Tests of independence in incomplete multi-way tables using likelihood functions

Kang (2006) and Kang and Larsen (in press) used the log likelihood function with Lagrangian multipliers for estimation of cell probabilities in two-way incomplete contingency tables. This paper extends results and simulations to three-way and multi-way tables. Numerous studies cross-classify subjects by three or more categorical factors. Constraints on cell probabilities are incorporated through Lagrangian multipliers. Variances of the MLEs are derived from the matrix of second derivatives of the log likelihood with respect to cell probabilities and the Lagrange multiplier. Wald and likelihood ratio tests of independence are derived using the estimates and estimated variances. In simulation results in Kang and Larsen (in press), for data missing at random, maximum likelihood estimation (MLE) produced more efficient estimates of population proportions than either multiple imputation (MI) based on data augmentation or complete case (CC) analysis. Neither MLE nor MI, however, lead to an improvement over CC analysis with respect to power of tests for independence in two-way tables. Results are extended to multidimensional tables with arbitrary patterns of missing data when the variables are recorded on individual subjects. In three-way and higher-way tables, however, there is information relevant for judging independence in partially classified information, as long as two or more variables are jointly observed. Simulations study three-dimensional tables with three patterns of association and two levels of missing information.