Simulating clade diversification
To do these analyses we conducted birth-death simulations of diversification in which rate of origination (λ) and extinction (μ) change over time (see Table 5). The code for this program is contained in Additional file 1 and was run on a Unix (Mac OSX) operating system. In all of our simulations, λ starts out high and then decays exponentially through time. We ran simulations under a variety of conditions, featuring exponentially decreasing (Fig. 3a), constant (Fig. 3b), or exponentially increasing (Fig. 3c) extinction rates (Table 6). Even in the case of decreasing extinction, the rate at which extinction decays is slower than the rate at which speciation decays, with the result that speciation eventually drops below extinction, and the clade begins to lose diversity.
Table 5 List of abbreviations used referring to evolutionary rate parameters Table 6 Schematic of the data analyses. Test type refers to whether the simulation was run to test for the propensity of BiSSE to indicate a most likely model with higher diversification rates in the ancestral state when the character has no effect (“asymmetry bias”) or to test the propensity of BiSSE to indicate a most likely model with higher diversification rates in the derived state when the character does have an effect (“Asymmetry detection”). The simulated phylogenetic trees were analyzed with BiSSE, which takes a tree with branch lengths and character states mapped onto the tips and fits models across the tree using maximum likelihood [2]. Because both our computer simulations and BiSSE are model-based, we make the following distinction for the sake of clarity: in this paper, we use the word “simulation” to refer to the simulated phylogeny incorporating different diversification parameters, the word “model” to refer to likelihood models fit using the BiSSE method, and the word “run” to represent an individual experiment from simulation to model-fitting. BiSSE models have six parameters: rate of origination (λ), extinction (μ), and state transition (q), each of which have values for derived (1) and ancestral (0) character states (Table 5). We use the term “neutral” to describe simulated characters that do not affect evolutionary rates in any way (i.e. λ0 = λ1, μ0 = μ1, q01 = q10). We likewise use the term “non-neutral” to describe a character that is not neutral by the above definition.
We conducted two sets of analyses, the first to determine if the BiSSE model with the highest likelihood includes a non-neutral character when the simulated character is neutral (e.g., λ0 = λ1 in the simulation, but the most likely parameter estimates from the BiSSE analyses have \( \widehat{\uplambda} \)0 ≠ \( \widehat{\uplambda} \)1, and likewise for μ and q). The second group of runs was to determine if declining speciation rate results in the most likely BiSSE model having the character being neutral when the character in the simulation is non-neutral (i.e. λ0 ≠ λ1, but \( \widehat{\uplambda} \)0 = \( \widehat{\uplambda} \)1). For this second set of analyses, we simulated two different ratios of λ1 to λ0, one in which λ1 = 1.33 x λ0, and one in which λ1 = 1.05 x λ0.
We began all simulations with extinction rate at half the speciation rate, and defined a single time unit as the time required for speciation and extinction rate to become equal (the point of peak richness). For real clades in the fossil record, the number of years to peak richness varies widely [10]. In our shortest runs (ended at 0.25 time units), speciation rate still well exceeds extinction rate and the clade is rapidly diversifying. At the end of the longest (two time units) simulations, speciation rate is half that of extinction rate and the clade is declining.
Our speciation rate asymmetries between λ0 and λ1 for non-neutral characters are smaller than those that Davis et al. [25] investigated. We did this for the following reason: at time 0, λ0 is twice μ0. At time 2, λ0 is half μ0. Because λ1 is a flat multiple of λ0, increasing the ratio of the two rates to 2:1 or beyond would result in species with the derived state continuing to have net positive diversification rates even at the end of our longest runs. We therefore used smaller, but still paleontologically realistic (e.g. [10]) values of 1.33 and 1.05 as the ratio of λ1 to λ0. Additionally, this real rate asymmetry lies in the opposite direction (i.e. λ1 > λ0) from what we predicted the most likely BiSSE model to be (\( \widehat{\uplambda} \)0 > \( \widehat{\uplambda} \)1). We did this because we hoped that the direction of the rate asymmetry in the best model would enable us to distinguish between a model-fitting bias created by our simulations’ violation of BiSSE’s assumptions as opposed to detection of real asymmetry created by our simulated non-neutral character.
We ran 100 simulations with each combination of parameters. Simulations that had fewer than 50 surviving species were re-run until the simulation had a total of 50 surviving species (36 for our punctuated equilibrium simulation – see below). Thus, each combination of parameters had a total of 100 simulations for which there was at least the minimum quota of species, and in most of our runs the number of surviving species is considerably larger (Fig. 4). We excluded simulation runs with few surviving species because Davis et al. [25] demonstrated that the statistical power to distinguish among BiSSE models is reduced when examining small clades. By setting this arbitrary cutoff, we introduce a significant acquisition bias to our study. This bias also exists in real clades, however, as workers tend to analyze large clades and especially those that have an imbalance in richness among species bearing specific traits. Because this acquisition bias exists in analyses of real clades, we decided not to correct for it, as doing so would have been both methodologically difficult and would introduce yet additional biases.
Simulating morphological evolution
We ran two types of simulation of character evolution in the context of our declining speciation and extinction rates. The first (“continuous-time”) type of simulation allows the character state of a species to change at any time, not just at speciation events, and thus emulates anagenetic character evolution. We set the values of q01 and q10 to 0.6 changes per time unit. This number matches the state transition rates typical of the punctuated equilibrium simulation described below. In the continuous simulation, q10 and q01 are time-invariant, irrespective of how λ and μ may be changing. The expected number of changes along the line of any surviving species from the root of the tree to the tip is thus the same regardless of how many branching events took place. This simulation is implicit in the assumptions of BiSSE as implemented in diversitree [26], and is equivalent to “phyletic gradualism” sensu Eldredge and Gould [18].
The second (“punctuated”) type of simulation emulates cladogenetic character evolution in which state changes happen only at speciation events. As a consequence, q01 and λ0 are linked and change together, as are q10 and λ1. We used a probability of state change (in either direction) of 0.09 per branching event, which is typical of minimum steps parsimony-based phylogenetic reconstructions of fossil taxa [27]. The expected number of total state changes along any branch thus depends on the number of branching events, and is independent of time required for those branching events to occur. The shortest simulations yield a geometric mean of 4.7 changes and the longest 15.2 changes, but because of extinction, not all of these changes appear in terminal taxa. This punctuated equilibrium simulation represents yet another violation of BiSSE’s assumptions in addition to time-variant speciation and extinction rates. Newer models, such as ClaSSE [19] do allow for punctuated equilibrium scenarios. However, we chose to use BiSSE because we sought to quantify whether anagenetic and cladogenetic forms of character evolution impact the biases that we here investigate.
Assessment
We conducted BiSSE analyses of our simulated clades using the diversitree package in R [26]. BiSSE models in diversitree are compared using likelihood ratio tests in the ANOVA function in R to assess the significance level of asymmetries between \( \widehat{\uplambda} \)0 versus \( \widehat{\uplambda} \)1, \( \widehat{\upmu} \)0 versus \( \widehat{\upmu} \)1, and \( \widehat{\mathrm{q}} \)01 versus \( \widehat{\mathrm{q}} \)10. The null hypothesis of these likelihood ratio tests is that the character is neutral (i.e. \( \widehat{\uplambda} \)0 = \( \widehat{\uplambda} \)1, \( \widehat{\upmu} \)0 = \( \widehat{\upmu} \)1, \( \widehat{\mathrm{q}} \)01 = \( \widehat{\mathrm{q}} \)10). This is different from the null hypothesis of BiSSE itself, which is that there is no variation in evolutionary rates across the phylogenetic tree, which we violate already by the time-variance of speciation and extinction rates in our simulation. Thus, for the sake of clarity, we do not discuss these errors as type-I and type-II in this paper. Because of our continually decreasing speciation rates, there is no BiSSE model that correctly represents our simulated phylogeny, but some BiSSE models are more incorrect than others. We assessed the statistical significance of the frequency by which the best BiSSE model included a neutral or non-neutral character using these likelihood ratio tests (e.g., if λ0 = λ1, how often does AIC indicate that \( \widehat{\uplambda} \)0 = \( \widehat{\uplambda} \)1 is the best model and how often does AIC imply that \( \widehat{\uplambda} \)0 ≠ \( \widehat{\uplambda} \)1 is a better model?).