Approximate methods in calculating discriminant functions

Approximate methods of solving for discriminant functions have been tried on three sets of data. The principal illustration is the problem of finding a weighted sum of scores, on four psychological tests, so that men and women may be distinguished most clearly. The work starts from the complete solution, due to R. A. Fisher, where it is necessary to solve as many simultaneous equations, dependent on the standard deviations of the tests and their mutual correlations, as there are tests. It is proposed, by way of numerical simplification, that a set of equations be substituted where some one quantity replaces all the correlations. A solution is obtained where the weights to be assigned the tests are very simply expressed in terms of differences between the mean values of tests, the standard deviations of tests, and the said quantity. The difficulty remains of finding an estimate of the arbitrary constant that will give good discrimination. If an optimal solution is made a result is obtained which, in the three sets of data considered, is almost indistinguishable from that yielded by the complete solution. The calculation of this optimal common quantity is, however, itself so considerable that another estimate, previously suggested by R. W. B. Jackson, appears more profitable. This estimate is derived simply from the variability between the total scores for each subject and the variability of each test. Using this estimate, the discriminant functions can be rapidly calculated; the results compare very favorably, in the case of the data considered, with those from the complete solution.The present work was done while the writer was employed by the Ontario Department of Health.