Test of independence in a 2×22×2 contingency table with nonignorable nonresponse via constrained EM algorithm

Test of independence for 2×2 contingency tables with nonignorable nonresponses is discussed. Dependency assumption between two observed outcomes is required to achieve identification in many models with nonignorable nonresponses in the analysis of 2×2 contingency tables (e.g., [Ma, W.-Q., Geng, Z., Li, X.-T., 2003. Identification of nonresponse mechanisms for two-way contingency tables. Behaviormetrika 30, 125–144]). The assumption is, however, violated under the null hypothesis when implementing the test of independence. In this article, we introduce a new simple assumption to achieve identification. The assumption involves pre-specified parameters. EM algorithms for finding the MLE are numerically unstable when there are nonlinear constraints, which are created by models treating nonignorable nonresponses. In the analysis of contingency tables, estimated values often fall outside the admissible region. We propose a new EM type algorithm to stably calculate the constrained MLE, and apply it to make the test of independence for a real data set (crime data). We compare empirical performance among several testing procedures for independence. It turns out that the new EM type algorithm works well to calculate the MLE, and that the nonignorable model with the correctly specified parameters performs best while the conventional chi-square test of independence works fairly well.