Multivariate Statistical Methods and Classification Problems

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Abstract
In this paper the roles which several widely used multivariate techniques—factor analysis, canonical variate analysis and cluster analysis—can play in the problem of classification of individuals are appraised. Of these techniques duster analysis is seen to be the only one which enables a direct attack on the problem to be made. Unfortunately, fully efficient methods have yet to be invented. Those at present available are known to work well in cases in which a heterogeneous population contains reasonably distinct clusters, and they provide a satisfactory means of identifying these. Discriminant function and canonical variate analyses are seen not to be classificatory devices in their own right, as they require as their starting point an existing classification, but they are useful tools for checking on the validity of that classification. Factor analysis as ordinarily used, that is with tests, items, scale scores, etc., as variables, does provide means of separating the variables into contrasting subgroups. But it does not enable the individuals in the sample to be classified in any clear-cut and acceptable manner. This is so because the scores for the members of the sample on each of a set of factors (rotated or unrotated) are by expectation normally distributed so that the majority of the sample falls in the vicinity of the mean of each factor distribution, and all individuals have scores on all factors. For a given factor, rotated let us say to represent some readily recognizable feature or aspect of behaviour, the most we can say is that individuals lying in one tail of its distribution would tend to display this feature in a marked degree while those in the other would display it minimally. But this would not imply a categorical difference between the individuals: as a group they are normally distributed along a continuum where the factor is concerned. Finally, if the scores of the individuals on two factors (rotated or unrotated) are plotted using orthogonal axes, then when the factors are uncorrelated the contours of density of the points will be circles and when they are correlated the contours will be ellipses.

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