Congruence between phylogeny and stratigraphy: Randomization tests and the gap excess ratio

Stratigraphic congruence indices (e.g., stratigraphic consistency index [SCI] of Huelsenbeck, 1994, Paleobiology 40:563-569; relative completeness index [RCI] of Benton, 1994, Trends Ecol. Evol. 9:181-185 [not to be confused with the rescaled consistency index of Farris, 1989, Cladistics 5:417-419]) are increasingly being quoted for cladograms containing fossil taxa. However, like the character consistency index (CI, summed over all characters), these values cannot be compared for trees derived from different data sets. Just as the number of characters and taxa in data matrices affects typical C1 values, so tree balance and the distribution of stratigraphic ranges (and indirectly tree size) delimit the range of congruence indices that can be obtained. Because investigators often seek to compare the performance of cladograms from different sources (e.g., different taxa, habitats, periods in history), indices of fit insensitive to these factors are desirable. Two approaches are proposed here: (1) The gap excess ratio (GER) is a new metric that controls for the distribution of range data but is sensitive to differences in tree balance. The GER expresses the difference between the minimum implied gap (MIG; the total ghost range implied by a given set of stratigraphic ranges on a given tree) and G_min (the minimum possible ghost range for those data on any tree) as a fraction of the range of values possible for those stratigraphic data on any tree. Rather than reflecting inferred completeness of the fossil record (as does the RCI, which is only partially determined by cladistic constraints), the GER indicates congruence alone. (2) Randomization tests hold most potentially conflated parameters constant and compare the observed RC1 or GER index with the distribution of indices obtained by randomly reassigning range data over the tree. This enables us to deduce whether the M1G is significantly less (i.e., shows greater congruence) than for random permutations of the same range data. Stratigraphic congruence indices have been invoked as an ancillary criterion for assessing competing hypotheses of relationships derived from different sources or for choosing between equally parsimonous trees derived from the same character matrix. The first application is illustrated with comparisons of actinopterygian and eutherian phylogenies, and the second is illustrated with a study of trees for fossil and Recent arthropods.