The article focuses on the interpretation of discriminant functions following a significant MANOVA test. Based on a geometric interpretation of MANOVA, new indices called discriminant ratio coefficients are derived which will aid in the identification and interpretation of that subset of variables that essentially contribute to a significant group discrimination. They will also aid in identifying situations where the relative importance of individual response variables can be simply quantified. In such cases, discriminant ratio coefficients provide natural measures of relative importance. Their efficacy as data analytic tools will be compared to several commonly used techniques, both theoretically and through examples. A modified approach for dealing with cases of coefficient instability is also described.

Interpreting Discriminant Functions: A Data Analytic Approach.

In canonical correlation analysis a hypothesis concerning the relevance of a subset of variables from each of the two given variable sets is formulated. The likelihood ratio statistic for the hypothesis and an asymptotic expansion for its null distribution are obtained. In discriminant analysis various alternative forms of a hypothesis concerning the relevance of a specified variable subset are also discussed.

A test for additional information in canonical correlation analysis

Feature Selection for Classification of Polar Regions Using a Fuzzy Expert System

We assessed the sampling stability of classification rates in discriminant analysis by using a factorial design with factors for multivariate dimensionality, dispersion structure, configuration of group means, and sample size. A total of 32,400 discriminant analyses were conducted, based on data from simulated populations with appropriate underlying statistical distributions. Simulation results indicated strong bias in correct classification rates when group sample sizes were small and when overlap among groups was high. We also found that stability of the correct classification rates was influenced by these factors, indicating that the number of samples required for a given level of precision increases with the amount of overlap among groups. In a review of 60 published studies, we found that 57% of the articles presented results on classification rates, though few of them mentioned potential biases in their results. Wildlife researchers should choose the total number of samples per group to be at least 2 times the number of variables to be measured when overlap among groups is low. Substantially more samples are required as the overlap among groups increases. J. WILDL. MANAGE. 54(2):331-341 Researchers often analyze biological data with discriminant analysis, a statistical method useful for classifying individuals into groups and for highlighting group differences. Necessary features for applications are the separability of samples into distinct groups and the measurement of 1 or more attributes on each sample unit. Typical applications by wildlife biologists involve the identification of groups of distinct animal or plant species and the examination of group differences based on habitat attributes (Edge et al. 1987). Other applications include morphometric analysis of species differences (Troy 1985) and analysis of plant or animal assemblages based on species abundances (Matthews 1979). Williams (1981, 1983) characterized ecological applications of the methodology by their grouping indices and attributes, and Williams and Titus (1988) described more recent applications in terms of sample sizes, system dimensions, and other statistical attributes. In many biological applications of discriminant analysis, the classification functions are characterized by substantial variability associated with small sample sizes and high multivariate dimensionality. There have been several theoretical assessments of the statistical properties of classification rates, especially asymptotic properties under various distributional assumptions (Anderson 1973, Gordon and Olshen 1978). However, the few studies that focused on small sample properties were limited in scope (Van Ness 1979, Page 1985) and generally focused on biases in apparent classification rates (Moran 1975, McLachlan 1976). Because wildlife studies often are hindered by an inability to collect a large number of independent samp es, the sources of variability in classification rates based on small sample sizes need to be examined. We report classification success as part of a simulation study of discriminant analysis (see Williams and Titus [1988] for canonical variates analysis). Our objective was to identify sources of variability in classification rates and to determine minimum sample sizes necessary for es imating these rates. Our study focused on assessment of discriminant analysis in representing system structure known to underlie the data under investigation. Our results compliment the work of Rexstad et al. (1988), who used data with no apparent underlying system structure to document a tendency, under certain conditions, for discriminant analysis to generate spurious relationships. We thank G. W. Pendleton, E. F. Burton, and K. Boone for technical support. D. E. Capen, D. W. Sparling, and J. S. Hatfield provided preliminary reviews and helpful comments. 1 Present address: Office of Migratory Bird Management, U.S. Fish and Wildlife Service, 18th and C Street NW, Washington, DC 20240. 2 Present address: Alaska Department of Fish and Game, Division of Wildlife Conservation, P.O. Box 20, Douglas, AK 99824.

Stability and bias of classification rates in biological applications of discriminant analysis

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A criterion for variable selection in multiple discriminant analysis

SUMMARY The purpose of the procedures proposed in this paper is to provide information about the relative importance of variable subsets in the analysis of multivariate data. In multiple discriminant analysis the procedures isolate subsets of variables which provide essentially as much separation among the groups in each subset of groups as the original set of variables. Each procedure is a simultaneous test procedure in that the type I probability error rate for the family of hypotheses tested cannot exceed a specified value.

R. J. McKAY

Papers

Simultaneous procedures for variable selection in multiple discriminant analysis