Page:Proceedings of the Royal Society of London Vol 60.djvu/524

Rh was always less than the standard error when only x2 was taken into account, unless p\z — 0.

We may now prove the similar theorem that when we use three variables, x2, x3, Xi, on which to base the estimate, the standard error will be again decreased, unless pu = 0.

The condition that S(V), in onr present case, shall be less than S(r2) in the last, is, in fact, rr 122 + r132 + r f —r12ru ~ fas^u2—b s V "1 < — 2 (r13r147-34 + r12rl3r23 + r 12r14r24) >(l— L 4-2(r12r14r23r34 + ^Ur13^21r23 + r12r13’24^34) J > (rn + r132—2r12ri3r23) (l—r232—r2i2—r342 + 2r23r24r34). This may be finally reduced to— O'u—r13r34 — r12r2i — rur232 -f rnr23r2i + r 12r23r34) * > 0, that is pu~ > 0-

The treatment of the general case of n variables, so far as regards obtaining the regressions, is obvious, and it is unnecessary to give it at length.

We can now see that the use of normal regression formulae is quite legitimate in all cases, so long as the necessary limitations of inter-, pretation are recognised. Bravais’ r always remains a coefficient of correlation. These results 1 must plead as justification for my use of normal formulae in two cases* where the correlation was markedly non-normal.

(1) If the ratio of two absolute measurements on the same or different organs be taken it is convenient to term this ratio an index.

If $$u - f_1\left( x, y \right )$$ and $$v = f_2 \left (z, y \right )$$ be two functions of the three variables x, y, z, and these variables be selected at random so that there exists no correlation between x,y, y,z, or z,x, there will still be found to


 * 'Economic Journal,’ Dec., 1895, and Dec., 1896, "On the Correlation of Total Pauperism with Proportion of Out-relief."