Methods for the Comparative Analysis of Variation Patterns

-Although comparisons of variation patterns with theoretical expectations and across species are playing an increasingly important role in systematics, there has been a lack of appropriate procedures for statistically testing the proposed hypotheses. We present a series of statistical tests for hypotheses of morphological integration and for interspecific comparison, along with examples of their application. These tests are based on various randomization and resampling procedures, such as Mantel's test with its recent extensions and bootstrapping. They have the advantage of avoiding the specific and strict distributional assumptions invoked by analytically-based statistics. The statistical procedures described include one for testing the fit of observed correlation matrices to hypotheses of morphological integration and a related test for significant differences in the fit of two alternative hypotheses of morphological integration to the observed correlation structure. Tests for significant similarity in the patterns and magnitudes of variance and correlation among species are also provided. [Morphometrics; comparative analysis; morphological integration; quadratic assignment procedures; Mantel's test; bootstrap. ] Comparing observed patterns of morphometric variation to theories of morphological integration (Olson and Miller, 1958; Cheverud, 1982) and among species, or subspecific populations (Arnold, 1981; Riska, 1985), has been a largely ad hoc procedure. Previously, a large body of methods has been used to analyze variation patterns, including various forms of cluster analysis, factor analysis, principal components, multi-dimensional scaling, matrix correlations, and visual inspection. The results of such analyses were then discussed relative to some theory of variation patterns or compared between species or populations. These comparisons might either be verbal or quantitative, but tests of statistical significance were rarely employed. More recently, there has been an increase in statistical rigor in the field, particularly involving the use of quadratic assignment procedures (QAP; sometimes referred to as Mantel's test) (Mantel, 1967; Deitz, 1983; Dow and Cheverud, 1985; Smouse et al., 1986; Dow et al., 1987a, b; Hubert, 1987) for testing the statistical significance of matrix comparisons (Cheverud and Leamy, 1985; Lofsvold, 1986; Kohn and Atchley, 1988; Cheverud, 1989a; Wagner, 1989) and the use of confirmatory factor analysis (Zelditch, 1987, 1988) for testing hypotheses concerning levels and patterns of variation. These new methods allow statistical inference for hypotheses of morphological integration and for comparisons across species. We will describe the use of several of these newer methods, especially those using randomization, for testing hypotheses of morphological integration and interspecific comparison and provide brief examples of their use. The procedures described below can be used to rigorously test hypotheses concerning the causes of morphological variation and covariation patterns. A closely related set of procedures can be directed towards comparative, cross-taxon, analyses of variation and correlation patterns. The systematic study of distinction among group means is well known and extensively represented in the literature. However, systematic studies of variation patterns (as measured by a multivariate variance/covariance or correlation matrix) have been relatively rare. This has been due, in part, to a lack of relevant theory and appropriate systematic methodology. Important theoretical advances over the