Dept of Mathematics, Keio University, Yokohama, 223-8522, Japan. e-mail: email@example.com
PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE
Article first published online: 13 DEC 2004
Australian & New Zealand Journal of Statistics
Volume 46, Issue 4, pages 657–664, December 2004
How to Cite
Baba, K., Shibata, R. and Sibuya, M. (2004), PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE. Australian & New Zealand Journal of Statistics, 46: 657–664. doi: 10.1111/j.1467-842X.2004.00360.x
- Issue published online: 13 DEC 2004
- Article first published online: 13 DEC 2004
- Received October 2002; revised November 2003; accepted February 2004.
- Cited By
- elliptical distribution;
- graphical modelling;
- monotone transformation
This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.