Power of the Concentrated Changes Test for Correlated Evolution

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Abstract
The concentrated changes test (CCT) calculates the probability that changes in a binary character are distributed randomly on the branches of a cladogram. This test is used to examine hypotheses of correlated evolution, especially cases where changes in the state of one character influence changes in the state of another character. The test may be sensitive to the addition of branches that lack either trait of interest (white branches). To examine the effects of the proportion of white branches and of tree topology on the CCT probability, we conducted a simulation analysis using a series of randomly generated 100-taxon trees, in addition to a nearly perfectly balanced (symmetrical) and a completely imbalanced (asymmetrical) 100-taxon tree. Using two models of evolution (gains only, or gains and losses), we evolved character pairs randomly onto these trees to simulate cases where (1) characters evolve independently (i.e., no correlation among the traits) or (2) all changes in the dependent character occur on branches containing the independent trait (i.e., a strong correlation among the traits). This allowed us to evaluate the sensitivity of the CCT to type I and type II errors, respectively. In the simulations, the CCT did not appear to be overly sensitive to the inclusion of white branches (low likelihood of type I error with both CCT probabilities < 0.05 and < 0.01). However, the CCT was susceptible to type II error when the proportion of white branches was < 20%. The test was also sensitive to tree shape and was positively correlated to Colless's tree imbalance statistic I. Finally, the CCT responded differently for simulations where only gains were allowed and those where both gains and losses were permitted. These results indicate that the CCT is unlikely to detect a correlation between characters when no such correlation exists. However, when a trait can be gained but not lost, the CCT is conservative and may fail to detect true correlations among traits (increased type II error). Determination of the sampling universe (the taxa included in the comparative analysis) can strongly influence the probability of making such type II errors. We suggest guidelines to circumvent these limitations.

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