Nonparametric bootstrap tests of conditional independence in two-way contingency tables

When analyzing a two-way contingency table, a preliminary question is often whether the categorical variables under study, say R and S, are independent or not. Suppose now that for each individual in the table, a continuous variable X is also known. It is then worth analyzing the table conditionally on X. Contrasting these "local" results to the global unconditional case allows one to go beyond the initial analysis and provide a better understanding of the underlying phenomenon. Recently, Geenens and Simar (2010) [11] have proposed two nonparametric procedures for testing whether R and S are conditionally independent given X, free of any constraining linearity assumptions. However, based on an average of kernel-based estimators, the asymptotic criterion they suggested shows an inflated Type I error (false positive) for small to moderate sample sizes. In this paper, we address this problem by proposing consistent bootstrap versions of the Geenens-Simar test procedures when testing for local independence. A comprehensive simulation study indeed shows the superiority of the bootstrap rejection criterion as compared to the asymptotic criterion in terms of Type I error. It also highlights the advantage of the flexibility guaranteed by the nonparametric Geenens-Simar tests when compared with parametric competitors, e.g.logistic models. The approach is finally illustrated with a real-data example.