Some Classification Problems with Multivariate Qualitative Data

Abstract

Since 1935, when Fisher's discriminant function appeared in the literature, methods for classifying specimens into one of a set of universes, given a series of measurements made on each specimen, have been extensively developed for the case in which the measurements are continuous variates. This paper considers some aspects of the classification problem when the data are qualitative, each measurement taking only a finite (and usually small) number of distinct values, which we shall call states. Our interest in the problem arose from discussions about the possible use of discriminant analysis in medical diagnosis. Some diagnostic measurements, particularly those from laboratory tests, give results of the form: -, + (2 states); or -, doubtful, + (3 states); or (with a liquid), clear, milky, brownish, dark (4 states). With qualitative data of this type an optimum rule for classification can be obtained as a particular case of the general rule (Rao, [1952], Anderson, [1958]). The rule is exceedingly simple to apply (Section 2). In practice, qualititative data are frequently ordered, as with -, doubtful, +. The classification rule discussed in this paper takes no explicit advantage of the ordering, as might be done, for instance, by assigning scores to the different states so as to produce quasi-continuous data. The best method of handling ordered qualitative data is a subject worth future investigation.