Comparative methods based on species mean values Colleen Kelly a, * , Trevor D. Price b a Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA b Department of Ecology and Evolution, University of Chicago, Chicago, IL 60637, USA Received 26 November 2002; received in revised form 20 October 2003; accepted 10 November 2003 Abstract Comparative methods that use simple linear regression based on species mean values introduce three difficulties with respect to the standard regression model. First, species values may not be independent because they form part of a hierarchically structured phylogeny. Second, variation about the regression line includes two sources of error: Ôbiological errorÕ due to deviations of the true species mean values from the regression line and sampling error associated with the estimation of these mean values [B. Riska, Am. Natural. 138 (1991) 283]. Third, sampling error in the independent variable results in an attenuated estimate of the regression slope. We consider estimation and hypothesis testing using two statistical models which explicitly justify the use of the species mean values, without the need to account for phylogenetic rela- tionships. The first (random-effects) is based on an evolutionary model whereby species evolve to fill a bivariate normal niche space, and the second (fixed-effects) is concerned with describing a relationship among the particular species included in a study, where the only source of error is in the estimation of species mean values. We use a modification of the maximum-likelihood method to obtain an unbiased estimate of the regression slope. For three real datasets we find a close correspondence between this slope and that obtained by simply regressing the species mean values on each other. In the random effects model, the P -value also approximates that based on the regression of species mean values. In the fixed effects model, the P -value is typically much lower. Simulated examples illustrate that the maximum-likelihood approach is useful when the accuracy in estimating the species mean values is low, but the traditional method based on a regression of the species mean values may often be justified provided that the evolu- tionary model can be justified. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Comparative methods; Errors in variables; Linear regression; Random effects; Fixed effects; Hypothesis testing; Maximum-likelihood estimation * Corresponding author. Tel.: +1-619 594 4170; fax: +1-619 594 6746. E-mail address: kelly@math.sdsu.edu (C. Kelly). 0025-5564/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2003.11.003 www.elsevier.com/locate/mbs Mathematical Biosciences 187 (2004) 135–154