer a graphical representation of tempo and mode in morphological macroevolution.

FIG2

Fig. 2A. The free model for the Ciconiiformes (based on Sibley & Ahlquist, 1990), where the tree represents the set of branch lengths which best explain the bodysize data. The tree retains some of the original topology.

FIG2

Fig. 2B. The free model for the Passeriformes. The tree resembles a pitchfork phylogeny. The tips are the lineages that arise at 10ΔT50 H units; tips with hyphenated names comprise all those families subtending that branch (see Mooers et al., 1994 for full explanation).

The results of this study have implications for comparative analyses. Many comparative methods rely on an a priori scenario of trait evolution (see Felsenstein, 1985; Harvey & Pagel, 1991; Martins, 1996). Common algorithms (e.g. CAIC, Purvis & Rambaut, 1995; the phylogenetic regression, Grafen, 1989) stipulate that the branch lengths on phylogenies express the expected amount of morphological change, so that they can be used to standardize comparisons. The branch lengths often represent time (e.g., see Berrigan et al., 1993) or are made to represent a speciational model (e.g., see Huey & Bennett, 1987; Bell & Mooers, 1997). Simulation studies have shown that methods work best in those situations that best meet the assumptions (Martins & Garland, 1991; DíazUriarte & Garland, 1996). As a guide to comparative biology, the results from both our data sets do not suggest that any mode of character change should be used a priori in the absence of other information. We therefore concur with Garland and others (Garland, 1992; DíazUriarte & Garland, 1996) that data sets should be explored on a casebycase basis. This same caution extends to those interested in using the various models when studying the relationship between character evolution and phylogenetic tree reconstruction (e.g. Rohlf et al., 1990; Heijerman, 1992, 1993; Mooers et al., 1995). No single model should be preferred a priori. However, the approach presented here can be used to decide on the appropriate model – the model which best fits the data would be that preferred in a subsequent comparative analysis.