Theory and methods a biplot based approach to discriminant analysis with categorical variables in the presence of reversals
Gower and Hand (1996) provide a unified biplot theory in which biplots are seen as multivariate analogues of ordinary scatterplots. In this paper a perspective is given on a biplot based approach to problems in discrimination. Various types of biplots are introduced for use when performing discriminant analysis in different situations. The focus is on discriminant analysis with categorical predictors. In particular it is shown how biplot based procedures can efficiently overcome the well-known problems that occur with binary predictors in the presence of reversals in log likelihood ratios, i.e. where the log likelihood ratio depends only on the number of positive coded elements of a random vector but it is not a monotone function of this number. It is argued that generalised biplot methodology allows the inclusion of categorical variables in a natural way by defining a distance matrix where both continuous and categorical variables contribute to the inter-point distances. A proposal is made where such distance matrices together with non-parametric procedures such as MARS and BRUTO are used as input to the non-linear biplot method to perform discriminant analysis. The potential of this proposal to address the reversal problem is investigated by means of a simulation study. Conditions favourable to linear discriminant analysis as well as non-linear discriminant analysis are considered. Examples of the associated biplots are provided.