Understanding the relative influence of various abiotic and biotic variables on diversification dynamics is a major goal of macroevolutionary studies. Recently, phylogenetic approaches have been developed that make it possible to estimate the role of various environmental variables on diversification using time-calibrated species trees, paleoenvironmental data, and maximum-likelihood techniques. These approaches have been effectively employed to estimate how speciation and extinction rates vary with key abiotic variables, such as temperature and sea level, and we can anticipate that they will be increasingly used in the future. Here we compile a series of biotic and abiotic paleodatasets that can be used as explanatory variables in these models and use simulations to assess the statistical properties of the approach when applied to these paleodatasets. We demonstrate that environment-dependent models perform well in recovering environment-dependent speciation and extinction parameters, as well as in correctly identifying the simulated environmental model when speciation is environment-dependent. We explore how the strength of the environment-dependency, tree size, missing taxa, and characteristics of the paleoenvironmental curves influence the performance of the models. Finally, using these models, we infer environment-dependent diversification in two empirical phylogenies: temperature-dependence in Cetacea and δ 13C-dependence in Ruminantia. We illustrate how to evaluate the relative importance of abiotic and biotic variables in these two clades and interpret these results in light of macroevolutionary hypotheses. Given the important role paleoenvironments are presumed to have played in species evolution, our statistical assessment of how environment-dependent models behave is crucial for their utility in macroevolutionary analysis.

#### Supplemental Figure 1

Recovered parameter estimates for trees simulated with a constant λ and an exponential dependency of μ on temperature. Simulations with: (A) varying λ0, constant μ0, and constant αμ; (B) constant λ0, varying μ0, and constant αμ; and (C) constant λ0, constant μ0, and varying αμ. Dashed red lines mark the simulated parameter value.

Figure_Supplemental_1.pdf

#### Supplemental Figure 2

Recovered parameter estimates for trees simulated with an exponential dependency of λ and μ on temperature. Dashed red lines mark the simulated parameter value.

Figure_Supplemental_2.pdf

#### Supplemental Figure 3

The effect of undersampling on parameter estimation. Parameter estimates for trees simulated with (a) a positive exponential dependency of λ on temperature and constant extinction (μ = 0.05) and (b) a positive exponential dependency of μ on temperature with constant speciation (λ = 0.5) by fitting the temperature-dependent model. Parameter estimates are shown for trees with increasingly smaller sampling fractions, which were achieved by jackknifing the simulated trees by a fixed % and then fitting the environment-dependent model. Simulated parameters are marked by dashed red lines. Model selection determined by (c) AICc and (d) Akaike weights for trees simulated with a positive exponential dependency of λ on temperature and constant extinction (grey, temperature- dependent model; light blue, time-dependent model; black, constant-rate model). In (d), weights are averaged across all trees within each undersampled bracket.

Figure_Supplemental_3.pdf

#### Supplemental Figure 4

The ability to correctly recover trees simulated under a time-dependent model or temperature-dependent model where the dependence is on extinction. Columns in each plot show the percentage of trees recovered (A,B) or Akaike weights (C,D) for constant-rate (black), time-dependent (light blue), or temperature-dependent extinction (brown) models for a set of trees simulated under (A,C) time-dependence or (B,D) temperature-dependence. Trees simulated under time-dependent extinction (B,D) were also fit with time-dependent speciation models (grey). The x-axis shows the strength of the dependencies (i.e., $\alpha$ value) used to simulate each set of trees. Akaike weights are averaged over all trees simulated with the same $\alpha$.}

Figure_Supplemental_4.pdf

#### Supplemental Figure 5

Parameter estimates for trees simulated with a positive (left) and negative (right) exponential dependency of $\lambda$ on temperature by fitting the temperature-dependent model. Parameter estimates are shown for trees with different species richness. Simulated parameters are marked by dashed red lines. Akaike weights are shown for trees fitted with temperature-dependent models (grey), time-dependent models (light blue), and constant-rate models (black). Weights are averaged across all trees within each species richness bracket.

Figure_Supplemental_5.pdf

#### Supplemental Figure 6

Parameter estimates for (a) λ0, (b) μ0, (c) αμ for trees simulated with a positive exponential dependency of μ on temperature by fitting the temperature- dependent model. Parameter estimates are shown for trees with different species richness. Simulated parameters are marked by dashed red lines. (c, inset) A magnified plot of αμ estimates. (d) Akaike weights averaged over all trees for each species richness cohort for models fitted with temperature-dependent models (grey), time-dependent models (light blue), and constant-rate models (black).

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#### Supplemental Figure 7

Model selection and parameter estimation in temperature- dependent trees under various spline smoothings in the fitted model. (a) Plots of temperature curve splines smoothed by different degrees of freedom. The time-series of temperature data are shown in grey dots and the smoothed curves in black lines. (b,c) Parameter estimates for trees simulated with an exponential dependency of speciation on temperature (b, λ = 0.2e0.05·T(t); C, λ = 0.3e−0.05·T(t)) with temperature curves determined using generalized cross-validation (degrees of freedom=208), where the fitted temperature-dependent models have temperature curve splines smoothed by different degrees of freedom. Simulated parameters are marked by dashed red lines. (d) The percentage of temperature-dependent trees, simulated with a temperature curve determined using generalized cross-validation, best sup- ported by models fit with temperature curve splines smoothed by different degrees of freedom versus constant-rate models and time-dependent models with an exponential dependence on speciation.

Figure_Supplemental_7.pdf

#### Supplemental Figure 8

Mean of the Akaike weights for constant-rate models (black), time-dependent models (light blue), and environment-dependent models (colors correspond to Figure 1) across all environment-dependent λ trees with the same αλ (see Figure 4).

Figure_Supplemental_8.pdf

#### Supplemental Figure 9

Effect of different characteristics on the rate of recovery of environmental curves. The ability to correctly recover the simulated model at different values of αλ for (a) abiotic and biotic variables (see Figure 4) and (b) linear and non-linear environmental curves. (c) Barplots for the slope and intercept for regression models fit to autocorrelation functions for multiple lag-times for each environmental curve; the correlation, slope, and degrees of freedom estimated for generalized least squares (GLS) linear fits to each environmental curve; and the average rate of change of each environmental variable with respect to time. Bars are colored according to panel a.

Figure_Supplemental_9.pdf

#### Supplemental Figure 10

Rate of recovery for trees simulated with an exponential de- pendency of μ on different paleoenvironments, X, (μ = 0.02e^{αμX(t)}) for varying values of αμ and constant λ (λ0 = 0.15). (Top) Simulated trees fitted with models with an exponential dependency of μ on the paleoenvironment and a constant λ. (Bottom) Simulated trees fitted with models with an exponential dependency of λ on the paleoenvironment and a constant μ. Colors correspond to Figure 1.

Figure_Supplemental_10.pdf

#### Supplemental Figure 11

Environment-dependency in Ruminantia computed from the Ruminantia supertree (11). AICc support for different environment-dependent models, a constant-rate birth-death model, and an exponential time-dependent model (without extinction) on a distribution of 5000 posteriorly sampled probabilities of the Ruminantia supertree. All environment-dependent models have an exponential dependency on the environmental variable. Colors correspond to Figure 1.

Figure_Supplemental_11.pdf