The curious case(s) of tree-like matrices with no synapomorphies

(This is a joint post by Guido Grimm and David Morrison)

Phylogenetic data matrices can have odd patterns in them, which presumably represent phylogenetic signals of some sort. This seems to apply particularly to morphological matrices. In this post, we will show examples of matrices that are packed with homoplasious characters, and thus lead to trees with a low Consistency Index (CI), but which nevertheless have high tree-likeness, as measured by a high Retention Index (RI) and a low matrix Delta Value (mDV). We will also try to explore the reasons for this apparently contradictory situation.

Background

A colleague of ours was recently asked, when trying to publish a paper, to explain why there were low CI but high RI values in his study. This reminded Guido of a set of analyses he started about a decade ago, using an arbitrary selection of plant morphological matrices he had access to.

The idea of that study was to advocate the use of networks for phylogenetic studies using morphological matrices, based on the two dozen data sets that he had at hand. The datasets were each used to infer trees and quantify branch support, under three different optimality criteria: least-squares (via neighbour-joining, NJ), maximum likelihood, and maximum parsimony. This study was was never wrapped up for a formal paper, for several reasons (one being that 10 years ago Guido had absolutely no idea which journal could possibly consider to publish such a paper, another that he struggled to find many suitable published matrices).

The signals detected in the collected matrices were quite different from each other. The set included matrices with very high matrix Delta Values (mDV), nontree-like signals, and astonishingly low mDVs, for a morphological matrix. Equally divergent were the CI and RI of the inferred equally most-parsimonious trees (MPT) and the NJ tree. The data for the MPTs and the primary matrices are shown in the first graph, as a series of scatterplots, where each axis covers the values 0-1. (Note: in most cases the NJ topologies are as optimal as the MPTs, and have similar CI and RI values.)

As you can see, the CI values (parsimony-uninformative characters not considered) are not correlated with either the RI or mDV values, whereas the latter two are highly correlated, with one exception.

The most tree-like matrix (mDV = 0.184, which is a value typically found for molecular matrices allowing for inference of unambiguous trees) was the one of Hufford & McMahon (2004) on Besseya and Synthyris. The number of MPTs was undetermined —using a ChuckScore of 39 steps (the best value found in test runs), PAUP* found more than 80,000 MPTs with a CI of 0.39 (third-lowest of all of the datasets), but an RI of 0.9 (highest value found).

A strict consensus network of the 80,003 equally parsimonious solutions, the network equivalent to the commonly seen strict consensus tree cladograms. Trivial splits are collapsed. Colours solely added for orientation (see next graph).

Oddly, the NJ tree had the same number of steps (under parsimony), but a much higher CI (0.69). The proportion of branches with a boostrap support of > 50% was twice as large in a distance-based framework than using parsimony.

Bootstrap consensus networks based on 10,000 pseudoreplicates each. Left, distance-based and inferred using the Neighbour-Joining algorithm; right, using a branch-and-bound search under parsimony as optimality criterion (one tree saved per replicate). Edge-lengths reflect branch support of sole or competing alternatives; alternatives found in less than 20% of the replicates not shown; trivial splits are collapsed. Same colour scheme than above for orientation.

The Neighbour-net based on this matrix has quite an interesting structure. Tree-like portions are clearly visible (hence, the low mDV) but the branches are not twigs but well developed trunks. The large number of MPTs is mainly due to the relative indistinctness of many OTUs from each other.

Neighbour-net based on simple mean (Hamming) morphological distances. Same colour scheme as above.

This distance-based 2-dimensional graph captures all main aspects of the tree inferences and bootstrap analyses, with one notable exception: B. alpina which is clearly part of the red clade in the tree-based analyses. We can see that the orange group, B. wyomingensis and close relatives, is (morphology-wise) less derived than the red species group. Although B. alpina is usually placed in a red clade, it would represent a morphotype much more similar to the orange cluster as it lacks most of the derived character suite that defines the rest of the red clade. In trees, B. alpina is accordingly connected to the short red root branch as first diverging "sister" with a very short to zero-long terminal branch, but in the network it is placed intermediate between the poorly differentiated but morphologically inhomogenous oranges and the strongly derived reds — being a slightly reddish orange. This reddishness may reflect a shared common origin of B. alpina and the other reds, in which case the tree-based inferences show us the true tree. Or just a parallel derivation in a member of the B. wyoming species aggregate, in which case the unambiguous clade would be a pseudo-monophylum (see also our recent posts on Clades, cladistics, and why networks are inevitable and Let's distinguish between Hennig and cladistics).

Interpretation, what does low CI but high RI stand for?

The distinction between the Consistency Index and the Retention index has been of long-standing practical importance in phylogenetics. For a detailed discussion, you can consult the paper by Gavin Naylor and Fred Kraus (The Relationship between s and m and the Retention Index. Systematic Biology 44: 559-562. 1995).

For each character, the consistency index is the fraction of changes in a character that are implied to be unique on any given tree (ie. one change for each character state): m / s, where m = the minimum possible number if character-state changes on the tree, and s = the observed number if character-state changes on the tree. The sum of these values across all characters is the ensemble consistency index for the dataset (CI).

The retention index (also called the homoplasy excess ratio) for each character quantifies the apparent synapomorphy in the character that is retained as synapomorphy on the tree: (g - s) / (g - m), where g = the greatest amount of change that the character may require on the tree. Once again, the sum of these values across all characters is the ensemble retention index for the dataset (RI).

Both CI and RI are comparative measures of homoplasy — that is, the degree to which the data fit the given tree. However, CI is negatively correlated with both the number of taxa and the number of characters, and it is inflated by the inclusion of parsimony-uninformative characters. RI is less sensitive to these characteristics. However, RI is inflated by the presence of unique states in multi-state characters that have some other states shared among taxa and, therefore, are potentially synapomorphic.

It is these different responses to character-state distributions (among the taxa) that apparently create the situation noted above for morphological data. Neither CI nor RI directly measures tree-likeness, but instead they are related to homoplasy. So, it is the relative character-state distributions among the taxa that matter in determining their values, not just the tree itself.

For example, increasing the number of states per character will, in general, increase CI faster than RI. Increasing the number of states that per character that occur in only one taxon will, in general, increase RI faster than CI.

Take-home message

This is just another example demonstrating that morphological data sets should not be used to infer (parsimony) trees alone, but analysed using a combination of Neighbour-nets and support Consensus Networks. No matter which optimality criterion is preferred by the researcher, the signal in such matrices is typically not trivial. It calls for exploratory data analysis, and inference methods that are able to capture more than a trivial sequence of dichotomies.

[Update 10/9/2018] Related data files can now be found in my Collection of morphological matrices (some including extinct taxa) and related phylogenetic inferences (Version 2) on figshare.