## Background: Discriminant Procedures

The term discriminant analysis (Fisher, 1936; Cooley and Lohnes, 1971; Tatsuoka, 1971; Kshirsagar, 1972; Lachenbruch, 1975, 1979; Gnanadesikan, 1977; Klecka, 1980; Hand, 1981, 1982; Silverman, 1986) refers to several different types of analyses. Classificatory discriminant analysis is used to classify observations into two or more known groups on the basis of one or more quantitative variables. Classification can be done by either a parametric method or a nonparametric method in the DISCRIM procedure. A parametric method is appropriate only for approximately normal within-class distributions. The method generates either a linear discriminant function (the within-class covariance matrices are assumed to be equal) or a quadratic discriminant function (the within-class covariance matrices are assumed to be unequal).

When the distribution within each group is not assumed to have any specific distribution or is assumed to have a distribution different from the multivariate normal distribution, nonparametric methods can be used to derive classification criteria. These methods include the kernel method and nearest-neighbor methods. The kernel method uses uniform, normal, Epanechnikov, biweight, or triweight kernels in estimating the group-specific density at each observation. The within-group covariance matrices or the pooled covariance matrix can be used to scale the data.

The performance of a discriminant function can be evaluated by estimating error rates (probabilities of misclassification). Error count estimates and posterior probability error rate estimates can be evaluated with PROC DISCRIM. When the input data set is an ordinary SAS data set, the error rates can also be estimated by cross validation.

In multivariate statistical applications, the data collected are largely from distributions different from the normal distribution. Various forms of nonnormality can arise, such as qualitative variables or variables with underlying continuous but nonnormal distributions. If the multivariate normality assumption is violated, the use of parametric discriminant analysis might not be appropriate. When a parametric classification criterion (linear or quadratic discriminant function) is derived from a nonnormal population, the resulting error rate estimates might be biased.

If your quantitative variables are not normally distributed, or if you want to classify observations on the basis of categorical variables, you should consider using the CATMOD or LOGISTIC procedure to fit a categorical linear model with the classification variable as the dependent variable. Press and Wilson (1978) compare logistic regression and parametric discriminant analysis and conclude that logistic regression is preferable to parametric discriminant analysis in cases for which the variables do not have multivariate normal distributions within classes. However, if you do have normal within-class distributions, logistic regression is less efficient than parametric discriminant analysis. Efron (1975) shows that with two normal populations having a common covariance matrix, logistic regression is between one-half and two-thirds as effective as the linear discriminant function in achieving asymptotically the same error rate.

Do not confuse discriminant analysis with cluster analysis. All varieties of discriminant analysis require prior knowledge of the classes, usually in the form of a sample from each class. In cluster analysis, the data do not include information about class membership; the purpose is to construct a classification. See Chapter 11: Introduction to Clustering Procedures.

Canonical discriminant analysis is a dimension-reduction technique related to principal components and canonical correlation, and it can be performed by both the CANDISC and DISCRIM procedures. A discriminant criterion is always derived in PROC DISCRIM. If you want canonical discriminant analysis without the use of a discriminant criterion, you should use PROC CANDISC. Stepwise discriminant analysis is a variable-selection technique implemented by the STEPDISC procedure. After selecting a subset of variables with PROC STEPDISC, use any of the other discriminant procedures to obtain more detailed analyses. PROC CANDISC and PROC STEPDISC perform hypothesis tests that require the within-class distributions to be approximately normal, but these procedures can be used descriptively with nonnormal data.

Another alternative to discriminant analysis is to perform a series of univariate one-way ANOVAs. All three discriminant procedures provide summaries of the univariate ANOVAs. The advantage of the multivariate approach is that two or more classes that overlap considerably when each variable is viewed separately might be more distinct when examined from a multivariate point of view.