Concordance factors

Concordance measures the strength of support that characters in a dataset present for each split (=edge/branch) in a tree (Minh et al. 2020; Smith 2026) .

Usage

ClusteringConcordance(tree, dataset, return = "edge", normalize = TRUE)

MutualClusteringConcordance(tree, dataset)

QuartetConcordance(tree, dataset = NULL, weight = TRUE, return = "edge")

PhylogeneticConcordance(tree, dataset)

SharedPhylogeneticConcordance(tree, dataset)

Arguments

tree

A tree of class phylo.

dataset

A phylogenetic data matrix of phangorn class phyDat, whose names correspond to the labels of any accompanying tree. Perhaps load into R using ReadAsPhyDat(). Additive (ordered) characters can be handled using Decompose().

return

Character specifying the summary to return. Options are:

"edge": average concordance of each tree split across all characters;
"char": concordance of each character averaged over all splits, weighted by each character's information content;
"tree": an overall tree‑level concordance score;
"all": a full array of MI components and normalized values for every split–character pair.

Matching is case‑insensitive and partial.

normalize

Controls how the expected mutual information (the zero point of the scale) is determined.

FALSE: no chance correction; MI is scaled only by its maximum.
TRUE: subtract the analytical expected MI for random association.
<integer>: subtract an empirical expected MI estimated from that number of random trees.

In all cases, 1 corresponds to the maximal attainable MI for the pair (hBest), and 0 corresponds to the chosen expectation.

weight

Logical specifying whether to weight sites according to the number of quartets they are decisive for.

Value

ClusteringConcordance(return = "all") returns a 3D array where each slice corresponds to a character (site), each column to a tree split, and each row to a different information measure. The normalized row gives the normalized mutual information between each split-character pair, scaled so that 1.0 corresponds to hBest (the theoretical maximum mutual information, being the minimum of hSplit and hChar) and 0.0 corresponds to miRand (the expected mutual information under random association). hSplit gives the entropy (information content) of each split's bipartition; hChar gives the entropy of each character's state distribution; hJoint gives the joint entropy of the split-character confusion matrix; mi gives the raw mutual information; and n records the number of informative observations. Negative normalized values indicate observed mutual information below random expectation. NA is returned when hBest = 0 (no information potential).

ClusteringConcordance(return = "edge") returns a vector where each element corresponds to a split (an edge of the tree) and gives the normalized mutual information between that split and the character data, averaged across all characters. When normalize = TRUE (default), values are scaled relative to random expectation; when FALSE, raw mutual information normalized by hBest is returned.

ClusteringConcordance(return = "char") returns a vector where each element corresponds to a character (site) and gives the entropy-weighted average normalized mutual information between that character and all tree splits. Characters with higher information content receive proportionally more weight from splits that can potentially convey more information about them.

ClusteringConcordance(return = "tree") returns a single value representing the overall concordance between the tree topology and the character data. This averages the fit of the best-matching split for each character. This is included for completeness, though it is not clear that this is a useful or meaningful measure.

MutualClusteringConcordance() returns the mutual clustering concordance of each character in dataset with tree. The attribute weighted.mean gives the mean value, weighted by the information content of each character.

QuartetConcordance(return = "edge") returns a numeric vector giving the concordance index at each split across all sites; names specify the number of each corresponding split in tree.

QuartetConcordance(return = "char") returns a numeric vector giving the concordance index calculated at each site, averaged across all splits.

PhylogeneticConcordance() returns a numeric vector giving the phylogenetic information of each split in tree, named according to the split's internal numbering.

SharedPhylogeneticConcordance() returns the shared phylogenetic concordance of each character in dataset with tree. The attribute weighted.mean gives the mean value, weighted by the information content of each character.

Details

ClusteringConcordance() measures how well each tree split reflects the grouping structure implied by each character (Smith 2026) . Characters and splits are treated as clusterings of taxa, and their agreement is quantified using mutual information (MI). All reported values are scaled so that 1 corresponds to the maximum possible mutual information for each split–character pair (hBest).

The normalize argument specifies how the zero point is defined.

If normalize = FALSE, zero corresponds to zero MI, without correcting for the positive bias that arises because MI is rarely exactly zero in finite samples.
If normalize = TRUE, the expected MI is computed using an analytical approximation based on the distribution of character tokens. This is fast and generally accurate for large trees (~200+ taxa), but does not account for correlation between splits.
If normalize is a positive integer n, the expected MI is estimated empirically by fitting each character to n uniformly random trees and averaging the resulting MI values. This Monte Carlo approach provides a more accurate baseline for small trees, for which the analytical approximation is biased. Monte Carlo standard errors are returned.

MutualClusteringConcordance() provides a character‑wise summary that emphasises each character’s best‑matching split(s). It treats each character as a simple tree and computes the mutual clustering information between this character‑tree and the supplied phylogeny. High values identify characters whose signal is well represented anywhere in the tree, even if concentrated on a single edge.

QuartetConcordance() is the proportion of quartets (sets of four leaves) that are decisive for a split which are also concordant with it (the site concordance factor (Minh et al. 2020) ). For example, a quartet with the characters 0 0 0 1 is not decisive, as all relationships between those leaves are equally parsimonious. But a quartet with characters 0 0 1 1 is decisive, and is concordant with any tree that groups the first two leaves together to the exclusion of the second.

By default, the reported value weights each site by the number of quartets it is decisive for. This value can be interpreted as the proportion of all decisive quartets that are concordant with a split. If weight = FALSE, the reported value is the mean of the concordance value for each site. Consider a split associated with two sites: one that is concordant with 25% of 96 decisive quartets, and a second that is concordant with 75% of 4 decisive quartets. If weight = TRUE, the split concordance will be 24 + 3 / 96 + 4 = 27%. If weight = FALSE, the split concordance will be mean(75%, 25%) = 50%.

QuartetConcordance() is computed exactly, using all quartets, where as other implementations (e.g. IQ-TREE) follow Minh2020) in using a random subsample of quartets for a faster, if potentially less accurate, computation. Ambiguous and inapplicable tokens are treated as containing no grouping information (i.e. (02) or - are each treated as ?).

PhylogeneticConcordance() treats each character in dataset as a phylogenetic hypothesis and measures the extent to which it supports the splits of tree. Each character is first interpreted as a tree (or set of trees) in which taxa sharing the same token form a clade. Only splits for which the character contains at least four relevant taxa can contribute information.

For each split, the function identifies which characters could potentially support that split (i.e. those for which the induced subtrees contain informative structure), and among these, which characters are actually compatible with the split. The concordance value for each split is the proportion of informative characters that support it. A value of 1 indicates that all characters informative for that subset of taxa support the split; a value of 0 indicates that none do. Characters that contain only ambiguous or uninformative states for the relevant taxa do not affect the result.

SharedPhylogeneticConcordance() treats each character as a simple tree. Each token in the character corresponds to a node whose pendant edges are the taxa with that token. The Shared Phylogenetic Concordance for each character in dataset is then the Shared Phylogenetic Information (Smith 2020) of this tree and tree.

References

Minh BQ, Hahn MW, Lanfear R (2020). “New methods to calculate concordance factors for phylogenomic datasets.” Molecular Biology and Evolution, 37(9), 2727–2733. doi:10.1093/molbev/msaa106 .

Smith MR (2020). “Information Theoretic Generalized Robinson-Foulds Metrics for Comparing Phylogenetic Trees.” Bioinformatics, 36(20), 5007–5013. doi:10.1093/bioinformatics/btaa614 .

Smith MR (2026). “Which characters support which clades? Exploring the distribution of phylogenetic signal using concordant information.” Forthcoming.

Author

Martin R. Smith (martin.smith@durham.ac.uk)