Generalized Bayesian dating.

Automated cognate detection. There is a rich literature on developing automated cognate detection methods [28, 29] for the purpose of detecting cognates and inferring phylogenetic trees [30, 31]. The automated cognate detection methods compute a similarity between two words based on hand-crafted phonetic similarity measures [32, 33], linear classifiers using word similarity scores [34, 35] or phoneme n-grams as features for training [36, 37] on hand-annotated training data, and neural networks [38]. Subsequently, the systems apply different clustering algorithms such as InfoMap [39] or UPGMA [40] to infer cognate clusters. Research on the performance of different clustering algorithms at the task of cognate detection [29, 41] shows that tuning the threshold can improve the results even for the simplest one, which is the average linkage clustering algorithm.

A recent paper [42] introduced a new clustering algorithm for cognate detection inspired by the Chinese Restaurant Process (CRP) [43], which does not require any threshold for forming cognate clusters. The CRP algorithm forms cognate clusters by linking a word to the cluster to which it has the highest similarity or to itself, where the self-similarity in a singleton cluster is penalized by multiplying it by a discount parameter α(> 0). The α parameter is tuned through a Metropolis-Hastings step for each meaning separately. Overall, the paper shows that the clusters formed by the CRP algorithm are of higher quality than the clusters using the Infomap algorithm, which works with a single threshold for all the meanings. The CRP algorithm has been evaluated on word lists belonging to the Austronesian, Austro-Asiatic, Pama-Nyungan, Indo-European, and Sino-Tibetan language families. Approaches [44] that infer trees based on an explicit model of sound change do not perform cognate detection at all and therefore cannot be applied to Swadesh list data but only to cognate data from etymological dictionaries; or else, they are not yet scalable for datasets consisting of hundreds of languages [45].

Recent research in computational historical linguistics [30, 46] has shown that the trees inferred using cognates inferred from automated methods are as good as those inferred from expert annotated cognate judgments. We believe that the next area for application of these cognate detection methods is in linguistic dating, because the dating process has traditionally been heavily dependent on manual cognate detection, which is time consuming, potentially biased, and not yet available for most of the world’s language families.

Using both cognate and sound class data. Bayesian phylogenetic studies typically use cognate classes to infer phylogenies, but it has been shown [46] that using both cognates and sound classes give better results for phylogenetic inference when drawing upon (an earlier version of) the ASJP database. We employ the automated cognate identification system described above [42] to assign cognate judgments to word lists. The choice of using such a system is supported by the results from two studies [30, 46], which show that automatically inferred cognates can yield high quality phylogenetic trees. In addition to the cognate characters, we operate with sound class characters, which are extracted as follows. Each word representing a given meaning is transformed into a set of characters representing the presence or absence of each of the sound classes found across words belonging to the given meaning. For instance, Swedish hund and English “dog” would yield a total of 7 characters, each representing a sound class in the set {o, u, d, g, h, n}.

Bayesian phylogenetic inference. All phylogenetic analyses were performed with MrBayes [47]. Cognate characters and sound class characters were treated as separate partitions and were subjected to a partition analysis. We used a birth-death model derived from the fossilized Birth-Death tree prior [48] with fossilization rate set to 0 since there are no extinct languages in the data. The birth-death model handles incomplete language sampling through a parameter ρ = n/N, where n is the number of languages in the sample and N is the total number of extant languages in the family. For families represented by more than 400 doculects we restricted the sample size to 400 doculects for reasons of processing time. The families in question and the number of selected vs. available doculects are: Austronesian (355/1254), Atlantic-Congo (265/1500), Afro-Asiatic (151/364), and Indo-European (205/462). In these cases the doculects were selected so as to maximize the phylogenetic diversity. Specifically, we applied the following procedure: first we selected the best documented doculect (the one with the largest number of words or a random one in case of ties) from the subgroups at the deepest level of the Glottolog classification and checked the size of the sample. If it was still greater than 400 we went up one level, again selecting one doculect per subgroup. This continued until the sample was below 400. Moreover, we corrected for extant taxa sampling bias through diversified sampling correction [49] implemented in [48]. We used the number of extant doculects in the ASJP database as representing the number of extant taxa, N, and fixed n (the number of sampled taxa) to be the number of doculects in the sample. In the case of families of size less than 400, we included all the doculects and assumed ρ = 1. Setting N to the number of extant doculects in the database and using ρ = 1 for smaller families are both approximations. But they are motivated by the facts that the database contains 7655 doculects and the number of languages in the world is 7570 by a widely used estimate [50], so it would be a fair approximation on average. Moreover, a consistent and objective criterion for counting languages vs. dialects [51] has not yet been applied in any language catalogue, so setting these parameters will in any case involve approximations. The birth-death prior used in this paper is conditioned on the root height, which has to be specified beforehand. Since the tree is uncalibrated, the root height is drawn from a Gamma distribution with rate and shape parameters set to 1.

Lexical substitution model. We model variation across sites for each partition separately. The Bayesian partition analysis assumes that each partition has a different rate (drawn from a symmetric Dirichlet distribution), which enters the calculation of the likelihood for a site through its multiplication with the branch lengths in the tree. In this paper, we employ a continuous time Markov chain process model for binary states (binCTMC) with site rates drawn from a four category discrete gamma distribution (binCTMC+Γ model) in all phylogenetic analyses, correcting for all-absence sites [52]. We prefer binCTMC model over other alternative substitution models such as covarion and stochastic Dollo for the following reasons. The binCTMC model is a single parameter model whereas the covarion model has three parameters, with binCTMC being a special case of the covarion model. In the stochastic Dollo model, a cognate can be gained only once, which prohibits regaining of a cognate through a process such as internal borrowing. The suitability of these substitution models has been shown to be variable, with binCTMC being the best suited for Indo-European [12] and covarion the best suited for Sino-Tibetan [17], and a comparative study of different substitution models on four different language families, including Indo-European [53], suggest that there is no clear winner between binCTMC and covarion models. Thus, we selected the binCTMC+Γ model for its simplicity and competitiveness against the more complex covarion model.

Branch lengths. A branch length in a time tree is the product of calendar time units and clock rate. The clock rate, c, measures the average number of substitutions that can occur per year per site and is assigned a prior. If the clock rate is assumed to be constant for the whole tree the model it is a strict clock rate. It is normal to relax this assumption since branches can have varying rates of evolution. The branch lengths, then, are further multiplied by a factor drawn from a distribution which is parameterized in terms of the original branch length. There are multiple relaxed clock rate models [54] employing different assumptions for deriving the multiplicative factor. In this paper, we use an Independent Gamma Rate [54] relaxed clock model (IGR) where each branch’s rate comes from different Gamma distributions whose mean is 1.0 and variance is proportional to the inverse of the branch length. We fix the clock rate to 1 but allow the branches to have different rates drawn from the IGR model, thereby allowing the branch rates to deviate from the strict clock assumption.

In this paper, we assume that the joint prior probability of the branch lengths under a birth-death process is conditioned on the tree height which in turn is drawn from a Gamma distribution with mean and variance set to 1. Does the tree prior assumption have any bearing on distribution of tree lengths? While we could not find any research showing a relation between tree length (in a birth-death process) and a known probability distribution, there are results [55] showing that the tree length in a Yule process (pure birth process) conditioned on tree height is dependent on a Gamma distribution, whereas the same Yule process, if conditioned on the number of extant languages, follows a Gamma distribution. (The Yule process, incidentally, is not a realistic model for languages since it is well known that languages may die out.) We acknowledge that the statistical properties of tree length in a birth-death tree conditioned on tree height is not yet known.

Topological constraints. As in most previous Bayesian phylogenetic studies we employ topological constraints, but unlike our predecessors we introduce as many constraints as possible since we are not concerned with language classification but only with the dating of recognized subgroups. A recent study [46] using the maximum likelihood phylogenetic software RAxML [56] showed that trees inferred from word lists in the ASJP database largely agree with the Glottolog [19] classification in terms of quartet distances [57]. We therefore find it justifiable to not take the extra step of inferring all nodes in the phylogenies but rather to fix most of the nodes of the tree based on the Glottolog classification. The limit on constrained nodes are determined by the data: as mentioned earlier, we require that each constrained node should suspend at least three doculects. All in all we extracted 1171 topological constraints from Glottolog.

Monte-Carlo Markov chain settings. For each family, we did two independent runs and sampled parameters by running one cold and two hot chains in parallel. The hot chains allow the MCMC chain to explore the parameter landscape efficiently by moving across the peaks and not get stuck in a local optimum. We ran the chains for different numbers of generations depending on the number, N, of doculects in a family: 10⁷ for N < 100, 3 × 10⁷ for 100 < N < 200 and 6 × 10⁷ for N > 200. The chain was sampled at every 1000th generation in order to reduce auto-correlation. We assessed the convergence of branch lengths using the Potential Scale Reduction Factor [58], whose value should approach 1 across independent runs should the runs converge. The independence between the samples for parameters such as tree height, birth-death rates, and the IGR variance parameter is assessed using Estimated Sample Size, which is normally required to be at least 100 for all our parameters [58]. The rate variation across sites is modeled using a discrete Gamma distribution with four rate categories [59].

The Nexus files and MrBayes command files are provided in the folder Nexus_MrBayes folder in S1 File. The output of MrBayes is available for download here: https://figshare.com/s/a2ba6dbb656b6cb0b9bb.

Gamma regression. Gamma regression is a special case in the Generalized Linear Model (GLM) framework [60], where the response variable follows a Gamma distribution. An exploratory data analysis showed that a log-log plot of known age vs. median tree length (MTL) is linear. We choose Gamma distributed ages over normally distributed ages for the following reasons: (1) The age of a language group is always positive, (2) The assumption of constant variance is not realistic because there is a difference in the variance in the ages of the language families, (3) A linear regression model on a log-log scale implies that the response variable has constant variance and is not on the original scale. These problems can be solved with a Gamma distribution with ν and λ as shape and rate parameters whose mean (μ = ν/λ) and variance are μ²/ν.

The Gamma distribution has positive support, non-constant variance, and removes the need for logarithmic transformation. The GLM framework features a smooth, invertible function that links the response variable with the predictor, which is a log function here. The parameters (coefficients: β₀ and β₁, and the shape parameter ν) can be estimated using an MCMC sampler in the Bayesian framework. The priors for β₀ and β₁ are uniform distributions in the range of −100 to 100. The ν parameter is always positive and the prior is a uniform distribution ranging from 0 to 100. The posterior distribution of the parameters are obtained by running the MCMC chain for 10⁶ generations and sampling at every 100th generation to reduce auto-correlation. The initial 25% of the generations are discarded as burn-in and the independence of the samples is assessed using the Estimated Sample Size (> 1000) for all the parameters in our experiments. All MCMC samplers are implemented using the RevBayes programming language [61].

In the following, we evaluate the choice of the Gamma regression model against the linear regression model using AICM (Akaike’s Information Criterion through MCMC [62]), where the best model has the lowest AICM. The prior for variance (in the linear regression model) is exponentially distributed with mean 1. The AICM values are estimated using the Tracer software [63] from the post-burnin (25% discarded) samples for the four models. The model with the lowest AICM (shown in bold) is the best fit for the data. The first three models assume that the age of a family follows a Gamma distribution whereas the last line shows a model where a known age is distributed normally. The null model for Gamma regression has a poor fit whereas the regression model with normally distributed ages shows the worst fit to the data. The results of the model evaluation are presented in Table 2. The AICM values show that Gamma distributed age model is better than a normally distributed age model.

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Table 2. AICM values for different models.

The first four lines show Gamma regression models using different predictors. The last line shows a linear model with logarithm of MTL as predictor.

https://doi.org/10.1371/journal.pone.0236522.t002

Inferring ages. The MCMC procedure yielded 7500 Gamma regression parameters sets. We employed each set of parameters to compute the age distributions using the median tree length for the topological constraint. Subsequently we report the median and 95% HPD intervals for the ages from the computed ages.

Although the sum of the branch lengths is used to compute the tree length of a language group, the relaxed clock setting (Independent Gamma Rates; IGR) applied in the tree inference procedure works with a slightly modified branch length called effective branch length, which is obtained by multiplying a branch length with a rate drawn from a Gamma distribution. The effective branch length is used to compute the likelihood of the tree using the pruning algorithm. We experimented with regard to the relation between the median effective branch lengths and the median branch lengths through a scatterplot for all the topological constraints. A linear trendline between both types of branch lengths shows a slope of 1.031, an intercept of −0.004 and R² = 0.94. We therefore use the unmodified branch lengths to compute the tree lengths.

The choice of median tree length as opposed to mean tree length is motivated by the fact that median tree length corresponds to the length of one actual tree, whereas the mean is a more abstract number, being computed from the sample of tree lengths. The two are in any case highly correlated for the 116 language families (R² = 0.997). We always report the median statistic for all ages and branch lengths.

The exponential function’s parameters, a = 1843 and b = 0.35, follow from the median estimates of β₀, β₁ which are 7.52 and 0.35. The formula is t = 1843 × MTL^0.35. When MTL tends towards 0, as is the case of close dialects, the age of the most recent common ancestor also tends towards 0.

In earlier exploratory experiments (not reported in detail here), R² is almost zero when tree height is used as a predictor for the 30 groups rather than tree length. So, we adopted tree length which is clearly to be preferred over tree height.

The Means, medians, standard deviations and 95% HPD intervals for these dates are given in S1 Table. In addition, consensus trees for all language families annotated with inferred dates are provided in Consensus_Trees_Ages folder in S2 File.

Non-Bayesian approaches.

We tested two non-Bayesian approaches: the published version of ASJP chronology [7] and a version of traditional glottochronology [64] which is modernized so as to draw upon automated rather than manual cognate counts and where the calibration of cognate percentages and age comes from the same 30 calibration points used in our Bayesian dating experiment rather than a few language pairs from the literature [65]. In both cases, for each group, a flat classification structure consisting of the same major subgroups assumed in the paper introducing ASJP chronology [7] was defined. These subgroups are usually the same as in Ethnologue [50], but in a few cases a date is found in the literature for a language group not considered a subgroup in that classification although it is still compatible with it. In such cases the classification followed here will differ from [50], but it will be the same as in [7]. The classification divides doculects into sets for the purpose of computing an average cognate count or average twice-modified Levenshtein distance (LDND) between all pairs of doculects whose members belong to different sets. When there is no internal classification of a group, such as in the case of Chinese, the sets are defined following the ISO-639-3 standard. Thus, for instance, for Wakashan an average is found for all pairs where one doculect pertains to Northern and another one pertains to Southern Wakashan. For Chinese, however, the sets are defined as [cdo], [cmn], [cpx], etc., and the average is found for all doculects pertaining to different sets of this kind.

ASJP chronology. The procedure for ASJP chronology is described in [7] and will only be briefly summarized here. The basic input to the method is the twice-modified Levenshtein distance called LDND (Levenshtein Distance Normalized Divided) used in many papers of the ASJP project and extensively discussed in [66]. This is turned into a similarity, LSND (Levenshtein Similarity Normalized Divided), by subtracting LDND from 100%. LSND can be abbreviated s. The LSND (or s) can be directly converted into an age estimate through a formula t = (log(s) − log(s₀))/2log(r), where s₀ is similarity at t = 0, r is a retention rate, and s is the measured similarity. Here we use the values s₀ = 92% and r = 0.72 that were established in [7] using 52 calibration points. Thus, our application of ASJP chronology is strictly orthodox. We use the original, published software for computing dates—Holman’s asjp62c.exe at https://asjp.clld.org/software (alternatively our own R software at https://github.com/Sokiwi/InteractiveASJP01 could be used). Holman’s program requires standard ASJP-style input files and looks for the Ethnologue classification, which is located between | and @ in the first line of metadata for each language. Since we use the Glottolog classification here, we have moved that to between | and @. Moreover, the program takes as part of its input the level of a given classification for which one wants to produce dates. This level can be set to 1 for the top level provided that the classification string starts at the level one is interested in. This is the practice followed for files given as input for asjp62c.exe. For the sake of replicability these input files are provided in the S3 File along with the outputs. The dates extracted from the outputs are listed in Table 6.

Automated glottochronology. Automated glottochronology is highly similar to traditional glottochronology except that the cognate counts are based on automatically inferred cognates. Moreover, while we infer a constant rate of change through a linear model calibration, as did [65], we allow the intercept to not be equal to 0, taking the insight from [7] that languages have internal variation such that lexical similarity at 0 years of separation is not necessarily 100%. Finally, we do not adopt a retention rate from the glottochronological literature, something which would probably disadvantage the method. Instead, for each of the 30 language groups we fit a linear model in a leave-one-out fashion, basing the linear model on the remaining 29 calibration points. The cognate proportions for each language group represent the number of cases where two doculects have the same cognate class for a given meaning divided by the number of meanings for which both doculects have attested words. The final score for a language group represents an average over pairs whose members belong to the different Glottolog subgroups. All relevant files and an R-script executing each step towards computing the final dates are included in S4 File.

The purpose of the exercise is not to revive glottochronology but to evaluate the method and derive a baseline for evaluating the performance of the other dating methods.