Impacts of Taxon-Sampling Schemes on Bayesian Tip Dating Under the Fossilized Birth-Death Process

Abstract

Evolutionary timescales can be inferred by molecular-clock analyses of genetic data and fossil evidence. Bayesian phylogenetic methods such as tip dating provide a powerful framework for inferring evolutionary timescales, but the most widely used priors for tree topologies and node times often assume that present-day taxa have been sampled randomly or exhaustively. In practice, taxon sampling is often carried out so as to include representatives of major lineages, such as orders or families. We examined the impacts of different densities of diversified sampling on Bayesian tip dating on unresolved fossilized birth-death (FBD) trees, in which fossil taxa are topologically constrained but their exact placements are averaged out. We used synthetic data generated by simulations of nucleotide sequence evolution, fossil occurrences, and diversified taxon sampling. Our analyses under the diversified-sampling FBD process show that increasing taxon-sampling density does not necessarily improve divergence-time estimates. However, when informative priors were specified for the root age or when tree topologies were fixed to those used for simulation, the performance of tip dating on unresolved FBD trees maintains its accuracy and precision or improves with taxon-sampling density. By exploring three situations in which models are mismatched, we find that including all relevant fossils, without pruning off those that are incompatible with the diversified-sampling FBD process, can lead to underestimation of divergence times. Our reanalysis of a eutherian mammal data set confirms some of the findings from our simulation study, and reveals the complexity of diversified taxon sampling in phylogenomic data sets. In highlighting the interplay of taxon-sampling density and other factors, the results of our study have practical implications for using Bayesian tip dating to infer evolutionary timescales across the Tree of Life. [Bayesian tip dating; eutherian mammals; fossilized birth-death process; phylogenomics; taxon sampling.]

Estimating the timescale of the Tree of Life has been a long-standing goal in evolutionary biology. This endeavor is dominated by analyses based on the molecular clock, which began as a simple, idealized model of evolutionary rate constancy (Zuckerkandl and Pauling 1965) but has undergone considerable developments over the past six decades (Ho 2020). Nevertheless, all molecular clocks need to be calibrated using external time information, which is usually obtained from fossil evidence. In this context, node-dating approaches use fossil data indirectly to inform calibration prior densities on internal node ages (e.g., Drummond et al. 2006; Yang and Rannala 2006; Nguyen and Ho 2020), whereas tip-dating approaches enable fossils to be included as sampled tips in the tree and directly make use of the temporal information attached to the fossils (Pyron 2011; Ronquist et al. 2012).

Bayesian methods of molecular-clock dating now hold particular appeal, because they can integrate diverse sources of information about lineage diversification, nucleotide substitution, and other aspects of the evolutionary process into a unified statistical framework, while taking into account their uncertainty (dos Reis et al. 2016; Bromham et al. 2018). In the Bayesian framework, a tip-dating approach treats the non-zero ages of the fossils as calibrating information, thereby eliminating the need to specify calibration prior densities for internal nodes (Ronquist et al. 2012; Zhang et al. 2016). Bayesian tip dating is usually implemented in conjunction with a tree prior based on the fossilized birth-death (FBD) process. This tree prior typically assumes a continuous process of fossil recovery through time (modeled by a Poisson process) in the lineage diversification history (modeled by a birth-death process of speciation and extinction), with extant taxa of the focal group ideally being completely or uniformly sampled (Stadler 2010; Heath et al. 2014). Several extensions have been made to the FBD process to increase its flexibility. For example, it can accommodate piecewise rates of lineage diversification and fossil recovery over time (Gavryushkina et al. 2014; Zhang et al. 2016), and be coupled with the multispecies coalescent (Ogilvie et al. 2022).

The FBD process can also account for nonuniform sampling of extant taxa in the group of interest. In practice, sampling of extant taxa is rarely exhaustive or random, especially for larger, species-rich clades. Instead, researchers might undertake a “diversified” sampling strategy that aims to include representatives of certain taxonomical ranks such as families and orders to maximize diversity (Höhna et al. 2011; Zhang et al. 2016; Matschiner 2019). This approach ensures that the deep divergence events are captured in the phylogeny (e.g., Fig. 1b), so it has been adopted by a range of prominent genome-sequencing initiatives (e.g., Jarvis et al. 2014; Lewin et al. 2018; Fan et al. 2020). For example, the Bird 10,000 Genomes project (Jarvis et al. 2014) has taken a four-phase approach that involves sampling major branches (orders and families) before proceeding to the twigs (genera and species) (i.e., a “branches-and-twigs” sampling strategy). Zhang et al. (2016) have modeled the diversified sampling strategy (see also Lambert and Stadler 2013) in the piecewise-constant FBD process. Specifically, the model assumes that exactly one representative extant taxon per clade is sampled after a cutoff time, x_cut, and that no fossil is sampled from x_cut to the present (Fig. 1b–e). The latter assumption is made for the sake of mathematical convenience (Zhang et al. 2016). The diversified-sampling FBD process has been able to correct overestimation of divergence times when the assumption of random sampling has been violated (Ronquist et al. 2016; Zhang et al. 2016). However, its performance with various taxon-sampling densities has not been explicitly evaluated, nor has the interaction of taxon-sampling density with the choice of priors, model misspecification, and other key factors. Addressing this gap has become increasingly important for gaining insights into evolutionary timescales while taking into account the taxon-sampling strategies that have been adopted by major genome-sequencing initiatives.

Figure 1.

a) A fossilized birth-death (FBD) tree depicting the reconstructed history of all present-day species (the number of extant taxa, N, is 50 in this case) and all sampled fossil occurrences (denoted by solid circles). The original species tree was generated under the birth-death process, on which fossil occurrences were simulated using a specified fossil recovery rate. Sampling fraction ρ for extant species is 1.0 for the complete FBD tree. b) An FBD tree after diversified sampling with ρ = 0.1 and pruning fossils younger than the cutoff time x_cut (i.e., only fossils of Category I are kept). The x_cut (dashed vertical line) is equal to (or slightly younger than) the (n – 1)th oldest divergence, where n is the number of sampled extant species $n=N\rho$. FBD trees with ρ = 0.2, 0.5, and 0.8 are shown in (c), (d), and (e), respectively. f) Flowchart showing the main steps in our simulation study. Details are provided in the Materials and Methods. Briefly, we simulated species trees, fossil occurrences, and nucleotide sequence evolution (outline arrows), performed diversified sampling on the extant species with sampling fraction ρ, and used the synthetic data for Bayesian tip dating on unresolved FBD trees (solid arrows), as listed in Table 1. For the core analyses in Table 1, we also simulated morphological characters for additional analyses using total-evidence tip dating for comparison.

Bayesian tip dating under the FBD process can be conducted as a combined analysis of molecular data for extant taxa and morphological data for extant and fossil taxa. This approach, known as total-evidence tip dating (Ronquist et al. 2012; Zhang et al. 2016), has a demonstrated potential to estimate evolutionary timescales accurately and precisely (Gavryushkina et al. 2014; Zhang et al. 2016; Luo et al. 2020). However, total-evidence tip dating also faces challenges, including less developed models of morphological evolution (Lewis 2001; Wright et al. 2016; Pyron 2017; Goloboff et al. 2019), the extent of among-lineage rate heterogeneity and convergent evolution (Kimura 1983; dos Reis et al. 2016), and typically high proportions of missing character data (Sansom and Wills 2013; O’Reilly and Donoghue 2021; Spasojevic et al. 2021). These can contribute to poor phylogenetic resolution for fossil taxa (Zhang et al. 2016; Luo et al. 2020) and overestimated divergence times (e.g., Ronquist et al. 2016; Arcila and Tyler 2017; Kealy and Beck 2017; Spasojevic et al. 2021).

Fortunately, Bayesian tip dating under the FBD process can allow fossil taxa to be included even if they are lacking in morphological data. In this case, the fossils only provide calibrating information for the phylogeny of extant taxa, while the placement of each fossil is sampled conditioned on a constrained portion of the phylogeny and the FBD process (Heath et al. 2014). Because the exact fossil placement is not known a priori or informed by morphological data, the FBD tree (i.e., topology of the fossil and extant lineages) is regarded as “unresolved” (Heath et al. 2014). By eliminating the need to include morphological characters, which cannot always be obtained for taxa of interest, the unresolved FBD tree potentially allows the inclusion of large amounts of fossil occurrence data. Bayesian tip dating on unresolved FBD trees has been performed for a range of organisms, including mammals (Gavryushkina et al. 2014; Heath et al. 2014; Law et al. 2018; Presslee et al. 2019), reptiles (Card et al. 2016; Grismer et al. 2016), fishes (Arcila et al. 2015), insects (Economo et al. 2018; O’Reilly and Donoghue 2020; Bossert et al. 2022), and plants (Grimm et al. 2015; Saladin et al. 2017). Although fossil sampling has been shown to have an impact on estimates of divergence times (O’Reilly and Donoghue 2020), the potential effects of diversified sampling of extant taxa have not been examined in detail for Bayesian tip dating on unresolved FBD trees.

Here we examine the impacts of taxon-sampling density on Bayesian tip dating on unresolved FBD trees. We use synthetic data produced by simulations of nucleotide sequence evolution, fossil occurrence times, and diversified taxon sampling. Following the “branches-and-twigs” sampling strategy used by major genome-sequencing projects, we also reanalyze a published eutherian mammal data set (Ronquist et al. 2016). Our results provide general insights into the impacts of diversified taxon sampling on Bayesian tip dating on unresolved FBD trees, with potential implications for the use of this approach in the phylogenomic era.

Materials and Methods

Simulation Study

Simulation of species trees.

We generated complete species trees under the birth-death process using TreeSim (Stadler 2011) in R (R Core Team 2018). Following Luo et al. (2020), we arbitrarily chose a constant speciation rate λ = 0.05 per Myr and extinction rate μ = 0.02 per Myr to ensure trees of reasonable size for computation, while conditioning on a root age (i.e., time of the most recent common ancestor [MRCA] of all sampled taxa) t_mrca = 100 Ma. The chosen root age reflects the evolutionary characteristics of groups such as eutherian mammals and modern birds. After generating 1000 trees, we retained 100 trees with moderate numbers of extant taxa (50 ≤ N ≤ 100; median 73.5) also for the sake of computational tractability. These trees all have extant descendants on both sides of the root bifurcation, and so t_mrca is equivalent to the crown age t_c (i.e., time of the MRCA of all extant taxa).

Based on the complete species trees, we sampled extant species using a diversified sampling strategy (Lambert and Stadler 2013; Fig. 1). With N extant species in the complete tree and a sampling fraction ρ, the number of sampled extant species is $n=N\rho$ (rounding to the nearest integer). These n species were chosen such that they were descended from the oldest n – 1 divergences in the reconstructed tree of extant species. According to Zhang et al. (2016), this is equivalent to setting the cutoff time, x_cut, to the (n – 1)th oldest divergence in the reconstructed tree and choosing only one representative species per clade after x_cut. We set ρ to 0.1, 0.2, 0.5, 0.8, and 1.0 (i.e., complete sampling) to emulate an increasingly dense taxon-sampling process (Fig. 1a–e). This led to 5–10 (median 7), 10–20 (median 15), 25–50 (median 36.5), 40–80 (median 58.5), and 50–100 (median 73.5) extant species sampled across the 100 trees.

Simulation of fossil occurrences.

Fossils were sampled along each of the complete species trees with a constant recovery rate ψ, which models fossil occurrence as a continuous Poisson process and allows fossils to be sampled ancestors (i.e., tips with zero-length terminal branches) of extant species (Heath et al. 2014). To reflect the typically limited number of fossils that are seen in practice, and to maintain computational feasibility, we set ψ = 0.003 which yielded 2–15 (median 7) fossil occurrences across the 100 trees. We ensured that at least one sampled fossil was older than x_cut when ρ = 0.1, to meet the assumptions of the diversified-sampling FBD process (Ronquist et al. 2016; Zhang et al. 2016). According to x_cut defined by sampled extant species under each ρ value, we divided all sampled fossils into two categories: those sampled not younger than x_cut (Category I; Fig. 1), and those sampled younger than x_cut (Category II). After sampling the extant species (with ρ and n as above), keeping all sampled fossils (i.e., fossils of both Category I and II) or only those of Category I, and pruning all unsampled lineages, we had a total of 1000 FBD trees (100 species trees × 5 extant sampling fractions × 2 combinations of fossil categories). Among the 1000 FBD trees, there are 142 duplicates (Table 1), because Category II can be empty and Category I then includes all fossils, especially under complete sampling of extant species (i.e., ρ = 1.0).

Table 1.

Scenarios and settings explored in our simulation study

Scenario	Clock models used for simulation and inference	t mrca prior	Tree topology	Fossil recovery rate	Morph. data	No. of data sets
Core analyses
Fossils of Category I, diversified sampling, matched sampling fraction ρ = 0.1, 0.2, 0.5, 0.8, and 1	Strict–strict	U(0,200)	Estimated	0.003	No	500
	Strict–strict	U(0,200)	Estimated	0.003	Yes	500
	Relaxed–relaxed	U(0,200)	Estimated	0.003	No	500
	Relaxed–relaxed	U(0,200)	Estimated	0.003	Yes	500
Comparative analyses
Fossils of Category I, diversified sampling, matched ρ = 0.1, 0.2, 0.5, 0.8, and 1	Relaxed–relaxed	N(100,10)	Estimated	0.003	No	500
	Relaxed–relaxed	N(120,10)	Estimated	0.003	No	500
	Relaxed–relaxed	N(80,10)	Estimated	0.003	No	500
	Relaxed–relaxed	U(0,200)	Fixed	0.003	No	500
	Relaxed–relaxed	N(100,10)	Fixed	0.003	No	500
	Relaxed–relaxed	N(120,10)	Fixed	0.003	No	500
	Relaxed–relaxed	U(0,200)	Estimated	0.01	No	500
	Relaxed–relaxed	N(100,10)	Estimated	0.01	No	500
Analyses with model mismatches
Fossils of Category I and II	Relaxed–relaxed	N(100,10)	Estimated	0.003	No	358
Random sampling	Relaxed–relaxed	N(100,10)	Estimated	0.003	No	400
Fossils of Category I and II, random sampling	Relaxed–relaxed	N(100,10)	Estimated	0.003	No	358
Mismatched ρ = 0.5	Relaxed–relaxed	N(100,10)	Estimated	0.003	No	400

Notes: t_mrca = root age, that is, time of the most recent common ancestor of all extant and extinct species; fossils of Category I = fossils not younger than the cutoff time x_cut; fossils of Category II = fossils younger than x_cut. The number of data sets varies because fossils of Category II can be empty and duplicated data sets were eliminated.

In addition, we adopted a fossil recovery rate ψ = 0.01 to evaluate the effects of including more fossils, which produced 10–46 (median 21.5) fossil occurrences across the 100 complete species trees. The ρ-sampling assumptions and fossil categories followed the procedures above. However, unless noted otherwise, the sampled fossils described below exclude these simulated under ψ = 0.01.

Simulation of character evolution.

Based on the complete species trees, we simulated the evolution of nucleotide sequences along the reconstructed trees of extant species only. We used either a constant evolutionary rate of 10⁻³ substitutions/site/Myr (i.e., a strict clock), or a lognormal relaxed clock (Drummond et al. 2006) via NELSI v0.2 (Ho et al. 2015) to characterize among-lineage rate variation with a mean of 10⁻³ substitutions/site/Myr and a standard deviation of 2 × 10⁻⁴ (reflecting the nuclear substitution rates of organisms such as mammals and Drosophila; Kumar and Subramanian 2002; Keightley et al. 2014). After sampling relative substitution rates among loci from a Dirichlet distribution with α = 3, we then simulated sequence evolution using Seq-Gen v1.3.4 (Rambaut and Grassly 1997) to generate alignments of five “loci,” with each locus having a length of 1000 bp; we used an HKY + G substitution model, with base frequencies {A:0.35, C:0.15, G:0.25, T:0.25}, transition/transversion ratio κ = 4.0, and gamma shape parameter of 0.5 with four rate categories (Hasegawa et al. 1985; Yang 1994; Posada and Crandall 2001). These settings directly followed Luo et al. (2020). The proportion of the generated sequence alignments used in subsequent analyses depended on the sampling fraction ρ of extant taxa.

To allow comparisons between Bayesian tip dating on unresolved FBD trees and total-evidence tip dating, we additionally simulated the evolution of morphological characters under the Mkv model (Lewis 2001) for both extant species and sampled fossils along the complete FBD trees. The same strict- and relaxed-clock models as for molecular sequences were applied, although the rate of morphological evolution is often more variable among lineages in practice (dos Reis et al. 2016). For simplicity, we began by simulating the evolution of nucleotide sequences under the Jukes-Cantor model (Jukes and Cantor 1969) and then converted the nucleotides into numerical character states (A to 0, C to 1, G to 2, and T to 3) (following the procedures described by Luo et al. 2020). After extant and fossil taxa were sampled according to ρ and x_cut as above, nonvariable characters were removed from the character matrix with an original size of 500 characters, resulting in 162–287, 214–377, 307–445, 328–464, and 332–466 characters under the strict clock, and 158–289, 217–376, 299–447, 319–461, and 324–462 characters under the relaxed clock for ρ = 0.1, 0.2, 0.5, 0.8, and 1, respectively.

Settings for core analyses.

We first analyzed the synthetic data sets derived from the FBD trees using settings that are typically employed in practice, while matching relevant models or priors (including for the sampling fraction ρ) to those used for simulation (Table 1, core analyses). The analyses were intended to provide a baseline and a reference for further comparisons in our subsequent analyses.

Specifically, we used the FBD model with diversified sampling as the tree prior (Zhang et al. 2016), assuming constant rates of speciation, extinction, and fossil recovery, and included fossil occurrence times of Category I for analyses (because of the assumption of zero fossil sampling after x_cut). We assigned diffuse priors for FBD model parameters: beta(1,1) for both turnover r = μ / λ and fossil sampling proportion s = ψ / (ψ + μ) (as they range between 0 and 1), and exponential(10) for net diversification rate d = λ − μ (assuming μ ≤ λ), which are reparameterizations of speciation rate λ, extinction rate μ, and fossil recovery rate ψ (Heath et al. 2014). Fossil occurrence times were directly used as point values as in our previous simulation study (Luo et al. 2020). Each fossil was topologically constrained so that it descended from its parent node in the FBD trees, thus allowing it to be placed in a crown or stem position among its extant relative(s). As the FBD process requires extra conditions to be properly defined, our analyses were conditioned on sampling at least one species and the root age. We adopted a diffuse uniform(0,200) prior for the root age t_mrca. The sampling fraction of extant species ρ was always fixed to its true value. For analyses employing nucleotide sequences simulated under the strict clock, a strict-clock model with a uniform(10⁻⁶,1) prior for the evolutionary rate was shared by the five loci; for those with data generated under the relaxed clock, we used an uncorrelated lognormal relaxed-clock model with a uniform(10⁻⁶,1) prior for the mean evolutionary rate. An independent HKY + G substitution model with four rate categories was applied to each locus. Default priors were adopted for parameters not mentioned here.

In our total-evidence tip-dating analyses used for comparison, we combined morphological characters with the molecular data. The phylogenetic positions of fossil taxa were not constrained but inferred from the morphological data. Morphological characters generated under the strict clock were combined with nucleotide sequences generated under the strict clock, while morphological characters generated under the relaxed clock were coupled with nucleotide sequences generated under the relaxed clock. In addition to following the settings described above, we used the Mkv model (Lewis 2001) for the variable morphological characters.

Settings for comparative analyses.

We built on the results of our core analyses by performing comparative analyses in which we explored the impacts of varying the root-age prior, constraining the tree topology, and increasing the number of fossils (Table 1, comparative analyses). We focused on the synthetic data that had evolved under a relaxed clock. The root-age prior potentially has a large influence on inferred divergence times, particularly when there are few fossils near the root. To explore the effects of the t_mrca prior, we replaced the diffuse uniform prior with an informative normal prior with a mean smaller than (80 Ma), equal to (100 Ma), or larger than (120 Ma) the true value and a standard deviation of 10 Myr.

Then, under the uniform(0,200), normal(100,10), and normal(120,10) t_mrca priors, we fixed tree topologies to match the FBD trees used for simulation. This mimics a scenario where the molecular and morphological characters are highly informative and the evolutionary models are perfectly matched, such that the tree topologies can be inferred without error while leaving the node times to be inferred. Other settings for these data sets followed those in the core analyses as appropriate.

Increasing the number of fossils for taxa of interest could potentially enhance both accuracy and precision of inferred divergence times (Luo et al. 2020). To investigate the specific effects of including more fossils, we used the fossil occurrence times of Category I simulated under a higher fossil recovery rate ψ = 0.01 for analyses. The molecular data and other settings matched the core analyses except for both uniform(0,200) and ­normal(100,10) t_mrca priors.

Settings for analyses with model mismatches.

In analyses of real data, the phylogeny and the values of other evolutionary parameters are almost always unknown and so some degree of model mismatch is inevitable. Thus, we further explored three situations involving mismatched models of taxon sampling (Table 1) based on data generated by simulation under a relaxed clock. While specifying the optimal normal(100,10) prior for t_mrca, we first examined a situation where all fossil occurrences were included while presuming diversified sampling of extant taxa. In this case, we kept only data sets with non-empty fossil occurrences in Category II (358 out of 500). Second, when fossils of Category I (or both Category I and II) were involved, we adopted an FBD process that assumed random sampling rather than diversified sampling of extant species, with the sampling fraction of extant species ρ < 1.0 fixed to its true value (because random sampling is identical to diversified sampling if ρ = 1.0). Third, because the true value of ρ is not always known, we fixed ρ to 0.5 under diversified sampling so that it is misspecified for most of the data sets. Other settings for each of the three situations matched those in our comparative analyses described above.

Data analyses.

Our focal Bayesian tip-dating analyses on unresolved FBD trees were performed using the BDSKY v1.4.5 package (Stadler et al. 2013) in BEAST v2.6 (Bouckaert et al. 2019), with posterior distributions estimated by Markov chain Monte Carlo (MCMC) sampling. The MM v1.1.1 package (https://github.com/CompEvol/morph-models) was also adopted for total-evidence tip dating. We ran MCMC sampling in duplicate at least, typically with samples drawn every 5000 steps over 100 million steps, and with a discarded burn-in fraction of 0.25; when tree topologies were fixed, MCMC samples were drawn every 2000 steps over a total of 40 million MCMC steps. Our simulated data sets and input files are in Supplementary Appendix 1, while script code used for the simulation study is provided in Supplementary Appendix 2 available on Dryad. By inspecting the combined MCMC samples, we checked that the effective sample sizes of parameters were greater than 100 using LogAnalyzer v2.6, and assessed convergence between chains using Tracer v1.7 (Rambaut et al. 2018). We then used TreeAnnotator v2.6 to summarize the maximum-clade-credibility tree among the sampled trees, with fossil taxa pruned beforehand using FullToExtantTreeConverter. With each of the 100 (or fewer) maximum-clade-credibility trees treated as an independent replicate under each scenario (Table 1), we examined the estimates of evolutionary parameters across our various scenarios, while evaluating the impacts of increasing taxon-sampling densities represented by ρ values.

Based on the maximum-clade-credibility trees without fossils, we focused on divergence times estimated for the crown age t_c to compare with the true root age t_mrca (which always has t_c = t_mrca during simulation). We measured accuracy and precision using both absolute metrics (i.e., posterior median and 95% credible interval [CI]) and relative metrics (i.e., relative bias and relative 95% CI width). Posterior tree length was used to summarize the global node times, with either its standard form (sum of branch lengths) or its variants (sum of internal branch lengths, or sum of branch lengths except those of the two branches descending from the root). We also examined total evolutionary change (the product of clock rate and tree length) and clock rate because these parameters are relevant to the inference of divergence times. To allow comparisons to be made more readily, we only adopted standardized measures (distances between the estimates and the true values, divided by the true values) for tree length and total evolutionary change. Topological distance between the maximum-clade-credibility tree and the true topology was measured by the Robinson-Foulds metric (Robinson and Foulds 1981) in the R package ape (Paradis et al. 2004; Popescu et al. 2012), corrected by the total number of extant taxa.

To examine the priors specified for the FBD parameters, additional MCMC sampling was carried out without data (no nucleotide sequences or morphological characters) but with topological constraints on fossils for the core analyses and some of the comparative analyses (see Results). As conditioning on sampling at least one species did not necessarily enforce extant descendants on both sides of the root bifurcation (i.e., estimates of t_mrca did not necessarily equal those of t_c), we focused on the marginal priors on the root age t_mrca from these analyses to compare with the marginal posterior distributions.

Analyses of Empirical Data

Our simulation study was designed with certain simplifications and with true evolutionary parameters already known. However, those settings might not sufficiently reflect the complex processes that have produced real biological data. To further explore the effects of taxon-sampling densities in phylogenomic data, we performed taxon sampling and dating analyses based on a data set of eutherian mammals from a study by Ronquist et al. (2016). In contrast with the completely sampled trees in our simulation study, the extant taxa included in this data set are generally diversified representatives of eutherians. However, the data set allows us to relatively efficiently evaluate the effects of taxon-sampling density in an empirical case.

Data set.

The data set from Ronquist et al. (2016) was originally assembled by O’Leary et al. (2013). It includes 41 extant species, representing the four eutherian superorders (Xenarthra, Afrotheria, Laurasiatheria, and Euarchontoglires) and all living eutherian orders, as well as 33 eutherian or potentially eutherian fossils ranging from the 35.3-Myr-old Leptictis dakotensis to the 123.3-Myr-old Eomaia scansoria. Following Ronquist et al. (2016), we used point values for the ages of the fossils. The data set comprises 36,860 nucleotide sites from 22 nuclear protein-coding genes and five nuclear untranslated regions, along with 4541 discrete morphological characters.

Total-evidence tip dating.

We first performed total-evidence tip dating with both the molecular sequences and morphological characters, as done by Ronquist et al. (2016) (Fig. 2). While accounting for diversified sampling of extant species, we conditioned the FBD process on the origin time t_o of eutherians using an offset (100 Ma) exponential prior with an expected mean of 145 Ma (based on the age of Eomaia scansoria). We specified weakly informative lognormal(−2.5,0.7) priors for λ and μ (with mean 0.1 and standard deviation 0.08), and a lognormal(−5,0.8) prior for ψ (with mean 0.01 and standard deviation 0.01). Both distributions are more diffuse than the posterior estimates of these parameters in Ronquist et al. (2016). Variable morphological characters were analyzed using the Mkv + G model. The molecular sequence data were partitioned into four subsets (three corresponding to the codon positions of the 22 protein-coding genes, and one comprising the five untranslated nuclear regions), with each subset assigned a separate GTR + G model. Following Ronquist et al. (2016), we applied a single uncorrelated lognormal relaxed-clock model to the morphological and molecular data, with a lognormal(−6,0.5) prior (expectation ~3 × 10⁻³ substitutions per site per Myr) for the mean rate and an exponential(1.0) prior for the standard deviation. The extant species sampling fraction ρ was fixed to 0.01. For our three independent runs of MCMC sampling using the BDSKY package in BEAST v2.6, samples were drawn every 10,000 steps over a total of 400 million steps, with a burn-in fraction of 0.25. To address problems with rooting and MCMC mixing, we enforced a monophyly constraint on Boreoeutheria (Euarchontoglires + Laurasiatheria) (Ronquist et al. 2016).

Figure 2.

Flowchart showing the analysis pipeline in our analysis of eutherian mammals. The steps are described in detail in the Materials and Methods. Briefly, we 1) performed total-evidence tip dating using molecular sequences from extant species and morphological characters from both extant species and 33 fossils. 2) Based on the maximum-clade-credibility tree, representatives of extant species at the superorder, order, and species levels were then sampled, along with selected fossil occurrences. 3) Using molecular sequences of sampled extant species and fossil times, we carried out Bayesian tip dating on unresolved FBD trees and compared the posteriors. Fossil occurrences are indicated by columns along the geological timescale, with different heights denoting 1, 2, 3, and 4 occurrences at point ages. Only three repeats of sampled extant species are shown for the superorder-level and order-level analyses, but our study involved ten repeats.

Taxon sampling.

Based on the maximum-clade-credibility tree from the total-evidence tip-dating analysis described above, we performed diversified sampling of extant species using a hierarchical approach similar to those used in major phylogenomic initiatives. To construct a superorder-level data set, we chose two representative species from each of the three superorders Afrotheria, Laurasiatheria, and Euarchontoglires, and one species from Xenarthra, with 10 random repeats. To construct 10 replicates of order-level data sets, we randomly selected a representative of each order of eutherian mammals. For each random replicate of species sampling at the superorder and order levels, we ensured that the MRCA of the chosen species from each group was as old as possible. Finally, we constructed a “species”-level data set comprising all extant species in the maximum-clade-credibility tree.

Considering x_cut relative to the posterior medians of divergence times in the maximum-clade-credibility tree, we first chose four (Eomaia scansoria at 123.3 Ma, Maelestes gobiensis at 77.8 Ma, Ukhaatherium nessovi at 77.8 Ma, and Zalambdalestes lechei at 77.8 Ma), five (adding Protungulatum donnae at 64.6 Ma), and all 33 fossil occurrence times at the superorder, order, and species levels, respectively. All of them are older than the corresponding x_cut in the maximum-clade-credibility tree under extant species sampling at different levels (cf. fossils of Category I in our simulation study). Second, in light of our simulation conditions (i.e., no stem fossil occurrence) and the first scenario explored in our analyses with model mismatches (i.e., including fossils from both Category I and II under diversified sampling), we eliminated the five stem fossil occurrences (Eomaia scansoria, Leptictis dakotensis, Maelestes gobiensis, Ukhaatherium nessovi, and Zalambdalestes lechei) in the maximum-clade-credibility tree, and used the remaining 28 fossil occurrence times at the three sampling levels. Specifically, the 28 fossil times then belong to Category II, both Category I and II, and Category I (corresponding to different x_cut values) under sampling at the superorder, order, and species levels, respectively.

Bayesian tip dating on an unresolved FBD tree.

We then evaluated the effects of taxon-sampling densities for this empirical case. We carried out analyses using molecular sequences for the sampled extant species and fossil occurrence times (i.e., without morphological characters), with other settings and priors as in our total-evidence tip-dating analyses. We drew samples every 10,000 MCMC steps over a total of 300 to 400 million steps, with the first 25% discarded as burn-in. During MCMC sampling, we either put monophyly constraints on the fossil placements, or fixed the topology of all sampled taxa based on the maximum-clade-credibility tree inferred from total-evidence tip dating. While using posterior medians directly for analyses at the species level, we recorded medians across the 10 repeats for analyses at the superorder and order levels. Input files for eutherian tip-dating analyses are available in Supplementary Appendix 3 available on Dryad.

Results

Core Analyses

In our Bayesian tip-dating analyses on unresolved FBD trees with synthetic molecular data, we unexpectedly found that increasing sampling density generally worsened estimates of the divergence times. For the crown age t_c, under the strict clock, a larger sampling fraction ρ led to more overestimated node times and wider 95% CIs (Fig. 3a). Estimates of the standardized tree length exhibited similar patterns to those of t_c, whether or not the two branches descending from the root or the terminal branches were excluded (Fig. 3a; Supplementary Appendix 4 on Dryad). Unlike the time estimates, the standardized total evolutionary change (using the clockRate parameter under a strict clock, multiplied by the tree length) was invariably estimated to approach the expected value of zero whether ρ was 0.1 or 1.0, though with wider 95% CIs when ρ was larger (Supplementary Appendix 4). As a result, the substitution rate was underestimated given overestimated node times. We found that our estimates under the relaxed clock were nearly identical to the estimates under the strict clock (Fig. 3b; Supplementary Appendix 4).

Figure 3.

Summary of estimates from our core analyses. a) Time estimates from Bayesian tip-dating on unresolved FBD trees under the strict clock, with black circles (representing posterior medians) and gray lines (representing 95% credible intervals, CIs) jittered to show 100 replicates at each discrete level of sampling fraction ρ (x-axis). Crown age t_c and tree length (sum of all branch lengths) are measured by absolute and standardized estimates, respectively. b) Boxplots summarizing posterior estimates of t_c from the unresolved FBD and total-evidence tip-dating (“+morpho”) analyses under strict and relaxed clocks, measured by relative bias (distance between the posterior median and the true value, divided by the true value) and relative 95% CI width (posterior 95% CI width divided by the true value). c) Violin-plot summaries of root-age t_mrca estimates from the marginal tree priors (“no data”), unresolved FBD (“mol only”), and total-evidence tip-dating (“+morpho”) analyses under the strict clock. d) Violin-plot summaries of topological distances from the unresolved FBD and total-evidence tip-dating analyses, measured by corrected Robinson-Foulds (R-F) distance between the maximum-clade-credibility tree of extant taxa and its true counterpart. For (b)–(d), 100 replicates are summarized into an independent boxplot or violin plot at each level of sampling fraction ρ (x-axis). For (a)–(d), dashed horizontal lines (blue or black) indicate the true or expected values.

Estimates from the total-evidence tip-dating analyses under the relaxed clock were also similar to those under the strict clock (Fig. 3b; Supplementary Appendix 4). In contrast with the results of our analyses on unresolved FBD trees, however, our total-evidence tip-dating analyses produced time estimates with good accuracy and precision, regardless of the crown age t_c or the global node times. The time estimates varied little across different values of ρ, except that larger ρ led to slightly reduced variation among repeats and increased precision.

Although we specified a uniform(0,200) prior for root age t_mrca across the core analyses, the effective (marginal) prior on t_mrca (measure by the parameter TreeHeight) favored values larger than 100 as ρ increased. While this had little impact on our total-evidence tip-dating analyses, we found that the effective t_mrca prior had a strong influence on posterior estimates of t_mrca in our analyses on unresolved FBD trees (Fig. 3c). Neither among-lineage rate variation nor inclusion of morphological data greatly altered the inferences of tree topologies of extant taxa: topological distances tended to be less variable with smaller medians across replicates at higher ρ values (Fig. 3d).

Impacts of Altering the Root-Age Prior

Altering the prior on t_mrca had noticeable impacts on time estimates from tip-dating analyses on unresolved FBD trees under the relaxed clock. In contrast with the overestimated node times under the uniform(0,200) prior, estimates of the crown age t_c and the tree length were generally accurate and precise under the ­normal(100,10) prior, which concentrates the prior density around the true value of t_mrca (Fig. 4a; Supplementary Appendix 5 on Dryad). Although these estimates had better precision than under the uniform(0,200) prior, using the normal(120,10) and normal(80,10) priors led to over and underestimated node times, respectively. The effects of the former on accuracy were similar to those of using the uniform(0,200) prior, but the posterior medians were less variable among repeats.

Figure 4.

Posterior estimates from our comparative analyses under alternative root-age t_mrca priors, with corresponding estimates under the uniform(0,200) prior from the core analyses also shown. a) Jittered posterior medians (circles) of the crown age t_c, trends indicated by smoothed lines (by R package ggplot2; Wickham 2016). b) Scatterplot for posterior medians of mean clock rates (measured by ucldMean) against those of rate variation among lineages (measured by coefficientOfVariation). c) Scatterplot for posterior medians of mean clock rates against standardized tree length. Ellipses represent variations around centroids of the estimates under a confidence level of 0.95 (Fox and Weisberg 2011). d) Violin-plot summaries of topological distances (measured by the corrected Robinson-Foulds distance). For (a) and (d), 100 replicates are shown or summarized as an independent violin plot under each t_mrca prior for each sampling fraction ρ (x-axis). For (b) and (c), 500 replicates across all ρ levels are presented under each t_mrca prior. For (a)–(d), dashed lines indicate the true or expected values.

Nevertheless, due to the sensitivity of mean clock rate (measured by the ucldMean parameter) to the t_mrca prior (Fig. 4b) and its inversely proportional relationship with estimated tree length (Fig. 4c), estimates of the total evolutionary change (using ucldMean multiplied by the tree length) were relatively robust to the choice of root-age prior, with medians of the standardized metric around zero (Supplementary Appendix 5). Topological distances under the three normal t_mrca priors were also similar to those under the uniform(0,200) prior (Fig. 4d). Molecular data thus appeared to be informative of the relationship and genetic distances among extant species, while estimates of divergence time and evolutionary rate were influenced by the root-age priors.

Compared with estimates under the uniform(0,200) prior, increasing sampling density had only small impacts on the estimates of node times under the normal priors (Supplementary Appendix 5; Fig. 5 could also be referred to). Under the normal(100,10) prior, estimates of node times were accurate and precise whether the sampling fraction ρ was 0.1 or 1.0, despite slight changes among ρ values; medians of t_c among repeats were close to the true value of 100 Ma across all sampling fractions. Under the normal(120,10) and normal(80,10) priors, estimates were less variable with larger ρ; for example, under the former, standard deviations of t_c declined from 4.4 Myr for ρ = 0.1 to 2.2 Myr for ρ = 1.0. It appeared that increasing ρ pushed posterior estimates further from the true value of 100 Ma under the uniform(0,200) and normal(120,10) t_mrca priors, but closer to the true value under the normal(80,10) prior.

Figure 5.

Comparison of posterior estimates of the crown age t_c from Bayesian tip dating on unresolved FBD trees under different root-age priors, with tree topologies estimated or fixed to those for simulation (denoted by “fixed”). a) Boxplots summarizing estimated accuracy, measured by relative bias. b) Boxplots summarizing estimated precision, measured by relative 95% CI width. For (a) and (b), each boxplot summary is based on 100 replicates for each sampling fraction ρ (x-axis), and dashed lines indicate the true or expected values.

Impacts of Fixing the Tree Topology

Based on the consistency of estimates of the crown age t_c with those of tree length (described above; Supplementary Appendix 6 on Dryad), we focus only on time estimates of t_c with relative metrics here, while providing other estimates in Supplementary Appendix 6. Fixing tree topologies to those used for simulation, as well as increasing sampling density, improved both accuracy and precision of estimates (Fig. 5). The positive effects were most prominent under the uniform(0,200) prior: estimates of t_c had much better accuracy and precision, in contrast with the tendency of further deviation with larger ρ in the core analyses. Relative bias did not change clearly with increasing ρ, but relative 95% CI width decreased with ρ.

Fixing tree topologies led to limited improvements in the accuracy and precision of t_c under the normal(100,10) and normal(120,10) priors. Under the former, the accuracy of t_c was slightly better for all ρ values when compared with the estimates made with non-fixed tree topologies, while precision increased slightly with ρ. Under the latter prior for the root age, both relative bias and relative 95% CI width decreased as ρ increased, different from the increasing tendency with ρ when tree topologies were estimated. Given fixed tree topologies, unlike overestimates under the normal(120,10) prior, estimates of t_c under the uniform(0,200) prior were similar to those under the normal(100,10) prior in accuracy, but were more variable among repeats.

Impacts of Sampling More Fossil Occurrences

We found that including more fossil occurrences increased the precision of t_c estimates for tip dating on unresolved FBD trees, though with different degrees of refinement for analyses under the uniform(0,200) and normal(100,10) t_mrca priors (Fig. 6b; Supplementary Appendix 7 on Dryad). However, it only improved accuracy of t_c estimates under the uniform(0,200) prior, whereas under the normal(100,10) prior, estimated accuracy of t_c was generally similar to that with fewer fossil occurrences involved (Fig. 6a). These were consistent with the patterns of marginal priors on t_mrca (measure by the TreeHeight parameter) (Fig. 6a).

Figure 6.

Posterior estimates of the crown age t_c from Bayesian tip dating on unresolved FBD trees under two root-age priors, with fossil occurrences simulated under fossil recovery rate ψ= 0.003 (as in previous analyses) or ψ= 0.01 (denoted by “0.01”). a) Boxplots summarizing estimated accuracy of t_c (measured by relative bias), embedded with boxplot summaries of root-age t_mrca estimates from the marginal tree priors. b) Boxplots summarizing estimated precision of t_c (measured by relative 95% CI width). For (a) and (b), each boxplot summary is based on 100 replicates for each sampling fraction ρ (x-axis), and dashed lines indicate the true or expected values.

When larger numbers of fossil occurrences were involved, effects of increasing sampling density ρ on estimated accuracy and precision of t_c were similar to their counterparts with fewer fossil occurrences, that is, negative and negligible effects under the uniform(0,200) and normal(100,10) priors, respectively. Increasing the number of fossil occurrences appeared to have little impact on the inferred tree topologies of extant species (Supplementary Appendix 7).

Impacts of Mismatched Species-Sampling Models

Our final analyses of synthetic data with mismatched models of species sampling led to posterior medians of t_c generally close to the expected values, across all ρ values (Fig. 7; Supplementary Appendix 8 on Dryad). When we used a diversified sampling model to analyze data sets that involved all fossils, t_c was underestimated given smaller ρ. However, when a random sampling model was assumed in analyses of data sets that involved fossils of Category I only, t_c was overestimated across ρ values; compared with the control, the degree of overestimation increased with smaller ρ. When all fossils were included and random sampling was assumed, the slight overestimates of t_c were intermediate between those from the first two scenarios, being roughly similar to those of the control. Under misspecified ρ as 0.5, t_c tended to be slightly overestimated; for data with actual ρ > 0.5, overestimation of t_c was greater than those from the control, and for data with ρ < 0.5, estimates were no different from those of the control. Except for the scenario of mismatched ρ, estimates of t_c across these scenarios were expected to reach similar values at ρ = 1.0.

Figure 7.

Posterior time estimates of the crown age t_c from Bayesian tip-dating on unresolved FBD trees, with model mismatches. In a) and b), accuracy and precision of estimates were measured by relative bias and relative 95% CI width, respectively. For each sampling fraction ρ (x-axis), results are shown for analyses of different scenarios if appropriate, with “control” referring to “relaxed–relaxed, N(100,10), estimated, 0.003” in Table 1 (first row under comparative analyses), “all” indicating including all fossils, “random” indicating random sampling assumption, and “biased rho” indicating mismatching ρ (fixed at 0.5). Boxplot summaries are based on varying numbers of replicates (Table 1). Note that when ρ = 0.5, “biased rho” does not apply, and when ρ = 1.0, only “biased rho” applies. Dashed lines indicate the true or expected value.

Among these analyses with generally limited effects on the precision of t_c estimates, including all fossils under diversified sampling had different impacts, in that the relative 95% CI widths were much smaller. In contrast, when random sampling was assumed or ρ was fixed to a value larger than the true value, the relative 95% CI widths were larger (Fig. 7). With reference to the control, ρ mismatches had the most limited effects on the precision of estimated t_c. For all analyses, precision of t_c improved or remained roughly constant when true ρ increased. Little impact on estimates of tree topologies of extant species was also seen in these analyses (Supplementary Appendix 8).

Analysis of Eutherian Mammals

Our total-evidence tip-dating analysis of eutherian mammals produced estimates of relationships that are consistent with previous inferences (Ronquist et al. 2016). These include monophyly of each of the four mammalian superorders and monophyly of Euarchonta (primates with tree shrews and flying lemurs). Posterior probabilities for most groupings are greater than 75%. However, our tree also shows some differences from the previous estimate by Ronquist et al. (2016). For extant taxa, we found a sister relationship between Caviidae (Cavia) and Sciuridae (Ictidomys) in rodents, rather than a grouping of Castoridae (Castor), Muridae (Rattus), and Caviidae (Cavia). Our tree resolved some polytomies involving fossil taxa such as Hyopsodus paulus. The placements of some fossil taxa also differ, such as the position of Sinopa rapax in crown Carnivora rather than on the stem lineage (Fig. 8a).

Figure 8.

a) Maximum-clade-credibility tree of eutherian mammals from our total-evidence tip-dating analysis. The tree is drawn with posterior medians divergence times; blue horizontal bars indicate 95% CIs. Colored circles represent posterior probabilities: ≥ 95% (black), < 95% but ≥ 75% (gray), and < 75% (white). The red square denotes the origin time of eutherian mammals. b) Posterior estimates of node times from all analyses on eutherian mammals. Circles represent posterior medians and lines represent 95% CIs. For each panel, from left to right, estimates from total-evidence tip dating (“TED”) are shown first; the next six estimates are from tip dating on unresolved FBD trees using different numbers of selected fossil occurrences, with extant species sampled at the superorder level and tree topologies estimated (“Superorder, unfixed”) or fixed (“Superorder, fixed”), with extant species sampled at the order level and tree topologies estimated (“Order, unfixed”) or fixed (“Order, fixed”), and with extant species sampled at the species level and tree topologies estimated (“Species, unfixed”) or fixed (“Species, fixed”); the rightmost six estimates are from tip dating on unresolved FBD trees using 28 fossil occurrences.

To compare our estimates with those from other tip-dating analyses, we focused on time estimates of the origin, Placentalia, Boreoeutheria, Euarchontoglires, Laurasiatheria, and Afrotheria. Our estimates are generally younger than those under uninformative priors but older than those under informative priors by Ronquist et al. (2016). For example, our tree placed the placental crown radiation in the Cretaceous with median 95 Ma and 95% CI 85–108 Ma (vs. median 118 Ma, 95% CI 95–178 Ma or median 85 Ma, 95% CI 76–93 Ma by Ronquist et al. 2016) and estimated that Boreoeutheria diverged into Euarchontoglires and Laurasiatheria about 92 Ma (vs. 113 Ma or 81 Ma by Ronquist et al. 2016) when fossils were pruned. Additionally, we found that all boreoeutherian orders diverged from each other before the Cretaceous-Paleogene mass extinction (66 Ma), whereas divergences among some afrotherian orders occurred afterwards (Fig. 8a).

We then examined the impacts of taxon sampling on divergence-time estimates in the maximum-clade-credibility trees with fossils pruned. When fossil occurrences were also sampled at corresponding densities (i.e., 4, 5, and 33 fossils at superorder, order, and species levels, respectively), the effects on time estimates were mixed (Fig. 8b). However, except for the origin time, there was a tendency for posterior medians to provide younger ages for each node at each sampling level when topologies of all sampled taxa were fixed than when only constraints on fossil placements were imposed. In comparison, analyses including 28 fossils at each sampling level produced clearer patterns, where the posterior medians at the superorder level were younger than those from the corresponding analyses with four sampled fossil occurrences. Posterior medians increased with taxon-sampling density regardless of whether tree topologies were fixed or not (with the exception of Afrotheria when tree topologies were not fixed), although the degree differed among nodes (e.g., the origin vs. Placentalia). Fixing tree topologies led to younger estimates at each taxon-sampling level, except for the origin time. In addition, we found that taxon-sampling density had almost no effect on the estimate of the origin time. Instead, fossil occurrences (or even the stem fossil occurrences) seemed to be more important for the estimate of the origin time, but they had limited effects on estimates of other node times.

Discussion

We have performed a detailed study to examine the effects of diversified taxon-sampling schemes on Bayesian tip dating on unresolved FBD trees. This is an approach that can estimate both time and topology and is likely to be more widely applicable than, but not necessarily distinct from, total-evidence tip dating. Our simulation study has shown that although it generally improves inference of the tree topology for extant taxa, increasing the density of diversified taxon sampling does not necessarily lead to better estimates of divergence times (Table 2); under favorable conditions, the dates of key nodes can be estimated accurately and precisely with very sparse taxon sampling. Our results thus did not reveal a substantial benefit to increasing the density of taxon sampling. At first glance, this conflicts with previous suggestions that dense taxon sampling should be able to mitigate some of the negative effects of tree imbalance (Duchêne et al. 2015), rate variation among lineages (Soares and Schrago 2015), and bare-branch attraction (Spasojevic et al. 2021) on Bayesian molecular-clock dating. However, this outcome is partly expected, due to our use of the diversified-sampling FBD process that matched the generating models for the data (cf. Matschiner 2019). Our results thus underscore the importance of accurately modeling extant taxon sampling in Bayesian tip-dating analyses. With regard to inference of the tree topology for extant taxa, our results lend support to the preference for denser taxon sampling in phylogenetic studies (Hillis 1998; Pollock et al. 2002; Brinkmann et al. 2005; Baurain et al. 2007; Heath et al. 2008), although other factors such as data type, model choice, analytical methods, and missing data can be important (e.g., Peloso et al. 2016; Streicher et al. 2016; Reddy et al. 2017; Prasanna et al. 2020).

Table 2.

Overview of the effects of taxon-sampling density under the diversified strategy and other factors on posterior estimates of divergence times in our simulation study

Factor	Effect	Interaction with sampling fraction ρ	Analysis
Main factor a
Sampling fraction ρ	Larger ρ had negative, positive, or negligible effects, depending on other factors	–	All
Others
Among-lineage rate variation	Negligible effects	–	Core
Including morphological characters	Strong and positive effects	Larger ρ had positive effects	Core
Root-age tmrca prior	Strong effects	Larger ρ had negative, positive, or negligible effects, depending on specific tmrca prior	Comparative
Fixing tree topologies	Positive effects	Larger ρ had positive effects, still depending on specific tmrca prior	Comparative
Increasing number of fossils	Positive or negligible effects, depending on specific tmrca prior	Larger ρ had negative or negligible effects, depending on specific tmrca prior	Comparative
Misspecified extant taxon sampling strategyb	Weak and negative effects	Larger ρ had positive effects	With model mismatches
Violation of diversified strategy by including inappropriate fossilsb	Weak and negative effects	Larger ρ had positive effects	With model mismatches
Two above factors combinedb	Negligible effects	–	With model mismatches
Mismatched ρb	Weak and negative effects for actual ρ > 0.5	–	With model mismatches

Notes: “–” no interaction or not applicable; t_mrca = root age.

^a The main factor examined in the simulation study.

^b Interpretations of the effects are based on comparisons with estimates from the control analyses under the normal(100,10) t_mrca prior (Fig. 7).

In our simulation study, other favorable conditions, which were expected to produce good estimates of times with sparse taxon sampling for Bayesian tip dating on unresolved FBD trees, included a root-age t_mrca prior that concentrated probability around the true value (i.e., normal(100,10) in our case). When we adopted a diffuse t_mrca prior that was weakly informative (i.e., uniform(0,200)), fixing the tree to the correct topology was beneficial. Moreover, our results underscore the importance of carefully selecting deep fossil occurrence times, although this is likely to be challenging for many groups of organisms. In practice, the evolutionary models used in phylogenetic analysis are likely to be misspecified to some degree. Hence, despite the results from our simulation study, there are likely to be considerable challenges to obtaining accurate estimates of divergence times in empirical studies with sparse diversified taxon sampling, even when the diversified-sampling FBD process is applied.

Our simulation study has shown that the time estimates obtained were variably influenced by increasing taxon-sampling density. In addition to its negligible effect for some analyses, denser sampling was seen to improve or reduce the accuracy and precision of time estimates, being contingent on other factors (e.g., alternative t_mrca priors). Regardless of the expected values, however, increasing sampling density was always accompanied by an increase in time estimates (represented by posterior medians) when the tree topology was jointly inferred, although the extent of this increase varied with the choice of the root-age prior. This result can largely be attributed to the weaker influence of molecular data compared with the larger impacts of the effective priors of the diversified-sampling FBD process. As additional fossil times towards the younger cutoff time x_cut are included, there is a greater tendency of the effective tree prior to push time estimates up but not to push them down. This can be caused, for example, by a stem fossil being wrongly placed as a crown fossil.

Given the mixture of results, we cannot categorically state whether sparse or denser diversified sampling of extant taxa should be preferred for Bayesian tip dating on unresolved FBD trees. Sparse sampling can be adequate for time estimation under some favorable conditions, but denser sampling leads to better estimates of tree topology. We note that, although we have evaluated the impacts of taxon-sampling density on estimates of divergence times and tree topology, it is also likely to have an influence on estimates of macroevolutionary parameters such as diversification rate and turnover (Supplementary Appendix 9 on Dryad; Höhna et al. 2011; Chang et al. 2020).

Factors with Greatest Impact on Bayesian Tip Dating

While fossil tips provide calibrating information, Bayesian tip dating usually needs the FBD tree prior to be conditioned on an informed starting point such as the origin time t_o or the root age t_mrca (Gavryushkina et al. 2014; Zhang et al. 2016). Our analyses on unresolved FBD trees have shown that the root-age t_mrca prior had a much greater impact than taxon-sampling density and other factors. This was most evidently demonstrated by analyses with joint estimation of time and topology under the three normal-distribution priors, where the t_mrca prior led to time estimates near the prior expectations, despite slight changes across the sampling fraction ρ values. Particularly when we specified a prior that placed high probability near the true value (i.e., normal(100,10)), divergence times tended to be estimated accurately and showed little variation across ρ values. Nevertheless, there were slight overestimations for both marginal priors and posteriors (Fig. 7a); regardless of ρ values, further work is needed to resolve the tendency of time estimates to be older than expected under the FBD process.

The impact of the t_mrca prior is not particularly surprising, given the substantial influence of such calibrations in node-dating approaches (e.g., Ho and Phillips 2009; Duchêne et al. 2014; Warnock et al. 2015; Bromham 2019). Although both total-evidence tip dating and Bayesian tip dating on unresolved FBD trees treat fossils as tips or sampled ancestors (Heath et al. 2014; Zhang et al. 2016), the latter method is more similar to the node-dating approaches with regard to topological constraints for fossil affinities (Zhang et al. 2016). The t_mrca prior effectively functions as an additional calibration under the FBD process, with potential impacts on posterior time estimates. It is also expected to influence the inferred placements of fossil taxa, with consequences for time and rate estimates. This potential effect is suggested by the improvements seen in our analyses with fixed tree topologies.

Although we explored the effects of specifying normal-distribution priors for the root age, our core analyses used a uniform-distribution prior which is more likely to be adopted for real data because of scarce prior knowledge. Under the uninformative uniform(0,200) t_mrca prior which effectively had a minimum bound corresponding to the oldest fossil occurrence involved, Bayesian tip dating on unresolved FBD trees considerably overestimated divergence times, especially given larger ρ. These results are in contrast with the accurate estimates obtained using total-evidence tip dating (this study and Luo et al. 2020), indicating different degrees of the effects of the root-age or origin-time prior on different approaches to Bayesian tip dating. Yet, when we fixed the tree topology in our analyses, the estimates of divergence times were generally accurate under the uniform(0,200) prior. This points to better resilience of the uniform prior than the normal prior for tip dating on unresolved FBD trees.

Although our results have indicated the importance of an informative root-age prior, in practice, the root age (or the origin time) is often highly uncertain and specifying an informative prior is usually impossible. Under these circumstances, by including morphological characters to inform both phylogenetic positions of the fossil taxa and evolutionary parameters (e.g., evolutionary rates), total-evidence tip dating is perhaps a better approach to Bayesian tip dating. This is supported by the results of our core analyses, where total-evidence tip dating was robust to the diffuse uniform(0,200) t_mrca prior and produced accurate time estimates across all ρ values. However, the evolution of morphological characters was simulated under simplified conditions and analyzed under the Mkv model which perfectly matched the data. As total-evidence tip dating has been applied to various groups of taxa (e.g., Larson-Johnson 2016; Gavryushkina et al. 2017; Upham et al. 2019; Spasojevic et al. 2021), the suitability of real morphological characters for Bayesian tip dating deserves further study.

Other Factors Influencing Bayesian Tip Dating

In combination with the substantial influence of the root-age prior, there are several other factors that interact with taxon-sampling density in Bayesian tip dating on unresolved FBD trees. Our core analyses showed that date estimates became worse with increasing sampling density of extant species. However, we found that the presence or absence of among-lineage rate variation had little impact on the outcomes, regardless of the sampling fraction ρ. This is most likely due to the molecular-clock models being generally matched between our simulations and inference. If the uncorrelated lognormal relaxed clock (Drummond et al. 2006) is used to analyze an empirical data set for which, for example, a white-noise model (Lepage et al. 2007) fits better, interaction between taxon-sampling density and among-lineage rate variation could be possible. In that case, denser taxon sampling should be beneficial for Bayesian molecular dating (Soares and Schrago 2015), because there will be a greater amount of information about among-lineage rate variation.

One appealing feature of Bayesian tip dating is the capacity to resolve the phylogenetic placement of a fossil as a tip or sampled ancestor. For our focal Bayesian tip dating on unresolved FBD trees, however, the fossil placement is a stochastic variable conditioned on a topological constraint, and so the fossil is normally pruned from the dated tree (Heath et al. 2014). Thus, we only computed topological distances of maximum-clade-credibility trees without fossils from their true counterparts, and showed that our generally good topological inferences for extant species always improved with increasing taxon-sampling density. However, the importance of fossil phylogenetic positions was evident in our core analyses where we compared time estimates from tip dating on unresolved FBD trees with those from total-evidence tip dating (where fossil phylogenetic positions were informed by morphological characters). It was also evident in the comparative analyses where we fixed tree topologies to those used for simulation and found improved accuracy and precision of time estimates. However, our results also showed that the positive effects of fixing tree topologies as well as denser taxon sampling varied with alternative root-age priors. These indicate that in practice, setting both accurate and precise topological constraints of fossil placements should be beneficial for analyses on unresolved FBD trees, with the most prominent effects for analyses under a diffuse t_mrca prior.

Another attractive feature of Bayesian tip dating, compared with node dating, is that it can potentially make use of larger numbers of fossils for the group being studied (Ronquist et al. 2012). This is particularly the case for analyses on unresolved FBD trees. Our results showed that sampling more fossil occurrences led to an improvement in the accuracy and precision of time estimates, but that this improvement varied with alternative root-age priors and was much greater under a diffuse root-age prior. Based on our results, it is expected that including a larger number of fossil occurrences can further improve time estimates particularly when the root-age prior is diffuse, although this could be computationally impractical (Matschiner et al. 2017; O’Reilly and Donoghue 2020).

In most of the treatments examined in our simulation study, we matched the models used for simulation and analysis. In reality, however, macroevolutionary processes are unobserved and phylogenetic models are invariably violated to some extent. In our analyses involving mismatched models of species sampling, we found that wrongly assuming random taxon sampling led to slight overestimates of divergence times, particularly under sparse taxon sampling. This echoes the results of a study that dated the early radiation of hymenopteran insects to the Triassic and Permian (~252 Ma) under a model of diversified taxon sampling versus the Carboniferous (~347 Ma) under a model of random taxon sampling (Zhang et al. 2016), and a huge gap (~200 Myr) estimated for the crown age of Placentalia (Ronquist et al. 2016). For some groups of organisms such as fishes (Matschiner et al. 2017), fossil occurrence records are abundant, such that fossil subsampling can be carried out to improve computational feasibility (Matschiner 2019; O’Reilly and Donoghue 2020). In contrast, in tip-dating analyses of groups that have a more limited fossil record, it might be necessary to include even very young fossil occurrences. Our analyses involving fossils younger than the cutoff time x_cut produced slight underestimates of node times, particularly with sparse taxon sampling. Compared with those from our control analyses, estimates from these two scenarios thus suggest that one benefit of denser taxon sampling is that the mismatch with the tree prior is reduced. In all of our analyses of Bayesian tip dating, we fixed the sampling fraction ρ for the sake of mathematical feasibility, because it should be the easiest parameter of the FBD process to be inferred beforehand. Even when it was over or underestimated, we found that the induced effects were generally limited. We note that our analyses with model mismatches were only performed under the optimal normal(100,10) t_mrca prior, which did not produce very large discrepancies from the expected node times. There is likely to be a greater risk and/or different patterns of model misspecification under suboptimal conditions (cf. Fig. 8c of Zhang et al. 2016) or in analyses of real data.

Diversified Sampling Strategy and Taxonomic Ranks

Our exploration of different densities of taxon sampling was inspired by the “branches-and-twigs” sampling strategy adopted by major genome-sequencing initiatives (e.g., Jarvis et al. 2014; Lewin et al. 2018; Fan et al. 2020). To mimic this strategy, we sampled extant species along a complete tree using the diversified sampling strategy with a series of levels for the sampling fraction ρ (Höhna et al. 2011; Lambert and Stadler 2013; Zhang et al. 2016). Under diversified sampling schemes (at each ρ < 1.0), the reconstructed tree has long terminal branches that maximize the tree length of the chronogram of extant taxa. In effect, this tree balances the number of extant descendants on either side of each of the deep divergence events. For example, subject to the cutoff time x_cut in Fig. 1b, it is possible that we actually sample five extant species that represent either five orders or two orders (with four species from one order and one species from the other order). In reality, however, taxa are usually sampled on the basis of their taxonomic ranks and other factors (e.g., specimen availability and a balance sampling between various groups). These sampled lineages would not necessarily be the same as those defined by a single cutoff time in the chronogram. For instance, our empirical analyses sampled two representatives from each of the three superorders Afrotheria, Laurasiatheria, and Euarchontoglires, and one representative from Xenarthra. This is not fully consistent with the seven exemplars sampled under the diversified sampling strategy based on the dated eutherian phylogeny in Fig. 7a. Therefore, although the diversified sampling strategy can account for incomplete taxon sampling to a large extent (Zhang et al. 2016), it cannot completely model the taxonomically “diversified” sampling schemes that are used in reality (Donoghue and Yang 2016).

Evolutionary Timescale of Eutherian Mammals

In our analysis of a phylogenomic data set from eutherian mammals (Ronquist et al. 2016), we examined the effects of taxon sampling at various levels and found several patterns that were generally consistent with those from our simulation study. These included the influence of fixing the tree topology, the interaction between taxon sampling and fixing the tree topology and, when all crown fossils were included, an increase in estimated node times with denser sampling of extant taxa. However, we are unable to gauge the accuracy of these estimates, given that the true evolutionary timescale of eutherian mammals is not known.

The diversification history of eutherian mammals continues to be debated, with different sources of data or analytical methods resulting in various scenarios (e.g., dos Reis et al. 2012; O’Leary et al. 2013; Phillips 2016; Ronquist et al. 2016; Davies et al. 2017; Upham et al. 2019; Álvarez-Carretero et al. 2022). For instance, an analysis of ghost lineages found support for an explosive model of placental divergences with a single ancestor around the Cretaceous-Paleogene boundary (O’Leary et al. 2013), whereas our estimates from Bayesian tip dating were more consistent with a fuse model in which some of the ordinal divergences occurred in the Cretaceous (Phillips 2016). Our total-evidence tip dating with morphological characters yielded older estimates than did tip dating on an unresolved FBD tree without morphological characters at the species level (Fig. 8b), a result that is consistent with previous findings of potential date overestimation by total-evidence tip-dating (e.g., Ronquist et al. 2016; Spasojevic et al. 2021). However, perhaps due to inadequate modeling morphological character evolution (e.g., Goloboff et al. 2019), this conflicts with the patterns from our simulation study; specifically, we obtained younger estimates when morphological characters were included in the data set than when they were excluded in our core analyses. In addition, informative priors that placed penalties on ghost lineages had profound effects on Bayesian tip dating of eutherian mammals, helping to close the gap between paleontological and molecular evidence (Ronquist et al. 2016). Additional fossil data and further exploration of the factors affecting Bayesian tip dating will lead to better resolution of eutherian evolutionary history.

Conclusion

Inspired by the taxon-sampling strategies that have been adopted by a range of genome-sequencing initiatives, we have investigated the impacts of increasing taxon sampling on Bayesian tip dating on unresolved FBD trees. Our results have shown that denser taxon sampling does not always lead to better estimates of evolutionary parameters, despite its apparent benefits for topological inference. Instead, performance is also influenced by the prior on the root age, the positions of fossil taxa, the number of fossils involved, and model misspecification. Among these, we found that the prior on the root age had a strong impact, reflecting the limitations of Bayesian tip dating on unresolved FBD trees as well as the potential advantages of total-evidence tip dating involving morphological characters in practice.

Our results are based on evolutionary simulations that do not fully capture the complexities of real data. As such, we recommend that future studies should explore additional simulation conditions that mimic those of biological reality (Barido-Sottani et al. 2020), such as species diversification rates informed by real data, lower sampling fractions, and uncertainties in fossil ages (O’Reilly et al. 2015; Barido-Sottani et al. 2019). Analyzing a wider range of phylogenomic data sets will also provide further insights into the benefits of denser taxon sampling in practice. With a greater understanding of the impacts of taxon sampling and other factors, Bayesian tip dating can be applied more effectively to phylogenomic data sets to resolve evolutionary timescales.

Supplementary Material

Data available from the Dryad Digital Repository: http://dx.doi.org/10.5061/dryad.zw3r2288t

Funding

This work was supported by the National Science Fund for Excellent Young Scholars (32122016) and the National Key Research Development Program of China (2022YFF0802300). A.L. was funded by the National Natural Science Foundation of China (32070465) and the National Science & Technology Fundamental Resources Investigation Program of China (2018FY100401). C.Z. was funded by the Hundred Young Talents Program of the Chinese Academy of Sciences (Y902061), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB26030300), and the National Natural Science Foundation of China (42172006). Q-S. Z. was funded by the National Natural Science Foundation of China (31801998). S.Y.W.H. was supported by the Australian Research Council. C-D. Z. acknowledges the support of the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB31000000).