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I'm a linguist interested in phylogenetic tree inference using language data. I'm posting here because I'm using Bayesian phylogenetic methods in my work (probably using BEAST and/or RevBayes). For the purposes of this question, please accept the assumption that phylogenetic trees are generally applicable to languages (I'm aware this premise may be subject to dispute). My question is intended to be methodological and not necessarily domain specific. You could imagine I'm constructing biological species trees using morphological data.

I have two datasets: A set of binary traits ($D_{b}$) and a set of continuous traits ($D_{c}$).

I'd like to compare the results of phylogenetic tree inference with (a) binary data alone vs. (b) binary data and continuous data together. Model parameters are the same, except of course (b) requires some extra parameters to model the evolution of continuous traits. In other words, both the data and model of (a) is a subset of (b).

I know this probably sounds like a strange research question. The reason it's of interest to me is that the binary data represent a data type that is well-established in historical linguistics and often used to infer language trees. The continuous data come from different language structures and haven't been used to infer language trees before. I want to know if adding this new kind of data can help aid tree inference significantly or if it has a neutral effect (or even just introduces more noise).

I'm familiar with phylogenetic model comparison using Bayes factors. But in the standard use case I'm familiar with, the dataset stays exactly the same and Bayes factors are used to evaluate the fit of different evolutionary models.

My question: Is there a standard way of evaluating the effect of adding more data to a phylogenetic tree model? Can marginal likelihoods be compared still even though the data differs in each model?

If not, would it make any sense to reframe the question as one of model comparison? I'm imagining, for example, running both (a) and (b) with identical datasets ($D_b + D_c$ for both). In (a), there would be a trivial model of trait evolution that doesn't aid tree inference at all—something like a model where traits can jump from any value to any other value with equal probability(?). In (b), there would be a more standard model of continuous trait evolution, e.g. Brownian motion.

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    $\begingroup$ I think there is certainly a way to wangle marginal likelihoods here, although I am not hugely familiar with it. Brown & Thomson, 2017 might offer some leads. Alternative, I can envision a relatively straightforward comparison of the tree prior, likelihood, and posterior - if the new data offers more information it should add more information to the posterior relative to the prior. $\endgroup$ – NatWH Mar 16 '20 at 12:32
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    $\begingroup$ You may also find this and this interesting. $\endgroup$ – NatWH Mar 16 '20 at 12:48
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The key issue is the data is subject to reticulation, languages are seldom clean breaks. In nucleotide data this can be identified due to deviations from physical linkage and call it recombination. Linkage doesn't apply to languages.

The fundamental assumption is one tree - the same tree - describes all the data, with reticulation that may not be a good assumption. Thus English is a hybrid language, and lots of English words find their way back into French and German 'parents'. Or perhaps a better example is Gaelic mingles with English over time. There are methods used to monitor reticulation in languages.

Thus id keep it simple. The branch length is not only directly comparable it provides linguistical information that is useful to understand. This is the amount of 'evolutionaey' change per word (it that is what you binary data is describing) and the relationship to the node ('breakpoiint') is of primary interest.

The second step is to simply draw a tree of data set A and compare it to data set A+B and data set B and the probabilities describing the sister groups. Are the trees congruent, i.e. are they statistically the same?

The area you describing is more given to parametric bootstrapping, using custom resampling, but you need to be certain that the model being used is appropriate for the data.

The final point is whether a reversible or directional matrix is being used, which impacts on the specific result.

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    $\begingroup$ Thanks—appreciate your perspective on this. The presence of horizontal transmission is well understood in linguistics. The general applicability of a phylogenetic tree model to language data is a big and interesting topic, but it's beyond the scope of my question here. I will edit the question to make this clearer. It's debatable whether English qualifies as a 'hybrid' language, though certainly it has a lot of borrowed vocabulary as you point out (though again, this is beyond the scope of the question at hand). $\endgroup$ – JaydenM-C Mar 18 '20 at 2:33
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    $\begingroup$ I agree there are many interesting questions of evolutionary dynamics that can be answered by looking at branch lengths. However, it's not clear to me how looking at branch lengths will help answer my specific question (comparing the fit of a phylogeny inferred with $D_b$ vs a phylogeny inferred with $D_b + D_c$). Your second step may be one way to go, simply inferring trees with $D_b$, $D_c$ and $D_b+ D_c$ then evaluating their congruence and comparing posterior values of various nodes. $\endgroup$ – JaydenM-C Mar 18 '20 at 2:52
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    $\begingroup$ @Michael with all due respect, all of this is very unhelpful advice and clearly out of scope of what the question is asking, which is a very clear question on the application of phylogenetic combinability and information content. OP, I agree that inferring the datasets separately and combined is important; there are also methods to formally test if these should be combined: see here and here. $\endgroup$ – NatWH Mar 18 '20 at 13:14
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    $\begingroup$ The additional test of if inclusion of the continuous data improves the fit should be answered with information theoretic methods, in likelihood form or their Bayesian equivalents. $\endgroup$ – NatWH Mar 18 '20 at 13:15
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    $\begingroup$ In particular, you may be misled by comparing branch support values for certain branches - inclusion of data but model misspecification can lead to positively misleading branch support, so you may increase support for erroneous splits. Formal tests using information criteria or equivalent are better. $\endgroup$ – NatWH Mar 18 '20 at 13:20

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