Testing conditional independence for continuous random variables ## AbstractA 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 a concept of Hoeffding to demonstrate that testing conditional independence with a continuous control variable is in a well-defined sense harder than testing the unconditional independence hypothesis. In the second part of the paper, a new methodology is introduced for nonparametric testing of conditional independence of continuous responses given an arbitrary, not necessarily continuous, control variable. The method is based on kernel estimation of what we term the partial copula, and allows arbitrary tests of unconditional independence to be used for testing conditional independence. Hence, robust tests and tests with power against broad ranges of alternatives can be used. These two favorable properties are not shared by the most commonly used test, namely the one based on the partial correlation. The feasibility of the approach is demonstrated by an example with medical data.
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