Questionable Multivariate Statistical Inference in Wildlife Habitat and Community Studies: A Comment

Abstract

Rexstad et al. (1988) questioned use of certain multivariate techniques in studies of wildlife habitat, based on analyses of a single set of meaningless In addition to the lack of replication, their study was flawed by use of ill-defined or irrelevant hypotheses, by analyses conducted in violation of a statistical assumption, and by interpretation unsupported by their own analyses. J. WILDL. MANAGE. 54(1):186-189 Rexstad et al. (1988) criticized 3 multivariate techniques commonly used in wildlife habitat and community studies. They performed principal component analyses (PCA), canonical correlation analysis (CC), and stepwise discriminant function analysis (DFA) on a data set composed of 15 functionally unrelated variables (e.g., liquor prices and telephone numbers). They concluded that Each [analysis] produced seemingly significant results and suggested strong relationships between the variables measured when applied to their meaningless data set (Rexstad et al. 1988:794) and that Our data, containing no inherent relationships, had relationships manufactured by the statistical (Rexstad et al. 1988:797). This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:08:13 UTC All use subject to http://about.jstor.org/terms J. Wildl. Manage. 54(1):1990 COMMENT ON MULTIVARIATE TECHNIQUES * Taylor 187 My objective is to point out that these conclusions are not supported by their results. The statistical methods did not manufacture relationships. Indeed, their summaries indicate that PCA did not detect significant results or strong relationships between their variables, that any relationship suggested by CC was at best weak, and that the data set was for the most part inappropriate for analysis by DFA. Their conclusions were based on rejection of irrelevant or illdefined hypotheses, on failure to compare their results with those expected due to chance, and on analysis of data that violated a basic statistical assumption. I thank L. D. Meeker, P. F. Sale, and 2 anonymous reviewers for their comments on various drafts of this manuscript. Rexstad et al. (1988:795) did not propose a testable hypothesis about PCA, but merely stated that [they] evaluated the extent to which PCA would explain the variance of the data set and reduce its dimensionality. They mentioned no criterion for the seemingly significant result specified in their conclusion. They did find that PCA of the covariance matrix required only 2 components to explain 99.5% of the variance. However, each component was dominated by a single variable with a large variance relative to the other 13 variables. Principal components dominated by single variables indicate no relationships between variables, contrary to Rexstad et al.'s conclusion. This PCA did not reduce dimensionality or explain a high proportion of the variance. It merely mirrored the essentially 2-dimensional nature of the original data, dominated by 2 uncorrelated variables contributing most of the variance. Rexstad et al. (1988) concluded that PCA of the correlation matrix of their data reduced dimensionality (from 15 original variables to 7 PC's with eigenvalues >1) while explaining a high proportion (69%) of the total variance. This reduction in dimensionality is trivial and well within the range of behavior expected from uncorrelated random variables. If the eigenvalues are themselves no more than random variables symmetrically distributed around a mean of 1 (the average contribution of a single original variable to the total variance), then about half, or around 7, should be >1. Also, any 10 of the original variables would, on average, explain about 67% (10/15) of the total variance even if they were totally unrelated. Such explanatory ability should be expected often in only 7 random variables, simply due to chance. Rexstad et al. (1988:796) actually confirmed this point with a sphericity test which failed to reject the hypothesis that all eigenvalues were of equal magnitude, i.e., that all PC's explained an equal amount of variation in the data, characteristic of random data. Another interpretation of the sphericity test is that no relationships (correlations) between variables other than those expected from chance were found; a nonsignificant sphericity test should lead to a decision not to perform PCA (Cooley and Lohnes 1971:103). The correct inference from both the covariance PCA and the sphericity test is that there was no structure in the The inferences of reduced dimensionality, seemingly significant results, and strong relationships between variables are in-