Ordinary least-squares regression allows one to investigate relationships between two variables in order to ask if change in one of these variables is associated with change in the other. One may ask, for example, how is variation in brain size related to body size, ecological role (predator vs. prey), climate, life history mode, or locomotion (Albert et al., 2000; Safi and Dechmann, 2005). The least-squares fitting procedure is commonly used in data analysis in comparative studies, and conventional regression analysis has been one of the main tools available to comparative neurobiology and ecological physiology to study form-function relationships and adaptation (Garland and Carter, 1994). However, it is now widely recognized that interspecific observations generally do not comprise independent and identically distributed data points, thus violating fundamental assumptions of conventional parametric statistics (Felsenstein, 1985, 1988; Pagel and Harvey, 1989; Harvey and Pagel, 1991).
Phylogenetically based statistical methods allow traditional topics in comparative neuroanatomy and physiology to be addressed with greater rigor, including the form of allometric relationships among traits and whether phenotypes vary predictably in relation to behavior, ecology, or environmental characteristics (Brooks and McLennan, 1991; Frumhoff and Reeve, 1994; Losos, 1996). In a conventional regression analysis the data points represent terminal taxa. In a phylo-genetic regression the data points represent sister-taxon comparisons (Grafen, 1989). These two methods are compared in Figure 5, in which identical data are analyzed using conventional and phylogenetic regression methods. The phylogeny of Figure 5 includes six terminal taxa (TA-TF) and two outgroup taxa (O1 and O2), which are represented by two continuously distributed characters (C1 and C2). The tree topology has been determined from data other than characters 1 and 2, and the branch lengths are treated as equal (under a model of punctuated equilibrium). There are seven internal tree nodes represented by ancestral taxa (AG-AM) with trait values estimated by least-square parsimony. By removing psuedorepilcates, the phylogenetic regression compares fewer taxa, has fewer degrees of freedom, and has a lower correlation coefficient (R2 value) than does the conventional regression. The phylogenetic regression, therefore, provides a better quantitative measure of correlated evolution between the two traits, and is a more conservative measure of the strength of adaptive pressures.
Relationships between brain size and the volume of frontal and visual cortices in mammals have recently been studied using the methods of phyloge-netic regression analysis (Bush and Allman, 2004a, 2004b). These studies found that size has a profound effect on the structure of the brain, and that many brain structures scale allometrically; that is, their relative size changes systematically as a function of brain size. They also conclude that the three-dimensional shape of visual maps in anthropoid
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