Evolutionary change is remarkably time-independent across scales

Apparent time-scaling of evolutionary rates presents an evolutionary dilemma. Rates of

negative scaling relationship across different types of evolutionary rates.Highly unlikely relationships between timescale and accumulated evolutionary change are required to produce anything other than negative rate-time relationships.Empirical rate-time scaling relationships reflect unscaled magnitudes of evolution that are either unrelated to time and/or vary in their relationship with time.Over 99% of variation in rate-time relationships across five datasets can be explained by time variation alone, and simulations suggest a range of rate-time scaling exponents can be generated by similar evolutionary processes.These results raise challenges in the interpretation of evolutionary rate-time relationships, and highlight segmented regression as one useful approach to understanding temporal variation in evolution.Time-independence of evolutionary change raises new questions about the factors that generate temporal consistency in evolution.
Main Text: Evolutionary rates vary with the time interval over which they are measured.A seemingly general pattern with faster evolutionary rate over short timescales, and slower rates over longer timescales has been recognized for decades (Gingerich 1983(Gingerich , 1993;;Hendry & Kinnison 1999;Gingerich 2001;Harmon et al. 2021).Such apparent time-dependence has subsequently been documented consistently for nearly every possible measure of evolutionary rate (Rolland et al. 2023), including rates of morphological and molecular evolution (Ho et al. 2011), and rates of lineage formation on phylogenies and in the fossil record (Figure 1).Negative rate-time scaling relationships were initially proposed as an artifact that needed to be taken into account, because evolutionary rates over different time intervals were not directly comparable (Gingerich 1983).Yet these negative relationships between evolutionary rate and time continue to generate discussions and stimulate new hypotheses about the possible shared features of the underlying evolutionary processes and ecological environment that could explain such general time-scaling (Henao Diaz et al. 2019; Rolland et al. 2023).Recent technical and statistical explanations to these apparent negative rate-time scaling relationships focus on sampling biases (Louca et al. 2022), such as a the "push of the past" (Rolland et  We here focus on elaborating the simplest explanation for such a pattern: the inevitable negative correlation generated by plotting a ratio against its own denominator (Pearson 1897).This is a well-known statistical issue across a range of scientific fields (Atchley et al. 1976;Gingerich 1983; Prarie & Bird 1989; Jackson & Somers 1991; Arnold 2014), including research on evolutionary rates (e.g., see Box 1 in Harmon et al. 2021).By revisiting the functional dependence of rates and time, analyzing large existing datasets and using new simulations, we show that a negative relationship between evolutionary rate and time is nearly inevitable, regardless of the underlying evolutionary process.We show that the key feature underlying negative rate-time relationships in empirical data is a weak, null, or variable relationship between magnitude of evolution and time.While some of these points have been well-appreciated in the past, our revisit to this phenomenon emphasizes that rate-time plots will nearly always appear to indicate qualitative consistency of the evolutionary process, in the form of negative scaling relationships, across types of evolutionary change.Our analysis also highlights new challenges in interpreting quantitative estimates of rate-time scaling patterns.We highlight the use of segmented regression techniques that can identify significant breakpoints in evolutionary time series

The negative rate-time association is nearly inevitable
Estimates of evolutionary rate can be modelled as a function of evolutionary change scaled by time: where y is the rate-measure, d is some measure of evolutionary change (e.g., change in population mean phenotype, base pair substitutions, log change in number of species in a clade) and t is some measure of time interval over which change was observed or inferred (e.g.The inevitable mathematical dependence between rate and time is an obvious challenge in interpreting the time-dependence of evolutionary rate (Gingerich 1983).For example, a random association between evolutionary change and time will result in a strong negative scaling relationship between rate and time (Kenney 1982).The strength of this negative relationship will inevitably increase when a greater range of timescales is sampled (Pearson 1897).The problem of interpreting such inevitable negative relationships between a rate and time interval have been discussed in several fields outside evolutionary biology, including geology (Schlager et al. 1998) and paleontology (Sheets & Mitchell 2001).The problem has also been raised in evolutionary biology, both in the context of rates of morphological evolution (Gingerich 1983 Where y is evolutionary rate (as in eqn 1).We also consider the amount, or magnitude, of accumulated evolution (rate numerator) as a function of time, where d is the accumulated evolutionary change.This could be the change in trait means between allochronic or synchronic populations, difference in log number of lineages between two sampling points in time (e.g., the stem age of a clade and present), or any other measure of evolutionary change.Rates,  = /, will be constant over time when d scales linearly with t, i.e.  M () is a positive constant over the timescale in question, with prime denoting the first derivative.Rates will increase with time, i.e.  M () > 0 when  MM () > 0; that is, rates increase over time when the amount of evolutionary change is an accelerating function of time.
Noteworthy is that under a null relationship between d and t, such that  M () = 0, the logarithmic regression of rate versus time (log() ~ µ + b*log(t)) will yield a negative slope estimate of b = -1 (Gingerich 1983).This can be seen most easily by expressing equation 3 as a power function, e.g.,  =  5 , in which case we can describe the regression of log() ~ b*log(t) as log(t 0 /t) ~ b*log(t); solving for b = log(t -1 )/log(t) = -1 .
This negative slope estimate provides one cutoff of interest in assessing rate-time scaling relationships: log-log regressions of rate vs time that are shallower than or not significantly different from -1 are consistent with either a null relationship between evolutionary change and time (time independence) or an increasing relationship between accumulated change and time, respectively.For some cases, such as morphological evolution, it is possible to develop predictions for rate-time scaling parameters under various models of evolutionary change (Hansen 2024;Holstad et al. 2024).
The relationships between equations 1-3 are illustrated in Figure 2, which shows the relationship between rate and time.Figure 2A illustrate differing relationships between rate numerator and time illustrated in Figure 2B for a range of power functions.There are few theoretical models that predict the type of accelerating relationship between magnitude of evolution and time that are needed to produce a positive relationship between evolutionary rate and time.For example, neutral and adaptive models of phenotypic and molecular evolution predict either no relationship with time or linear accumulation (constant change) with time (Morgan 1998).Similarly, birth-death models of lineage accumulation also predict either a linear or a decelerating accumulation of log lineage number through time under most conditions.
Even models of time-dependent lineage diversification make no prediction about accelerating log lineage number versus interval time across a sample of clades.Moreover, lineage-throughtime (LLT) plots for single clades diversifying under such a process can produce complex lineage through time plots (Paradis 2015) that are consistent with several different underlying diversification processes (Louca & Pennell 2020).
Some biological and ecological scenarios could potentially lead to accelerating change with time, and thus generate positive relationships between evolutionary rates and time (Figure 2).Such scenarios include "diversity begets diversity" models of lineage diversification (  Which shows that the relationship between rate and time is governed both by the variance in time interval as well as the relationship between accumulated change (d) and time.

Data and simulation reveal persistence of inevitable scaling relationships
Here we revisit five published empirical datasets of evolutionary rate.Reanalysis of these empirical datasets of evolutionary rate-time associations reveals little consistent patterns in the relationship between amount of evolutionary change and time, despite a pervasive negative relationship between rate and time.Interestingly, we even recover a nonlinear negative relationship between evolutionary rate and time when examining phenotypic change that occurred within a timeframe less than the average generation time of the study organism (in this case, primarily long-lived fish; Figure 1E).
This within-generation dataset had a log-log regression of rate and time with a slope that was not significantly different from -1 (b = -0.83,95%CI -1.08, -0.58), indicating that there is no accumulation of change with interval time for intervals less than one generation, as we may expect since such change is likely an outcome of non-accruing environmental effects.This final example demonstrates the appearance of a rate-time scaling relationship for a case that obviously defies evolutionary explanation based on between-generation change.This conclusion is further strengthened by the fact that the slope estimate does not significantly differ from the expectation under a random relationship between rate numerator and time.That is, scaling phenotypic change by time in order to calculate rate generates a non-linear negative relationship with time even for this within-generation change (Figure 1E, J), that is likely to reflect phenotypic plasticity, culling, or other non-evolutionary within-generational environmental effects.
We can then decompose the covariance between rate and time for each of these datasets into its components associated with variation in time and covariance between accumulated change and time, as in equation 4. A caveat in this exercise is that not all of the rate estimates are calculated directly as in equation 1, for example phylogenetic lineage diversification rates in(Henao Diaz et al. 2019), and so we do not expect all of the variance in rate-time relationships to be explained by these two components.Nonetheless, this analysis reveals that 99.7% (54% with morphology data dropped) of the variance in rate-time relationships can be explained by variance in time alone (F1,3 = 916, P = 7.9x10 -5 , Figure 3A), while change-time covariance alone explains only 14.6 % (25% with morphology data dropped) of the variance in rate-time relationships (F1,3 = 0.51, P = 0.52, Figure 3B).Placing both predictors in a multiple regression reveals support for variance in time (t = -25.97,P = 0.0015, df = 2), but not covariance between change and time (t = 0.764, P = 0.52, df = 2), as a significant predictor of rate-time relationships.
While it is true that all covariance relationships will depend strongly on variance in their constituent variables, this analysis reveals the noteworthy contrast in lack of meaningful contribution of covariance between accumulated change and time.
In summary: the striking negative rate-time relationship shared by four independent evolutionary datasets and one non-evolutionary (within generation) dataset reflects, primarily, a lack of strong or consistent association between the amount of evolutionary change (rate numerator) and the timescale over which this change took place (Figure 1 F-J).These results largely recapitulate the findings of original studies from which these datasets were obtained, although we have shown that a decoupling of amount of evolution and timescale of measurement is likely a general feature across a range of evolutionary datasets, from morphological change to lineage diversification, that together describe the pace of life.
Simulations of classical constant-rate evolutionary models reveal that negative rate-time associations readily emerge from these existing models.Phenotypic evolution by random genetic drift results in a Brownian motion process wherein phenotypic variance across lineages increases linearly with time (Figure 4A), and evolutionary rate is time-independent when calculated for an entire clade (4E inset panel).Even this simple constant rate model results in a negative scaling relationship between lineage-specific estimates of evolutionary rate and time (Figure 4E), resulting in a scaling exponent of -½ in a log-log regression of rate versus time (Hansen 2024).
Lineage specific rates are even more strongly associated with time under adaptive model of natural selection towards a stationary optimum (an Ornstein-Uhlenbeck process), where a null relationship between time and divergence (Figure 4B) generates a strong negative relationship between per-lineage rate and time (Figure 4F) with a log-log slope of -1 over long timescales (Hansen 2024), reminiscent of empirical data from vertebrate body size evolution (Figure 1).
However, we note that a stationary OU model readily produces negative rate-time scaling exponents that deviate from -1 when sampling lineages evolving under different values of genetic variance and strength of stabilizing selection (Figure 5).Adding any movement of the optimum in an OU process produces dramatic shifts in the form of rate time scaling, yet also indicates that a range of models incorporating stabilizing selection readily produce scaling relationships similar or identical to that expected under pure genetic drift (Figure 5).
Finally, the simplest model of lineage accumulation, a constant-rate pure-birth process, also leads to these types of associations.The log number of lineages accumulates approximately linearly through time in a pure-birth process, as seen when sampling pure-birth phylogenies differing in age (Figure 4C, 4D).Yet, pure-birth estimators of diversification rate from these phylogenies (Kendall-Moran Yule rates, Figure 4G, S1) as well as maximum likelihood estimates of net diversification from a birth-death diversification model (Figure S2) nonetheless show a negative scaling relationship with time.In principle Yule rates do not inherently show time dependence when estimates of zero rates, which rarely result in phylogenies of more than a single taxon, are included.For example this can be shown by the crown-age estimates using clade age -time data (Magellon-Sanderson rate estimates, Figure 4H).However, even in this case non-zero rate estimates show negative time scaling (Figure 4H).Similarly, sampling directly from the theoretical distribution of pure-birth richness and age yields zero-rate estimates that can cancel out non-zero rates for young clades (O'Meara & Beaulieu 2024), we note that these zero rates are largely unobservable in actual data.Our simulations and others (O'Meara & Beaulieu 2024) show that non-zero rates show negative time-scaling even under a constant-rate pure-birth diversification process.This negative scaling relationships is driven largely by the inclusion of high evolutionary rates estimated over short timescales (Figure 4G), yet the scaling pattern can sometimes remain even when focusing only on larger and older clades (Figure S1).
This illustrates the inherent problems of interpreting evolutionary change scaled by time.
Moreover, we recover the same general pattern when simulating phylogenies of varying age under an accelerating time-dependent diversification process (Figure S3).These simulation results are not in any way meant as evaluations of these models, rather a demonstration that constant rate processes readily and almost inevitably will produce negative rate-time scaling relationships at least in some circumstances.

Discussion
Here we have explored the problem of why all evolutionary rates appear to scale negatively with time, and what information may be gleaned from these rate-time relationships.
First, it is readily apparent there will only exist a limited range of relationships between amount of evolution and time that generate anything other than negative rate-time relationships.Second, revisiting several independent datasets of evolutionary rates demonstrate that existing, widelydocumented negative rate-time scaling relationships clearly suggest weak and inconsistent relationships between the amount of evolution and timescale of measurement, with no evidence for a systematic slowdown in accumulated evolutionary change or time-dependency in evolutionary change.Our analysis of these datasets reveals that variation in rate-time scaling pattern is primarily driven by variation in timespan of the dataset rather than patterns of accumulated evolutionary change.Finally, simulation of some of the most common processbased models of constant-rate evolutionary change reveals that all can readily generate a range of negative rate-time scaling relationships.Although many of these points have been discussed in regards to specific datasets, our results unequivocally demonstrate the challenge in interpreting widespread (Gingerich 1983 Clapham & Karr 2012) that can be used to identify significant breakpoints in evolutionary change across time.Such an approach avoids the problem of statistical confounders between rate and time interval that was highlighted by Gingerich (Gingerich), at least when applied to analysis of change-time relationships, and also allow for the possibility of identifying shifts in the form of rate-time scaling relationships (Hansen 2024).These and related statistical approaches (Uyeda et al. 2011) can be used to identify temporal shifts in the evolutionary process over time.Although analysis of rate-time and change-time plots present the same total information, uninformed comparisons of rate-time relationships will readily generate the appearance of shared features of the evolutionary process.Our analysis revealed that variation in evolutionary rate-time relationships across datasets revealed little information about variation in accumulated evolutionary change.
A main conclusion from our analysis is that we should generally be surprised when evolutionary rates do not show negative time scaling.However, it is likely that some estimators of evolutionary rate do not show the types of inevitable time scaling relationships we have reported; for example, clade-level estimates of Brownian motion rate appear to be less affected.
Indeed, in their review, Harmon et al. (2021) draw a distinction between "summary rates" and "parameter rates"; the former being descriptive summary statistics, such as haldanes, while the latter are inferences of instantaneous rate parameters estimated from a process-based model of evolutionary change, with Brownian motion rate or birth-death lineage diversification rate presenting two examples.In this distinction, and barring inadequate models or data, parameter rates from constant rate models should be free from inevitable time-scaling (Hunt 2012;Harmon et al. 2021).Yet a challenge remains in that many maximum likelihood estimators of parameter rates are ultimately simple functions of amount of evolution and time, and noteworthy is that even most summary rates could in fact be considered parameter rates under appropriate evolutionary models (Harmon et al. 2021).The haldane, for example is a parameter rate under one very simple model of directional trait change (constant directional selection), but note that it shows positive and negative time scaling in different process-based manifestations of directional evolution models (see Fig. 5C).Given that we can never know the true process model giving rise to an evolutionary dataset, and thus our models will always be wrong to some degree (Box 1976), it remains to be seen if there is a general distinction between these categories of rate estimate in regards to prevalence of time scaling.
Our analyses of rate numerators versus time reveal lack of consistent time-dependency in accumulated evolutionary change across datasets (Figure 1).This is also reflected in variation in rate-time scaling slopes, where estimates of rate-time slope range from near unity (morphological body size evolution, lineage net diversification rate) to modestly negative (fossil lineage diversification).The former indicates that over some very broad time scales evolutionary change is largely decoupled from time, while the later case indicates that there is some meaningful accumulation of evolutionary change with time over some time intervals.We note that these diverse datasets contain information from many thousands of species of extant and extinct plants, animals, and prokaryotes, range in time interval from less than one generation to hundreds of millions of years, and together span the entire history of life.One conclusion is that the qualitative persistence of negative rate-time relationships do not provide evidence of shared features of evolution across these disparate data.Existing data (Fig. 1F-J (Rabosky 2009).We emphasize that while there is little consistent relationships between amount of evolution and interval time, evolutionary events can still be substantial, as shown in Figure 1.We also note that timing of key events in the history of life has, of course, played a major role shaping the origins of diversity (Gould 1989).These include shifts in body variance.However, it is noteworthy that even if phenotypic divergence in deep time was generally consistent with drift under observed levels of standing variance, we would still expect to see a negative scaling relationship between lineage specific rates of evolution and time interval (Figure 3).Our analysis suggests that the decoupling of amount of evolution and time applies not only to morphological change, but also to lineage accumulation.For morphological evolution, it is indeed possible to construct explicit quantitative hypotheses for the form of negative scaling relationship expected between rate and time (Holstad et al. 2024).However, as noted above, currently we lack quantitative predictions for other forms of rate-scaling relationships.
Time-invariant evolutionary change still poses other challenges for reconciling microand macroevolution (Uyeda et al. 2011).Constancy of evolutionary rates, and thus linear  Covariance between change and time, also expected to be a component of the covariance between rate and time, is not significantly associated (Panel B).Lines and 95% confidence intervals are from linear models fit separately to each; a multiple regression with both var(time) and cov(change, time) as predictors yielded equivalent conclusions (see text).For each dataset, we plot the standard measure of rate against the timeframe over which that rate was measured, fitting exponential linear models by nonlinear least squares (Rolland et al. 2023).We also plot a measure of the rate numerator against time for each of these datasets, and fit segmented linear regressions to identify significant linear relationships between amount of evolution and time, as well as any potential shifts in the relationship between evolution and time.
For each dataset, we fit nonlinear models of the form rate = a*time b using nonlinear least squares (nls()) in R, as well as segmented regression using the segmented package.We compared models fits of an ordinary linear regression to a model with a single breakpoint, and also when appropriate to a null model containing only an intercept, using the anova() function in R. Data presented in Figure 1  Simulation We simulated three different constant rate evolutionary process.The aim of our simulations was not to evaluate the performance of statistical models used to estimate rates.
Rather, our aim was simple to examine scaling patterns observed in the simplest cases of simulated constant rate evolution.For morphological evolution, we simulated random genetic drift for 1-1000 generations assuming a heritability of 0.50 and a population size of 100.We also simulated evolution under genetic drift and stabilizing selection, with the same demographic and genetic assumptions but assuming weak selection (w 2 = 15) towards an optimum.Each simulation was started with the population at its optimum.For morphological evolution, we plotted rates in units of Haldanes (Gingerich 1993), which was natural as we simulated evolution on the scale of phenotypic standard deviations.We then repeated our simulations of OU evolution but exploring a range of parameter values for genetic variance, the strength of selection (curvature of the fitness surface) and the stability of the optimum.For the later, we sampled changes in the optimum from a normal distribution centered at zero and exploring different values of the variance, producing a random walk of the optimum.
For lineage diversification, we simulated pure-birth phylogenies with a constant speciation rate of 0.15, corresponding to a expected number of 2.3 taxa at timepoint 1 under a continuous time pure birth process.We simulated pure-birth diversification because, while biologically unrealistic, it excludes explanations for rate-time scaling patterns that are based on sampling biases generated by extinction.We also note that although many challenges exist in sampling all possible trees in a simulation, these are identical to challenges observed in actual data; namely, inclusion of trees with zero diversification rate.Given the aims of our simulations noted above, we proceeded by first simulating trees conditioned on number of tips as opposed to clade age, focusing first on trees with at least 3 tip taxa (conclusions remain unchanged by focusing instead on trees with 20 or more tips; Figure S3).We then calculated the Kendall-Moran estimator (Baldwin & Sanderson 1998;Nee 2001) for the Yule process for these trees using the diversitree package (FitzJohn 2012).We also estimated maximum likelihood birth-death diversification rates for the same set of trees constraining diversification rates to be positive.Finally, we also generated trees with ages ranging from 1-35 million years, and plot the maximum likelihood estimator of the crown-age Yule diversification rate, [log(N)-log(2)]/t (Magallon & Sanderson 2001).Our simulations of phenotypic evolution were performed using nested for() loops in R and parameterized as described above.Our simulations of constant rate pure birth phylogenetic diversification was performed using the pbtree() function in the phytools package (Revell 2012).
Our simulations of time-dependent pure birth diversification (Figure S3) were performed using the TESS package (Hoehna 2013).
For our first example, vertebrate body size evolution (Figure 1A, E), past work (Estes & Arnold 2007; Uyeda et al. 2011; Arnold 2014) has shown that body size changes are largely unrelated to time, and in time-series of extant populations found no detectable change (Sanderson et al. 2021).Although a log-log regression of rate vs time yields a slope that is nominally steeper than -1 (b = -1.019,95% CI -1.031, -1.007), which suggests no accumulation of change with interval time, there is a breakpoint in the relationship between morphological change and time (F2, 5388 = 207.2,P < 2.2 x 10 -6 , breakpoint = -2.63log MY (0.072 MY), SE = 0.147).The result is an estimated weak negative relationship between time and evolutionary change, followed by an increasing slope after the breakpoint (Fig. 1A, F; or shallower negative slope on the log scale, Figure 1A inset panel), identical to that found by (Uyeda et al. 2011).Thus, for body size evolution, amount of change is largely unrelated to time interval except over the deepest periods of evolutionary time (Uyeda et al. 2011).Similar results are seen in patterns of lineage accumulation in the fossil record (Figure 1B, G), where a strong negative rate-time relationship (found by (Henao Diaz et al. 2019) and replotted here) reflects a weak and variable relationship between evolutionary change and time (segmented regression F2, 86 = 3.199, P = 0.046, breakpoint = -109 MY, SE = 33.13).However, this segmented regression model was not a significantly better fit than a model with only an intercept; F3, 86 = 2.6, P = 0.057.Consistent with this finding, a log-log regression between rate and time yields a slope that is significantly shallower than -1 (b = -0.24,95%CI -0.36, -0.11) and no significant evidence for a breakpoint (F2, 75 = 1.71,P = 0.18), indicating that there is some accumulation of diversity with interval time.On phylogenetic trees (Figure1C, D, G, H), there is also a consistent striking negative relationships between rate and time (as found by the original studies; Scholl & Wiens 2016; Henao Diaz et al. 2019) that corresponds to underlying variable association between diversity and time(Rabosky et al. 2012; but seeScholl & Wiens 2016).We recover a weak positive linear relationship between diversity and time in one dataset (F1, 112 = 14.17,P = 0.000267; Figure1H) corresponding to a log-log regression that is significantly shallower than -1 (b = -0.53,95%CI -0.68, -.39), indicating some accumulation of diversity with time, and a segmented linear relationship in another dataset (segmented regression F2, 430 = 12.67, P = 4.474 x10 -6 , breakpoint = 4.6 log MY, SE = 0.108) corresponding to a log-log regression of rate and time with a slope that is not significantly different from -1 (b = -1.04,95%CI -1.1, -0.98), indicating the accumulated diversity and interval time are decoupled.
size evolution(Uyeda et al. 2011) or rapid jumps in bill evolution in birds(Cooney et  al. 2017).There is also evidence of substantial variation in evolutionary change in response to different selective agents, including human disturbances and harvesting(Sanderson et al. 2021), but such change in response to different environmental agents does not appear to be systematically linked to time interval over which evolution was recorded.Our results and other recent findings (Voje 2016; O'Meara & Beaulieu 2024) call into question the value of the mere existence of rate-time scaling patterns as empirical evidence of a fundamental disparity between micro and macroevolution (Rolland et al. 2023), since the qualitative existence of a negative relationship is readily explained by many constant-rate processes.For example, consider the widely-appreciated fact that microevolutionary quantitative genetic models of random genetic drift are inconsistent with the amount of evolutionary change in deep time (Hansen & Houle 2004; Houle et al. 2017); we often find far less evolutionary change in deep time than expected via drift alone, based on patterns of standing extant genetic

Fig. 1 .Fig. 2 .
Fig. 1.Evolutionary rate-time relationships reflect null or variable underlying relationships between evolutionary change and time.Panels A-E show evolutionary rate plotted against the timescale of measurement (Rolland et al. 2023), with dashed red lines showing the least squares fit of the function rate = a*time b and inset panels showing the linear relationship on a log-log scale.Panels F-J plot the amount of evolutionary change (i.e., a rate numerator) against time frame for the same datasets, with dashed red lines showing the linear segmented regression, if statistically significant.Panels A and F show vertebrate body size evolution (Uyeda et al. 2011), Panels B and G show diversification of genera from fossil lineages (plants, mammals, and marine animals)(Henao Diaz et al. 2019), Panels C and H show diversification data estimated from time-calibrated molecular phylogenies (Henao Diaz et al. 2019), while Panels D and I show a larger database of phylogenetic diversification rates(Scholl & Wiens 2016) where rates were inferred directly from age and richness data(Magallon & Sanderson 2001).Panels E and J show within-generation morphological changes (mainly from fish harvesting).All nonlinear negative relationships plotted in A-E were statistically significant, whereas relationships in F-J were

Fig. 3 .
Fig. 3. Time interval determines rate-time relationships across datasets.The covariance between rate and time in the five empirical datasets shown in Figure 1 is almost entirely explained by variance in time (Panel A).Covariance between change and time, also expected to be a component of the covariance between rate and time, is not significantly associated (Panel B).Lines and 95% confidence intervals are from linear models fit separately to each; a multiple regression with both var(time) and cov(change, time) as predictors yielded equivalent conclusions (see text).

Fig 5 .
Fig 5. Heterogeneity in rate-time scaling exponents under different forms of morphological evolution.Panels A and B show simulated evolution under drift and stabilizing selection; an OU process.When there is zero variance in the optimum, this is a stationary OU process generating

Fig S2 .
Fig S2.Birth-death estimator diversification rate for trees plotted in Figure 3D, H. Shown (Haldane 1949)r years) and noting that rates are undefined at t = 0. Some classical estimates of evolutionary rate are calculated directly by dividing phenotypic change with time, the latter typically measured in generations or years.For example, Darwins(Haldane 1949) (Nee et al. 1994)14)zJohn 2012 phylogenetic net diversification rate(Magallon & Sanderson 2001).Other estimates are obtained from more sophisticated statistical models of evolution(Nee et al. 1994;FitzJohn 2012;Rabosky et al. 2014).Regardless, all evolutionary rate measures are, per definition, scaled by time (e.g., waiting times between observed branching events on a phylogeny are a direct component of the likelihood in maximum likelihood models of birth-death diversification(Nee et al. 1994)).