Analysis of Contingency Tables with Clustered Observations

A new approach is presented for testing independence in contingency tables with clustered observations. The approach is based on the framework of generalized linear mixed models. Under the multinomial logistic link function, the category counts are modelled with random cluster effects and a modified likelihood ratio statistic is used for testing independence. The method is applicable to multi-way tables, and can accommodate multiple levels of clustering. It is illustrated using a benchmark dataset.     

Contingency tables with structural zeros

In this paper, the method of Hocking and Oxspring (1971) to estimate multinomial probabilities when full and partial data are available for some cells is extended to estimate the cell probabilities of a contingency table with structural zeros. The estimates are maximum likelihood, and the process is sequential. The gain in precision is due to the use of partial data and the bias of the estimates is also investigated.     

A Bayesian test of independence in a two-way contingency table using surrogate sampling

We consider a Bayesian approach to the study of independence in a two-way contingency table which has been obtained from a two-stage cluster sampling design. If a procedure based on single-stage simple random sampling (rather than the appropriate cluster sampling) is used to test for independence, the p-value may be too small, resulting in a conclusion that the null hypothesis is false when it is, in fact, true. For many large complex surveys the Rao–Scott corrections to the standard chi-squared (or likelihood ratio) statistic provide appropriate inference. For smaller surveys, though, the Rao–Scott corrections may not be accurate, partly because the chi-squared test is inaccurate. In this paper, we use a hierarchical Bayesian model to convert the observed cluster samples to simple random samples. This provides surrogate samples which can be used to derive the distribution of the Bayes factor. We demonstrate the utility of our procedure using an example and also provide a simulation study which establishes our methodology as a viable alternative to the Rao–Scott approximations for relatively small two-stage cluster samples. We also show the additional insight gained by displaying the distribution of the Bayes factor rather than simply relying on a summary of the distribution.     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

Maximum likelihood estimation for the negative multinomial log-linear model

A log-linear model is defined for multiway contingency tables with negative multinomial frequency counts. The maximum likelihood estimator of the model parameters and the estimator covariance matrix is given. The likelihood ratio test for the general log-linear hypothesis also is presented.     

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.     

Estimation under generalized sampling of cell proportions for contingency tables subject to marginal constraints

Under simple random (multinomial) sampling the problem of estimating cell proportions for a contingency table subject to marginal constraints has been well explored. We briefly review methods that have been considered; then we develop a general method, for more complicated sampling, which reflects the variance structure of the estimated cell proportions. For stratified and cluster sampling we compare our method against earlier methods for the 2×2 table and find it potentially advantageous.     

Positive association in a two-way contingency table:likelihood ratio tests

Given a two-way contingency table in which the rows and columns both define ordinal variables, there are many ways in which the informal idea of positive association between those variables might be defined. This paper considers a variety of definitions expressed as inequality constraints on cross-product ratios. Logical relationships between the definitions are explored. Each definition can serve as a composite alternative against which the null hypothesis of no association may be tested. For a broad class of such alternatives a decomposition of the log-likelihood gives both an explicit likelihood ratio statistic and its asymptotic null hypothesis distribution. Results are derived for multinomial sampling and for fully conditional sampling with row and column totals fixed.     

Maximum likelihood estimation for incomplete multinomial data via the weaver algorithm

In a multinomial model, the sample space is partitioned into a disjoint union of cells. The partition is usually immutable during sampling of the cell counts. In this paper, we extend the multinomial model to the incomplete multinomial model by relaxing the constant partition assumption to allow the cells to be variable and the counts collected from non-disjoint cells to be modeled in an integrated manner for inference on the common underlying probability. The incomplete multinomial likelihood is parameterized by the complete-cell probabilities from the most refined partition. Its sufficient statistics include the variable-cell formation observed as an indicator matrix and all cell counts. With externally imposed structures on the cell formation process, it reduces to special models including the Bradley–Terry model, the Plackett–Luce model, etc. Since the conventional method, which solves for the zeros of the score functions, is unfruitful, we develop a new approach to establishing a simpler set of estimating equations to obtain the maximum likelihood estimate (MLE), which seeks the simultaneous maximization of all multiplicative components of the likelihood by fitting each component into an inequality. As a consequence, our estimation amounts to solving a system of the equality attainment conditions to the inequalities. The resultant MLE equations are simple and immediately invite a fixed-point iteration algorithm for solution, which is referred to as the weaver algorithm. The weaver algorithm is short and amenable to parallel implementation. We also derive the asymptotic covariance of the MLE, verify main results with simulations, and compare the weaver algorithm with an MM/EM algorithm based on fitting a Plackett–Luce model to a benchmark data set.     

The limiting distribution of a test for multivariate structure

We define a chi-squared statistic for p-dimensional data as follows. First, we transform the data to remove the correlations between the p variables. Then, we discretize each variable into groups of equal size and compute the cell counts in the resulting p-way contingency table. Our statistic is just the usual chi-squared statistic for testing independence in a contingency table. Because the cells have been chosen in a data-dependent manner, this statistic does not have the usual limiting distribution. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic when the data is sampled from a multivariate normal distribution. The chi-squared statistic is useful in detecting hidden structure in raw data or residuals. It can also be used as a test for multivariate normality.     

Modeling data with a truncated and inflated Poisson distribution

Zero inflated Poisson regression is a model commonly used to analyze data with excessive zeros. Although many models have been developed to fit zero-inflated data, most of them strongly depend on the special features of the individual data. For example, there is a need for new models when dealing with truncated and inflated data. In this paper, we propose a new model that is sufficiently flexible to model inflation and truncation simultaneously, and which is a mixture of a multinomial logistic and a truncated Poisson regression, in which the multinomial logistic component models the occurrence of excessive counts. The truncated Poisson regression models the counts that are assumed to follow a truncated Poisson distribution. The performance of our proposed model is evaluated through simulation studies, and our model is found to have the smallest mean absolute error and best model fit. In the empirical example, the data are truncated with inflated values of zero and fourteen, and the results show that our model has a better fit than the other competing models.     

A Bootstrap Approach for Testing Marginal Independence Between Two Categorical Variables When Subjects Have Repeated Responses

To assess independence in two-way contingency tables, the Pearson chi-square test or Fisher’s exact test are typically used. These tests assume that each subject contributes at most one count to only one table cell (e.g., sex versus blood type). In other situations, each subject may have more than one count contributing to the table and these counts may occur in different cells of the table. One may wish to test independence, adjusting for the within-subject correlation. We provide a simple nonparametric bootstrap approach and assess its performance through simulation studies. The method is illustrated on subjects with multiple mental health presentations to Emergency Departments.     

Random effects regression models for count data with excess zeros in caries research

We extend the family of Poisson and negative binomial models to derive the joint distribution of clustered count outcomes with extra zeros. Two random effects models are formulated. The first model assumes a shared random effects term between the conditional probability of perfect zeros and the conditional mean of the imperfect state. The second formulation relaxes the shared random effects assumption by relating the conditional probability of perfect zeros and the conditional mean of the imperfect state to two different but correlated random effects variables. Under the conditional independence and the missing data at random assumption, a direct optimization of the marginal likelihood and an EM algorithm are proposed to fit the proposed models. Our proposed models are fitted to dental caries counts of children under the age of six in the city of Detroit.     

Some remarks on the chi-squared test with both margins fixed

Under the hypothesis of independence, the chi-squared test statistic for independence in a two-way contingency table follows an asymptotic chi-squared distribution under both a multinomial and a product-multinomial models. Alalouf(1987) showed the same result holds for the third case where both margins are fixed. In this paper an intuitively easier way of proof using the conditional limit theorems is suggested and some points are discussed.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

Exact inference in contingency tables via stochastic approximation Monte Carlo

Monte Carlo methods for the exact inference have received much attention recently in complete or incomplete contingency table analysis. However, conventional Markov chain Monte Carlo, such as the Metropolis–Hastings algorithm, and importance sampling methods sometimes generate the poor performance by failing to produce valid tables. In this paper, we apply an adaptive Monte Carlo algorithm, the stochastic approximation Monte Carlo algorithm (SAMC; Liang, Liu, & Carroll, 2007), to the exact test of the goodness-of-fit of the model in complete or incomplete contingency tables containing some structural zero cells. The numerical results are in favor of our method in terms of quality of estimates.     

Exact conditional tests for incomplete contingency tables: estimating attained significance levels

A Monte Carlo exact conditional test of quasi-independence in two-way incomplete contingency tables is proposed. The null distribution of a random table under quasiindependence is derived. This distribution depends only on the counts in the cells of interest and not on the counts in the remaining cells. This result is used to improve the efficiency of a proposed simulate-and-reject Monte Carlo procedure for estimating the attained significance level.     

Log-linear model analysis of the semi-symmetric intraclass contingency table

This paper addresses the problem of analyzing a three-way contingency table that is upper-triangular, and a priori symmetric within layers. The log-linear model is modified to handle this kind of table, and maximum likelihood estimation is carried out for the modified log-linear model. This leads to an expression of the maximum likelihood estimates exclusively in terms of the observed cell counts. It is skin this analysis is equivalent to an application of the gone log-linear model to an artificially complete table, obtain. by splitting the off-diagonal cells in half within layers. This analysis is used in analyzing the results of a study done to determine the effect of the sex-linked dwarfing gene in male chickens on resistance to E. coli infection; the conclusion differs from that of a previous analysis of the same data (see Norwood and Hinkelmann 1978). It is found, in fact, that the structure of association among the two allele variables and the disease variable is somewhat more complex than previously proposed. A second example is taken from Ishii (1960). Finally, collapsibility conditions for the modified log-linear model, as well as various other sampling plans and limitations to the testing procedure, are discussed.     

Accuracy of Power-Divergence Statistics for Testing Independence and Homogeneity in Two-Way Contingency Tables

The small-sample accuracy of seven members of the family of power-divergence statistics for testing independence or homogeneity in contingency tables was studied via simulation. The likelihood ratio statistic G ² and Pearson's X ² statistic are among these seven members, whose behavior was studied at nominal test sizes of.01 and.05 with marginal distributions that could be uniform or skewed and with a set of sample sizes that included sparseness conditions as measured through table density (i.e., the ratio of sample size to number of cells). The likelihood ratio statistic G ² rejected the null hypothesis too often even with large table density, whereas Pearson's X ² was sufficiently accurate and only presented a minor misbehavior when table density was less than two observations/cell. None of the other five statistics outperformed Pearson's X ². A nonasymptotic variant of X ² solved the minor inaccuracies of Pearson's X ² and turned out to be the most accurate statistic for testing independence or homogeneity, even with table densities of one observation/cell. These results clearly advise against the use of the likelihood ratio statistic G ².     

A comparison of tests of homogeneity for sparse contingency tables

Five tests of homogeneity for a 2x(k+l) contingency table are compared using Monte Carlo techniques. For these studiesit is assumed that k becomes large in such a way that thecontingency table is sparse for 2xk of the cells, but the sample size in two of the cells remains large. The test statistics studied are: the chi-square approximation to the Pearson test statistic, the chi-square approximation to the likelihood ratio statistic, the normal approximation to Zelterman's (1984)the normal approximation to Pearson's chi-square, and the normal approximation to the likelihood ratio statistic. For the range of parameters studied the chi-square approximation to Pearson's statistic performs consistently well with regard to its size and power.