On the Performance of the Linear Discriminant Function

This paper considers the question: can the probability of misclassification given by a discriminant function, when used to classify specimens into one of two populations, be predicted accurately from the probabilities given by the variates when used individually? Theoretical consideration of the role of correlations between variates shows that (i) there is no mathematical reason why such predictions should be accurate and (ii) positive correlations (in the sense defined) are generally harmful and negative correlations helpful. Examination of 12 well-known numerical examples from the literature suggests that in practice (i) most correlations are positive (ii) it is usually safe to exclude from a discriminant, before computing it, a group of variates whose individual discriminatory powers are poor, except for any such variate that has negative correlations with most of the good discriminators (iii) the performance of the discriminant function can be predicted satisfactorily from a knowledge of individual powers plus the average correlation coefficient.