Testing conditional independence for continuous random variables

Abstract: 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

Maximum augmented empirical likelihood estimation of categorical marginal models for large sparse contingency tables

Categorical marginal models (CMMs) are flexible tools for modelling dependent or clustered categorical data, when the dependencies themselves are not of interest. A major limitation of maximum likelihood (ML) estimation of CMMs is that the size of the contingency table increases exponentially with the number of variables, so even for a moderate number of variables, say between 10 and 20, ML estimation can become computationally infeasible. An alternative method, which retains the optimal asymptotic efficiency of ML, is maximum empirical likelihood (MEL) estimation. However, we show that MEL tends to break down for large, sparse contingency tables. As a solution, we propose a new method, which we call maximum augmented empirical likelihood (MAEL) estimation and which involves augmentation of the empirical likelihood support with a number of well-chosen cells. Simulation results show good finite sample performance for very large contingency tables

Beyond Pearson’s correlation: modern nonparametric independence tests for psychological research

When examining whether two continuous variables are associated, tests based on Pearson’s, Kendall’s, and Spearman’s correlation coefficients are typically used. This paper explores modern nonparametric independence tests as an alternative, which, unlike traditional tests, have the ability to potentially detect any type of relationship. In addition to existing modern nonparametric independence tests, we developed and considered two novel variants of existing tests, most notably the Heller-Heller-Gorfine-Pearson (HHG-Pearson) test. We conducted a simulation study to compare traditional independence tests, such as Pearson’s correlation, and the modern nonparametric independence tests in situations commonly encountered in psychological research. As expected, no test had the highest power across all relationships. However, the distance correlation and the HHG-Pearson tests were found to have substantially greater power than all traditional tests for many relationships and only slightly less power in the worst case. A similar pattern was found in favor of the HHG-Pearson test compared to the distance correlation test. However, given that distance correlation performed better for linear relationships and is more widely accepted, we suggest considering its use in place or additional to traditional methods when there is no prior knowledge of the relationship type, as is often the case in psychological research

Marginal models for categorical data

Citation for published version (APA)