# Testing Hypotheses of Diversification in Panamanian Frogs and Freshwater Fishes Using Hierarchical Approximate Bayesian Computation with Model Averaging

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## Abstract

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## 1. Introduction

_{e}), mutation rate, and generation time create high variation in gene tree topologies and their depths (correlated with timing of their divergences; e.g., [11,19]), what appear as multiple divergence events may actually reflect the obscured signal of a single event [20]. Thus, comparing independent gene-tree depths and divergence time estimates can lead to erroneous inferences of multiple vicariance events.

## 2. Materials and Methods

#### 2.1. Taxon Sampling and Molecular Data

#### 2.2. Genetic Diversity and Neutrality

#### 2.3. Estimating Gene-Tree Depths Using Bayesian Dating

_{MRCA}s) and their Bayesian credible intervals for each species/lineage using BEAST version 1.8.2 [65]. BEAST runs (100 million generations, sampled every 4000; ‘burn-in’ = 10%) started from UPGMA (unweighted pair group method with arithmetic mean) tree topologies, employed HKY + Γ + I site models, and used coalescent constant size tree priors [66]. We evaluated the fit of different molecular clock models and investigated clock-likeness of the data by comparing results of strict-clock and relaxed-clock (uncorrelated lognormal model; [67]) runs on each dataset. Due to uncertainty over frog mtDNA mutation rates, we specified a range of 0.1–1.3% lineage

^{−1}Myr

^{−1}(per million years) in the frog runs spanning rates of protein-coding mtDNA gene evolution documented in a broader mtDNA dataset from one of our focal taxa [51]. This ‘frog rate’ spans Macey et al.’s [68] Mongolian toad (Bufo bufo) rate of 0.69% lineage

^{−1}Myr

^{−1}, which is commonly used to date patterns of herpetofaunal diversification [51]. Likewise, we specified a ‘fish rate’ of 0.17–1.4% lineage

^{−1}Myr

^{−1}in fish runs, a range spanning mutation rates estimated in 15 previous studies of teleost freshwater fish mtDNA (refs. in [69,70]). Rate ranges were supplied to the program as uniform priors. We summarized posterior distributions and ensured convergence and adequate effective sample sizes (all ESS >> 200) using Tracer version 1.5 (available at: http://beast.bio.ed.ac.uk/Tracer). In TreeAnnotator version 1.8.2, we summarized the posterior distribution of trees from each run by calculating a maximum clade credibility (MCC) tree annotated with median node ages from a sample of 5000 post-burn-in trees obtained using the ‘BEASTPostProc.sh’ script in PIrANHA version 0.1.4 [71].

#### 2.4. Tests for Synchronous Diversification

_{1}–M

_{4}), to compare using model averaging. Each model class consisted of one of two uniformly distributed priors on population divergence times (τ), ancestral population size (θ

_{A}), and daughter population size parameters (θ

_{D}; Table 2). Second, we obtained 5 million random (simulated) samples from each model class specified by a discrete uniform hyperprior distribution Pr(M

_{k}) = 1/4. We visually checked for efficient prior sampling by conducting principal components analysis on 1000 prior draws from each model class in R. Third, we obtained the ABC joint posterior distribution using the default summary statistic vector (D) from MTML-msBayes and rejection sampling to identify the 1000 closest Euclidean distances between the observed summary statistics (D*) for the data and D

_{i}calculated from 20 million random draws across all four priors (M

_{1}–M

_{4}). However, prior to rejection sampling, we rescaled the dispersion index of population divergence times Ω (=Var[τ]/E[τ]; the ratio of variance to the mean of the divergence times) and the mean assembly-wide divergence time, E[τ], from models with smaller upper θ

_{D}prior bound values (frog and fish M

_{1}, M

_{2}, and M

_{4}) to have the same coalescent units as the other model, M

_{3}[24]. Resulting estimates of Ω and E[τ] were weighted by Bayesian model averaging [24]. Last, we conducted hypothesis testing by comparing hyperposterior probability distributions of Ω estimates, to determine whether the data supported single or multiple diversification periods.

_{10}Bayes factors under the parameter thresholds above while accounting for prior support for the hypotheses, using B

_{10}“weight of evidence” criteria in Jeffreys [72] and Kass and Raftery [73]. We estimated mean assemblage-wide divergence times by converting model-averaged E[τ] estimates (in coalescent units of 4N

_{ave}generations) to absolute time (T

_{div}) using the equation T

_{div}= E[τ] × (θ

_{ave}/μ), where μ is the assumed mutation rate per site per generation and θ

_{ave}(per site) is the mean of the upper θ prior. Conversions used mutation rates equivalent to 0.7% lineage

^{−1}Myr

^{−1}and 0.785% lineage

^{−1}Myr

^{−1}, the median rates of uniform ‘frog rate’ and ‘fish rate’ priors used in our BEAST analyses. We assumed an average generation time of 1 year, which was also used in previous studies of our focal taxa ([51,52,54]; Michael J. Ryan, pers. comm.), and we acknowledge that divergence time estimates are sensitive to generation times. Shell and R scripts used during our MTML-msBayes analyses are accessioned in Mendeley Data [74].

## 3. Results

#### 3.1. Genetic Diversity and Neutrality

#### 3.2. Estimating Gene-Tree Depths Using Bayesian Dating

_{MRCA}s for each of the seven population-pairs were consistent across multiple BEAST runs, which had ESS values >500 for nearly all parameters. Results also were similar across strict-clock and relaxed-clock runs; however, we only present the results of the strict-clock analyses because 95% highest posterior densities (HPDs) of ‘ucld.stdev’ (standard deviation of the relaxed clock) abutted zero, indicating that the data could not reject strict clock models. The mitochondrial MCC time trees had variable gene-tree depths (Figure 3), and geometric mean t

_{MRCA}estimates (closer to peak likelihood values than the means) varied substantially across species/lineages, ranging from 1.71 Ma in A. coeruleopunctatus to 10.79 Ma in C. crassidigitus (Figure 4; frog geometric mean t

_{MRCA}range: 3.78–10.80 Ma; fish geometric mean t

_{MRCA}range: 1.71–4.88 Ma). Consistent with patterns of DNA polymorphism above, t

_{MRCA}s of the fish lineages exhibited less variation with a much narrower region of overlap in their coalescence times (1.92–5.57 Ma) as compared with that of the frog lineages (3.88–14.39 Ma), although the frog t

_{MRCA}s also overlapped substantially (Figure 4). Hereafter, geometric mean values are relied upon because they were closer to peak likelihood parameter estimates.

#### 3.3. Tests for Synchronous Diversification

_{10}= 1.14 for Ω < 0.01 versus Ω > 0.01; fish B

_{10}= 1.33 for Ω < 0.01 versus Ω > 0.01) as well as the asynchronous diversification model (frog B

_{10}= 0.88 for Ω > 0.01 versus Ω < 0.01; fish B

_{10}= 0.75 for Ω > 0.01 versus Ω < 0.01). Bayes factors were technically less than 1 for two or more divergences, but at best this provides very weak negative evidence for asynchronous divergence [72,73]. Likewise, posterior probabilities for the best-supported models were low (<0.5) and corresponding Bayes factors for synchronous diversification were weak, being approximately less than or equal to 1.

## 4. Discussion

#### 4.1. Comparative Phylogeography of Panamanian Frog and Fish Assemblages

_{MRCA}s for the taxon-pairs (Figure 4), strong peaks in the Ω posteriors near zero (Figure 5) and credible intervals including zero (Table 2), and peak-likelihoods and credible intervals of assemblage E[τ]. By contrast, Bayes factor model selection indicated that the data provide only a marginal accumulation of evidence in favor of synchronous diversification and against the null scenario of asynchronous diversification. The msBayes approach has been shown to correctly reject synchronous diversification using only mtDNA and summary statistics such as those used herein (e.g., [23,24,25]), and visual checks suggested that our priors were efficient samplers of the data (Figure S1). Moreover, we avoided the problem of overly broad or narrow τ priors (e.g., in [26,32]), which can cause ABC samplers to explore parameter space exceeding saturation effects on mitochondrial genes [24], by matching the upper bounds of these priors to empirical estimates from the data. As a result, we conclude that additional genetic data from unlinked loci or additional species, or improved methods for hABC or Bayes factor estimation, are needed to more confidently assess the timing and number of events at the WPI break in these taxa. Nevertheless, the much narrower credible intervals of our assemblage divergence times relative to the Bayesian gene-tree depths inferred in BEAST (Figure 3 and Figure 4) indicate that accounting for coalescent processes and changes in population sizes through time in our models yielded much more precise estimated divergence times across the WPI break than were previously available. Assuming that peaks in Ω and E[τ] contain one or multiple clusters of population divergence events, and acknowledging limitations and caveats of our mtDNA data (see Introduction and Section 4.2 below), we use our Bayesian assemblage E[τ] estimates to draw broad conclusions about the timing of diversification in these two assemblages and conduct hypotheses tests. We also discuss implications of our results for understanding the historical biogeography and diversification of Panamanian frog and freshwater fish species assemblages.

_{MRCA}s slightly older than that of E. pustulosus but younger than that of C. crassidigitus (Figure 3). In turn, C. crassidigitus (like C. talamancae) is considered more ecologically specialized, preferring wet forest habitat more so than its dry-forest congeners (i.e., C. fitzingeri; [42]). That the C. crassidigitus gene tree extends farthest backward in time thus seems to suggest this species experienced long-term isolation and persistence in preferred habitats, rather than an ability to tolerate climatic or vegetational shifts. However, whereas all the other frog species are egg-layers, Craugastor dirt frogs are direct-developing species that readily reproduce, and their local population sizes can therefore be notoriously large [82]. Given the direct relationship between N

_{e}and time to coalescence from coalescent theory, this contrast in life-history strategies would suggest that on one hand C. crassidigitus may have been superior at colonizing open niches or patches and spread throughout isthmian wet forest habitats more easily than the other taxa, while on the other hand its t

_{MRCA}estimate may also be inflated due to large ancestral N

_{e}[19].

_{MRCA}estimates for all three fish species/lineages fell within the predicted interval of diversification (Figure 3 and Figure 4). Peak posterior distributions from hABC model-averaging (e.g., Figure 5) also suggested that the focal fish species/lineages most likely diversified across the WPI break 1.36 Ma in the early Pleistocene, with Bayesian credible intervals ranging from early–late Pleistocene. This time period correlates best to the ‘Calabrian’ age (1.806–0.781 Ma; [83]), a time of 41-kyr periodicity of Pleistocene glaciations with drier and cooler-than-present conditions but less-extreme climatic oscillations than those following 800 ka [75,84]. Nevertheless, glacial periods vastly dominated the Calabrian to present, such that the Panama isthmus would have experienced many glacio-eustatic cycles but spent the majority of time since 1.8 Ma under glacial conditions with lowered sea levels and exposed continental shelf habitats. Eustatic sea-level curves give no convincing evidence that the oceans reached modern sea levels for any substantial period of time (e.g., >10–20 kyr) since the Calabrian, and the next eustatic sea-level highstand is not registered in the geological record until ~550–390 ka [76,81]. Geological patterns and processes are also consistent with decreased likelihood of fish dispersal across the WPI break zone since the Calabrian. In the break zone, the Pacific continental shelf becomes narrower (~0–40 km), tapering to the western Azuero peninsula coastline before being bisected by Cébaco and Coiba islands at the nearby Gulf of Montijo draining Soná peninsula (Figure 1). To evaluate the impact of lowered sea levels during Pleistocene glaciations on drainage connectivity in this area, we obtained a GIS model predicting paths of LGM paleo-drainages over modern bathymetry using ArcMap (courtesy of Peter J. Unmack, University of Canberra). The GIS model suggests that rivers draining to the west versus east of Soná peninsula did not anastomose over the continental shelf during the LGM (Figure 1), and possibly also preceding glaciations. Overall, our results combined with external environmental data suggest that a relatively stable geological setting at the Soná peninsula barrier has aided the historical isolation of drainage basins, maintaining fish lineage divergences at the WPI break during lower seas of the Calabrian to present.

#### 4.2. Caveats and Potential Limitations

#### 4.2.1. Mitochondrial DNA

#### 4.2.2. Migration and “Secondary Contact”

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Supplementary Methods

#### Appendix A.1.1. Taxon Sampling, Molecular Data, and Outgroup Details

_{MRCA}s in BEAST, tests for simultaneous diversification in MTML-msBayes, and our IMa2 analyses of community divergence. However, it was necessary to specify outgroup taxa in our analyses evaluating whether the data were consistent with predictions of the neutral model of molecular evolution [91]. To this end, we conducted MK tests, which test for deviations of the ratio of replacement (nonsynonymous fixed; RI) to synonymous fixed (SI) substitutions within species from that between species. Specifically, the SI:RI ratio from patterns of substitutions within one species (e.g., one of our focal population-pair datasets) is compared with that of the between-species patterns yielded from comparison to a close relative (e.g., a congeneric outgroup or sister species of one of our focal taxa). In the implementation of the MK test in DnaSP, substitution patterns in an intraspecific dataset are compared against the one or multiple sequences in an outgroup species dataset, and a non-significant difference between the two rations based on two-tailed Fisher’s exact tests is taken as evidence of neutrality. Next, we list outgroups used in MK tests for each population-pair dataset as well as references to studies used to determine the appropriate outgroups. We also provide GenBank accession numbers for outgroup sequences in parentheses, and all outgroup sequences were homologous to those of the focal taxon dataset. Based on phylogenetic analyses in Robertson et al. [54] and Robertson & Zamudio [53], we used one 16S-ND1 sequence of the congener Agalychnis saltator (GenBank GQ366296, [92]) in our MK test of the A. callidryas dataset. Based on phylogenetic analyses in Crawford et al. [42], we used 12 composite cytb-cox1 sequences of the congener Craugastor fitzingeri (GenBank DQ350193–198, DQ350236–241, [42]; EF635371, EF629419–423, EF629462, EF629458, EF629459, EF629455, EF629453, other studies cited in [42]) in our MK test of the C. crassidigitus dataset. Based on Robertson et al. [54], we used one composite sequence of 16S, tRNA-Leu, and ND1 genes from the congener Dendropsophus microcephalus (GenBank AY819503, [93]) as the outgroup in our MK test of the D. ebraccatus dataset. Based on Weigt et al. [51], we used one congeneric Engystomops petersi cox1 sequence from that same study (GenBank DQ120042) as the outgroup in our MK test of the E. pustulosus dataset. Based on McCafferty et al. [54], we used eight ATPase6/8 sequences from congener Andinoacara rivulatus (GenBank JX677777–784, [94,95]) as the outgroup sequences in our MK tests of the A. coeruleopunctatus dataset. Based on results in Bermingham and Martin [9] and Martin and Bermingham [55], we used composite ATP6/8, cox1 sequences from six individuals of the “type B” Pimelodella chagresi lineage (a cryptic species lineage discovered in that study; GenBank AF040388, AF040407–409, AF040412, AF040415, AF040484–489) as the outgroup sample in our MK test of the P. chagresi population-pair dataset. Last, based on Bermingham and Martin [9], we used composite ATP6/8, cox1 sequences from three individuals of the congener Roeboides meeki (GenBank AF040522–524 and AF040559–561 [9]) as the outgroup sample in our MK test of the R. occidentalis–R. guatemalensis population-pair dataset.

#### Appendix A.1.2. Supplementary MTML-msBayes Methods

#### Appendix A.2. Supplementary Results

#### Appendix A.2.1. Additional Genetic Diversity and Neutrality Results and Discussion

#### Appendix A.2.2. Additional MTML-msBayes Results and Discussion

_{2}mode Var[τ] = 3.89 × 10

^{−4}; model-averaging Var[τ] = 3.85 × 10

^{−4}, estimated from hyperparameter modes) and freshwater fishes (M

_{2}mode Var[τ] = 0.0015; model-averaging Var[τ] = 9.63 × 10

^{−5}, estimated from hyperparameter modes). However, variance of the modal τ values from model averaging was greater than that from the best-supported models.

Parameter | Description | Prior Distribution |
---|---|---|

A. Hyperparameters | ||

Ω | Dispersion index of divergence times, Var[τ]/E[τ] (ratio of variance to mean), across Y taxon/population-pairs | Set by Y, τ_{max}, and Ψ |

E[τ] | Mean multi-species (i.e., assembly-wide) divergence time across Y | Set by Y, τ_{max}, and Ψ |

Ψ | Number of distinct divergence times across Y | Discrete uniform [1, Y] |

S | Vector of mutation rate scalars to accommodate among-locus mutation rate heterogeneity | Gamma (with shape and scale parameters α and 1/α, respectively) |

α | Shape parameter of the gamma distribution | Uniform (1, 20) |

B. Parameters | ||

τ | Divergence time in units of 4N generations | Uniform (0.0, τ_{max}) |

θ = 4Nµ | Population size (mutation) parameter where N is average population size of two daughter populations, µ is the per-site mutation rate | Uniform (θ_{min}, θ_{max}) |

θ_{A} | Ancestral effective population size (mutation) parameter | Uniform (0.01, θ_{max} N_{anc-max}) |

θ_{A1}, θ_{A2} | Ancestral effective population size (mutation) parameters for daughter populations 1 and 2 over ancestral period (from time τ to τ_{B}) | Uniform (0.01, θ_{B1}), (0.01, θ_{B2}) |

θ_{B1}, θ_{B2} | Effective population size (mutation) parameters for daughter populations 1 and 2 over recent period (from time τ_{B} to 0.0, the present time) | Uniform (0.01 θ, 1.99θ), and where θ_{B2} = 2θ − θ_{B1} |

τ_{B1}, τ_{B2} | Times at which θ_{A1} and θ_{A2} experience exponential growth into populations with sizes of θ_{B1} and θ_{B2} | Uniform (0.0, τ), with τ_{B1} = τ_{B2} |

M = Nm | Effective migration rate, where m is the probability of symmetric migration between pairs of daughter populations | Uniform (0.0, M_{max}) |

r | Per-locus mutation rate scalar | fixed |

p | Per-locus ploidy and/or generation time scalar | p |

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**Figure 1.**Map of the study area. The western Panama isthmus (WPI) break zone is shaded gray, and major physiographic features including the continental divide, peninsulas, and mountain ranges are shown over a digital elevation layer; GC, Golfo de Chiriquí; GM, Golfo de Montijo. Paleo-bathymetric river paths modeled assuming a 135 m eustatic sea level drop during the Last Glacial Maximum using ArcMap (ESRI, Redlands, CA, USA; courtesy of Peter J. Unmack, University of Canberra) are shown with the −135 m bathymetric contour (dashed line) as a reference.

**Figure 2.**Geographical locations of WPI phylogeographic breaks registered in different species/lineages of Panamanian (

**A**,

**B**) frogs and (

**C**,

**D**) freshwater fishes evaluated in this study. Map features and lines are identical to those in Figure 1.

**Figure 3.**BEAST maximum clade credibility (MCC) time trees for all seven focal species/lineages split across the WPI break. Tip labels are sequence codes used or modified from the original studies, colored according to geographical position east versus west of the break zone. Horizontal bars show 95% highest posterior densities (HPDs) for node ages, numbers along nodes are mean node ages, and numbers at bar tips are upper 95% HPD values for truncated bars. Scale bars: 1 million years.

**Figure 4.**Comparison of divergence time estimates for species/lineages, and species assemblages, diverged across the WPI break. Species/lineage gene-tree depths (t

_{MRCA}s) from BEAST [65] are shown as geometric means (dots) and 95% HPDs, with regions of overlap in coalescence times shaded gray. Estimated times of assemblage co-divergences (E[τ]) are shown as modal peak posterior estimates (diamonds) and 95% HPDs from approximate Bayesian computation (ABC) model averaging in MTML-msBayes [25].

**Figure 5.**Hierarchical approximate Bayesian computation (hABC) results. Joint hyperposterior probability distributions of the mean divergence time, E[τ] (left x-axis, coalescent time; right x-axis, absolute time), and the dispersion index of divergence times, Ω, from MTML-msBayes [25] are presented for (

**A**) frogs and (

**B**) freshwater fishes based on ABC model-averaging across model classes. Inset graphs show the posterior densities of Ω from each analysis.

**Figure 6.**GIS sea-level model for lower Central America. This map is based on the 250 m NASA Shuttle Radar Topographic Mission digital elevation model and shows predicted eustatic sea levels in the study area potentially resulting from highstands of +50 m (light blue), +100 m (blue), and +250 m (dark blue) above present sea level (a.s.l.), relative to current elevation. The approximate WPI break zone is mapped in red.

**Table 1.**List of taxa used to examine temporal diversification patterns across the western Panama isthmus.

Taxon | mtDNA Genes | Length (bp) | n_{total} (West/East) | π | Ti/Tv | % div | Source |
---|---|---|---|---|---|---|---|

Frogs | |||||||

Agalychnis callidryas | 16S, ND1 | 1149 (118, 1031) | 28 (16/12) | 0.0337 | 11.373 | 8.0 | Robertson & Zamudio (2009) |

Craugastor crassidigitus | cytb, cox1 | 1353 (714, 639) | 13 (5/8) | 0.0645 | 14.55 | 13.9 | Crawford et al. (2007) |

Dendropsophus ebraccatus | ND1, tRNAs | 1877 | 18 (11/7) | 0.0245 | 12.04 | 5.9 | Robertson et al. (2009) |

Engystomops pustulosus | cox1 | 564 | 15 (2/13) | 0.0139 | 31.67 | 4.6 | Weigt et al. (2005) |

Freshwater Fishes | |||||||

Andinoacara coeruleopunctatus | ATP6/8 | 842 | 10 (2/8) | 0.0093 | 20.77 | 2.4 | McCafferty et al. (2012) |

Pimelodella chagresi | ATP6/8, cox1 | 1471 (842, 629) | 14 (6/8) | 0.0210 | 12.19 | 3.5 | Bermingham & Martin (1998) |

Roeboides occidentalis–R. guatemalensis | ATP6/8, cox1 | 1493 (842, 651) | 11 (8/3) | 0.0311 | 10.36 | 6.4 | Bermingham & Martin (1998) |

**Table 2.**Prior model classes and results of tests for synchronous diversification using ABC model averaging in MTML-msBayes.

Prior | P(τ) | P(θ_{D}) | P(θ_{A}) | P(M_{k}|D)^{1000} | Ω Mode | Ω 95% HPDs |
---|---|---|---|---|---|---|

WPI frogs (Y = 4) | 0.0036 | [0.000, 0.0565] | ||||

M_{1} | ~U(0, 1.75) | ~U(0, 0.1) | ~U(0, 0.25) | 0.2666 | – | – |

M_{2} | ~U(0, 1.75) | ~U(0, 0.1) | ~U(0, 0.5) | 0.4990 | – | – |

M_{3} | ~U(0, 1.75) | ~U(0, 0.4) | ~U(0, 0.25) | 0.0000 | – | – |

M_{4} | ~U(0, 0.875) | ~U(0, 0.1) | ~U(0, 0.25) | 0.2344 | – | – |

WPI fishes (Y = 3) | 0.0017 | [0.000, 0.0423] | ||||

M_{1} | ~U(0, 0.8) | ~U(0, 0.1) | ~U(0, 0.25) | 0.3124 | – | – |

M_{2} | ~U(0, 0.8) | ~U(0, 0.1) | ~U(0, 0.5) | 0.3766 | – | – |

M_{3} | ~U(0, 0.8) | ~U(0, 0.4) | ~U(0, 0.25) | 0.0000 | – | – |

M_{4} | ~U(0, 0.4) | ~U(0, 0.1) | ~U(0, 0.25) | 0.3110 | – | – |

_{k}) run for each of two analyses of Y population-pairs used to test for synchronous diversification across the western Panama isthmus (WPI) break. Prior models had varying τ, θ

_{D}, and θ

_{A}prior distributions, P(x) for random variable x, but assumed zero post-divergence migration. Approximate posterior probabilities, P(M

_{k}|D)

^{1000}, of each model are given based on 1000 accepted simulated draws from 20 million random draws from the four prior models, with that of the best-supported model underlined. Modal Ω hyperparameter estimates and their 95% highest posterior densities (HPDs) from model averaging over all four prior models are given in the first row of each section. Abbreviations: M, model class (candidate prior); P, probability (density); U, uniform distribution.

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**MDPI and ACS Style**

Bagley, J.C.; Hickerson, M.J.; Johnson, J.B. Testing Hypotheses of Diversification in Panamanian Frogs and Freshwater Fishes Using Hierarchical Approximate Bayesian Computation with Model Averaging. *Diversity* **2018**, *10*, 120.
https://doi.org/10.3390/d10040120

**AMA Style**

Bagley JC, Hickerson MJ, Johnson JB. Testing Hypotheses of Diversification in Panamanian Frogs and Freshwater Fishes Using Hierarchical Approximate Bayesian Computation with Model Averaging. *Diversity*. 2018; 10(4):120.
https://doi.org/10.3390/d10040120

**Chicago/Turabian Style**

Bagley, Justin C., Michael J. Hickerson, and Jerald B. Johnson. 2018. "Testing Hypotheses of Diversification in Panamanian Frogs and Freshwater Fishes Using Hierarchical Approximate Bayesian Computation with Model Averaging" *Diversity* 10, no. 4: 120.
https://doi.org/10.3390/d10040120