Independence In Two-way Tables Research Articles

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.

Residual-Based Shadings for Visualizing (Conditional) Independence

Residual-based shadings for enhancing mosaic and association plots to visualize independence models for contingency tables are extended in two directions: (a) perceptually uniform Hue-Chroma-Luminance (HCL) colors are used and (b) the result of an associated significance test is coded by the appearance of color in the visualization. For obtaining (a), a general strategy for deriving diverging palettes in the perceptually based HCL space is suggested. As for (b), cutoffs that control the appearance of color are computed in a data-driven way based on the conditional permutation distribution of maximum-type test statistics. The shadings are first established for the case of independence in two-way tables and then extended to more general independence models for multiway tables, including in particular conditional independence models.

The Analysis of Categorical Data from Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables

Abstract The effect of stratification and clustering on the asymptotic distributions of standard Pearson chi-squared test statistics for goodness of fit (simple hypothesis) and independence in a two-way contingency table, denoted as X 2 and XI 2, respectively, is investigated. It is shown that both X 2 and XI 2 are asymptotically distributed as weighted sums of independent χ1 2 random variables. The weights are then related to the familiar design effects (deffs) used by survey samplers. A simple correction to X 2, which requires only the knowledge of variance estimates (or deffs) for individual cells in the goodness-of-fit problem, is proposed and empirical results on the performance of corrected X 2 provided. Empirical work on XI 2 indicated that the distortion of nominal significance level is substantially smaller with XI 2 than with X 2. Some results under simple models for clustering are also given.

The Analysis of Categorical Data From Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables

The Analysis of Categorical Data From Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables