### Abstract

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

**Abstract** The theory of ‘punctuated equilibrium’ hypothesises that most morphological change in species takes place in rapid bursts triggered by speciation. Eldregde and Gould postulated the theory in 1972, as an alternative to the idea that morphological change slowly accumulates in the course of time, a then common belief they dubbed ‘phyletic gradualism’. Ever since its introduction the theory of punctuated equilibrium has been the subject of speculation rather than empirical validation. Here I present a method to detect punctuated evolution without reference to fossil data, based on the phenotypes of extant species and on their relatedness as revealed by molecular phylogeny. The method involves a general mathematical model describing morphological differentiation of two species over time. The two parameters in the model, the rates of punctual (cladogenetic) and gradual (anagenetic) change, are estimated from plots of morphological diversification against time since divergence of extant species.

### Introduction

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

In an influential 1972 paper, Eldredge and Gould proposed their theory of punctuated equilibrium: the theory that phenotypic evolution is concentrated in short periods of rapid change, interspersed with long periods of stasis. Their theory challenged what they called ‘phyletic gradualism’, the belief that morphological difference between species accumulates gradually over time – a belief that apparently prevailed amongst geneticists at the time (Mayr, 1982; Gould & Eldredge, 1993). The issue whether evolution proceeds according to the punctual mode or in a more gradual way has since been a source of uncertainty, as it proved hard to test.

The original evidence for punctuated equilibrium came from the fossil record. Although gradual morphological change has been reported in a range of fossil taxa (Gould & Eldredge, 1993 and references therein), much of the observed variation in phenotypes seems to arise in short bursts (Eldredge & Gould, 1972) often associated with speciation. The conclusion that morphological diversification of species is sudden rather than gradual therefore seems an observation rather than a theory. The conclusion is logically flawed, however, as fossils are considered to belong to separate species *only if* their phenotypes differ considerably (Stebbins & Ayala, 1981). On the other hand, punctuated equilibrium cannot be entirely a semantic issue either, as fossils, independent of the names assigned to them, do display long periods of stasis (Williams, 1992). Nevertheless, many features of extant populations that define them biologically or genetically as separate species cannot be identified from fossil specimens. It has been suggested that fossil species do actually represent biological species (e.g. Jackson & Cheetham, 1990), but many extant, morphologically similar species would presumably not be distinguished if encountered as fossils, so uncertainty remains. Thus, observations on the mode of evolution drawn from fossil data may not apply to extant species.

Similarly, the theory excludes neither punctuated nor gradual evolution. In fact, sudden, punctuated diversification, prolonged morphological stasis as well as gradual differentiation can be the outcome of theoretical models (Williams, 1992). Considerable cladogenetic change (sudden phenotypic change upon speciation) may occur when, upon reproductive isolation, differential selection regimes dissipate the phenotypes of populations that were until then prevented from adaptation to local conditions by gene flow (Lande, 1980; Garcia-Ramos & Kirkpatrick, 1997). Stasis has been explained as the inertia of natural populations consisting of large, coadapted gene complexes (Lerner, 1954) to respond to differential selection. On the other hand, there is also theoretical support for anagenetic change (gradual change over time between speciation events). Stochastic processes like genetic drift alter the average phenotypes of populations, which may result in considerable changes on an evolutionary time scale (Lande, 1976; Lynch & Hill, 1976; see also Williams, 1992). Furthermore, if the phenotype is subject to selection the optimum phenotype may change gradually over time as a result of environmental change.

Here I introduce a method to estimate rates of cladogenetic and anagenetic evolution from data on phenotype and evolutionary relationships of extant species. The method allows estimation of how important punctuated and gradual evolution have been in the diversification of biologically or genetically identified species. The test is based on a mathematical formalization of some simple implications of the theory of punctuated equilibrium: If evolution is mainly anagenetic (Fig. 1a), morphological differentiation of species gradually increases over time. Recently diverged species would be expected almost similar, unlike distantly related species (Fig. 1c). On the other hand, if evolution proceeds according to the punctual model, phenotypic evolution is concentrated in speciation events (Fig. 1b). Then, recently diverged species would already show considerable morphological differentiation (Fig. 1c). Species that shared a common ancestor long ago would show greater differentiation as they are separated by more speciation events. This paper presents a method to estimate rates of ana- and cladogenesis from plots of phenotypic difference between species pairs against time since divergence (Fig. 1c).

### Model derivation

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

If the average phenotype *z* of a species is subject to genetic drift, neutral mutation, or any other set of selectively neutral factors, the average phenotype changes over time in an anagenetic fashion, which can be described as ‘Brownian motion’ (Felsenstein, 1985; see also Lande, 1976; Lynch & Hill, 1986), i.e.

where *z*_{0} is the average phenotype before change and *a* is a normally distributed random number with mean 1 and a standard deviation that increases over time. Similarly, the average phenotype may change upon speciation in a cladogenetic fashion so that:

where *c* is a normally distributed random number with mean 1 and standard deviation effectively independent of time. It is assumed that the standard deviation of *c* is equal through time and across species. The average phenotype of an extant species can therefore be written:

If we compare the average phenotypes of two species *A* and *B* since their latest shared ancestor, we thus obtain:

- (1)

where *c*_{i} refers to independent cladogenetic changes that take place at each speciation event, and *a*_{i} corresponds to independent anagenetic changes between speciations. Notice that because *z*_{0} is the average phenotype of the latest common ancestor of both species *A* and *B*, *z*_{0} does not contribute to the difference between the average phenotypes *z*^{A} and *z*^{B}*.* That is why *z*_{0} does not occur in (1). If *c*_{i} and *a*_{i} are all random numbers from a normal distribution, it follows that the probability density function of ln *c* is:

where *f* is the normal probability density function of *c*, and *σ*_{c} is the standard deviation by which *c* is distributed. The same goes for the probability density function of ln *a*, of course. For small *σ*_{c} a good approximation of *φ (*ln *c*) is the normal distribution with mean 0 and standard deviation *σ*_{c} which greatly facilitates estimation of the probability density function of the sums of ln *c* and ln *a*. According to the central limit theorem, the probability density function of the sum of independent, identically distributed random variables approaches a normal distribution whatever the probability density function of the random variables. As ln *c* and ln *a* themselves are approximately normally distributed, for present purposes good approximation of the probability density functions of Σln *c* and Σln *a* is normal with mean 0 and standard deviations *σ*_{c} and ζ_{a}, respectively. Alternatively, it is often assumed that the evolution of metric characters is described by Brownian motion on a logarithmic scale (Felsenstein, 1985). In any case the distribution of the logarithmic ratio of phenotypes of species *A* and *B* becomes:

where *N*(*µ*,*σ*^{2}) denotes a normal distribution with mean *µ* and variance *σ*^{2}. As ζ_{a} depends on the time *τ* over which anagenetic change of *z* has accumulated (Lande, 1976) as:

where *σ*_{a} is the standard deviation of anagenetic change per unit time. This simplifies to

As the sums of *τ* must both equal the time *t* since species *A* and *B* shared a common ancestor, we finally obtain:

- (2)

where the parameters *k* and *l* are the number of speciation events that affected *z*^{A} and *z*^{B}, respectively, after their diversification (1). The expected morphological diversification of the species as a result of cladogenesis increases with the number of such events.

### Hidden speciations

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

It is necessary to take into account the uncertainty in the number of speciations that affected a pair of species since they diverged from each other. This uncertainty is because of the fact that species may go extinct, and do not have any extant descendants. Estimates based on fossil as well as on molecular data suggest that extinctions may occur at over half the rate of speciation (Kubo & Iwasa, 1995), which makes it very important to take uncertainty on the number of speciations into account. From a phylogeny it is clear that some species are separated by more than one speciation event, like *Pan troglodytes* and *Homo sapiens* that are separated by three ‘observed’ speciations (Fig. 2). The difficulty is to take into account the additional number of ‘hidden’ speciations–speciations that are not drawn in the phylogeny. Denoting the number of speciations observed in the phylogeny *S*°, we find for the probability that the actual number of speciations that separates a pair of species *S*^{a}:

where *S*^{a} ≥ *S*° as there cannot have been less speciations than observed if we agree on the shape of the phylogeny.

Hidden speciations may occur anywhere in the phylogeny on the path between the two species. The total number of hidden speciations between two species *S*^{h} can therefore be written: *S*^{h}=*S*^{h}_{I} + *S*^{h}_{II} + . . . + *S*^{h}_{V} where the subscripts refer to the segments of the phylogeny in Fig. 2. Similarly *P*(*S*^{h})=*P*(*S*^{h}_{I} + *S*^{h}_{II} + . . . + *S*^{h}_{V}) and as the segments are all statistically independent, we can write for this probability for the general case of *n* segments:

- (3)

This equation is the summation of the probability that *n* segments hide *S*^{h} speciations over all possible distributions over the segments. For example, if the number of segments is *n*=3 and the number of hidden speciations is *S*^{h}=3, the possible combinations are [300], [210], [201], [120], [111], [102], [021], [012], and [003]. Notwithstanding the visual complexity of (3), it is straightforward to evaluate it numerically on a computer. The possible number of combinations *Q*, however, increases dramatically with *n* and *S*^{h}:

which makes the procedure slow for large phylogenies.

In order to evaluate the function, however, we must know how to calculate *P*(*S*^{h}_{i}), the probability of the number of hidden speciations in any given segment. In other words, we must calculate the probability that a segment of length *t*_{w} and age *t*_{op} (time between its origin and present) has no extant representatives, although *S*^{h} speciations occurred. In order to do so, two fundamental probabilities must be taken into account: the speciation and extinction probabilities *λ* and *µ* (speciation and extinction rates can sometimes be estimated from molecular phylogenies: Nee *et al*., 1994a,b; Kubo & Iwasa, 1995). On the assumption that the probability of speciation is constant over time, the probability of *S*^{h} speciations during time interval *t*_{w} is a Poisson probability:

The probability that out of *S*^{h} species originated at time *t*_{op} before present none has any extant representatives is binomial:

where *α* is the probability that a single lineage has no extant descendants after time *t*_{op}, which is according to Bailey (1964):

Combining these probabilities, the probability that either no speciations occurred during interval *t*_{w} or that none of the species originated during that interval has any extant descendants is, according to Bayes rule:

Fortunately, this reduces to:

which reads as follows: the probability of *S*^{h} hidden speciation events given *λ*, *µ* and given that none of the resulting species is currently extant (which defines them as hidden). This equation does not take into account that within a segment over time the probability of no present descendants *α* changes, but for relatively short segments *t*_{w} this should not affect the probability distribution of *S*^{h} too much. The larger change in *α* between segments is taken into account by the use of (3).

As a next step, we must consider that at the instant of speciation only the newborn species undergoes cladogenetic change. We do not know whether that species is the one that we are looking at in present nature or one that has no extant representatives. We can only state in retrospect that the probability that a speciation event has affected the phenotype of an extant species is ½. In addition, we know that the first speciation that separated two species certainly affected either of them. Thus, we are interested in the probability that out of *S*^{a} − 1 speciation events *S*^{e} affected the species pair we are studying. As *S*^{a} − 1=*S*^{h} + *S*° − 1 this probability becomes the binomial probability:

and the unconditional probability of *S*^{e} effective speciations becomes, by summation of the conditional probability over all possible actual speciation numbers:

Actual evaluation of the above equation from 0 to infinity is impossible, but *P*(*S*^{h}) drops rapidly for *S*^{h} > 5 if the extinction rate is not very high compared with the speciation rate, so that 10 is often an adequate approximation for infinity in numerical evaluation.

Now we are at the point that we can take into account the possibility of hidden speciations in the consideration of the morphological difference between species. Namely, we know from (2) that:

where 1 + *S*^{e} signifies that the first speciation that divided a species pair certainly affected either of the species, and that of the subsequent speciations *S*^{e} were affecting the diversification of the pair. Summation of overall possible numbers of effective speciations yields the unconditional probability to observe morphological differentiation *z*^{A}/*z*^{B}

which can now be used to evaluate the likelihood function.

### Parameter estimation

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

Application of the above derivations will generally include the estimation of *σ*_{c} and *σ*_{a} by maximum likelihood. The likelihood function *L* to be maximized with respect to *σ*_{c} and *σ*_{a} is:

- (4)

where *u* is the number of species, and *t* is still the time since species *A* and *B* shared a common ancestor. This likelihood function is maximized when its logarithm is maximized, and it can be shown that the maximum likelihood estimates σ_{c} and σ_{a} also maximize:

from which estimates are readily obtained. Many phylogenies can be calibrated so that real dates can be assigned to branches in the phylogeny. In that case the phylogeny has a time scale, and the estimates of *σ*_{c} and *σ*_{a} can be assigned units. A straightforward choice of units for *σ*_{c} would be phenotypic standard deviations, and for *σ*_{a} phenotypic standard deviations per million years.

To test hypotheses about the cladogenetic and anagenetic variability of the average phenotype it is necessary not only to have the best estimates of *σ*_{c} and *σ*_{a}, but also estimates of their variability. As the number of combinations of *u* species is ½(*u*^{2} − *u*), the number of data from which to estimate *σ*_{c} and *σ*_{a} will be typically quite large (even if *u* is moderate). Therefore, approximate confidence regions for the parameters *σ*_{c} and *σ*_{a} can be determined assuming that the distribution of −2 times the log-likelihood ratio approximates a chi-square distribution with 2 degrees of freedom. Alternatively, estimates of variability can be obtained by jack-knifing. However, if the likelihood is to be used for other purposes than just obtaining the best estimates of *σ*_{c} and *σ*_{a} (for example to construct a confidence interval around the estimated parameters or to test for significant differences between a parameter and an expected value), the replication of data in the summation in (4) should be corrected for. The summation of log-likelihoods is namely over ½(*u*^{2} − *u*) pairs, although there are only *u* species, so that the sum of log-likelihoods in (4) should be multiplied by 2/(*u* − 1) to obtain corrected values and, subsequently, likelihood ratios. As each likelihood is multiplied by the same factor, this correction does not affect what the best parameter estimates are.

### Hominoid body size evolution

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

Purvis (1995 ) constructed a composite phylogeny of primates and used this phylogeny to test several macroevolutionary hypotheses ( Purvis *et al*., 1995 ). I used the phylogeny of Hominoidea from this composite tree, because for Hominoidea we can be relatively sure that all species are known, because for all species good phenotypic data are readily available and because the nodes in the phylogeny have been assigned dates from which Purvis *et al*. (1995 ) estimated * λ * =0.134 and * µ * =0.037 per million years. Although different dates might be more appropriate ( Yoder & Yang, 2000 ), for the present, illustrative purpose I used Purvis *et al*.'s (1995 ) estimates. Body size divergence of species pairs was calculated as in (1) as ln *z*^{A} − ln *z*^{B} using the average of male and female body size. Morphological (body size) divergence is plotted against time since divergence (Fig. 4). As expected, the largest morphological differentiation occurs between species that have diverged long ago. However, more recently diverged species also show considerable differentiation, indicating some cladogenetic (punctual) evolution. Rates of cladogenetic (*σ*_{c}) and anagenetic (*σ*_{c}) evolution of Hominoid body size were estimated by maximization of the likelihood function (4). The best estimate is shown with likelihood ratio-based confidence region (Fig. 5).

It can now be tested whether there is a punctual component in evolution of the trait under investigation. That is, it can be assessed whether there is a statistically significant cladogenetic variability *σ*_{c}. The best estimate of *σ*_{c} for Hominoid body size is 0.13, indicating a large change of body size upon speciation. Nevertheless, *σ*_{c}=0 lies within the approximate 95% confidence region of *σ*_{c} (Fig. 5). Therefore the null hypothesis that body size evolution in Hominoids has been entirely anagenetic (gradual) cannot be rejected. In this particular, illustrative case, that is not surprising, as the low number of five extant Hominoid species offers little statistical power. With a larger phylogeny containing more species *σ*_{c} and *σ*_{a} could be estimated with less variance.

Similarly, it is possible to test for significant differences between *σ*_{a} from a theoretically predicted value. Lynch & Hill (1986) derived the expected rate of change of the average phenotype of a population under neutral evolution. According to their theoretical work, the expected asymptotic rate of population divergence with respect to the average phenotype of a quantitative trait is:

per generation, where *V*_{m} is the variance in the phenotype as a result of mutation, *V*_{e} is the environmental variation and *N*_{e} is effective population size. As the divergence rate approaches *N*_{e}^{−0.5} as *N*_{e} approaches infinity, the estimate obtained for large *N*_{e} is independent of the ratio *V*_{m}/*V*_{e}. So, a conservative estimate of the expected per year rate of anagenetic body size divergence in Hominoids can be obtained assuming that generation time is 20 years and that *N*_{e}=10^{6}. The rate of anagenetic change would then be 0.22 phenotypic standard deviation per million years. If *N*_{e} is smaller this rate would be higher. It can now be tested whether the rate of anagenetic body size evolution estimated from empirical data is significantly lower than expected. It can be seen that *σ*_{a}=0.22 lies far outside the approximate 95% confidence region of *σ*_{a} (Fig. 4). Therefore, it is concluded that anagenetic change has been significantly lower than expected from neutral evolution without stabilizing selection. (Because the actual rate of anagenesis is not expected to be higher than under purely neutral evolution a one-sided hypothesis would have been appropriate.)

Other hypotheses that can be tested with the method proposed are whether a certain trait evolved faster, or in a different way, in one lineage than in another. It might, for instance, be hypothesized that rates of cladogenesis have been different in Hylobates (gibbons) and Hominoids. In order to do so, rates of cladogenesis would have to be estimated in both clades, together with an estimate of their variance as in Fig. 5, from which statistical comparisons can be made.

### Discussion

- Top of page
- Abstract
- Introduction
- Outline of the test
- Model derivation
- Hidden speciations
- Parameter estimation
- Procedure test
- Applications
- Hominoid body size evolution
- Discussion
- Acknowledgments
- References

The present is not the first technique proposed to extract rates of evolution from molecular phylogenies, and also not the first to use molecular phylogenies to distinguish between punctuated equilibrium and phyletic gradualism. Mooers & Schluter (1998) used maximum likelihood techniques comparable with the one used here to determine which macroevolutionary model fitted best the observed body sizes of extant vertebrates. With a similar approach, Mooers *et al.* (1999) fitted different macroevolutionary models to observed phenotypes of extant crane species (Gruinae). These approaches, however, compared the goodness of fit of different models, and were not primarily concerned with estimating the parameters for a certain model, as the present procedure. Thus, although the idea and methodology of both approaches are quite similar (and proper parameter estimation is as crucial in Mooers and Schluter's studies as in the present one), the applications are fundamentally different. Another closely related approach by Pagel (1997, 1999) uses generalized least squares (GLS) techniques to test whether a character covaries with evolutionary time. Pagel's approach is, like the present, concerned with parameter estimation (i.e. both *σ*_{c} and *σ*_{a}) and testing of the significance of this parameter. However, these methods did not take explicitly into account that phylogenies hide many speciation events. Mooers *et al.* (1999) wrote, ‘Complete trees do not include information on extinct lineages, of course, so we must assume that there has been negligible extinction or that extinctions have been random on the tree.’ Considering the probability of hidden speciations at all branches of the tree would therefore be expected to make the present procedure perform better than previous ones. This consideration, which renders the present approach rather computation-intensive and thereby slow, is expected to have the same effect on the GLS approach used by Pagel (1997, 1999), if it is otherwise possible to take hidden speciations into account in the variance–covariance matrix necessary for GLS. Using independent contrasts is not an option, as punctuated equilibrium assumes that phenotypic variance is not proportional to branch length (Björklund, 1995).

The present procedure to detect phyletic gradualism vs. punctuated equilibrium is analytically relatively simple and therefore easily applied. There are some limitations to its applicability and use, however. If estimates of average phenotypes of species are unreliable, so is the test. In the example of Hominoid body size evolution, data on average phenotypes came from moderately large samples. Nevertheless, sample averages deviate from the species actual average phenotype. For many species extensive data are available on body size (Dunning, 1993), but for some characters (e.g. encephalization, left ventricular volume, alarm call frequency) it may be hard to obtain reliable estimates from poorly studied species. Especially in such cases, sampling variation in the average should be taken into account. Extensive data available for avian body size (Dunning, 1993) indicate that the coefficient of intraspecific phenotypic variation is approximately invariant across species of largely different average body sizes. If this also applies to other metric traits and to other taxa, a simple way to take sampling variation into account is by considering that if *σ*_{c}=0 one would estimate at most *σ*_{c}=*CV*_{m}, where *CV*_{m} measures the sampling variation in the average phenotype. This measure, which is the coefficient of variation of the average phenotype within species, is related to the coefficient of variation of the phenotype by the number of individuals sampled in each species. As *CV*_{m} drops rapidly with increasing sample size, sampling variation in the average phenotype is negligible even for small samples if the interspecific variation in the trait under consideration is much larger than the intraspecific variation. In cases where interspecific variation is small compared with intraspecific variation, sufficiently large samples should be collected.

Other factors that affect the performance of the procedure are the correctness of the phylogeny and the estimates of speciation and extinction probabilities *λ* and *µ*. An incorrect phylogeny will inevitably lead to erroneous results. In the present case, *λ* and *µ* are estimated from the same phylogeny (Nee *et al*., 1992, 1994a,b; Kubo & Iwasa, 1995; Purvis *et al*., 1995), so both will be incorrect if the phylogeny is incorrect. In fact, for the Hominoid example used here, the dates assigned to branching points might be incorrect (Yoder & Yang, 2000). In addition, *λ* and *µ* may be misestimated even if the phylogeny is correct, as is the case with estimated statistics in general. Speciation rates may also be estimated, with similar precaution, from fossil data rather than a phylogeny. Alternatively a value for a higher taxon may be used, in which case the exact rates remain unknown and it is recommended to perform multiple tests with different estimates of this ratio. If the species turnover rate is overestimated, the frequency of speciations that affect diversifying species is also overestimated, and consequently *σ*_{c} will be underestimated. The other way round, if the ratio is underestimated, less morphological differences are accounted for by cladogenesis and *σ*_{c} will be overestimated. Phenotypic variation is ascribed to *σ*_{a} if not to *σ*_{c} (Fig. 5), so that underestimation of *σ*_{c} is accompanied by overestimation of *σ*_{a} and, vice versa, overestimation of *σ*_{c} leads to underestimation of *σ*_{a}.

The mathematics behind the estimation of cladogenetic and anagenetic rates of evolution assume Brownian motion of average phenotypes. Thus, as one reviewer mentioned: ‘a fairer assessment of the model is that it measures pure Brownian motion and deviations from pure Brownian’. It is often assumed that the long-term evolution of phenotypes can be described as (branching) Brownian motion (Felsenstein, 1985). The bell-shape of the normal distribution most likely reflects the nature of phenotypic change quite well, and even if phenotypic change is not normally distributed, the central limit theorem says that the outcome of an additive series of non-normal phenotypic changes will still be approximately normal. Nevertheless, the mathematical derivation of models of long-term evolutionary change from population genetic models (Lande, 1976; Lynch & Hill, 1986) that can be verified only in short-term experiments is an orders of magnitude extrapolation. The present procedure would be severely biased if, for example, the phenotype is more likely to change in one direction, i.e. if body size increase is (*a priori*) more likely than decrease. However, empirical data to falsify models of long-term evolution are typically lacking, and it is through studies like the present, fitting different models to empirical data (e.g. Mooers & Schluter, 1998; Mooers *et al*., 1999), that insight can be gained.

It is possible that quick morphological changes also occur when there is no speciation going on. For example, a population may, at some time, have been more susceptible to new selection pressures or subject to stochastic processes like drift. Anagenetic change might, in fact, entirely consist of such rapid changes. If so, it seems reasonable to assume that such events happen every now and then (rather than once), and in that case their effect can be thought of as included in the estimate of *σ*_{a}. That is, the estimate of *σ*_{a} indicates the morphological differentiation that accumulates over time excluding speciation, but it does not imply that anagenetic morphological change is constant over time.

Application of the test in its present form is limited to relatively closely related species. Its mathematical derivation requires that rates of morphological evolution are, on average, the same for all species in the phylogeny. This seems a reasonable assumption for relatively closely related species. If one would want to estimate rates of body size evolution from a phylogeny containing mice and elephants, the assumption of equal evolutionary rates would almost certainly be violated. Mice namely have much shorter generation times and much larger population sizes than elephants. Those population parameters, generation time and effective population size, may critically determine rates of evolution (Lande, 1976; Lynch & Hill, 1986). In order to compare distantly related species, it would be necessary to incorporate those effects in the test which would require additional assumptions on effective population size and generation time, which I anticipate would make the test far more complicated. (What could be tested with the test in its current form is whether rates of evolution have been different in mice and elephants.)

Molecular phylogenies become available for more and more taxa. The presented method allows relatively simple estimation of rates of evolution from those phylogenies. So it will become possible to test in a variety of taxa and for a variety of traits whether evolution is punctuated or gradual. Taking into account population genetic theory, most probably evolution is both punctuated and gradual. The determination of their relative importance from empirical data will provide insight in how speciation proceeds in different taxa, and in the main causes of interspecific phenotypic variation.