Univariate Analyses of Morphometric Variation Do Not Emulate the Results of Multivariate Analyses

In a recent paper, Willig et al. (1986) compared the empirical results of univariate (ANOVA) and multivariate (MANOVA) analyses of morphometric data from a large suite of external and cranial characters that were obtained from 21 species of bats from Northeast Brazil. Recognizing that multivariate analysis is the methodological approach of choice when asking systematic questions, we stated that "the degree to which univariate results are coincident with multivariate results will assess the utility of the univariate approach" (p. 196). The critique of Corruccini (1987) contends that we "assume a priori that the existence of differences shows the multivariate approach superior and preferable" (p. 396). For fundamental theoretical reasons, outlined in our original contribution and reiterated herein, we consider it axiomatic that for a multivariate question, a multivariate analysis is preferable. We pointed this out in the original article, and cited a number of biometric references that hold similar contentions. We reiterate that it is well understood that the MANOVA is preferable to a sequence of ANOVAs; the point of our previous work was to evaluate empirically whether the commonly used univariate ANOVAs emulate the multivariate results. Our results indicated that no consistent application of a univariate criterion (a priori number of significant ANOVAs required before declaring a significant multivariate difference between groups) effectively emulated the results of the MANOVAs. Generally speaking, Willig et al. (1986) tested the null hypothesis that no difference in systematic conclusion exists between multivariate and univariate methodologies in evaluating group differences. Moreover, on a case by case basis, the probability of correctly emulating multivariate conclusions, or making a Type I or Type II error with respect to the multivariate conclusions (not to be confused with a or f error in the strict sense), was documented for cranial, external, and combined character suites. The figure in Willig et al. (1986) shows the distribution of those errors based on a variety of univariate criterion values; however, it must be emphasized that these distributions were only known a posteriori (after the MANOVAs were performed for each species) and that they may only apply to these bat taxa from Northeast Brazil. An error rate of more than 15% occurred when the "best" univariate criterion was chosen (see fig. 1B in Willig et al., 1986). We consider this error rate to be unacceptable for systematic conclusions. Although Corruccini's second paragraph is true, it is incomplete and not germane to the issue that we examined in our original article. A correlation is a measure of association between variates in different distributions; the existence of a correlation does not logically or pragmatically imply a predictive relationship. That a significant association between MANOVA and ANOVA results exists is not surprising or interesting; that the correlation coefficient (r) was so low (0.68) for the combined cranial and external character suites attests to the lack of correspondence between univariate and multivariate methodologies. Moreover, given a number of significant ANOVA effects for a previously unanalyzed biological population, we have no idea of the relative distribution of significant univariate and multivariate results. Even if we knew the correlation in advance, which we would not, it would be uninformative for any particular systematic comparison. The extent to which predictive uncertainty is characteristic of our results is shown