Mixtures of Continuous and Categorical Variables in Discriminant Analysis: A Hypothesis-Testing Approach

SUMMARY In the classical case of discrimination between two multivariate normal populations, an allocation rule based on a hypothesis-testing approach (John's Z statistic) has long been in competition with the rule based on simple parameter estimation and substitution in the Bayes procedure (Anderson's W statistic). Recent evidence suggests that the Z statistic may be preferable in certain circumstances. The purpose of this paper is to extend this approach to mixed discrete and continuous variables. Accordingly, an allocation rule based on the location model is derived by use of a hypothesis-testing argument. A method for the estimation of error rates with this rule is outlined. Some examples are given, and the new rule is compared with its direct competitors; some of its possible merits are discussed briefly. Let V be a p-component random vector which has probability densities f1(v) and t2(v) in two distinct populations 7r1 and 7r2, respectively, and suppose that it is required to use V in discriminating between the two populations. More particularly, suppose that it is necessary to decide from which of the two populations an observed value v of V has come. This is the classical problem of discriminant analysis, which has been described by many authors; a standard treatment is given by Anderson (1958, Chapter 6). Anderson shows that, if the probability densities f1(v) and t2(v) are completely specified, the optimal decision rule is given by the Bayes procedure, which is to allocate v to w1 or 7r2 according to fl(v)lf2(v) s k. Ae cut-off point k is determined by the prior probabilities of an observation coming from each population, and by the costs incurred when an individual is misclassified from each population. In practice the densities t1(v) and f2(v) are rarely known, and the only information about 7r1 and 7r2 which is usually available comes from two training sets: v(ll), v21), . . ., v(l) known to come from 7r1, and v(l2), v22), . . ., v(m2) known to come from 7r2. In such circumstances, estimation is necessary; various methods have been proposed, each involving a different level of assumptions. At one extreme, no assumptions at all are made about the functional form of the densities ti(v) (i = 1, 2), and these densities are estimated from the training sets by some nonparametric method. A comprehensive bibliography of such methods is provided by Wertz and Schneider (1979). At the other extreme, exact functional forms which include unknown parameters are assumed for the densities. A classical approach to the problem involves estimation of the unknown parameters from the training sets. Replacing the unknown parameters by these estimates in fl(v)lt2(v) leads