In From Evolution to Geobiology: Research Questions Driving Paleontology at the Start of a New Century, Paleontolog- ical Society Short Course, October 4, 2008. Paleontological Society Papers, Volume 14, Patricia H. Kelley and Richard K. Bambach. (Eds.). Copyright ? 2008 The Paleontological Society. INTRODUCTION Of the many lessons that can be learned from the protracted Punctuated Equilibrium debates, two seem particularly lasting. First, paleontology is often data- limited. The debate initially focused much attention on the rather few cases that met the minimum require- ments to assess evolutionary mode within fossil lin- eages: numerous samples, good age control, and care- fully measured morphology. Controversy motivated many more studies documenting evolutionary changes in species-level lineages, and we now have a pool of relevant studies that is much larger (Levinton, 2001; Gould, 2002; Hunt, 2007), but still unfortunately thin for many taxa and depositional environments. A second lesson, obvious even in the midst of the controversy, is that different scientists can draw radi- cally incompatible conclusions from the same set of observations (Gould and Eldredge, 1977; Gingerich, 1985). Indeed, a subtext to much of the disagreement about fossils was the interplay between theory and ob- servation, and how the former frames perception of the latter (Eldredge and Gould, 1972; Fortey, 1988). It was recognized early on that battling verbal descrip- tions would not resolve these confl icts, and in response a variety of statistical methods were developed (Raup, 1977; Bookstein, 1987; Bookstein, 1988; Gingerich, 1993; Roopnarine et al., 1999; Roopnarine, 2001). These methods employed as a null model a random walk, which is a simple model in which trait increases and decreases are equiprobable. The effect of these developments was to inject much-needed quantitative rigor into discussions that were previously focused mostly on visual impressions. As useful as these approaches are, there are some limitations inherent to treating the random walk as a null model. Simulation studies showed that these tests can have very low statistical power, and therefore fail- ing to reject the null of a random walk?by far the most common outcome of these analyses?provides little information about the nature of evolutionary changes (Roopnarine et al., 1999; Sheets and Mitch- Abstract?Patterns of phenotypic change documented in the fossil record offer the only direct view scientists have of evolutionary transitions arrayed over signifi cant durations of time. What lessons should be drawn from these data, however, has proven to be rather contentious. Although we as paleontologists have made great progress in documenting the geological record of phenotypic evolution with greater thoroughness and sophisti- cation, these successes have been limited by the use of verbal models of how phenotypes change. Descriptive terms such as ?gradual? have been understood differently by different authors, and this has led to completely incompatible summary statements about the fossil record of morphological evolution. Here I argue that the solution to this ambiguity lies in insisting that different evolutionary interpretations be represented as explicit, statistical models of evolution. With such an approach, the powerful machinery of likelihood-based inference can be help resolve long-standing paleontological questions. Here I fi rst review this approach and some aspects of its implementation. Then, I show how this approach leads to new traction on important issues in evolutionary paleobiology, including: understanding modes of evolution and determining their relative importance, separating evolutionary mode from tempo, assessing the evidence for hypotheses of punctuated change, and detecting adaptive evolution in the fossil record. EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES: MODEL-BASED ASSESSMENT OF MODES, RATES, PUNCTUATIONS AND PROCESS GENE HUNT Department of Paleobiology National Museum of Natural History, Smithsonian Institution NHB, MRC 121, P.O. Box 37012 Washington DC 20013-7012 118 GENE HUNT ell, 2001; Hannisdal, 2006). Moreover, what was most urgently needed was the ability to measure statistical support for plausible alternative interpretations, but this is not easily implemented in the null hypothesis- testing framework. In this paper, I advocate a particular approach for guiding the interpretation of evolutionary change in fossil lineages. The key to this approach is that it in- sists that all candidate evolutionary explanations be represented as concrete statistical models, which are then evaluated using the standard machinery of like- lihood-based inference. A great many practical and theoretical benefi ts follow from this simple starting point. The structure of this argument will be as fol- lows. First, I start with an overview of this analytical framework and then address a few aspects of its im- plementation. Then, I will attempt to demonstrate the value of this analytical approach by using it to explore several outstanding issues in evolutionary paleobiol- ogy. In the fi nal section, I briefl y consider some of the most promising avenues for future work. FROM VERBAL TO STATISTICAL MODELS Since Raup?s pioneering papers (Raup, 1977; Raup and Crick, 1981), scientists examining evolu- tion within fossil lineages have mostly considered three canonical modes of evolution: directional evo- lution, random walks and stasis. Of these, however, only the random walk was defi ned explicitly. Support for the other modes was discerned from long-term evolutionary divergences that were either too great or too limited to be explained by a random walk. These departures from a random walk were attributed to di- rectionality and stasis, respectively (Raup and Crick, 1981; Bookstein, 1987; Gingerich, 1993; Roopnarine, 2001), but these two modes were never fi t and com- pared in their own right. Over the past few years, I have developed a statis- tical framework with which these and other kinds of evolutionary dynamics can be represented as statisti- cal models, each of which can be fi t to real paleon- tological data and compared to one another on equal footing (Hunt, 2006). The fi rst step in this procedure is to represent modes of change as fully statistical mod- els. It is most convenient to start with a general ver- sion of the random walk (the general random walk of Hunt, 2006). This model occurs in discrete time incre- ments, during each of which an evolutionary change is drawn at random from a distribution of evolutionary transitions or ?steps.? The long-term dynamics of this model can be shown to depend only on the mean (? s ) and variance (?2 s ) of this step distribution. The former determines the directionality of the sequence; on aver- age, positive ? s values generate increasing trends, and negative values generate decreasing trends. Increasing the step variance results in a greater range in the evo- lutionary increments and correspondingly more vola- tile evolutionary changes (Hunt, 2006). This model provides the basis for understanding both directional evolution and random walks. Direc- tional evolution results whenever the mean of the step distribution is not zero, although subtle trends can be obscured by the variability in evolutionary steps. Random walks are a special case of this more general model for sequences lacking directionality (? s = 0). The paleontological literature refl ects some inconsis- tency about what to call these two models. The model referred to here as directional evolution has elsewhere been termed a biased, directional or general random walk. Of these, the term ?directional random walk? is perhaps the clearest since it includes this model?s most salient feature (directionality) while indicat- ing its underlying similarity to random walks (there are other approaches for modeling directionality that are not random walk-like, e.g., Sheets and Mitchell, 2001). The term random walk is standard, although it is sometimes modifi ed as unbiased or symmetric to indicate its lack of directionality. It is also worth not- ing that in the phylogenetic methods literature, these models are usually referenced by their continuous time approximations: Brownian motion (or diffusion) for the unbiased random walk, and Brownian motion with a trend for directional evolution. This diversity of terminology has unfortunately obscured the concep- tual commonality of these models across and within disciplines. Several different approaches have been used to model stasis (e.g., Roopnarine, 2001). I follow Sheets and Mitchell (2001) in modeling stasis as normally distributed variation (with variance ?) around a stable phenotypic optimum (?). This model is simple and analytically tractable, and it produces trait trajecto- ries that conform well to recent qualitative accounts of stasis (Gould, 2002; Eldredge et al., 2005). Stasis 119EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES and random walks, while both lacking directionality, differ importantly in that total amount of evolutionary change does not increase with time under stasis. Ran- dom walks, by contrast, show increasing evolution- ary divergences over time; they drift in morphospace, whereas lineages experiencing stasis fl uctuate around a fi xed point. The above information allows the forward simula- tion of evolutionary sequences, but is not suffi cient for the inverse problem of fi tting the models, which re- quires an understanding of what these models predict for real evolutionary divergences. At present, there are two available parameterizations for these models, each of which employs paleontological data somewhat dif- ferently. The fi rst considers morphological differences in each ancestor ? descendant (AD) pair of popula- tions as separate observations (Hunt, 2006), while the second simultaneously weighs the joint distribution of trait values across all sampled populations (Hunt et al., 2008). The statistical models described above make predictions as to how these different kinds of data should be distributed. Considering separate AD differ- ences (the fi rst parameterization), these are expected to be normally distributed with means and variances that are functions of model parameters and elapsed time (Table 1). Considering all trait means jointly (the second parameterization), an entire sequence of trait means can be considered a single draw from a multi- variate normal distribution with a vector of means and covariance matrix that are also functions of the model parameters and the age model (Table 1; note that, in this parameterization, the directional and random walk models require an additional parameter, X 0 , equiva- lent to the intercept or root parameter in comparative methods, that represents the trait mean at the start of the sequence). In either case, the density function of the normal or multivariate normal distribution allows computation of the log-likelihood of observed data, and the best fi tting parameter set can be estimated by searching numerically through the space of possible parameter values and choosing those that maximize the likelihood of producing the observed data (Hunt, 2006; Hunt et al., 2008). While log-likelihoods provide a natural measure of how well data fi t models, they are not as well suited for choosing among models because log-likelihoods AD parameterization Joint Parameterization Mean AD difference Variance AD difference Joint Means Joint Covariance Matrix E[?X] = ?stAD Var[?X] =? s2tAD + ?A + ?D E[Xi] = X0 + ?sti Var[Xi] =? s2ti + ?i Cov[Xi,X j ] =? s2 ?min(ti, t j ) E[?X] = 0 Var[?X] =? s2tAD + ?A + ?D E[Xi] = X0 Var[Xi] =? s2ti + ?i Cov[Xi,X j ] =? s2 ?min(ti, t j ) E[?X] = ? ? XA Var[?X] =? + ?D E[Xi] = ? Var[Xi] =? + ?i Cov[Xi,X j ] = 0 Directional evolution Random walk Stasis Model Table 1?Mathematical basis for fi tting three evolutionary models (directional evolution, random walk, and sta- sis) to empirical paleontological sequences. Two different parameterizations of the problem are possible: one uses the phenotypic differences between ancestor ? descendant (AD) pairs of populations, the other considers the distribution of all sample means jointly (Joint). For all models, the expected difference between ancestor and descendant is normally distributed with means and variances that are functions of model parameters and elapsed time. Under the joint approach, each sequence of trait means can be treated as a single draw from a mul- tivariate normal distribution with a vector of means and covariance matrix given below. Model parameters: ? s , mean of the step distribution; ?2 s , variance of the step distribution; ?, phenotypic optimum; ?, variance around optimum; X 0 , estimated trait mean at the start of the sequence. AD abbreviations: t AD , time elapsed between an- cestor and descendant; X A , phenotypic mean of ancestral populations; ? A , ? D , sampling variances of the ancestor and descendant populations. Joint abbreviations: t i , t j , time elapsed from the start of the sequence to the ith and jth populations; ? i , sampling variance of the ith population; min, minimum; E[x], expected (mean) value of x; ?X, the difference between ancestor and descendant populations (X D ? X A ). 120 GENE HUNT generally increase with model complexity. Models with more tunable knobs (parameters) have an unfair advantage over simpler ones, and it is therefore nec- essary to balance model fi t against model complex- ity. One common metric for doing so is the Akaike Information Criterion (AIC), which is defi ned as AIC = ?2(log-likelihood) + 2(# model parameters) (Akaike, 1974). Lower AIC scores indicate higher model sup- port; this metric refl ects the amount of information lost in approximating reality with a model. In practice, it is usually better to use a version of the AIC with a bias correction for small sample sizes called the AIC C (Anderson et al., 2000). Relative model supports for a set of candidate models are conveniently summarized using Akaike weights, which result from a simple transformation of AIC or AIC C scores such that total support sums to unity across the models considered (Anderson et al., 2000). The outputs from these cal- culations are a series of weights that indicate the pro- portion of total empirical support each model receives (e.g., models A, B and C may receive, respectively, 80%, 15% and 5% of the evidential support). All the analyses described in this paper can be performed with functions provided in the R package paleoTS (Hunt, 2008b), which is publicly available at the Comprehensive R Archive Network (CRAN, see http://www.r-project.org/). This package is most easily downloaded and installed from within R in the usual way for CRAN packages; see software documentation for details. R is a free and cross-platform environment for statistical computing, graphics and programming (R Development Core Team, 2008). ANALYTICAL ISSUES Sampling Error is Surprisingly Large In practice, analyses of trait sequences almost always treat sample means as if they were known without er- ror. Of course they are not?all fi nite samples entail error in estimating mean values, and this sampling variance has a predictable magnitude equal to within- sample variance divided by the number of measured individuals (Sokal and Rohlf, 1995). In many fossil lineages, measurable individuals are scarce and mor- phological differences subtle, and thus sampling error may sometimes be quite large. Sampling error can be accommodated quite naturally in the expected evolu- tionary changes over time (Table 1), and it has been shown that this approach can accurately estimate evo- lutionary parameters even when data are noisy (Hunt, 2006). To my knowledge, all other methods so far pro- posed ignore this unavoidable fact that paleontologi- cal samples are fi nite. It is conceivable that sampling error is generally unimportant although there are indications otherwise (Kinnison and Hendry, 2001; Hunt, 2006). The cru- cial factor is the magnitude of sampling error relative to true evolutionary differences among fossil popu- lations. This latter quantity?true evolutionary vari- ance?can be estimated as the variance parameter (?) of the stasis model (this holds regardless of the under- lying mode of evolution). One simple way to assess the importance of sampling error is to compare this estimate of true evolutionary variance to the total ob- served variance of sample means. If sampling error is important, estimates of ? will be much less than the variance among sample means taken at face value. Judging from the sample of 251 empirical fossil sequences analyzed previously (Hunt, 2007), sam- pling error is quite often substantial (Fig. 1). On aver- age, about 44% of the variation in trait means is attrib- utable to sampling error, and it is perhaps surprisingly common for this proportion to be very close to 100% (Fig. 1). Ignoring sampling error causes one to overes- timate morphological divergence by mistaking noise for true evolutionary differences. This will bias pa- rameter estimates (Hunt, 2006) and has the potential to infl uence statistical inference (see below). Mun- dane though it may be, sampling error is nevertheless practically important and should be accounted for in analyzing evolutionary sequences. Model Selection Performance Hunt (2006) presented simulations showing that the AD approach can provide good estimates of model pa- rameters, even with noisy and incomplete fossil data. The performance of these methods in terms of model selection has been less explored, and previous results suggest that some models may be easier to detect ana- lytically than others (Sheets and Mitchell, 2001; Han- nisdal, 2006). In particular, the stasis model, because it is mathematically similar to sampling error (i.e., both produce uncorrelated Gaussian variation), may be both easier to detect and unduly favored in noisy sequences. In this section, I present some simulations 121EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES that explore the performance of AIC-based model se- lection criteria under three evolutionary scenarios: a moderate directional trend, an unbiased random walk, and stasis. These simulations were designed to investigate the relative performance of the two available parameter- izations (AD and joint) for a range of sequence lengths and magnitudes of sampling error. There is an enor- mous range of evolutionary scenarios one could con- sider and this investigation provides only preliminary guidance for a few empirically reasonable situations. Across all simulations, within-sample phenotypic vari- ance and the elapsed time between each sampled pop- ulations were set to unity. Absolute values for these do not matter because statistical inference is insensitive to changing units of both time and morphology (Hunt, 2006). Simulations were run at two different levels of sampling error?relatively low (n = 50 observations per sample), and relatively high (n = 5 observations per sample). The fi rst scenario investigated is that of a mod- est directional trend (Fig. 2, left column). The mean step was chosen to produce, over the entire sequence, a net increase of two units of within-population stan- Proportion of Trait Variance in Empirical Fossil Sequences due to Sampling Error Fr eq ue nc y 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 dard deviations. This is an average over many realiza- tions, some of which would show stronger and some weaker trends. The step variance was set so that 99% of simulated sequences would show net increase over time; if the step variance is too large, volatility in steps will swamp directionality. The second scenario is that of an unbiased random walk (Fig. 2, center column), with the step variance chosen such that the standard deviation of trait means at the end of the sequence was equal to twice the within-sample phenotypic standard deviations. The fi nal scenario considered is that of evolutionary stasis, with the evolutionary variance (?) set equal to the within-sample phenotypic variance (Fig. 2, right column). These particular scenarios were investigated in detail because they produce evolution- ary sequences with realistically moderate evolution- ary divergence?generally larger than sampling error, but not overwhelmingly so. Model selection performance under these three scenarios is summarized in Figure 2. I will focus here on three main results. First, while both parameteriza- tions generally perform similarly, the joint parameter- ization is better able to correctly identify directional trends. This advantage increases with sequence length, and is most pronounced when sampling error is high (Fig. 2). High sampling error obscures the point-to- point differences employed by the AD parameteriza- tion, rendering this approach less effective for time- series that are long and noisy. Sampling error does not accumulate, however, and so noise has less effect when considering the distribution of all sample means jointly. Second, sampling noise generally increases the support for the stasis model, and lessens support for the directional change and random walk models (Fig. 2). This is the expected effect because stasis and sampling noise have the same mathematical form. Third, stasis is relatively easy to correctly identify, particularly in sequences that are not short. This fi nd- ing is consistent with some previous simulation results (Sheets and Mitchell, 2001; Hannisdal, 2006). While the joint parameterization is better able to detect trends in noisy data, there are several respects in which the AD approach is superior. First, the joint estimation approach occasionally encounters compu- tational diffi culties. Calculating log-likelihoods for this approach requires inverting a covariance matrix that may be singular. This does not seem to be very common?it occurred in fewer than ten of the 251 se- Figure 1?Sampling error in empirical fossil sequenc- es. Histogram shows for 251 fossil time-series, the proportion of observed variance in trait means that is attributable to sampling error. Sampling error is often large?on average, nearly half of all variation in trait means observed in fossil sequences is noise. 122 GENE HUNT quences analyzed above. Nevertheless, the joint ap- proach may fail for some data sets, or at least require modifi cation to work in some circumstances. Second, these simulations indicate that the parameter estimates for the step variance parameter in the directional evo- lution model may be biased towards values that are too low, especially for time-series with high sampling error (results not shown). Third and fi nally, the AD approach is analytically simpler, and more easily ex- tendible to allow for punctuations, covariates and oth- er biologically interesting models (see below). Thus, both approaches have their strengths. Which is most Tr ai t M ea n 0 1 2 3 4 - 1 Low noise High noise 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 7 10 20 40 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 7 10 20 40 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Random Walk - High Noise 0 2 4 7 10 20 40 7 10 20 40 Low noise High noise 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0 2 4 7 10 20 40 7 10 20 40 Low noise High noise M ea n Su pp or t f or C or re ct M od el Directional Evolution Random Walk Stasis time time time Number of Sampled Populations in the Evolutionary Sequence Stasis RW Dir AD Joint Figure 2?Summary of model selection performance. The top row shows ten realizations for directional evolu- tion (left), random walks (center) and stasis (right). Shown are the true population means for sequences with ten sampled populations; bars in the upper left of each indicate the approximate size of 95% confi dence intervals on trait means under low and high noise simulations. These bars do not correspond to specifi c samples, but rather illustrate typical confi dence intervals for the two levels of sampling noise. The bottom two rows summarize the model selection performance for the models with low sampling noise (middle row, n = 50 observations per sample), and high sampling noise (bottom row, n = 5 observations per sample). Plotted under each scenario is the mean Akaike weight for the correct model, with open circles indicating the ancestor-descendant, and fi lled circles the joint parameterization. For each set of conditions and parameterization, the rectangles indicate the mean Akaike weight for each of the three models: directional evolution (black bar), random walk (white bar), and stasis (grey bar). Note that the horizontal axis of the bottom two rows is not to scale. 123EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES suitable will depend on the nature of the data and the goals of each particular study. EVOLUTIONARY INSIGHTS Relative Importance of Evolutionary Modes Even the most partisan voices involved in the Punctu- ated Equilibrium debate acknowledged that no evolu- tionary mode was universal. Accordingly, the central issue of this debate concerned the relative frequency of stasis versus gradual change. Reviews of the pub- lished paleontological literature, however, reached dramatically different conclusions about the domi- nance of stasis and gradual change (Gingerich, 1985; Erwin and Anstey, 1995; Jackson and Cheetham, 1999; Levinton, 2001; Gould, 2002), mostly because different authors held incompatible interpretations for the same fossil sequences. One avenue to resolve these disagreements is to apply each of the three canonical modes of evolution? stasis, directional change, and random walks?to all available empirical data sets and summarize support for these three models. Scouring the paleontological literature produced a set of 53 lineages for which the requisite data were published to allow these statistical models to be fi t (trait means, variances, and sample sizes, along with an age estimate for each sample in an evolutionary sequence). Some excellent and relevant studies could not be included because the original ref- erences did not publish summary statistics at the level of individual samples or stratigraphic levels (e.g., Jackson and Cheetham, 1994; Gingerich and Gunnell, 1995). Of the 53 included lineages, most were mea- sured for multiple morphological traits for a total of 251 evolutionary sequences (Hunt, 2007). Of these, only 13 (5%) were best fi t by the directional evolution model, and the remaining split approximately equally between random walks and stasis. Similar levels of support are indicated by the median Akaike weights for each mode (Table 2), suggesting that directional evolution is rarely observed in fossil sequences. This result, which was obtained using the AD parameter- ization of these models, also holds when the joint pa- rameterization is used instead (Table 2). As might be expected from the simulation results presented above, directional evolution garners slightly more support under the joint approach, but it is still infrequent. Even this relatively low incidence of directional evolution is almost certainly an overestimate because paleontolo- gists have focused greater attention on lineages and traits with prior evidence of gradual change (Gould, 2002). Incidentally, stasis in its strictest sense of no true evolutionary change (? = 0) is not very common; only about 9% of analyzed sequences (23/251) are consistent with true constancy of form. While the rarity of directional evolution confi rms one key claim of the Punctuated Equilibrium model, it is noteworthy that random walks are at least as com- mon as evolutionary stasis (Hunt, 2007)(Table 2). Qualitatively, fossil sequences seem to meander (ran- dom walk) at least as often as they show fl uctuations around a relatively stable mean (stasis). In retrospect, it seems that at least some of the disagreements about the relative frequency of stasis and gradual change stemmed from differences in how patterns similar to a random walk were classifi ed. To proponents of Punc- tuated Equilibrium, their lack of strong directionality rendered these sequences examples of stasis; to its critics, such meandering paths were a kind of gradual change. Neither lumping seems desirable because ran- dom walks and stasis are actually distinct, and a fair accounting of the relative importance of different evo- lutionary modes should refl ect this fact. These three evolutionary modes correspond to identifi able patterns in trait trajectories. How these patterns relate to process is a complex issue because each mode is consistent with a multitude of microevo- Table 2?Statistical support, measured as median Akaike weight, for three canonical modes of evolution. Re- sults are shown separately for the ancestor-descendant (AD) and joint parameterizations. 124 GENE HUNT lutionary scenarios (for recent attempts at inferring process from pattern, see Polly, 2004; Estes and Ar- nold, 2007). For example, populations evolving under neutral genetic drift will show trait trajectories that are random walks (Lande, 1976; Turelli et al., 1988), but so will populations experiencing randomly fl uctuating directional selection and those tracking an adaptive optimum that meanders over time. In general, it will be diffi cult to translate from pattern to microevolutionary process, but I will discuss below a specifi c example in which the signal of adaptive evolution is apparent in an exceptionally information-rich fossil sequence. Tempo and Mode Decomposed The foregoing sections follow current usage in label- ing as ?modes? qualitatively different kinds of evolu- tionary patterns. This usage descends from Simpson (1944), who separated how fast evolutionary transi- tions occurred (tempo) from the nature or mode of the change. Although the three modes commonly consid- ered today differ from Simpson?s suite of speciational, phyletic, and quantum modes (Simpson, 1944), the distinction between tempo and mode remains useful. The fundamental differences between stasis, ran- dom walks, and directional change are not in the mag- nitude or pace of evolutionary divergences, but rather in how constituent evolutionary changes are deployed in a sequence. With directional evolution, changes in the same direction are stacked together, generating persistent trends. For random walks, evolutionary in- crements are stacked with no preference for one direc- tion over another, and the resulting evolutionary tra- jectories show increasing but meandering divergence over time. Stasis, by contrast, results when evolution- ary increments are stacked antagonistically such that divergences in one direction are preferentially fol- lowed by opposing changes so that populations do not wander far from a fi xed point. Within each of these modes, the evolutionary tem- po can vary. Random walks, for example, can be faster or slower depending on the magnitude of the underly- ing step variance. Similarly, evolutionary fl uctuations under stasis can be small or large, depending on the value of the variance parameter (?). Thus, different models correspond to modes of change, with certain parameters of these models governing the tempo of change within that mode. These tempo-controlling pa- rameters are potentially useful as a means to measure evolutionary rates. Particularly promising in this re- gard is the random walk model, which has just a single parameter?the step variance?that determines the pace of evolutionary change. Using the estimated step variance of the random walk as a measure of evolutionary rate has a number of advantages over traditionally defi ned rates such as darwins or haldanes (Haldane, 1949; Gingerich, 1993; Gingerich, 2001). Notably, the step variance: allows for evolutionary reversals; it is estimated in a way that accounts for sampling error; and, at least for true unbi- ased random walks, its inference is unaffected by the time scale over which it is observed (Hunt, 2006). In addition, Lynch?s (1990) rate metric derived for purely neutral evolution is essentially a scaled version of the step variance. When time is measured in organismal generations, Lynch?s metric is equal to the estimated step variance standardized by within-sample pheno- typic variance. Therefore, the step variance even has a convenient benchmark?the neutral expectation? for judging what constitutes fast or slow evolution- ary change. Although parameters related to the step variance are increasingly used to measure phenotypic rates of evolution for phylogenetically related popula- tions (Martins, 1994; O?Meara et al., 2006), they are seldom applied to fossil data. This recommendation to use the step variance pa- rameter as a rate metric is in confl ict with Bookstein?s (1987) argument that rates of evolution are undefi ned for random walks. However, this claim is true only in the technical sense that, as discrete time models, the derivative of a random walk is undefi ned. How- ever, if differentiability is the sole relevant criterion, evolutionary rates never exist because they ultimate- ly change only with the origin or demise of discrete generations or individuals. A broader and more useful defi nition of evolutionary rate would encompass any model parameter that relates phenotypic divergence to elapsed time (see also Foote, 1991). Punctuations, and When they are Justifi ed Some of the most intractable disagreements during the Punctuated Equilibrium battles concerned the distinc- tion between gradual and punctuated change (Gould and Eldredge, 1977; Gingerich, 1985). Punctuations are quite easy to fi nd when looked for, but it is dif- fi cult to be sure that hypotheses of pulsed change are truly warranted (Fortey, 1988). Although tests were 125EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES developed to detect rate heterogeneity in evolutionary sequences (Charlesworth, 1984; Kitchell et al., 1987), these were seldom applied, and in any case of some- what limited usefulness because all reasonable models predict some heterogeneity in point-to-point rates of change. The key to gaining traction on this issue is to rec- ognize that the fundamental claim of punctuational hy- potheses is not that rates vary, but rather that evolution is not homogenous. Punctuations are thought to arise when the normal operation of stasis is temporarily sus- pended, allowing for a period of elevated change that differs qualitatively from stasis. Therefore, this claim should be tested by comparing support for this kind of non-uniform dynamic to that for models in which evo- lution operates by the same evolutionary rules through the entire sequence, as for example, in a random walk (Hunt, 2008a). Punctuational explanations posit that evolution- ary sequences can be divided into segments, each of which has its own set of evolutionary rules. In prac- tice, each segment can be fi t as described above as if it were a complete sequence, and the divisions be- tween segments can be determined by choosing the shift point or points that maximize the log-likelihood of the model (Hunt, 2008a). Punctuations can appear differently depending on their rapidity relative to the temporal resolution of samples; here I will focus only on punctuations that are so fast that intermediate mor- phologies are not observed. This kind of punctuation can be modeled as two separate intervals of stasis, each with its own evolutionary optimum (? 1 and ? 2 ; Fig. 3.1). The magnitude of the pulsed phenotypic change is simply the difference between the two opti- ma. Assuming separate evolutionary variance param- eters in each segment, this model has fi ve parameters: two phenotypic optima, each with its own variance parameter, and a parameter that determines the timing of the shift from one segment to the other (Fig. 3.1). AIC C scores can be used to decide if the log-likeli- hood advantage of this model more than compensates for its additional complexity, relative to uniform evo- lutionary models. Note that only the general form of the model is decided a priori (e.g., one punctuation or two punctuations); the actual values of the phenotypic optima and the timing of shifts between stasis regimes are free parameters of the model that are estimated by maximizing the likelihood of the observed sequence. Although a complete discussion of this class of models is presented elsewhere (Hunt, 2008a), a brief example should suffi ce to make the general ap- proach clear. Chiba (1996) documented a sequence of evolutionary changes in a suite of shell characters of Mandarina chichijimana, a land snail endemic to the Chichijima Islands in the western Pacifi c Ocean. Radiocarbon dating of shells collected from stratifi ed Quaternary deposits produced a precise and fi nely re- solved age model; the mean elapsed duration between 10 15 20 ?1 3 Tr ai t M ea n 0 10 20 30 40 24 25 26 27 28 29 M ea n Sh el l W id th 0 10 20 30 40 4. 6 4. 8 5. 0 5. 2 5. 4 5. 6 5. 8 M ea n N um be r o f W ho rls 0 5 Time Time (Kyr) Time (Kyr) 2 1 0 ?1 ?2 shift ?1 ?2 1 2 3 Figure 3?Punctuational models. 3.1, Schematic showing hypothetical punctuated pattern in which the pheno- typic optimum shifts suddenly between samples 10 and 11. 3.1 ? 3.2, Evolution in shell characters of Mandari- na chichijimana, a land snail species endemic to the Chichijimana Islands of the western Pacifi c Ocean (Chiba, 1996). The study interval spans approximately the last 40 Kyr. 3.2, Evolutionary trajectory of the number of shell whorls; a punctuated model decisively outperforms uniform models of change for this character (Table 3). 3.3, Relatively continuous evolution of shell width in the same lineage; a uniform model of change (unbiased random walk) is the best-supported model for this character. 126 GENE HUNT adjacent samples was less than three thousand years. Chiba argued that several of the traits changed in a discontinuous or punctuated manner, including the number of whorls in the shell (Fig. 3.2). Evolution in some other traits, such as overall shell width, seemed more smoothly continuous (Figure 3.3). To assess these interpretations, I fi t to these two traits the standard suite of uniform models (directional change, random walk, stasis), and compared their fi t to a model that posited a single, punctuated change. The results corroborate Chiba?s interpretations. The model of a single punctuated change accounts for the evolu- tion of whorl number far better than any of the uniform models (Table 3). In contrast, the model of a uniform random walk is best supported for shell width, with the punctuational model accounting for less than 10% of total model support (Table 3). It may be noted that, although Punctuated Equi- librium posits that punctuations are associated with lineage splitting events, the above example concerns a putative punctuation occurring within an unbranched lineage. This is a fair refl ection of the paleontologi- cal literature. The claim that morphological jumps co- incide with cladogenesis was based on the perceived consequences of allopatric speciation model, not on any direct reading of the fossil record (Eldredge and Gould, 1972). Because there are very few examples for which lineage splitting is thought to be captured in the fossil record (Gingerich, 1985), most discus- sion about punctuations has focused on examples like Chiba?s that document within-lineage evolutionary changes (Gould and Eldredge, 1977; Gould, 2002). If a splitting event is documented directly, however, it would be straightforward to test the Punctuated Equi- librium claim by fi tting a model in which the pace of evolutionary change increased during cladogenesis, and testing the fi t of this model versus one in which phenotypic evolution during splitting followed the same rules as those operative within lineages. Natural Selection in Fossil Sequences Inferring process from pattern is famously diffi cult, and fossil data offer few exceptions to this rule. Pale- ontologists have generally attributed patterns of direc- tional evolution to the action of natural selection (e.g., Bell et al., 2006). While it is diffi cult to imagine trends in fossil lineages arising without the intervention of natural selection, the relationship between patterns of long-term divergence and microevolutionary scenar- ios can be complex, with multiple processes capable of producing the same macroscopic pattern (Raup and Crick, 1981; Hansen and Martins, 1996). Given this situation, it is worth asking the question: What should a simple bout of adaptive evolution look like in the fossil record? While this question is a good place to start, it is not precise enough to answer because adaptive evolu- tion can take different forms. If we accept that long- term evolutionary trajectories are best considered in the context of adaptive landscapes (Simpson, 1944; Table 3?Performance of uniform and punctuated evolutionary models for two shell measurements in Man- darina chichijimana (Chiba, 1996). Punctuated evolution is strongly supported for whorl counts, but uniform models, especially the random walk, are favored for shell width. 127EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES Arnold et al., 2001; Estes and Arnold, 2007), it is possible to focus on one plausible and tractable adap- tive scenario: the evolution of populations climbing from suboptimal phenotypes to a nearby peak in the adaptive landscape. This is the expected dynamic for a population that invades an environment with some- what different selective conditions than its ancestral habitat, or for a population residing in an environment that changes suddenly. The expected evolutionary dynamic in this sce- nario is not a simple directional trend, but instead an exponential approach to the new phenotypic optimum (Fig. 4). Change is initially strongly directional, but this directionality tapers rapidly as the new optimum is approached, after which evolutionary stasis ensues. The statistical properties of this model were described by Lande (1976; see also Hansen and Martins, 1996; Hansen, 1997), and it has four key parameters: the ini- tial phenotype, the optimal phenotype, the strength of selection (which determines the rapidity with which the optimum is approached), and the step variance (Lande, 1976; Hansen, 1997; Hunt et al., 2008). This model, which is sometimes referred to as an Ornstein- Uhlenbeck process, can be fi t to fossil sequences via maximum likelihood just like the standard modes of change, and its success can be gauged in the normal way using AIC C scores (Hunt et al., 2008). This approach was applied to what is probably the most promising example yet described for detect- ing natural selection in a fossil lineage: Bell et al.?s (2006) study tracking skeletal armor reduction in a stickleback lineage from Miocene lake sediments. In this study, Bell and colleagues documented a tapering decrease in three skeletal traits, including the number of dorsal spines (Fig. 4). These authors applied several methods to test if evolutionary changes had been too rapid or too directional to result from neutral drift, but none of these tests revealed compelling evidence for the action of natural selection. These negative results are noteworthy because (i) the temporal resolution of 250 years?determined by counting yearly varves?is exceptionally fi ne for fossil studies, and (ii) consider- able circumstantial evidence for natural selection ex- ists, including the observation that skeletal reduction is common in modern stickleback that invade lakes with few predatory fi shes (as was apparently the case in the paleo-lake studied). 0 2000 4000 6000 8000 0. 6 0. 8 1. 0 1. 2 1. 4 Time (generations) M ea n Nu m be r o f D or sa l S pi ne s Figure 4?Evolution of dorsal spine counts in a fossil stickleback lineage. Shown is the mean number of dorsal spines (log-transformed as described in Bell et al 2006), with 95% confi dence intervals. Time is measured in stickleback generations (= 2 years), with the chronology determined from counting yearly varves. Dotted line shows the best fi t adaptive model with its characteristic exponential approach to a new optimal phenotype (see text). Grey region outlines the 95% probability region of the adaptive model. 128 GENE HUNT Rather than attempting to reject a null model of a random walk, a better approach would entail fi tting random walk and adaptive models, and then compar- ing their relative empirical support. When this is done, the adaptive model decisively outperforms the random walk model (the expected form under neutral genetic drift) for all three traits analyzed (Hunt et al., 2008; Fig. 4 shows the model fi t for dorsal spine counts). The adaptive model accounts for over 99% of the Akaike weight, and thus there is strong statistical evi- dence that natural selection has shaped the evolution of these traits, despite the non-signifi cant results from traditional tests (Hunt et al., 2008). These results demonstrate that, at least under fa- vorable circumstances, it is possible to document natural selection in fossil lineages. It is unclear how often this kind of inference will be possible because this case study benefi ted from several exceptionally favorable circumstances. Temporal resolution was excellent, and the evolutionary changes were slow enough to occur over several thousand years. While this is a geologically rapid change, much faster in- stances of adaptive evolution are known from extant populations, including other cases of skeletal reduc- tion in sticklebacks (Bell et al., 2004). Finally, this study benefi ts from a fortunate window of observa- tion, which happens to include the invasion of a pa- leo-lake by this particular stickleback lineage (Bell et al., 2006). Adaptive adjustments are expected to oc- cur quickly upon a population?s encounter with novel selective conditions, and thus it may be particularly fortuitous to sample the initial stages of an invasion. Thus, while it is possible to infer the action of natural selection in fossil lineages, the requisite depositional and biological conditions may be rather uncommon. Regardless, success in detecting adaptive evolution in the fossil record requires that we are actually looking for its proper signature. FUTURE DIRECTIONS The approach described here is not so much a method as it is a general approach that may be applied to evaluate almost any kind of hypothesis about the nature of phenotypic changes within lineages. In this paper, I described several different applications using this approach, but there is still ample room to expand the range of evolutionary models considered beyond those discussed here. One natural extension of this approach would be to incorporate putatively causal factors into evolutionary models. The resulting class of models could be used to assess, for example, the effects of climate or productivity on body size evolu- tion (Schmidt et al., 2004; Hunt and Roy, 2006; Finkel et al., 2007; Novack-Gottshall, 2008), and compare the success of these models to those that lack covari- ates. Another potentially fruitful class of models to explore are those Alroy (2000) refers to as ?structured state space models.? These models are characterized by having dynamics that vary with the phenotypic values of the lineage, allowing for the possibility of phenotypic values that attract or repulse evolutionary trajectories. The stasis model is a simple example of this kind of model?the optimum is essentially a very strong attractor?but a whole range of potentially in- teresting models has yet to be considered. A second area that could use analytical develop- ment is the effect of uncertain chronologies in infer- ring evolutionary models. These effects are likely to be strongly data and model-dependent, and simula- tions may help explore the sensitivity of analyses to these kinds of effects (Hunt, 2006; Hunt, 2008a). Some recent work (Hannisdal, 2007) has taken the ap- proach of integrating stratigraphic and evolutionary inferences into a single framework. This approach, though demanding of computational effort and data quantity, offers a very elegant means of accommodat- ing age model uncertainty. A fi nal area I would point to as profi table for future work is comparing evolutionary patterns within fossil lineages to similar analyses performed with phyloge- netically related populations. This area actually does not require much in the way of theory development; for the most part, the models applied in comparative studies are the same as those described in this paper. In particular, the random walk model is dominantly used to explore trait evolution (Felsenstein, 1985; Martins, 1999; Garland and Ives, 2000), although studies that employ directional (Pagel, 2002), adaptive (Hansen, 1997; Butler and King, 2004) and other models (e.g., Pagel, 1999) are increasingly common. Yet at pres- ent, two complementary data sets about phenotypic divergence?fossils and phylogenies?employ almost entirely non-overlapping sets of methods. If we are to ever reconcile these two views of evolution, it will 129EVOLUTIONARY PATTERNS WITHIN FOSSIL LINEAGES begin with analytical approaches able to evaluate dis- parate kinds of evidence in comparable terms. ACKNOWLEDGMENTS For discussion about many of the ideas presented here, I thank M. Foote, S. Wang, K. R. Thomas, B. Hannisdal, P. Wagner, M. 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