Partial correlation coefficient

2020 Mathematics Subject Classification: Primary: 62-XX [MSN][ZBL]

A partial correlation coefficient is a measure of the linear dependence of a pair of random variables from a collection of random variables in the case where the influence of the remaining variables is eliminated. More precisely, suppose that the random variables $X_1,\dots,X_n$ have a joint distribution in $\R^n$, and let $X^*_{1;3\dots n}$, $X^*_{2;3\dots n}$ be the best linear approximations to the variables $X_1$ and $X_2$ based on the variables $X_3,\dots,X_n$. Then the partial correlation coefficient between $X_1$ and $X_2$, denoted by $\rho_{12;3\dots n}$, is defined as the ordinary correlation coefficient between the random variables $Y_1 = X_1 - X^*_{1;3\dots n}$ and $Y_2 = X_2 - X^*_{2;3\dots n}$:

$$\rho_{12;3\dots n} = \frac{\mathrm{E}\{(Y_1- \mathrm{E}Y_1)(Y_2- \mathrm{E}Y_2)\}}{\sqrt{\mathrm{D}Y_1\mathrm{D}Y_2}}.$$ It follows from the definition that $-1 \le \rho_{12;3\dots n}\le 1$. The partial correlation coefficient can be expressed in terms of the entries of the correlation matrix. Let $P=\|\rho_{ij}\|$, where $\rho_{ij}$ is the correlation coefficient between $X_i$ and $X_j$, and let $P_{ij}$ be the cofactor of the element $\rho_{ij}$ in the determinant $|P|$; then

$$\rho_{12;3\dots n} = - \frac{P_{12}}{\sqrt{P_{11} P_{22}}}.$$ For example, for $n=3$,

$$\rho_{12;3} = - \frac{\rho_{12}\rho_{33} - \rho_{13}\rho_{23}}{\sqrt{(1-\rho_{13}^2)(1-\rho_{23}^2)}}.$$ The partial correlation coefficient of any two variables $X_i,\; X_j$ from $X_1,\dots,X_n$ is defined analogously. In general, the partial correlation coefficient $\rho_{12;3\dots n}$ is different from the (ordinary) correlation coefficient $\rho_{12}$ of $X_1$ and $X_2$. The difference between $\rho_{12;3\dots n}$ and $\rho_{12}$ indicates whether $X_1$ and $X_2$ are dependent, or whether the dependence between them is a consequence of the dependence of each of them on $X_3,\dots,X_n$. If the variables $X_1,\dots,X_n$ are pairwise uncorrelated, then all partial correlation coefficients are zero.

The empirical analogue of the partial correlation coefficient $\rho_{12;3\dots n}$, the empirical partial correlation coefficient or sample partial correlation coefficient is the statistic

$$r_{12;3\dots n} = - \frac{R_{12}}{\sqrt{R_{11}R_{22}}},$$ where $R_{ij}$ is the cofactor in the determinant of the matrix $R=\|r_{ij}\|$ of the empirical correlation coefficients $r_{ij}$. If the results of the observations are independent and multivariate normally distributed, and $\rho_{12;3\dots n}$, then $r_{12;3\dots n}$ is distributed according to the probability density

$$\frac{1}{\sqrt{\pi}} \frac{\Gamma((N-n+1)/2)}{\Gamma((N-n)/2)}(1-x^2)^{(N-n-2)/2}, \quad -1<x<1$$ ($N$ is the sample size). To test hypotheses about partial correlation coefficients, one uses the fact that the statistic

$$t=\sqrt{N-n}\frac{r}{\sqrt{1-r^2}},\quad \textrm{where}\ r = r_{12;3\dots n},$$ has, under the stated conditions, a Student distribution with $N-n$ degrees of freedom.

References

[Cr]	H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014
[KeSt]	M.G. Kendall, A. Stuart, "The advanced theory of statistics", 2. Inference and relationship, Griffin (1979) MR0474561 MR0243648 Zbl 0416.62001
[Mu]	R.J. Muirhead, "Aspects of multivariate statistical theory", Wiley (1982) MR0652932 Zbl 0556.62028