Comparative studies of the relationship between two phenotypes, or between a phenotype and an environment, are frequently carried out by invalid statistical methods. Most regression, correlation, and contingency table methods, including nonparametric methods, assume that the points are drawn independently from a common distribution. When species are taken from a branching phylogeny, they are manifestly nonindependent. Use of a statistical method that assumes independence will cause overstatement of the significance in hypothesis tests. Some illustrative examples of these phenomena have been given, and limitations of previous proposals of ways to correct for the nonindependence have been discussed. A method of correcting for the phylogeny has been proposed. It requires that we know both the tree topology and the branch lengths, and that we be willing to allow the characters to be modeled by Brownian motion on a linear scale. Given these conditions, the phylogeny specifies a set of contrasts among species, contrasts that are statistically independent and can be used in regression or correlation studies. The considerable barriers to making practical use of this technique have been discussed.