The binary-state speciation and extinction (BiSSE) model has been used in many instances to identify state-dependent diversification and reconstruct ancestral states. However, recent studies have shown that the standard procedure of comparing the fit of the BiSSE model to constant-rate birth–death models often inappropriately favours the BiSSE model when diversification rates vary in a state-independent fashion. The newly developed HiSSE model enables researchers to identify state-dependent diversification rates while accounting for state-independent diversification at the same time. The HiSSE model also allows researchers to test state-dependent models against appropriate state-independent null models that have the same number of parameters as the state-dependent models being tested. We reanalyse two data sets that originally used BiSSE to reconstruct ancestral states within squamate reptiles and reached surprising conclusions regarding the evolution of toepads within Gekkota and viviparity across Squamata. We used this new method to demonstrate that there are many shifts in diversification rates across squamates. We then fit various HiSSE submodels and null models to the state and phylogenetic data and reconstructed states under these models. We found that there is no single, consistent signal for state-dependent diversification associated with toepads in gekkotans or viviparity across all squamates. Our reconstructions show limited support for the recently proposed hypotheses that toepads evolved multiple times independently in Gekkota and that transitions from viviparity to oviparity are common in Squamata. Our results highlight the importance of considering an adequate pool of models and null models when estimating diversification rate parameters and reconstructing ancestral states.

#### Figure S1. Plot of the posterior and prior distributions from BAMM analysis using an expected number of shifts prior of 1.

Figure S1. Plot of the posterior and prior distributions from BAMM analysis using an expected number of shifts prior of 1.

Fig. S1 - BAMM_prior_posterior.pdf

#### Figure S2. Ancestral state reconstruction of viviparity for Liolaemidae using the uncorrected sample fraction

Figure S2. Ancestral state reconstruction of viviparity for Liolaemidae using the uncorrected sample fraction from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S2 - Liolaemidae_uncorrected_frac_states.pdf

#### Fig. S3 - BAMM net diversification

Figure S3. Maximum shift credibility tree from BAMM showing diversification rate shifts across Squamata with tip labels. Rates shown are net diversification rates.

Fig. S3 - BAMM_NetDiv_labelled.pdf

#### Fig. S4 - BAMM speciation

Figure S4. Maximum shift credibility tree from BAMM showing diversification rate shifts across Squamata with tip labels. Rates shown are speciation rates.

#### Fig. S5 - BAMM Extinction

Figure S5. Maximum shift credibility tree from BAMM showing diversification rate shifts across Squamata with tip labels. Rates shown are extinction rates.

#### Fig. S6 - Gekkota Diversitree BiSSE states

Figure S6. Ancestral state reconstruction of gekkotan toepads using the BiSSE model from Diversitree. Colored circles at tips and nodes indicate the states of species and probability that ancestors were toepad-bearing (white) or padless (black).

Fig. S6 - Gamble_Diversitree_BiSSE_states.pdf

#### Fig. S7 - Gekkota BiSSE-like HiSSE states

Figure S7. Ancestral state reconstruction of gekkotan toepads from HiSSE using the BiSSE-like model. Colored circles at tips and nodes indicate the states of species and probability that ancestors were toepad-bearing (white) or padless (black). Branches are colored by model-averaged diversification rates.

Fig. S7 - Gekkota_BiSSE-like_HiSSE_states.pdf

#### Fig. S8 - Gekkota HiSSE toepad states

Figure S8. Ancestral state reconstruction of gekkotan toepads from HiSSE model-averaged across all tested models with tip labels. Colored circles at tips and nodes indicate the states of species and probability that ancestors were toepad-bearing (white) or padless (black). Branches are colored by model-averaged diversification rates.

Fig. S8 - Gekkota_toepad_states.pdf

#### Fig. S9 - Squamata HiSSE states

Figure S9. Ancestral state reconstruction of viviparity for Squamata with from HiSSE model-averaged across all tested models with tip labels. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S9 - Squamates_states.pdf

#### Fig. S10 - Scincoidea HiSSE states

Figure S10. Ancestral state reconstruction of viviparity for Scincoidea from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S10 - Scincoidea_states.pdf

#### Fig. S11 - Serpentes HiSSE states

Figure S11. Ancestral state reconstruction of viviparity for Serpentes from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S11 - Snake_states.pdf

#### Fig. S12 - Viperidae HiSSE states

Figure S12. Ancestral state reconstruction of viviparity for Viperidae from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S12 - Viper_states.pdf

#### Fig. S13 - Anguimorpha HiSSE states

Figure S13. Ancestral state reconstruction of viviparity for Anguimorpha from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S13 - Anguimorpha_states.pdf

#### Fig. S14 - Phrynosomatidae HiSSE states

Figure S14. Ancestral state reconstruction of viviparity for Phrynosomatidae from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S14 - Phrynosomatidae_states.pdf

#### Fig. S15 - Liolaemidae HiSSE states with corrected sampling fraction

Figure S15. Ancestral state reconstruction of viviparity for Liolaemidae using the corrected sampling fraction from HiSSE model-averaged across all tested models. Colored circles at tips and nodes indicate the states of species and probability that ancestors were oviparous (black) or viviparous (white). The legend in the lower left shows the range of rates, as well as the distribution of rates across the tree (upper histogram) and the frequency of tips in each state (lower bars).

Fig. S15 - Liolaemidae_states.pdf

#### Table S1. HiSSE model fits for Liolaemidae using the ucorrected sampling fraction

Table S1. HiSSE model fits for Liolaemidae using the ucorrected sampling fraction

Table S1_uncorrected_sampling_Liolaemidae.xlsx

#### Table S2. Diversitree and BiSSE-like HiSSE Gekkota parameter estimates

Table S2. Parameters estimated using the BiSSE model in Diversitree and the BiSSE-like model in the HiSSE package for Gamble et al. dataset.

Table S2 - Gamble parameters.xlsx

#### R code, tree, and data for Gekkota HiSSE analyses

This zip file contains the tree, trait data, and commented R script to run HiSSE analyses. The null-four nine-rate model requires the separately uploaded file containing the modified null-four function.

Gekkota_analyses.zip

#### HiSSE.null4.9rate.R

Null-four function modified to allow nine transition rates among hidden states. To run this with 9 transition rates, set trans.type = "All.no.dual", all other arguments are identical to the hisse.null4 in the HiSSE package.

#### Viviparity analyses R scripts

This zip file contains two commented R scripts. One fits HiSSE models to the full squamate dataset. The other fits models to subtrees. See comments in scripts for details. The null-four nine-rate model requires the HiSSE.null4.9rate.R file separately available Dryad. For the location of the tree and data used, see comments in scripts.

Viviparity analyses.zip