NOTES ON AN EXPERIMENTAL TEST OF ERRORS IN PARTIAL CORRELATION COEFFICIENTS, DERIVED FROM FOURFOLD AND BISERTAL TOTAL COEFFICIENTS

IN many biometric and social investigations we are obliged to deal with variables which cannot be expressed quantitatively, and yet nevertheless partial correlation is necessary to unravel the skein of their interrelations. The total correlation coefficients in these cases can only be found by using biserial, tetrachoric or other fourfold r's or coefficients of contingency, so that the question arises, how far are we justified in drawing conclusions from partial coefficients based on total coefficients of this kind ? The present note describes an experimental test of this question, which was ipade for the Committee of Industrial Health Statistics of the Medical Research Council. The greater part of the arithmetical coinputation has been done by my fellow workers on the staff of that Committee-Mr E. Lewis-Faning, Mr J. Martin and Miss C. Thomas-and our thanks are due to Dr Major Greenwood and Dr L. Isserlis for helpful suggestions and advice. In these tests we have confined ourselves chiefly to tetrachoric r's but have also included some biserials. As regards the values of total tetrachoric coefficients, tests have previously been made on approximately normal distributions by Professor Pearson * and also by the late W. R. Macdonell t. The agreement with product moment values was good. Professor Pearson found also that the probable error of tetrachoric r increases with the distance of the dividing lines from the mean, but not very rapidly, and that the probable error of the tetrachoric r is 15 to 2 times the probable error as found by the product moment imethod, and also that it is not necessarily the case, in various positions of the dividing lines, that the smaller the probable error, the more nearly will the tetrachoric agree with the product moment coefficient. Deviations between the tetrachoric and product moment values arise from want of normality and not fronm errors of sampling (except in so far as these may affect the normality of the sainple), hence the question of a suitable criterion for judging the reliability of these partial correlations needs consideration. It seems clear first that, since the ordinary method of partial correlation is equivalent to correlating deviations from regression straight lines, and so involves