Page:Proceedings of the Royal Society of London Vol 60.djvu/523

488 and we may again call

$$b_{21} \frac{ \begin{vmatrix} \gamma_{12} \gamma_{23} \gamma_{24} \\ \gamma_{13} 1 \gamma_{34}  \\ \gamma_{14} \gamma_{34} 1 \end{vmatrix} } {\begin{vmatrix} 1 \gamma_{13} \gamma_{14} \\ \gamma_{13} 1 \gamma_{34}  \\ \gamma_{14} \gamma_{34} 1 \end{vmatrix}}

\frac{\sigma_2}{\sigma_1} $$

and we may again call

$$\sigma_{12} = \sqrt{b_{12} b_{21,}}$$

the net coefficient of correlation between $$x_1$$ and $$x_2$$. Expanding the determinants, we have, in fact,

$$\sigma_{12}=\frac { \gamma_{12} \left ( 1 - {\gamma_{34}}^2 \right ) + \gamma_{13} \left ( \gamma_{34}\gamma_{24} - \gamma_{23} \right ) + \gamma_{14} \left ( \gamma_{23}\gamma_{34} - \gamma_{24} \right ) }{ \sqrt{ 1 \left [ \left ( 1 -{\gamma_{34}}^2 \right ) + \gamma_{23} \left ( \gamma_{34} \gamma_{24} - \gamma_{23} \right ) + \gamma_{24} \left ( \gamma_{23} \gamma_{34} - \gamma_{24} \right ) \right ]

\left [ \left ( 1 - {\gamma_{34}}^2 \right ) + \gamma_{13} \left ( \gamma_{34} \gamma_{14} - \gamma_{13} \right ) + \gamma_{14} \left ( \gamma_{13} \gamma_{34} - \gamma_{14} \right ) \right ]

} }$$ ............. (16).

There are six such net coefficients, $$\sigma_{12}, \sigma_{13},\sigma_{14},\sigma_{23},\sigma_{24},\sigma_{34}$$ The above values of the regressions are again those usually obtained on the assumption of normal correlation The net correlation $$\sigma_{12}$$ becomes, on that assumption, the coefficient of correlation for any group of the $$x_1,x_2 $$ variables associated with fixed types of x_3> and $$x_4$$. If we write $$u = x_1 - \left ( b_{12} + b_{13} x_{3} + b_{14} x_{4} \right )$$, we have, after some rather lengthy reduction,

$$\frac{1}{N} S \left ( u^2 \right ) = {\sigma_{1}}^2 = \left ( 1 - {R_1}^2 \right )$$,

where

$$

{R_1}^2 =

\frac{

\begin{Bmatrix} {\gamma_{12}}^2 + {\gamma_{13}}^2 + {\gamma_{14}}^2 - {\gamma_{12}}^2 {\gamma_{34}}^2 - {\gamma_{13}}^2 {\gamma_{24}}^2 \\ - 2 \left ( \gamma{13} \gamma{14} \gamma{34} + \gamma{12} \gamma{14} \gamma{24} + \gamma{12} \gamma{13} \gamma{23} \right ) + 2 \left ( \gamma{12} \gamma{14} \gamma{23} \gamma{34} + \gamma{13} \gamma{14} \gamma{23} \gamma{24} + \gamma{12} \gamma{13} \gamma{23} \gamma{34} \right ) \end {Bmatrix} } {1- {\gamma_23}^2 - {\gamma_34}^2 - {\gamma_24}^2 + 2{\gamma_23}{\gamma_34}{\gamma_24} }

$$

normal correlation, $$\sigma_1\sqrt{1- {R_1}^2 }$$ is the standard deviation of all arrays associated with fixed types of $$x_2$$, $$x_3$$ and $$x_4$$. In general correlation, it is most easily interpreted as the standard error made in estimating $$x_1$$ by equation (14), from its associated values of $$x_2$$,$$x_3$$ and $$x_4$$.

As in the case of three variables, the quantity R may be considered as a coefficient of correlation. It can range between ±+1, and can only become unity if the linear relation (14) hold good in each individual instance.

We showed at the end of the last section that the standard error made in estimating $$x_1$$ from the relation

$$x_1 = b_{12}x_2 + b_{13}x_3$$