# Partial correlation coefficient

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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 have a joint distribution in , and let , be the best linear approximations to the variables and based on the variables . Then the partial correlation coefficient between and , denoted by , is defined as the ordinary correlation coefficient between the random variables and :

It follows from the definition that . The partial correlation coefficient can be expressed in terms of the entries of the correlation matrix. Let , where is the correlation coefficient between and , and let be the cofactor of the element in the determinant ; then

For example, for ,

The partial correlation coefficient of any two variables from is defined analogously. In general, the partial correlation coefficient is different from the (ordinary) correlation coefficient of and . The difference between and indicates whether and are dependent, or whether the dependence between them is a consequence of the dependence of each of them on . If the variables are pairwise uncorrelated, then all partial correlation coefficients are zero.

The empirical analogue of the partial correlation coefficient , the empirical partial correlation coefficient or sample partial correlation coefficient is the statistic

where is the cofactor in the determinant of the matrix of the empirical correlation coefficients . If the results of the observations are independent and multivariate normally distributed, and , then is distributed according to the probability density

( is the sample size). To test hypotheses about partial correlation coefficients, one uses the fact that the statistic

has, under the stated conditions, a Student distribution with degrees of freedom.

#### References

 [1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)