Nonparametric test for conditional independence in two-way contingency tables

Consider a two-way contingency table, built on two categorical variables R and S. Testing for independence between these two categorical variables is a well-known problem : the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual of the sample we can also observe a set of p characteristics describing him. These extra covariates can provide more information on the dependence structure of the table, so that it could be interesting to test the independence between R and S conditionally to them. In this paper, we propose two nonparametric tests which generalize the chi-square test and the likelihood ratio test ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics is derived. The finite sample behaviour of the procedure is analysed through some Monte-Carlo experiments and the approach is illustrated with a real data example.