Paleobiology

Published by: The Paleontological Society

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Paleobiology 32(4):578-601. 2006
doi: 10.1666/05070.1

Fitting and comparing models of phyletic evolution: random walks and beyond

Gene Hunt

Gene Hunt.Department of Paleobiology, National Museum of Natural History, Smithsonian Institution, Washington, D.C. 20013-7012.

Abstract

For almost 30 years, paleontologists have analyzed evolutionary sequences in terms of simple null models, most commonly random walks. Despite this long history, there has been little discussion of how model parameters may be estimated from real paleontological data. In this paper, I outline a likelihood-based framework for fitting and comparing models of phyletic evolution. Because of its usefulness and historical importance, I focus on a general form of the random walk model. The long-term dynamics of this model depend on just two parameters: the mean (μstep) and variance (σ2step) of the distribution of evolutionary transitions (or “steps”). The value of μstep determines the directionality of a sequence, and σ2step governs its volatility. Simulations show that these two parameters can be inferred reliably from paleontological data regardless of how completely the evolving lineage is sampled.

In addition to random walk models, suitable modification of the likelihood function permits consideration of a wide range of alternative evolutionary models. Candidate evolutionary models may be compared on equal footing using information statistics such as the Akaike Information Criterion (AIC). Two extensions to this method are developed: modeling stasis as an evolutionary mode, and assessing the homogeneity of dynamics across multiple evolutionary sequences. Within this framework, I reanalyze two well-known published data sets: tooth measurements from the Eocene mammal Cantius, and shell shape in the planktonic foraminifera Contusotruncana. These analyses support previous interpretations about evolutionary mode in size and shape variables in Cantius, and confirm the significantly directional nature of shell shape evolution in Contusotruncana. In addition, this model-fitting approach leads to a further insight about the geographic structure of evolutionary change in this foraminiferan lineage.

Accepted: June 7, 2006



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Appendix


Parameter Estimates for the General Random Walk Model

For the special case in which samples are evenly spaced in time and sampling error for trait means is the same for all samples, simple equations can be derived for the maximum-likelihood estimates for the parameters of the general random walk, μstep and σ2step.

In order to simplify notation, let x denote an evolutionary transition in trait X (x = XD − XA). If all samples have the same phenotypic variance (Vp) and sample size (n), the sampling variance for each transition will be equal to ε = 2Vp/n. Making these substitutions into equation (6) yields the log-likelihood of a single evolutionary transition, x

Summing equation (A.1) over a vector (x) of N evolutionary transitions yields
Taking the constant and the next term outside the summation,
then expanding the remaining summation and substituting the sum of all xi as the mean x times N yields
Parameter estimates for μstep and σ2step are obtained by setting the partial derivatives of equation (A.2) equal to zero and solving. The resulting parameter estimates are
Within the parentheses, the first two terms constitute the maximum-likelihood estimate of the variance of x. As a result, equation (A.4) can also be written as
Thus, the sampling error associated with estimating trait means is subtracted from the observed variance in evolutionary steps in order to estimate σ2step. Note that maximum likelihood estimate of variance from equation (A.4) is downwardly biased, accounting for the (N − 1)/N bias observed in the estimating σ2step (Lynch and Walsh 1998, p. 810).



Parameter Estimates for the Stasis Model

Assuming constant sampling error, we can derive simple equations for maximum likelihood estimators of the parameters of the stasis model: θ, the trait optimum, and ω, the variance around this optimum. Again, let X denote a trait value, and x refer to an evolutionary transition in this trait (x = XD − XA). As explained in the text, the expected evolutionary transition (x) is a function of the ancestral trait value, such that the mean step is θ − XA. The step variance is equal to ω + εX, where εX is the sampling variance of the descendant population (the sampling error of XA does not contribute because we are conditioning on the observed ancestral trait value). Assuming that population means are normally distributed around θ, the log-likelihood of x is obtained by substituting the appropriate mean and variance into equation (4):

Because x + XA is equal to XD, the numerator of the right-most term becomes (XD − θ)2. From this point, the steps closely parallel the preceeding section and are not shown. Briefly, equation (A.6) is summed over all evolutionary transitions, and the partial derivatives of the resulting log-likelihood function are set to zero and solved. The resulting estimators are
These estimators are quite sensible: the optimum is estimated simply as the mean of all descendant trait values, and evolutionary variance around this optimum is estimated as the variance of descendant trait values with the contribution of sampling error removed.

Figure 1. Three example step distributions (top) used to generate corresponding evolutionary sequences of 100 steps (bottom). When the mean of the step distribution is zero, increases and decreases are equally likely and the overall dynamics are nondirectional (A, C). Step distribution B has a positive mean, and therefore will tend to produce positively trended evolutionary sequences. With increasing step variance, evolutionary sequences are more volatile, with larger positive and negative excursions (compare C with A)

Figure 2. Mean trait in an evolving lineage (black line), sampled at five points in time (gray open circles). Close-up shows trait evolution between the last two sampled populations. The true trait difference between these two samples is equal to the sum of all evolutionary steps (si) separating the sampled populations. Because of sampling error, the estimated trait means (+) will differ from the true means by an error term (e). The observed difference between two populations includes both evolutionary differences (Σsi) and sampling error (e)

Figure 3. The sampling distribution of μstep when estimated according to the maximum-likelihood method outlined in the text. Shown are the results for five different values of μstep, corresponding to sequences (n = 20 evolutionary transitions) that are strongly directional (−0.1, +0.1), weakly directional (−0.01, +0.01) and nondirectional (μstep = 0). Dotted lines shows the mean of the 1000 replicates for each value of μstep. In all cases the mean estimated μstep is very close to the true generating value, indicating that the estimation procedure is unbiased

Figure 4. The sampling distribution of σ2step when estimated according to the maximum-likelihood method outlined in the text. Shown are the results for five different values of σ2step, increasing in magnitude from the top (σ2step = 0.001) to the bottom panel (σ2step = 10). Dotted lines shows the mean of the 1000 replicates for each value of μstep (each sequence consisted of n = 20 evolutionary transitions). Means of the σ2step estimates tend to be very close to, but slightly less than, the true generating value, indicating a slight bias (see text for details)

Figure 5. Boxplots showing sampling distribution of μstep (left) and σ2step (right) when estimated from evolutionary sequences of varying levels of completeness. In all cases, the true sequence had 10,000 steps; of these, 0.1% (10 steps), 1% (100 steps), 10% (1000 steps), and 100% (10,000 steps) were sampled at random and used to estimate μstep and σ2step. True values of both μstep and σ2step were 0.1 for all simulations (dotted lines). Boxes indicate the middle two quartiles of the estimates, with the median indicated by a vertical bar, and the total range by the horizontal lines extending from the boxes

Figure 6. Analysis of tooth measurements in Cantius. A, Evolutionary sequences of a size-related trait (M1 length) and a shape trait (M1 L/W ratio). Dots indicate population means, with approximate 95% confidence intervals. Sequences were standardized by within-sample variance and shifted so the first sampled point has a mean of zero, as described in the text. Time scale is in Myr counting forward from the first sample. B, C, Log-likelihood surface for estimates of the parameters (μstep and σ2step) of the general random walk model. B, M1 length. C, M1 L/W. Cross (+) indicates position of the maximum-likelihood estimate, and thin contours indicate the decrease in log-likelihood from this optimuum. The thick contour outlines the 95% joint confidence region. Solutions corresponding to an unbiased random walk are indicate by the gray dotted line at μstep = 0

Figure 7. Evolutionary sequences in shell conicity for the foraminifera Contusotrucana at two sites: DSDP 384 (North Atlantic, black crosses), and DSDP 525 (South Atlantic, gray open circles). Note the higher volatility around the directional trend at site DSDP 525

table

Table 1. Reanalysis of Cantius data consisting of four size-related (lengths and widths of two molars) and nine shape-related measurements (length-to-width ratios, the X and Y shape coordinates for three cusps, and the hypocone angle). Shown are the number of samples in each sequence (N), the mean number of individuals measured per sample (n), followed by the maximum-likelihood parameter estimates for the general random walk and stasis models. AICC values and Akaike weights are given for three models: GRW (general random walk), URW (unbiased random walk), and stasis. Akaike weights for models with more than minimal support (>0.05) are in bold. Trait sequences were transformed prior to analysis by within-sample variation, converting the parameter estimates to a common scale (see text)

table

Table 2. The evolution of shell conicity in Contusotruncana at two sites (DSDP 384, North Atlantic Ocean; DSDP 525, South Atlantic Ocean). Each row corresponds to a model fit to the two evolutionary sequences. Models either allowed for directional evolution (GRW, general random walk), or not (URW, unbiased random walk). In addition, the models differed in terms of the homogeneity of dynamics across the two sites. In columns 3 and 4, “same” indicates that the parameter in question (μstep or σ2step) was constrained to be equal at the two sites, and “diff” means that the parameter was different, i.e., estimated separately at each site. K indicates the number of parameters in the model, and ℓ is log-likelihood. Parameter estimates are listed for the two models that provide reasonably good fits to the observed data (as indicated by bold Akaike weights). Subscripts for parameters indicate locality names

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