Page:Proceedings of the Royal Society of London Vol 60.djvu/521

486 zero if both net regressions are zero; (2) it is a symmetrical function 6f the variables; (3) it cannot be greater than unity; for t>y (13)* ri22 + r 132—2r12r2Sr31 < 1—r^2,

or adding r13V232 to both sides, and transferring r132 to the right-hand side (n2- n 3r23)2 < ( l—r132) ( l r 232).

If any two coefficients, say r12rj3, be supposed known, the inequality we have used above will give us limits for the value of the third. Throwing it into the form (r23—rl2r13)2 < 1 -f n»Yu2—rls2—r132, we have r23 must lie between the limits ri2^i3 ± \/r 12V132—r122 - r132+ 1.

The values of- these limits for some special cases are collected in the following table :—

Values of rn and r,3. Limits of 2*33.

fi2 = r13 = 0 0 r 12 = r13 = +1 + 1 ri2 == + 1, r13 = — 1 —1 ria = 0, r13 = +1 0 rlt — 0, r13 = ±_r + \ / l—r* VlZ — ^13 = + r 1 and 2r2—1 n 2 = + r, ru = —r 2r2—1 and —1 n 2 = rX3 = ± \o*5 = 07070 and 1 n 2 = + \/0*5 r12 = — y^O-5 0 „ - 1

One is rather prone to argue that if A be correlated with B, and B with 0, A will be correlated with 0. Evidently this is not necessary. A may be positively correlated with B, and B positively correlated with 0, but yet A may, in general, be negatively correlated with 0. Only, if the coefficients (AB) and (BO) are both numerically greater than 0707, can one even ascribe the correct sign to the (AC) correlation.

It is evident that one would, in general, expect to make a smaller standard error in estimating ajj from the two associated variables z2 and a*3, than in estimating it from one only, say x2. But it seems desirable to prove this specifically, and to investigate under what conditions it will hold good. The necessary condition is—

ri22+ r13*—2ri2r23ri3 ^ _ 2 w ? — > n ”