Quasi local odds symmetry model for square contingency table with ordinal categories

This paper proposes a new model for square contingency tables. The proposed model tests the equality of local odds ratios between the one side of the main diagonal and corresponding other side and it represents the non-symmetric structure of the square contingency table. The proposed model is compared with twenty-five models introduced for analysing the square contingency tables for both symmetric and non-symmetric structures. The results show that the proposed model provides best fit performance than other existing models for square contingency tables.     

Partitioning Pearson's chi-squared statistic for a partially ordered three-way contingency table

This paper presents a generalization of the partition of the chi-squared statistic presented in Beh & Davy (1998). For a three-way contingency table with one or two sets of ordered categories, the chi-squared statistic partition is defined using orthogonal polynomials. Using this partition, information about the relationship between the variables can be obtained by identifying important associations in terms of the location (linear), dispersion (quadratic) and higher order components. The paper compares these partitions with log-linear models for ordinal data.     

Log-linear models for correlated marginal totals of a contingency table

The marginal totals of a contingency table can be rearranged to form a new table. If at least twoof these totals include the same cell of the original table, the new table cannot be treated as anordinary contingency table. An iterative method is proposed to calculate maximum likelihood estimators for the expected cell frequencies of the original table under the assumption that some marginal totals (or more generally, some linear functions) of these expected frequencies satisfy a log-linear model.In some cases, a table of correlated marginal totals is treated as if it was an ordinary contingency table. The effects of ignoring the special structure of the marginal table on thedistributionof the goodness-of-fit test statistics are discussed and illustrated, with special reference to stationary Markov chains.     

Bayesian reconstruction of contingency tables with partially categorized data

This paper suggests a Bayesian approach to the reconstruction of a 2 × 2 contingency table where some of the observations are only partially categorized and others are fully categorized. In contrast, most previous Bayesian and non-Bayesian analyses of the partially categorized data problem have been concerned with estimation of the parameters that generated the data. We show in an example that estimates may not be extremely sensitive to the weight placed on prior information relative to the sample data.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

Bayesian analysis of contingency tables: a simulation and graphics-based approach

In this paper we present a simulation and graphics-based model checking and model comparison methodology for the Bayesian analysis of contingency tables. We illustrate the approach by testing the hypotheses of independence and symmetry on complete and incomplete simulated tables.     

A bayesian analysis of the multinomial model for a dichotomous response with nonrespondents

This paper studies the mu1tinomial model 2x2 contingency table data with some cell counts missing .Various hypotheses of interest including row-column independence are tested by using Bayes factors which represent the ratio of the posterior odds to the prior odds for the null hypothesis. The Dirichlet-Beta family of prior distributions is considered for the multinomial parameters cond itional on the complement of the null hypothesis. The Bayes factor for the incomplete data is a mixture of the Bayes factors for different possibilities for the full data.     

Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but in fact it extends back at least into the 19th century. Moreover, it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. Roy played a crucial role in this development with two papers co-authored with his students, Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for log-linear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem.     

A pooled Bayes test of independence for sparse contingency tables from small areas

We study the association between bone mineral density (BMD) and body mass index (BMI) when contingency tables are constructed from the several U.S. counties, where BMD has three levels (normal, osteopenia and osteoporosis) and BMI has four levels (underweight, normal, overweight and obese). We use the Bayes factor (posterior odds divided by prior odds or equivalently the ratio of the marginal likelihoods) to construct the new test. Like the chi-squared test and Fisher's exact test, we have a direct Bayes test which is a standard test using data from each county. In our main contribution, for each county techniques of small area estimation are used to borrow strength across counties and a pooled test of independence of BMD and BMI is obtained using a hierarchical Bayesian model. Our pooled Bayes test is computed by performing a Monte Carlo integration using random samples rather than Gibbs samples. We have seen important differences among the pooled Bayes test, direct Bayes test and the Cressie-Read test that allows for some degree of sparseness, when the degree of evidence against independence is studied. As expected, we also found that the direct Bayes test is sensitive to the prior specifications but the pooled Bayes test is not so sensitive. Moreover, the pooled Bayes test has competitive power properties, and it is superior when the cell counts are small to moderate.     

A bayesian analysis of the intraclass correlations in the mixed linear model

This article is concerned with the study of intraclass correlations in the mixed linear model. A brief account of the shortcomings of the existing meth¬ods (frequentist. likelihood and Bayesian) is followed by alternative Bayesian parametrizations involving intraclass correlations and variance ratios. Our prior specifications accommodate a priori dependencies as well as situations which involve little or no prior information. We give examples of interval estimation and hypothesis testing using data from an animal breeding study.     

The validity of the chi-squared test when expected frequencies are small:A list of recent research references

A bibliography is given containing about 60 recent (chiefly post-1968) items on the validity of the chi-squared test for categorical data when the expected frequencies are small.     

An Example of the Large Sample Behavior of the Midrange

A data set in the form of a 2 × 2 × 2 contingency table is presented and analyzed in detail. For instructional purposes, the analysis of the data can be used to illustrate some basic concepts in the loglinear model approach to the analysis of multidimensional contingency tables.     

A model of uniform association plus two-diagonalsparameter and its application to occupational mobility table data

For the analysis of square contingency tables with ordered categories, Agresti (1988) introduced a model having the structure of uniform association plus a main-diagonal parameter. This paper extends that model. The extended model has the structure of uniform association plus two-diagonals-parameter, and it is a special case of the quasi-uniform association model introduced by Goodman (1979). The Danish occupational mobility table data are analyzed using the models introduced here.     

A more powerful unconditional exact test of homogeneity for 2 × c contingency table analysis

The classical unconditional exact p-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact p-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate the 5th percentiles of the distributions of the p-values of the exact test over a range of scenarios and implemented a regression model to predict the values for two-sample multinomial settings. Our results show that the new test is uniformly more powerful than Fisher's, Barnard's, and Boschloo's tests with gains in power as large as several hundred percent in certain scenarios. Lastly, we provide a real-life data example where the unadjusted unconditional exact test wrongly fails to reject the null hypothesis and the corrected unconditional exact test rejects the null appropriately.     

On testing and estimating the interaction between treatments and environmental conditions in binomial experiments: The case of two stations

Estimating confidence intervals for the interaction between treatments and environmental conditions in binomial experiments is analyzed. Testing the interaction is studied also. The problem is reduced to that of estimating or testing the interaction parameter in 2 × 2 × 2 contingency tables with given marginals. Programs for determining the exact conditional tests and their power functions are provided for sample of size not exceeding 100. Large sample approximations based on maximum likelihood (ML) and on the arcsin transformation for proportions are studied.     

On the Hypothesis of No Interaction on a Linear Scale in Contingency Tables

The purpose of this paper is to review briefly the three main formulations of no Interaction hypotheses in contingency tables and to consider the formulation on a linear scale in some detail.More specifically we (i) present a situation in 2×2 tables where such a formulation may be more appropriate than others, (ii) study the geometry for this problem, (iii) give contrast-type or parametric ANOVA type formulations in the general n-dimensional tables, (iv) discuss estimation and testing procedures and (v) consider collapsibility of contingency tables in relation to the hypotheses of no interaction on a linear scale.     

A class of statistics based on the information concept

The methodic use of Shannon's entropy as a basic concept, complementing probability, leads to a new class of statistics which provides, inter alia, a measure of mutual dissimilarity y between several frequency distributions. Application to contin-gency tables with any number of dimensions yields a dimension-less, standardised contingency coefficient which depends on the direction of inference and will combine multiplicatively with the number of observed events. This class of statistics further in-cludes a continuous modification W of the number of degrees of freedom in a table, and a measure Q of its overall information content. Numerical illustrations and comparisons with former re-sults are worked out. Direct applications include the optimal partition of a quasicontinuum into cells by maximizing Q, the ordering of unordered tables by minimising local values of y, and a tentative absolute weighting of inductive inference based on the minimal necessary shift, required by an hypothesis, between the actually observed data and a set of assumed future events.     

On the inferential use of predictive sample reuse

The predicitive sample reuse (PSR) data analysis technique proposed by Geisser and Eddy (1979) is applied to the analysis of categorical data. This application yiclds a new approach which has a number of advantages over classical methods for analysis of such data. A PSR technique for comparing linear or nonlinar regressino relationships, for two or more populations, and a PSR ailernative to certain nonparaemetri statistical tesis are also proposed.