Delayed-response phylogenetic correlation: an optimization-based method to test covariation of continuous characters.

A new phylogenetic comparative method is proposed, based on mapping two continuous characters on a tree to generate data pairs for regression or correlation analysis, which resolves problems of multiple character reconstructions, phylogenetic dependence, and asynchronous responses (evolutionary lags). Data pairs are formed in two ways (tree-down and tree-up) by matching corresponding changes, Delta x and Delta y. Delayed responses (Delta y occurring later in the tree than Delta x) are penalized by weighting pairs using nodal or branch-length distance between Delta x and Delta y; immediate (same-node) responses are given maximum weight. All combinations of character reconstructions (or a random sample thereof) are used to find the observed range of the weighted coefficient of correlation r (or weighted slope b). This range is used as test statistic, and the null distribution is generated by randomly reallocating changes (Delta x and Delta y) in the topology. Unlike randomization of terminal values, this procedure complies with Generalized Monte Carlo requirements while saving considerable computation time. Phylogenetic dependence is avoided by randomization without data transformations, yielding acceptable type-I error rates and statistical power. We show that ignoring delayed responses can lead to falsely nonsignificant results. Issues that arise from considering delayed responses based on optimization are discussed.