Figure 1.
Example of tree simulated under mixture of three distinct evolutionary processes.
(A) Clade diversification under constant-rate “background” diversification process with λ = 0.032 and μ = 0. (B) Shift to new adaptive zone with subsequent diversity-dependent regulation of speciation and diversity-independent extinction (blue branches; λ0 = 0.395; K = 66; μ = 0.041). (C) Another lineage shifts to diversity-dependent speciation regime (red branches; λ0 = 0.21; K = 97; μ = 0.012). Total tree depth is 100 time units. Despite undergoing two distinct diversity-dependent slowdowns in the rate of speciation, the overall gamma statistic [63] for the tree is positive (γ = 2.51) and provides no evidence for changes in the rate of speciation through time. Note that a tree with three distinct processes contains two distinct transitions between processes.
Figure 2.
BAMM analysis of example tree (Figure 1).
Example tree was simulated under three distinct processes (one constant rate and two diversity dependent processes; two transitions in total). The tree was analyzed under (i) the full multi-process BAMM model with time-variable speciation; (ii) a constrained multi-process BAMM with time-constant speciation; and (iii) a fully constrained 1-process constant-rate birth-death model. (A) Log-likelihoods for thinned MCMC chains for the constant rate birth-death process (bottom), the time-constant multi-process model (middle), and the full BAMM model with time-varying speciation (top). (B) Numbers of transitions during rjMCMC sampling when model is constrained to time-constant speciation rates; sidebar gives frequency distribution of sampled states. (C) Numbers of transitions under full BAMM model with time-variable speciation processes. Black sidebar denotes true number of transitions in generating model. The true number of transitions was estimated correctly only when the assumption of time-constant rates was relaxed.
Figure 3.
BAMM analyses of constant-rate phylogenies and prior on Poisson rate parameter (γ).
Histograms in (A–C) display the frequency distribution of the estimated number of processes in the model with the maximum a posteriori (MAP) probability as a function of three different priors on the Poisson rate parameter Λ (γ = 1; γ = 5; γ = 10). This “best-fit” model was simply the model that was visited most often during the MCMC simulation of the posterior. (D–F) show the distribution of posterior probabilities for the true model (M0). With a relatively flat prior on models (γ = 10), the MAP model is biased towards a model with 2 processes ( = 1 transition). However, the posterior probability of the true model M0 remains substantial (F), and M0 nonetheless had a posterior probability greater than 0.10 for the vast majority of simulations. Results are based on 500 simulated phylogenies per γ scenario.
Figure 4.
Frequency distribution of evolutionary rate regimes estimated using BAMM, compared with true number of processes.
For each simulation, the estimated number of processes was simply the model that was most frequently sampled during MCMC simulation of the posterior distribution. Black bars denote true number of processes in generating model. For example, 84% of trees simulated under a single-shift exponential change model (exp2; top panel; two processes in the generating model) were correctly inferred to have been generated under a two-process model. For a diversity-dependent model with five processes (DD5), power to detect the true number of processes is lower, although though most analyses (80.4%) recovered either 4 or 5 process models as the MAP model. Results for each model are based on 500 simulated phylogenies and used a conservative γ = 1 prior on the expected number of non-root processes (see Figure 3).
Figure 5.
BAMM estimates of speciation and extinction rates for phylogenies simulated under constant-rate birth-death process.
(A) Relationship between speciation rate in generating model and reconstructed mean rate across the tree under BAMM. Solid black line: identity line, expected if λTRUE = λESTIMATED. Solid gray line: fitted OLS regression to estimates (black points) obtained using full BAMM model (multiple processes with time-variable speciation rates). (B) Corresponding extinction rate estimates for same set of trees.
Figure 6.
Precision and bias of BAMM in the estimation of branch-specific rates of speciation.
Phylogenies were simulated under 5 distinct evolutionary scenarios. For each simulated phylogeny, I reconstructed branch-specific speciation rates using BAMM and modeled these as a function of the true branch rates from the generating model. Frequency distributions of the estimated slope of this relationship are shown in the left column for each simulation scenario. Center column denotes corresponding r2 values from the same OLS regressions. Right column is distribution of mean relative rate differences (RRD) for each scenario. A value of 1 implies that, on average, branch-specific speciation estimates are unbiased; a value of 0.5 would imply that branch-specific estimates are, on average, equal to 50% of the true value. Results for each simulation scenario are based on 500 simulated phylogenies (thus giving 500 slopes, r2 values, and RRD values for each simulation scenario).
Table 1.
Relationship between branch-specific BAMM estimates of extinction and true rates in the simulation model.
Figure 7.
Frequency distribution of evolutionary rate regimes estimated using MEDUSA, compared with true number of processes.
Phylogenies were simulated under 5 distinct evolutionary scenarios. For each simulation, the number of distinct rate partitions was estimated using the stepwise AICc algorithm as implemented in MEDUSA. Black bars denote true number of processes in generating model. MEDUSA consistently underestimates the true number of processes in simulated datasets when rates of speciation vary through time. Comparable results for BAMM using the same set of simulated datasets are shown in Figure 4. A total of 500 simulated datasets were analyzed per diversification scenario.
Figure 8.
Precision and bias of MEDUSA in the estimation of branch-specific rates of speciation.
For each simulated phylogeny, MEDUSA was used to estimate the number, location, and parameters of diversification rate shifts. The resulting branch-specific rates of speciation were compared with the true branch rates from the generating model. Results are based on the same simulated datasets analyzed with BAMM and can be directly compared to those shown in Figure 6. Branch-specific speciation rates estimated with MEDUSA show little correspondence with true rates when rates vary through time, at least in comparison to rates estimated with BAMM.
Figure 9.
Dynamics of cetacean diversification through time as revealed by BAMM analysis.
(A) Phylogeny of cetaceans [51], with branch lengths drawn proportional to their marginal speciation rate as estimated using BAMM. A large increase in the rate of speciation (>6-fold) occurred in one of the ancestral branches leading to the Delphinidae (including or excluding the killer whale, Orcinus orca). Despite this increase, the overall trend is towards decelerating rates through time. (B) Cetacean phylogeny with branch lengths scaled by the posterior probability that they contain a rate shift. Numbers above branches denote branch-specific shift probabilities. The probability that a rate shift occurred on at least one of these three branches was 0.975. No other branches had shift probabilities exceeding 0.02. (C) Posterior distribution of the number of distinct processes (including the root process) on the cetacean phylogeny. A two-process model vastly outperforms a one-process model. (D) Speciation rates through time during the extant cetacean radiation; distinct shaded regions denote (from bottom) 0.05, 0.25, 0.50, 0.75, and 0.95 quantiles on the posterior distribution of rates at a given point in time. Massive spike in mean speciation rates at 7.5 Ma corresponds to the early radiation of the Delphinidae clade. (E) Corresponding extinction through time curve.
Figure 10.
Sensitivity of marginal rate estimates to prior on Poisson rate parameter.
Each panel shows a pairwise plot comparing branch-specific (marginal) diversification rate estimates for two values of γ for the Cetacean dataset, with results for speciation and extinction separated by the diagonal. Speciation rate estimates for the cetaceans are remarkably robust to choice of prior: even γ = 10 and γ = 0.1 yield strikingly similar marginal distributions for branch-specific speciation rates. This is generally not true for extinction, where mean marginal rates for each branch were more sensitive to prior formulation. However, extinction was nonetheless estimated to be low overall regardless of the prior γ.