Fisher’s linear discriminant function

When Hotelling’s T 2 distribution (section 30.4) indicates that one should reject the hypothesis that two multivariate normal distributions have equal mean vectors, it may be useful to determine the linear combination of original variates Y=b 1 X 1 +b 1 X 1 +... + b 1 X 1 which brings out differences between observations belonging to the two populations as clearly as possible, and which may be used to estimate from which population a new individual observation X=[X 1,X 2,... X q] has come. This linear combination is called a discriminant function and was developed by Fisher (1936), whose attention was drawn to the problem by Edgar Anderson (see Anderson, 1954), an American botanist, during a study of iris species of the Gaspé Peninsula, Québec. In his paper, which he illustrated with data provided by Anderson, Fisher (1936) obtained his discriminant function by maximizing the ratio of the difference between the means of two groups to the within-groups standard deviation. Fisher also showed that the same function could be obtained by estimating the coefficients b = [b 1,b 2,...b q] through multiple regression (sections 23.3, 25.3 and 25.5) after attributing arbitrary Y values to individual observations, such arbitrary values being constant within each group but different from one group to the other. Fisher’s linear discriminant function can be useful not only in taxonomy but also in medical diagnosis, where the variates X=[X 1,X 2,... X q] may represent symptoms, and in various other fields where a conclusion must be drawn or a decision must be made on the basis of many quantitative data jointly.