Page:Proceedings of the Royal Society of London Vol 60.djvu/520

Rh gay. In normal correlation \ / l —R]2 is the standard deviation of an Xi-array, corresponding to auy given types of X2 and X3. In general correlation it may be regarded as the mean standard deviation of the Xi-arrays from the plane . Xi — 6 j2 ® 2 “h ^13®3>

or as the standard error made in estimating xl from x2 and Xs by relation (1*2).

The quantity R is of some interest, as it exactly takes the place of r in the residual expressions (7). Ri may, in fact, he regarded as a coefficient of correlation between Xi and ! R can only be unity if the linear relation (9) or (12) hold good in every case.

The quantities ,&«, &J3, &c. (the others may be written down by symmetry), may be termed the net regressions of *1 on a^, on x3, &c. If we write 2 for 1 and 1 for 2 in the value of bl2, we have t _ r12—r13rvi <x2 02i-- — — } 1 — a\

bn being the the net regression of x% on xx. In normal correlation, b\2 and bn are the regressions for any group of Xi s or X2 s associated with a fixed type of X3’s. Hence, in this case (normal correlation), the coefficient of correlation for such a group is the geometrical mean of the two regressions, or _____ T1 2 --^*13^*23,_____ M = \/ (1—ns2) (1—r232)

a quantity that may be called the net coefficient of correlation between and The similar net coefficients between xx and #3, X2 and #3, may be written down by interchanging the suffixes.

In normal correlation pn is quite strictly the coefficient of correlation for any sub-group of Xl’s and X2’s, whatever the associated type of X3’s. In generalised correlation this will not be so, and />i2 can only retain an average significance.

The method does not appear to be capable of investigating changes* in the net coefficient as we pass from one type to another, but it may be noted that whatever the form of the correlation, pi2 retains three of the chief properties of the ordinary coefficients : (1) it can only be

&c.,” 4 Phil. Trans.,’ A, yol. 187 (1896), p. 287),• “ Coefficients of double regression,” and quantities like bl2 —, &13 —, &c., “ coefficients of double correlation.” M y'
 * My quantities, &12, &i3, &c., were termed by Professor Pearson (“ Regression

quantities p he did not use. Having named the p’s “ net correlation,” it seemed most natural to rename the J’s “ net regressions,” as the 5’s and p’s are corresponding quantities.

Some of my results given above were quoted by Professor Pearson in his paper (loc. cit.} notes on pp. 268 and 287),