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

Marginal models for categorical data

Citation for published version (APA)

Advancements in marginal modeling for categorical data

Very often the data collected by social scientists involve dependent observations, without, however, the investigators having any substantive interest in the nature of the dependencies. Although these dependencies are not important for the answers to the research questions concerned, they must still be taken into account in the analysis. Standard statistical estimation and testing procedures assume independent and identically distributed observations, and they need to be modified for observations that are clustered in some way. Marginal models provide the tools to deal with these dependencies without having to make restrictive assumptions about their nature. In this paper, recent developments in the (maximum likelihood) estimation and testing of marginal models for categorical data will be explained, including marginal models with latent variables. The differences and commonalities with other ways of dealing with these nuisance dependencies will be discussed, especially with GEE and also briefly with (hierarchical) random coefficient models. The usefulness of marginal modeling will be illuminated by showing several common types of research questions and designs for which marginal models may provide the answers, along with two extensive real world examples. Finally, a brief evaluation will be given, including a discussion of shortcomings and strong point