Testing Conditional Independence for Continuous Random Variables

A common statistical problem is the testing of independence of two (response) variables conditionally on a third (control) variable. In the first part of this paper, we extend Hoeffding’s concept of estimability of degree r to testability of degree r, and show that independence is testable of degree two, while conditional independence is not testable of any degree if the control variable is continuous. Hence, in a well-defined sense, conditional independence is much harder to test than independence. In the second part of the paper, a new method is introduced for the nonparametric testing of conditional independence of continuous responses given an arbitrary, not necessarily continuous, control variable. The method allows the automatic conversion of any test of independence to a test of conditional independence. Hence, robust tests and tests with power against broad ranges of alternatives can be used, which are favorable properties not shared by the most commonly used test, namely the one based on the partial correlation coefficient. The method is based on a new concept, the partial copula, which is an average of the conditional copulas. The feasibility of the approach is demonstrated by an example with medical data.