Partial correlation and conditional correlation as measures of conditional independence
Partial correlation and conditional correlation as measures of conditional independence
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DOI：
10.1111/j.1467-842x.2004.00360.x
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发表时间：
2004-12-01
影响因子：
1.1
通讯作者：
Sibuya, M
中科院分区：
文献类型：
--
作者：
Baba, K;Shibata, R;Sibuya, M
本文研究了偏相关和条件相关作为衡量两个随机变量条件独立性的作用。它首先建立了偏相关与条件相关重合的充分条件。该条件不仅适用于多元正态分布，也适用于椭圆分布、多元超几何分布、多元负超几何分布、多项式分布和Dirichlet分布。这种分布族的特征是作为参数分布族的半群性质。给出了部分协方差与条件协方差重合的充要条件。然而，除了多元正态分布外，无法找到满足这一条件的已知多变量分布族。本文还表明，除多元正态分布外，条件独立性与零偏相关关系不密切，而与条件零相关关系较密切。证明了正态变量的条件协方差为零与条件独立性之间的等价性可以通过每个变量的任意单调变换来保持。结果表明，在使用这种相关性作为条件独立性的衡量标准时必须谨慎，除非已知联合分布是正态的。否则，可能需要引入条件独立性的新概念，以取代通过零条件相关性或其他统计量的条件独立性。
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.