Partial correlation coefficient
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) |
Comments
References
[a1] | R.J. Muirhead, "Aspects of multivariate statistical theory" , Wiley (1982) |
Partial correlation coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_correlation_coefficient&oldid=14288