Polytomies in Comparative Analyses of Continuous Characters

Abstract

ple species. One requirement of the meth? od is that the true phylogeny linking the species in the data set be known in full, including the lengths of the branches in units of expected variance of change for the character(s) being studied (see Harvey and Pagel, 1991; Martins and Garland, 1991; Garland et al., 1992). In practice, we cannot be certain of the true phylogeny for any set of species. Instead, we must use an es? timate of it. This working phylogeny (sensu Graf en, 1989) might differ from the real phylogeny in three ways: (1) taxa depicted as sister groups in the working phylogeny might not be so in reality, causing the wrong taxa to be compared when comput? ing contrasts; (2) the working phylogeny might not depict all of the real sister-group relationships, i.e., a series of bifurcations in the true phylogeny might be repre? sented as a polytomy in the working phy? logeny; and (3) the lengths of branches be? tween nodes (including tips) of the working phylogeny might be incorrect. Differences of the first type clearly must be minimized because they violate the as? sumptions of Felsenstein's method to an unknown and possibly very serious de? gree. Differences of the third type also weaken the method, but apparently not fatally (see Graf en, 1989; Martins and Gar? land, 1991). However, we still want the branch lengths on the working phylogeny to reflect the truth as accurately as possible (Garland et al., 1992). Differences of the second type?incomplete resolution of the working phylogeny?raise some interest? ing conceptual and analytical issues. Un? less we are prepared to assume that spe? ciation is always a strictly dichotomous process, then we must recognize that some polytomies in working phylogenies will represent true cases of multiway speciation events; these are termed polyto? mies (Maddison, 1989). In the original de? scription of the comparative method of phylogenetically independent contrasts, Felsenstein (1985:10) alluded to the correct way to analyze hard polytomies. However, other polytomies in working phylogenies simply reflect our ignorance of the true dichotomous branching pattern (soft po? lytomies, Maddison, 1989). Felsenstein's suggestion to set some internode branch lengths to zero when computing standard? ized independent contrasts leads to unbi? ased estimates of correlations and slopes for both hard and soft polytomies but does not indicate how many degrees of freedom are available for hypothesis testing with the latter. Grafen (1989), Harvey and Pagel (1991), and Pagel (1992) have offered other more complicated ways of dealing with soft polytomies that allow only one degree of freedom per node. Table 1 presents the differences between the two positions. We want to recognize any hard polytomies as such to claim the maximum degrees of freedom associated with them.