Detecting Lineage-Specific Shifts in Diversification: A Proper Likelihood Approach

Abstract

The branching patterns of molecular phylogenies are generally assumed to contain information on rates of the underlying speciation and extinction processes. Simple birth–death models with constant, time-varying, or diversity-dependent rates have been invoked to explain these patterns. They have one assumption in common: all lineages have the same set of diversification rates at a given point in time. It seems likely, however, that there is variability in diversification rates across subclades in a phylogenetic tree. This has inspired the construction of models that allow multiple rate regimes across the phylogeny, with instantaneous shifts between these regimes. Several methods exist for calculating the likelihood of a phylogeny under a specified mapping of diversification regimes and for performing inference on the most likely diversification history that gave rise to a particular phylogenetic tree. Here, we show that the likelihood computation of these methods is not correct. We provide a new framework to compute the likelihood correctly and show, with simulations of a single shift, that the correct likelihood indeed leads to parameter estimates that are on average in much better agreement with the generating parameters than the incorrect likelihood. Moreover, we show that our corrected likelihood can be extended to multiple rate shifts in time-dependent and diversity-dependent models. We argue that identifying shifts in diversification rates is a nontrivial model selection exercise where one has to choose whether shifts in now-extinct lineages are taken into account or not. Hence, our framework also resolves the recent debate on such unobserved shifts. [Diversification; macroevolution; phylogeny; speciation]

The literature abounds with examples of spectacular radiations, where specific clades seem to have an elevated diversification rate against a much slower background rate of diversification (Liem 1973; Mitter et al. 1988; Schluter 2000; Blount et al. 2008; Yoder et al. 2010). This phenomenon may occur due to increased mutation rates (Hua and Bromham 2017) but may also be caused by ecological opportunity (Simpson 1944, 1953; Wellborn and Langerhans 2015; Mahler et al. 2010 which may arise in three different ways: 1) antagonist extinction; 2) availability of a new environment, either due to dispersal to a new area or due to environmental change driven; or 3) a key innovation that enables escape from competition for niche space (Heard and Hauser 1995). The theoretical arena to study macroevolutionary diversification is the framework of stochastic birth–death models, where speciation is modeled as a birth event and extinction as a death event. These birth–death models allow for estimating speciation and extinction rates from phylogenetic trees. Nee et al. (1994) provided the mathematical tools to do so for the birth–death model with constant or time-dependent speciation and extinction rates. Their work has been the foundation for biologically more complex models. One example is the diversity-dependent birth–death model where speciation and extinction rates are influenced by the number of species in the same clade (Etienne et al. 2012). Another set of examples are the trait-dependent birth–death models, notably the State-dependent Speciation and Extinction (SSE) models (e.g., BiSSE by Maddison et al. (2007), QuaSSE by FitzJohn (2010), MuSSE by FitzJohn (2012), HiSSE by Beaulieu and O’Meara (2016) and SECSSE by Herrera-Alsina et al. (2018)). These models allow for trait shifts over macroevolutionary time and assign different diversification rate regimes to each trait value.

Methods to detect clades with elevated diversification rates without reference to traits or diversity have been developed and are available in a number of software programs. There are two types of approaches. The first type maps the rate shifts on the tree and then asks whether these rate shifts are statistically supported. Implementations of this type include MEDUSA (Alfaro et al. 2009), BAMM (Rabosky 2014), the Key Innovation model in DDD (Etienne and Haegeman 2012), and the split-SSE models in DIVERSITREE (FitzJohn 2010, 2012). The second type does not map the shifts explicitly on the phylogeny but assumes a multistate SSE model with each state having its own speciation and extinction rates, where the shifts in states (and hence in diversification rates) are modeled dynamically. Implementations of this type include the Lineage-Specific Birth–Death-Shift (LSBDS) models in RevBayes (Hoehna et al. 2019), the MultiState Birth–Death model (MSBD) in BEAST2 (Barido-Sottani et al. 2020), and ClaDS in RPANDA (Maliet et al. 2019). LSBDS and MSBD assume that lineages change to a different state along a branch, while ClaDS assumes that the state shift occurs during speciation. They are special cases of the SECSSE (Herrera-Alsina et al. 2018) and MISSE (Caetano et al. 2018) frameworks—which combine features of MuSSE (FitzJohn 2012), GeoSSE (Goldberg et al. 2011), ClaSSE (Goldberg and Igic 2012) and HiSSE (Beaulieu and O’Meara 2016)—applied to many concealed traits (and no examined traits). Here, we address implementations of the first type, that is, with shifts in diversification rates mapped on the phylogeny. These methods rely on the same framework as the SSE models but without state-dependence. We will refer to this framework as the D–E framework with mapped rate shifts, from the names of its core functions ( and ). However, here we show that the D–E framework, while mathematically sound in general (such as in applications to SSE models), cannot be applied to models with mapped shifts in rates. We demonstrate that it can lead to probabilities larger than 1. We propose a new, mathematically correct, analytical likelihood formula based on the functions originally introduced by Nee et al. (1994) for the constant-rate birth–death model of diversification without rate shifts. Using simulated data with a single shift, we show that our new likelihood performs better in parameter estimation than the incorrect likelihood. Furthermore, we extend our mathematical reasoning to multiple shifts and time-dependent and diversity-dependent models. Finally, we show that model selection in this framework requires making decisions on whether unobserved shifts (i.e., shifts on extinct lineages) are allowed or not.

Methods

The D–E framework

The D–E framework uses two variables: , the probability of observing the tipward part of the phylogeny at a given lineage at a given time in the phylogeny, and , the probability of a lineage alive at time to have no surviving descendants at the present. To compute the likelihood of the entire phylogeny, one computes and by integrating the following set of differential equations, for every branch from tip to the first rootward node:

(1)

(2)

which has the following solution for initial conditions and ,

(3)

(4)

At a node the two -values of the two daughter branches are multiplied with one another and the speciation rate to obtain the of the parent branch. This value will then serve as the new initial condition to further integrate the system back in time to obtain at the next rootward node. This is then continued until one reaches the crown or stem of the phylogeny. The value at the stem or crown is the likelihood of the phylogeny. For nothing changes at the node, as this extinction probability is independent of observed branching points. We refer to the original papers by Alfaro et al. (2009) and Rabosky (2014), and to Appendix A for more details. The likelihood computed in this way is correct as long as the same rates are used for all lineages (observed and extinct) at a particular time. Below, we show that the likelihood is no longer correct if there are lineage-specific rates.

The D–E framework applied to mapped rate shifts leads to probabilities larger than 1

Rate shifts have been accommodated in the D–E framework in the software packages mentioned in the introduction (e.g., MEDUSA, BAMM, DDD) by using different rates of speciation () and extinction () for the lineage that undergoes a rate shift. There has been some debate on the initial condition for the equation for at the shift point that will be used for further rootward integration of the equations. Rabosky et al. (2017) proposed a “recompute” and a “pass-up” algorithm. The first, “recompute,” recomputes the using the rootward rates, whereas “pass-up” uses as initial condition at the time of the shift the already computed with the shifted rates for the subclade experiencing the rate shift from the present until the time of the rate shift. The “pass-up” algorithm is incorrect because the extinction rate that is needed in the computation of is computed for lineages that will not shift. The “recompute” algorithm is the correct one to compute the extinction rate, but we show here that the D–E framework applied to mapped shifts still suffers from another problem that yields an incorrect likelihood.

We look at a simple example of a phylogeny with only a single extant branch. This ensures that the we obtain at the root is a real probability and not a probability density due to the multiplication by at the nodes (formally the multiplication is with for an infinitesimal ). We assume that at time a clade-wide rate shift in diversification rates occurs: all lineages present at that time undergo this shift. Subsequently, at time another shift occurs involving only one branch, which is the branch that we currently observe. Other branches become extinct before the present.

To study the full process, we divide it into three subprocesses (Fig. 1), each characterized by a set of speciation and extinction rates ():

Figure 1.

Example phylogeny leading to an incorrect likelihood in the D–E framework. The phylogeny consists of a single branch from to , with a lineage-specific rate shift at (indicated by a -mark). This rate shift initiates subclade not included in main clade . In addition, the rates in the main clade change at from to . Open circles indicate species that go extinct before the present and therefore are invisible in the phylogeny. From Eq. (5) onwards, we consider the limit of .

• for subprocess : rates and ;

• for subprocess : rates and . These rates do not only govern the diversification dynamics occurring in the interval , but also the diversification in the interval for all the lineages that do not undergo the lineage-specific rate shift at (which is why the “pass-up” algorithm is incorrect);

• for subprocess : rates and .

We have used and to denote these subprocesses, because the first two processes occur in the main () clade that does not undergo a clade-specific shift, but the last one occurs in the a subclade () after a shift in a single lineage.

Adopting the D–E framework and the “recompute” strategy an analytical formula for the likelihood can be derived (Appendix A). In the limit of , that is, when the lineage-specific rate shift occurs immediately after the clade-wide range shift, this solution can be written in a compact way using the functions and from Nee et al. (1994):

(5)

where

(6)

(7)

with the subscript referring to the rate regime (, or ).

Exploring likelihood (5)—which is a probability—numerically for different values of and we observe that it exceeds unity in a large part of parameter space (left panel of Fig. 2), and hence must be incorrect.

Figure 2.

Comparison of uncorrected and corrected likelihood for the example of Figure 1. We computed the likelihoods as a function of speciation rate and extinction rate and plotted the results as a heatmap in the plane. The other parameters are kept constant at . Left panel: The likelihood of the D–E framework is larger than one for a large part of parameter space (shades of red) and can reach values that are several orders of magnitude above one. Right panel: The corrected likelihood is always smaller than one. We have used , , in this numerical example.

We note that we need a clade-wide shift to get probabilities larger than 1. The reason is that to maximize the error in the likelihood, many species need to exist at the time of the shift, which requires a high speciation rate and a low extinction rate. However, in the sampled phylogeny only one lineage remains, so all lineages but one (namely the one that undergoes the shift) then need to go extinct, which requires low speciation rate and high extinction rate. This can only be achieved by having rates change over time, and a clade-wide shift is the simplest way of achieving that. This does not mean that the problem does not occur unless there are clade-wide shifts, but these clade-wide shifts are needed to obtain probabilities larger than 1 which is a clear signal that the likelihood calculation is incorrect. The likelihood calculation remains incorrect when the probabilities are smaller than 1.

Corrected likelihood—example

The D–E framework only prescribes how to compute the likelihood, but it does not explicitly state the model underlying this likelihood. What has been missing (as pointed out by May and Moore 2016) in applications of the D–E framework to mapped rate shifts, is a model for these shifts. Here, we propose a simple stochastic model for a lineage-specific rate shift, and we use it to yield the right likelihood formula for the example shown in the previous section (Fig. 1).

Our model runs from past to present. The simple stochastic model for the rate shift we propose is that at the rate shift time one of the extant species is chosen at random to undergo the rate shift. To construct the likelihood corresponding to this model, we use again functions (eq. (6)) and (eq. (7)) from Nee et al. (1994). We denote by the number of extant species immediately before the rate shift time. Then, the basic elements of the likelihood are:

1. The single species present at the initial time undergoes a diversification process with rates and . The probability that it has descendant species immediately before the rate shift time (i.e., the 1 in the probability corresponds to the single species at and the to the number of descendants at ), is

   (8)
2. The data (the reconstructed tree of Fig. 1) impose that the rate shifted species (governed by rates and ) survives until the present and has only one descendant species. The probability of this event is

   (9)
3. The other species extant at time are governed by rates and and should become extinct in the time interval . The probability of this event is

   (10)

Combining these elements we can write the corrected likelihood

(11)

which can be simplified to

(12)

It is instructive to rewrite the likelihood in a form similar to eq. (11),

(13)

showing that the difference with resides in the additional factor in the summand of eq. (13). This factor is erroneous, given the explicit rate shift model we consider here (see Appendix B for more details), and it also causes the probabilities to exceed unity for some parameter values (left panel of Fig. 2).

The corrected likelihood solves the problem with probabilities larger than 1. This can be seen from eq. (12) by noting that . We also checked this numerically (right panel of Fig. 2).

We stress that defining a model for how the rate shift occurs is crucial, which explains why SSE-based approaches that model shifts dynamically (Barido-Sottani et al. 2020; Hoehna et al. 2019; Maliet et al. 2019) yield correct likelihoods. However, these models require the introduction of (an arbitrary number of) states and a rate shift parameter and are therefore arguably more complex than the model we propose here.

Corrected likelihood—general case

The corrected likelihood can be extended to general trees, as we show in Appendix C. Suppose the rate shift occurs in the th lineage of the lineages present at the rate shift time . Denote by the subclade including all descendants of the shifted lineage , by the main clade excluding , and by and the parts of before and after the rate shift, respectively (Fig. 3). Furthermore, denote by the operation of breaking the tree sensu Nee et al. (1994) into separate branches each of which we will index with . Here can be either , , or . Strictly, needs not be a single tree, but may consist of several trees. For example, in the top-left panel of Figure 3, consists of three clades which have stem age at : the clades that arise from lineages 2, 3, and 4.

Figure 3.

Definition of subtrees used in likelihood formula (14). We consider a phylogeny with lineages at the rate shift time . Top-left panel: Suppose the rate shift occurs in the first lineage (). The subtree is the subclade containing all the descendant species of the shifted lineage (in red). The corresponding main clade is , which we decompose in a part before the shift (in green) and a part after the shift (in blue). Other panels: If the rate shift occurs in another branch (), the corresponding main and subclade are different (subtrees and ).

The likelihood that the rate shift occurs in branch of the main clade, at time , and that it is observed, that is, that the rate shift is visible in the observed tree,

(14)

where denotes the number of observed lineages in the phylogeny at time and the summation index denotes the number of species that come from branch in .

Conditional likelihoods

Likelihoods of diversification models are often conditioned on the existence of the phylogeny, that is, the survival of the two crown lineages (Nee et al. 1994), but also other conditionings have been discussed in the literature (Stadler 2012). Here, we discuss the standard conditioning on crown lineage survival () and two additional conditional probabilities. The two additional ones still require that the two crown lineages survive to the present, but we additionally require that the lineage with the rate shift survives to the present with () or without () requiring that there is at least one other unshifted surviving lineage of the same crown lineage. To describe these conditional probabilities we need to introduce some notation. We denote by and , the two distinct subprocesses arising from left and right crown species, respectively. We assume that undergoes the rate shift at . With we denote the process arising from the shifted species, as before.

For the standard conditioning on the survival of the two crown lineages, we require to survive from the crown to the present, and from the crown to the shift, and either or to survive from the shift to the present. The conditional probability is given by

(15)

where the factor of two arises from the symmetry of the system: swapping with yields a tree that is indistinguishable from the original one.

For the first new conditioning, we require to survive from the crown to the present, to survive from the crown to the shift, and to survive from the shift to the present, but is not required to survive to the present. The conditional probability is therefore given by

(16)

As a second new conditioning, we require the survival to the present of both left and right crown species descendants in clade , as well as the survival of clade . Its probability, , is given by

(17)

The observed tree satisfies the different conditionings. Hence, the conditional likelihood is obtained simply by dividing the likelihood (14) by either , , or .

Performance of the corrected likelihood in parameter estimation

We tested numerically the performance of the corrected likelihood formula versus the incorrect likelihood resulting from applying the D–E framework to mapped rate shifts for parameter estimation on phylogenies simulated under a constant-rate birth–death model with a single shift for various values of the generating parameters and two conditionings (Table 1). We did not use conditioning probability , because to be useful the simulations required the subclade to survive; using would have introduced biases unrelated to the quality of the estimation method.

Table 1.

Parameter settings used in the simulations. For each parameter setting we simulated 1000 trees. To simulate the subclades parameters and have been used; we did not vary them as they do not influence the inference outcome. We did not use conditioning because the simulations were conditioned on survival of the shited subclade.

Parameter	Values
	0.2, 0.3, 0.4, 0.5
	0, 0.05, 0.1, 0.15
	,
	,

We simulated 1000 trees for each parameter setting. We then maximized the likelihood for each tree to infer the best parameter values for , , , and . We find that the corrected likelihood produces less biased parameter estimates than the likelihood resulting from applying the D–E framework to mapped rate shifts (Fig. 4).

Figure 4.

Comparison of the parameter estimates of the main clade diversification parameters and inferred by the corrected likelihood versus the D–E likelihood. The boxplots show, for various parameter settings displayed on the -axis, the distributions of the error ratios and , where , , , and were obtained via likelihood maximization and and are the known true values used to simulate the trees. Boxplots below 1 imply that the corrected likelihood is closer to the true generating value than the D–E likelihood. Parameter values are given in Table 1; the results for shift times and are shown in the same boxplot. For each unique parameter setting, trees were simulated and hence parameter sets estimated for each likelihood. The two different colors represent the two parameters, while the two panels represent different conditionings ( and , see eqs. (16) and (17)).

Extending the corrected likelihood to time-dependent and diversity-dependent diversification rates

When the diversification rates depend on time, the basic structure presented in the previous section still holds. However, according to Nee et al. (1994), the core functions (6) and (7) have to be replaced with

(18)

(19)

where

(20)

The corrected likelihood can also be extended to diversity-dependent rates. To do so, we make use of the general framework first introduced in Etienne et al. (2012), which was later used to study the specific case of a single lineage rate shift due to the introduction of a key innovation (Etienne and Haegeman 2012. The correction comes down to a division by the number of lineages at the shift time (see Appendix E for details), and has been implemented in version 4.3 of the DDD package (Etienne and Haegeman 2020).

Extending the corrected likelihood to multiple shifts

The extension of the corrected likelihood (14) to the case with multiple shifts is relatively straightforward. Although the formulas become increasingly cumbersome, the correction is based on the same idea as in the single-shift case. To account correctly for the choice of the species undergoing the rate shift at the rate shift time, we have to divide by the number of species in which the rate shift can occur. In Appendix C, we explicitly work out the correction for the case of two rate shifts occurring in the main clade, leading to the likelihood formula (C11).

The conditioning probabilities (15)–(17) can be also extended to the multiple-shifts case. To keep the computations manageable, we consider only the simplest extension. We require that already shifted subclades cannot undergo another shift, that is, shifts only happen in the main clade. Note that this need not be a strong restriction, because most applications involve large trees with a handful of shifts. In addition, as in conditioning probability , we require that there are surviving species in both crown clades and in all shifted subclades. The corresponding conditional likelihood can be easily evaluated within the diversity-dependent framework of Etienne et al. (2012), and has been implemented in version 4.3 of the DDD package (Etienne and Haegeman 2020).

Detecting rate shifts in phylogenetic trees

The corrected likelihood (14) can be used to ask whether a given lineage underwent a rate shift at the designated shift time . To do so, we must compare the likelihood (14) with a version of (14) where the rates of the main clade M and the subclade S are the same. That is, we compare a model where the rates actually change at the shift time with a model where the rates remain the same at the shift time (which we will refer to as a dummy shift).

It is important to note that this dummy-shift likelihood is different from Nee et al.’s likelihood. This can be understood by observing that the former likelihood depends on the predetermined lineage in which the rate shift possibly occurred, while the latter does not distinguish between lineages but treats all lineages equally. To recover Nee et al.’s likelihood as the reference case in the model comparison, we should ask whether the data contain evidence for a rate shift at a specified time without specifying the rate shift lineage. To address this question, we have to account for two possibilities: either the rate shift occurred in one of the observed lineages of the phylogeny, or it occurred in a lineage present at time , but that has become extinct after . We refer to these two cases as observed and unobserved rate shifts, respectively. The likelihood of an observed rate shift is simply obtained by summing across all branches present at time ,

(21)

The likelihood of an unobserved rate shift is given by

(22)

The likelihoods for observed and unobserved shifts can be added to yield a (generalized) likelihood for a model with a rate shift, on any lineage in the phylogeny that was alive at :

(23)

If we take the same rates for main clade M and subclade S (i.e., a dummy shift), we get (see Appendix D)

(24)

where is the likelihood for a phylogeny produced under a constant-rate birth–death model without considering rate shifts, as provided by Nee et al. (1994). This shows that the generalized rate shift likelihood can be compared with the standard likelihood without rate shifts, if we do not specify a priori the lineage in which the rate shift might occur. Note that no such construction is possible when applying the D–E framework to mapped rate shifts, as this systematically overestimates the rate shift likelihood.

Discussion

We have shown that several methods developed for detecting shifts in diversification rates at particular points in phylogenies are based on an incorrect likelihood that could even yield probabilities larger than 1. Our new likelihood formulas—which even apply when diversification rates are time-dependent or diversity-dependent—can be used to correct these methods. This was already been done for the KI model in version 4.1 of the DDD package (Etienne and Haegeman 2020), which involved only a few lines of code (division by the number of species). For a single shift, DDD allows conditioning in three different ways, corresponding to the diversity-dependent extensions of Eqs. (15)–(17). For multiple shifts, only the last conditioning is feasible, under the additional assumption that shifts can only occur in the main clade. We noticed that the results for two different conditionings were similar for a single shift, so we are confident that this restriction is not very severe. For multiple shifts, the DDD package only contains the likelihood calculation, not a function to do likelihood optimization. This could be done by integrating this likelihood calculation with BAMM, MEDUSA, or related multishift methods.

Our framework has identified four ways to ask questions on rate shifts at a given shift time. One can ask whether a rate shift occurred in a particular observed lineage in the phylogeny at the shift time, whether it occurred in any observed lineage present at the shift time, whether it occurred in any unobserved lineage present at the shift time, or whether it occurred at all at the given shift time (either in an observed or unobserved lineage). We have provided likelihood formulas for all four cases. This was possible because we provided an explicit model for the choice of the lineage on which the rate shift occurs. Moore et al. (2016) already remarked that such a model was missing in BAMM which they considered problematic for estimating evolutionary rates, although the Moore et al. (2016) critique employed the problematic D–E framework applied to mapped rate shifts. With our formulas, we have offered a solution to these problems. In all four cases, we have also provided the appropriate null model for model comparison, that is, a model where there is a shift but rates do not actually change (dummy shift). We have also shown that we recover the well-established likelihood of a constant-rate or time-dependent birth–death model (Nee et al. 1994) in case we allow this dummy shift to occur anywhere in the phylogeny.

While we offer a likelihood that can account for unobserved rate shifts, the use of maximum likelihood to infer shifts on extinct branches is not especially useful in practice. Indeed, this will lead to the estimate of the shifted extinction rate being infinite, as this makes the observed phylogeny most likely. Even when fixing the extinction rate at a given value (or not allowing it to shift), maximizing the likelihood will lead to the shifted speciation rate being estimated to be 0 in unobserved lineages. Fortunately, the generalized likelihood developed to answer the question whether a shift occurred at all at the given shift time (either in an observed or unobserved lineage)—of which the likelihood for unobserved shifts is an integral part—will not suffer from this problem as any inferred shifted extinction rate applies to both observed and unobserved lineages. To assess whether estimates are biased, one should perform an extensive analysis of simulated trees which is beyond the scope of the current article. An alternative is that for a specific study one conducts simulations with the estimated parameters and checks whether the distribution of parameters estimated from the simulations is in line with the original estimated parameters used to generate the simulated phylogenies. In fact, this parametric bootstrap procedure is useful for any parameter estimation method.

Because the likelihoods implemented in most rate shift methods suffer from the largely unappreciated issue described in this article, we suggest that researchers interpret results from these methods with caution. The likelihoods obtained with these methods are systematically too large, and hence using these methods to detect rate shifts may be subject to false positives. Simulation studies have generally found that BAMM appears conservative with respect to the inference of rate heterogeneity (Maliet et al. 2019), and rate estimates have been shown to be reasonable across a broad range of parameter space (Title and Rabosky 2019). However, the fact that the likelihood expression is incorrect implies that BAMM and other approaches may behave unexpectedly in some areas of parameter space or on some data sets. We expect the errors to be largest when there are many species at the time of the shift which do not survive to the present, but the specific conditions under which this situation arises may be hard to identify. Researchers who use these methods should be vigilant in assessing whether results are sensible and should strive to cross-check inferences with alternative methods wherever possible. Our numerical results show how, on sets of simulated trees, the new likelihood (14) performs consistently better in estimating the rates (i.e., produces less bias) than the likelihoods based on applying the D–E framework to mapped rate shifts. In some simulations the difference was large, in others it seemed small, but it is difficult to say beforehand when estimates will deviate mostly between the corrected likelihood and the likelihood resulting from applying the D–E framework to mapped rate shifts.

As stated in the introduction, alternative approaches to detecting shifts in diversification rates have been developed that do not explicitly map shifts in diversification rates on the phylogeny, but rather look for evidence for multiple rate regimes in general across the phylogeny (Barido-Sottani et al. 2020; Hoehna et al. 2019). These approaches—which rely on the same mathematical equations—do not suffer from the problems we identified here, but they resort to ancestral state reconstruction to identify what parts of the phylogeny are governed by a rate regime, that is, the shifts in diversification are not fixed, but one obtains a (posterior) probability distribution for the shift positions. Another recently developed approach assumes that each speciation event is accompanied by (usually small) rate shifts in each descendant lineage and allows reconstruction of branch-specific rate estimates (Maliet et al. 2019). However, in each of these frameworks, it is not possible to directly link rate shifts to historical events that happened at specific times: shifts are either assumed to happen with each speciation event (Maliet et al. 2019) or the model determines where the most probable shift locations are (Barido-Sottani et al. 2020; Hoehna et al. 2019). When linking rate shifts to historical events such as glaciation (e.g., Weir et al., 2016) or mountain uplift (e.g., Chaves et al., 2011) is the ultimate goal of rate shift analyses, our likelihood framework is ideally suited.

Lastly, we emphasize that our approach can also be used for diversity-dependent diversification models, which is not the case for the three recent approaches mentioned above (Barido-Sottani et al. 2020; Hoehna et al. 2019; Maliet et al. 2019).

It has recently been suggested that making inference on diversification scenarios from phylogenies of extant species may be a futile enterprise, because this type of data cannot distinguish between models assuming constant speciation and extinction rates and an infinite set of models with time-dependent speciation and extinction models (Louca and Pennell 2020), which is a generalization of the results by Nee et al. (1994) that a model with constant speciation and extinction rates is equivalent to a model without extinction and time-dependent speciation rate. While this problem of nonidentifiability has not yet been mathematically shown to apply also to SSE models (Louca and Pennell 2020), diversity-dependent models (but see Etienne et al. 2016) or rate shift models as considered here, the results of Louca and Pennell (2020) have made clear that we can only draw conclusions on the models that we are comparing, and not on the existence of time-dependence, diversity-dependence or rate shifts in general.

In summary, we have described theoretical concerns with the likelihood expression used by a number of diversification rate shift methodologies, and we have provided a new likelihood formula for rate shifts that is mathematically consistent. We implemented this approach for macroevolutionary scenarios involving a single rate shift and found that the approach performed better than the incorrect D–E framework. We hope that our formulas or algorithms to compute the likelihoods will be applied in likelihood-based inference tools such as BAMM and MEDUSA.

 

Appendix A: D–E likelihood for example phylogeny of Fig. 1

Here, we compute the D–E likelihood for the tree of Fig. 1. We apply the “recompute” algorithm and use the solutions (3) and (4) for the functions and .

In the example, the phylogeny has only a single extant branch. This implies that the D–E likelihood we obtain at the root is a real probability and not a probability density due to the multiplication by at the nodes (formally the multiplication is with for an infinitesimal ). We assume that at time a clade-wide rate shift in diversification rates occurs: all lineages present at that time undergo this shift. Subsequently, at time another shift occurs involving only one branch, which is the branch that we currently observe. Other branches become extinct before the present.

The derivation of the D–E likelihood proceeds in several steps:

• We solve the D–E equations in the interval with rates and . For initial conditions, and the solution reads

  (A1)

  (A2)

  where the index 1 refers to the first computation in the interval .
• We solve the D–E equations a second time in the interval , this time with rates and . This is required for the “recompute” variant of the D–E framework. The initial conditions are again and , so that

  (A3)

  (A4)

  where the index 2 refers to the second computation in the interval .
• We solve the D–E equations in the interval with rates and . The initial conditions are (species that originate in and become extinct in are governed by rates and ) and (the observed branch in is governed by rates and ). We get

  (A5)

  (A6)
• We solve the D–E equations in the interval with rates and . Using as initial conditions the expressions for and , we get

  (A7)

Then, the likelihood as prescribed by the D–E framework (which we are going to show to be incorrect) is .

To make formulas clearer, we set , that is, the lineage-specific rate shift occurs immediately after the clade-wide range shift. Then, eqs. (A1)–(A7) can be expressed in terms of the functions and introduced by Nee et al. (1994), already stated in eqs. (6) and (7), which yields

(A8)

(A9)

(A10)

which establishes eq. (5).

Appendix B: Corrected likelihood for phylogeny of Figure 1

In the main text, we presented a short derivation of the corrected likelihood for the example phylogeny shown in Figure 1. Here, we provide an introduction to the approach of Nee et al. (1994) on which the derivation is based, and a comparison of the corrected likelihood with the incorrect likelihood computed within the D–E framework.

A short introduction to Nee et al. (1994)

Many properties of the constant-rate birth–death process can be expressed in terms of functions and introduced by Nee et al. (1994),

where is the speciation rate and the extinction rate. In particular, the probability that the process starting at time with a single species has descendant species at a later time is given by

(B1)

We see that the number of species is a geometric distribution with parameter with an added zero term (Nee et al. 1994). This formula can be generalized to the probability that species at time have species at time , denoted by . The computation exploits the fact that the species at time undergo independent dynamics, so that the number of species at time is the -fold convolution product of . For example, the formula for the case reads

(B2)

Nee et al. (1994) showed that under the constant-rate birth–death model the likelihood of a phylogeny can be computed as a product over branches. Each of these branches runs from a branching time to the present time . The corresponding likelihood contribution is equal to the probability that a speciation event occurs at multiplied by the probability that the branch has a single descendant species at the present time . Explicitly,

(B3)

The infinitesimal factor is required to impose that the branching time occurs in the infinitesimal interval . Because, this factor does not affect likelihood maximization, we will leave it out of the likelihood formulas (so that the likelihood is no longer a probability, but a probability density). Taking the product over branches, we obtain the (unconditioned) likelihood of the phylogeny

(B4)

where denotes the number of branching events in the phylogeny and denotes the crown age. Because the likelihood is obtained by decomposing the phylogeny into branches, the approach of Nee et al. (1994) is sometimes referred to as “breaking the tree.” In Appendix C, we present an alternative derivation of eq. (B4), which in contrast to the argument of Nee et al. (1994) can be easily generalized to phylogenies with rate shifts.

Comparison of D–E likelihood and corrected likelihood

From the expressions of the likelihoods and , we see that the difference resides in an additional factor . To interpret this difference, we isolate in both likelihoods the probability that, given that there are species at the rate shift time , one of them undergoes the rate shift and has surviving descendant species at the present time , while the other species has no descendant species at . This probability is

(B5)

(B6)

In Figure B1, we construct this probability by explicitly considering all possible full trees corresponding to a specific reconstructed tree. Note that in the figure we use simplified notation and set and .

Figure B1.

Demonstration of correct likelihood formula. a) Case of phylogeny without rate shift. We consider a reconstructed tree consisting of a single branch running from to . We assume that there are three extant species at an intermediate time (indicated by the middle vertical line) and consider the dynamics of these three species from to , all governed by rates and . There are three possible full trees, each having probability , so that the total probability is . b) Case of phylogeny with rate shift. The reconstructed tree consists of a single branch with an observed rate shift at the intermediate time . We assume again that there are three extant species at , each of them having probability of undergoing the rate shift. The rate shifted species has rates ; the other species have rates . We show the nine combinations obtained by first selecting the species undergoing the rate shift, and second selecting the species surviving until the present. Only three of these nine combinations have the correct reconstructed tree (consistent combinations are plotted in blue); the other six combinations correspond to unobserved rate shifts. Each of the three consistent full trees have probability , so that the total probability is . Symbols have same meaning as in other figures (-mark: rate shift; filled circle: species surviving until present; open circle: species going extinct before present).

It is instructive to first consider the case without rate shift (Fig. B1a). We consider a reconstructed tree consisting of a single branch. If there are species at the intermediate time , there are possible full trees, because each of the species extant at time can be chosen to survive. Each full tree has probability , so that the total probability is . Note that this probability cannot be larger than one; indeed, when adding the probabilities that none or more than one species survives, we get

(B7)

Next consider the case with rate shift (Fig. B1b). As in the example of Fig. 1, we consider a reconstructed tree with a single branch and an observed rate shift. We see that there are possible full trees, corresponding to two choices, each one having options. First, we have to choose the species that is undergoing the rate shift. Second, we have to choose the species that is going to survive, which can be either the species that has undergone the rate shift or one of the other species. Importantly, not all of these full trees are consistent with the reconstructed tree. In particular, for of them the rate shift is unobserved. The full trees for which the rate shift is observed in the corresponding phylogeny each have probability , so that the total probability is . This probability cannot be larger than one, because when adding the probabilities that none or more than one species survives,

(B8)

This computation demonstrates that formula (B6), and hence likelihood , is correct, and that formula (B5), and hence likelihood , is not. The latter, which can be seen as a naive generalization of the formula without rate shift, does not account correctly for unobserved rate shifts (i.e., rate shifts that occur in a species that is not represented in the phylogeny). Note that formula (B5) is also different from the probability of having either an observed or an unobserved rate shift. The correct probability for this case is

(B9)

Appendix C: Corrected likelihood for general phylogenies

In this appendix, we derive the likelihood formula for a general phylogeny with one or several lineage-specific rate shifts. We start the computation, which is related to the “breaking the tree” argument of Nee et al. (1994), by deriving the likelihood again for a general phylogeny without rate shift.

A useful identity

We will repeatedly use the following identity:

(C1)

This identity can be understood by introducing a sampling process at time , in which each extant species is sampled with probability . Multiplying the left-hand side of eq. (C1) by , we see that

is the probability that a species at time has descendant species at time and one sampled descendant species. We establish identity (C1) by computing this same probability in a different way. To do so, we consider the species extant at time . For this group of species to have one sampled descendant species at time , there should be one of the species with a single sampled descendant species, and all other species should have no sampled descendant species. Therefore, the probability can also be computed as

Here, is the number of species at (before sampling) descendant from the single species extant at with a sampled descendant species at ; and is the number of species at (before sampling) descendant from the other species extant at . From the species, one should be sampled; from the species none should be sampled. Dividing the last expression by , we obtain the right-hand side of eq. (C1).

Case without rate shift

We construct the likelihood of a tree under speciation–extinction dynamics without rate shift. This likelihood will be extended below to speciation–extinction dynamics with one or more rate shifts. The argument is related to the “breaking the tree” approach of Nee et al. (1994). As in the proof of eq. (C1), we consider a sampling process at the present time with sampling probability .

We start by decomposing the tree into simple branches, that is, branches of the reconstructed tree without branching events. See left panel of Figure C1, where the simple branches are labeled by letters “a” to “n.” We denote the time interval of branch by . We distinguish internal and boundary simple branches: internal branches are those for which , and boundary branches are those for which . We denote the set of internal simple branches by , and the set of boundary simple branches by . For the example of Figure C1 (left panel), we have

Figure C1.

Decomposing a phylogeny in simple branches. Left panel: Example tree without rate shift, decomposed in simple branches labeled by letters “a” to “n.” Right panel: Same example tree, but this time with rate shift at . Simple branches that contain the rate shift are split, resulting in a larger set of simple branches (labeled by letters “a” to “r”).

For each internal simple branch , we denote by the number of descendant species at the end of the branch (at time . One of these descendant species is represented in the tree. For the other descendant species, we keep track of the number of descendants at ; we denote this number by . For each boundary simple branch , we denote by the number of descendants at in addition to the species represented in the tree (hence, descendant species in total).

The numbers and allow us to write down the likelihood

(C2)

where we used short-hand notation for the multidimensional sums, for example,

Equation (C2) can be understood as follows. The product on the first line corresponds to internal simple branches. The term for branch contains the probability of having descendants at , a factor related to the selection of the descendant species that is represented in the tree, the probability that the other species at have descendants at , and the probability that none of the latter descendants is sampled. The product on the second line corresponds to boundary simple branches. The term for branch contains the probability of having descendants at , a factor related to the selection of the sampled descendant species, and the probability of sampling this species and not sampling the other species.

Equation (C2) can be simplified by combining simple branches. In particular, the terms relating to an internal branch can be incorporated in the terms relating to the boundary branch to which it is connected. For example, referring to Figure C1 (left panel), consider branches “a” and “e,” which are an internal and a boundary simple branch, respectively. The terms in eq. (C2) associated with branches “a” and “e” are

(C3)

where we have applied eq. (C1) in the last line. We see that the part of the likelihood corresponding to the composed branch “ae” (composed of simple branches “a” and “e,” with and , where ) has the same form as a boundary simple branch in eq. (C2). Hence, we can absorb an internal simple branch in the boundary simple branch to which it is connected.

By repeatedly absorbing internal simple branches, we obtain boundary branches that are increasingly composed, until there are no internal simple branches left in eq. (C2). Denoting the resulting set of boundary branches by , we obtain

(C4)

and for the case where all species are sampled (set )

(C5)

This is the “breaking the tree” likelihood of (Nee et al., 1994), see eq. (B4).

Note that there are several ways of combining simple branches into composed ones. For example, for the tree shown in Fig. C1 (left panel), two possible sets of boundary branches are

However, these sets lead to the same value of likelihoods (C4) and (C5). In fact, the likelihoods only depend on the branching times, and the latter do not depend on the specific choice of composed branches.

Case of a single rate shift

The “breaking the tree” approach can also be used to construct the likelihood of a tree with a rate shift. For the rate shift model described in the main text, we have to divide by the total number of species extant at the rate shift time . Hence, we have to keep track of the total number of species at .

First, note that the subclade with the rate shift can be dealt with separately from the main clade. For the subclade, we can apply the likelihood formula for a tree without rate shift. The subclade tree starts at the rate shift time and continues until the present time . Here, we derive the likelihood formula for the main clade. For the example phylogeny of Figure C1 (right panel) the main clade corresponds to the entire tree except branches .

We decompose the main clade into simple branches. In the case of a rate shift, all simple branches are split at the rate shift time, see Figure C1 (right panel). We distinguish internal simple branches before the rate shift (with ; set denoted by ), boundary simple branches before the rate shift (with ; set denoted by ), internal simple branches after the rate shift (with ; set denoted by ) and boundary simple branches after the rate shift (with ; set denoted by ). For the example of Fig. C1 (right panel),

As before, we introduce the numbers and for internal simple branch , and for boundary simple branch . For an internal branch, stands for the number of descendant species at . Only one of the species is represented in the phylogeny; the other species have descendant species at time (for a branch before the rate shift) or at time (for a branch after the rate shift). Similarly, for a boundary branch, stands for the number of descendant species not represented in the phylogeny at time (for a branch before the rate shift) or at time (for a branch after the rate shift).

Using the numbers and we can construct the likelihood for the main clade

(C6)

The first and second line impose the branching times before the rate shift, while keeping track of the total number of species at the rate shift. This number is equal to , by which we divide in the third line, where denotes the number of species represented in the phylogeny at the time of the shift (in Fig. C1 (right panel) ). The other factor on the third line imposes that the species that are not represented in the phylogeny, of which there are , do not have sampled species. The fourth and fifth line impose the branching times after the rate shift, and require that the species that are (not) represented in the phylogeny are (not) sampled.

As for the case without rate shift, the likelihood expression can be simplified. We absorb internal into boundary branches, until there are no internal branches left. This leads to a set of boundary branches before the rate shift, and a set of boundary branches after the rate shift, which we denote by and , respectively. For example, for the tree of Fig. C1 (right panel), these sets could be

Other ways of combining branches are possible, leading to different sets and , but result in the same likelihood value,

(C7)

If all species are sampled, we get (by setting )

(C8)

This is the likelihood formula reported in the main text, see eq. (14).

Case of multiple rate shifts

Consider two rate shifts at times and . We distinguish two cases. First, we assume that the second rate shift occurs in the subclade with the first rate shift (Fig. C2, left panel). The corresponding likelihood can be readily constructed from the single rate shift formula. Indeed, because the main-clade diversification dynamics after are unaffected by the second rate shift, the part of the likelihood dealing with the main clade follows directly from the one for a single rate shift. Similarly, because the subclade diversification dynamics are unaffected by the main clade, also the part of the likelihood dealing with the subclade follows directly from the one for a single rate shift.

Figure C2.

Example phylogenies with two rate shifts. Colors indicate the different rate regimes: main clade in blue, subclade initiated by first rate shift in green, subclade initiated by second rate shift in red. Left panel: The second rate shift occurs in the subclade of the first rate shift. The likelihood of this phylogeny is the product of the one-shift likelihood for the main clade, the one-shift likelihood for the first subclade, and the no-shift likelihood for the second subclade. Right panel: The second rate shift occurs in the main clade. The total likelihood is the product of the two-shifts likelihood for the main clade, the no-shift likelihood for the first subclade, and the no-shift likelihood for the second subclade.

Here, we work out the second, more complicated case, in which the second rate shift occurs in the main clade (Fig. C2, right panel). We denote the sets of simple branches by and (branches before ), and (branches between and ) and and (branches after ). We again introduce numbers and to keep track of the total number of species at the two rate shifts. Then, the part of the likelihood relating to the main clade (i.e., the blue clade in Fig. C2) is

(C9)

The latter expression can be simplified by incorporating internal branches into longer boundary branches. Denoting the resulting sets of boundary branches by , , and , we obtain

(C10)

If all species are sampled, we get

(C11)

Appendix D: Likelihood for unobserved rate shift

It is intuitively clear that the rate shift model in which the rate shift has no effect, that is, when rate shifted rates are equal to nonrate shifted rates , should be connected to the model without rate shift. Here, we show how the likelihood formula with rate shift should be combined to recover Nee et al.’s (1994) likelihood. More precisely, we prove that

(D1)

We start by noting that when the rate shift has no effect (i.e., when setting ), the likelihoods and can be rewritten as

(D2)

(D3)

with common factor given by

(D4)

Hence,

(D5)

Substituting the expression for ,

(D6)

where in the second equality we have applied eq. (C1) with (and hence and ). This concludes the proof of eq. (D1).

Appendix E: Rate shifts in diversity-dependent model

The corrected likelihood , eq. (14), can be extended to the case of diversity-dependent diversification. Here, we describe how this can be done within the framework of Etienne et al. (2012). In this approach the total number of species, including the unobserved ones, is tracked through time. Therefore, the likelihood correction of eq. (14), which basically consists in the division by the total number of species present at the rate shift, can be readily implemented.

The framework of Etienne et al. (2012) is built on the quantities , the probabilities that the diversification process is consistent with the observed phylogeny from the starting time (typically crown age ) to the current time , with visible (i.e., represented in the phylogeny) and invisible (i.e., not represented in the phylogeny, as they will go extinct before the present time or they are unsampled in the data) species at time . The probabilities are computed forward in time, from to . We introduce the vector with components for all . Then, for a phylogeny with tips and without rate shift,

(E1)

Starting from the initial vector at crown age, we repeatedly apply matrices between branching times and and matrices at branching time . Here, the branching times are time-ordered (i.e., ), so that at branching time the phylogeny transits from to branches. The (unconditioned) likelihood of the tree under the diversity-dependent diversification model is then obtained as the component of .

The matrices appearing in eq. (E1) is obtained by solving the linear differential equation

(E2)

so that , with the matrices given by

(E3)

Here, the speciation rates and extinction rates can depend on the number of extant species at the event times. The matrices are given by

(E4)

For the case of a lineage-specific rate shift, we apply the same computation from to the rate shift time . At the rate shift, we choose the species that is undergoing the rate shift, which corresponds to dividing by the total number of species (both visible and invisible) extant at . We then continue the computation from to , taking into account that the main clade has lost one branch at the rate shift. Hence, eq. (E1) has to be modified to

(E5)

with the matrices given by

(E6)

Note that at branching time the main-clade phylogeny goes from to branches, and at branching time it goes from to branches.

The likelihood contribution of the main clade is obtained as the component of computed with eq. (E5). The likelihood contribution of the subclade can be computed with eq. (E1) starting at the rate shift time . By multiplying these two contributions, we get the (unconditioned) corrected likelihood for a phylogeny with an observed rate shift in a specific branch

(E7)

Using techniques of Laudanno et al. (2020), it can be proven that this formula reduces to the likelihood (14) in the diversity-independent case.

In the same way as for the likelihood of an observed rate shift, diversity-dependent extensions can be derived for

• the likelihood of an unobserved rate shift. Note however that in the diversity-dependent case a dummy rate shift (i.e., rates before and after rate shift are the same) does have an effect on the diversification dynamics, because the subclade is not subjected to the diversity dependence of the main clade. Hence, there is no diversity-dependent extension of eq. (D1);

• the conditioning probabilities , , and , see eqs. (15), (16), and (17) of the main text; see also Etienne and Haegeman (2012), where explicit expressions are presented for the diversity-dependent analog of eq. (15);

• the likelihood for multiple rate shifts, that is, eqs. (C10) and (C11) for the case of two rate shifts in the main clade. In fact, while the explicit formulas are cumbersome for multiple rate shifts, the framework of Etienne et al. (2012) with the vector can easily incorporate an arbitrary number of rate shifts and can be used to numerically evaluate the multiple-shifts likelihood. This has been implemented in version 4.3 in the R package DDD (Etienne and Haegeman 2020).