Most phylogenetic comparative methods used for testing adaptive hypotheses make evolutionary assumptions that are not compatible with evolution toward an optimal state. As a consequence they do not correct for maladaptation. The "evolutionary regression" that is returned is more shallow than the optimal relationship between the trait and environment. We show how both evolutionary and optimal regressions, as well as phylogenetic inertia, can be estimated jointly by a comparative method built around an Ornstein-Uhlenbeck model of adaptive evolution. The method considers a single trait adapting to an optimum that is influenced by one or more continuous, randomly changing predictor variables.KEY WORDS: Adaptation, maladaptation, optimality, Ornstein-Uhlenbeck process, phylogenetic comparative method, phylogenetic effect, phylogenetic inertia.Optimality models and other adaptive hypotheses are often tested by comparing their predictions to the trait values of species in different environments (e.g., Ridley 1983;Harvey and Pagel 1991). Most comparative methods are, however, based on models that are inconsistent with evolution toward an optimum. For example, the method of independent contrasts makes the assumption that traits evolve according to a Brownian-motion process (Felsenstein 1985), but if evolution is governed by this process, the expected trait value of a descendant species must equal the trait value of its ancestor, and there can be no systematic evolution toward an optimal state. If the ancestral species' trait value does not match the optimum, then the trait value of the descendant species is not expected to match it any better. Hansen and Orzack (2005) 4 Current address: Department of Zoology, University of Hawaii at Manoa, Honolulu, Hawaii 96822 called this the problem of inherited maladaptation. A multivariate Brownian-motion process can be used to represent correlated evolutionary changes in two or more traits, but correlated evolution is not equivalent to adaptive evolution. For example, suppose we predict that the optimal relation between two traits, x and y, is y = x. This simple prediction is incompatible with evolution as a Brownian-motion process. Even if there is a positive correlation between changes in y and x, any deviation from the 1:1 line will be inherited by the descendant species and there will be no systematic tendency to evolve toward the predicted relationship.This lack of attention to the fundamental nature of adaptive evolution has influenced the application of phylogenetic comparative methods. Such methods are often used erroneously to "correct" for phylogeny when they should only correct for the residual effects of phylogeny that remain after adaptation has been accounted for. A phylogenetic signal in the data can arise both from the influence of ancestral character states due to a lag or inertia in adaptation to the current niche, and from the effects of adaptation to niche variables that are themselves phylogenetically structured. One should correct only for the former. Correcti...