Models For Contingency Tables Research Articles

Two-Dimensional Index of Departure from the Symmetry Model for Square Contingency Tables with Nominal Categories

In the analysis of two-way contingency tables, the degree of departure from independence is measured using measures of association between row and column variables (e.g., Yule’s coefficients of association and of colligation, Cramér’s coefficient, and Goodman and Kruskal’s coefficient). On the other hand, in the analysis of square contingency tables with the same row and column classifications, we are interested in measuring the degree of departure from symmetry rather than independence. Over past years, many studies have proposed various types of indexes based on their power divergence (or diversity index) to represent the degree of departure from symmetry. This study proposes a two-dimensional index to measure the degree of departure from symmetry in terms of the log odds of each symmetric cell with respect to the main diagonal of the table. By measuring the degree of departure from symmetry in terms of the log odds of each symmetric cell, the analysis results are easier to interpret than existing indexes. Numerical experiments show the utility of the proposed two-dimensional index. We show the usefulness of the proposed two-dimensional index by using real data.

Polynomial Columns-Parameter Symmetry Model and its Decomposition for Square Contingency Tables

This study proposes a polynomial columns-parameter symmetry model for square contingency tables with the same row and column ordinal classifications. In the proposed model, the odds for all i < j that an observation will fall in row category i and column category j instead of row category j and column category i depend on only the value of column category j . The proposed model is original because many asymmetry models in square contingency tables depend on the both values of row and column category. The proposed model constantly holds when the columns-parameter symmetry model holds; but the converse does not necessarily hold. This study shows that it is necessary to satisfy the polynomial columns-marginal symmetry model, in addition to the columns-parameter symmetry model, to satisfy the proposed model. This decomposition theorem is useful for explaining why the proposed model does not hold. Moreover, this study shows the value of likelihood ratio chi-square statistic for testing the proposed model is equal to the sum of that for testing the decomposed two models.

A Bivariate Index Vector to Measure Departure from Quasi-symmetry for Ordinal Square Contingency Tables

This study proposes a bivariate index vector to concurrently analyze both the degree and direction of departure from the quasi-symmetry (QS) model for ordinal square contingency tables. The QS model and extended QS (EQS) models identify the symmetry and asymmetry between the probabilities of normal circulation and reverse circulation when the order exists for arbitrary three categories. The asymmetry parameter of the EQS model implies the degree of departure from the QS model; the EQS model is equivalent to the QS model when the asymmetry parameter equals to one. The structure of the EQS model differs depending on whether the asymmetry parameter approaches zero or infinity. Thus, the asymmetry parameter of the EQS model also implies the direction of departure from the QS model. The proposed bivariate index vector is constructed by combining existing and original sub-indexes that represent the degree of departure from the QS model and its direction. These sub-indexes are expressed as functions of the asymmetry parameter under the EQS model. We construct an estimator of the proposed bivariate index vector and an approximate confidence region for the proposed bivariate index vector. Using real data, we show that the proposed bivariate index vector is important to compare degrees of departure from the QS model for plural data sets.

A study on Freeman-Tukey test statistic under the symmetry model for square contingency tables

The symmetry model is the basic model in the analysis of square contingency tables. Multiple test statistics have been developed for the goodness of fit test. Freeman-Tukey test statistics is appropriate to be used in large samples. However, the required sample size to use the Freeman-Tukey test statistics is not clear. In this paper, the asymptotic properties of Freeman-Tukey test statistic are discussed via extensive Monte-Carlo simulation study. The Freeman-Tukey test statistic is compared with members of power-divergence family test statistic under the symmetry model. The results of simulation study are evaluated based on the Type-I error and power of a test. The results of simulation study and artificial data study show that Freeman-Tukey’s T^2 test statistic does not converge to chi-squared distribution for both sparse and non-sparse square contingency tables.

Exploring temporal interactions of crash counts in California using distinct log-linear contingency table models

Temporal trait of crashes has huge impact on road crash occurrence and a large proportion of research have considered different time periods to determine the causes and features of crash occurrence or frequency. Compared with other safety studies based on a single time interval, considerably less research has relied on the use of multiple time units, especially for the time intervals of less than one year. The research aims to fill the gap by investigating the temporal distribution of crash counts using multiple time spans including hour, weekday and month. To illustrate the most accurate results possible, both the Chi-square test and Cochran–Mantel–Haenzel tests were employed to explore the independence of various time units based on two-way and three-way contingency tables. Eight contingency table models were developed which can be classified into four groups including Complete Independence, Joint Independence, Conditional Independence and Homogeneous Association. Finally, a set of evaluation criteria were utilized for evaluation of the model performance. The results revealed the significant association existence in all time variables (hour, weekday, month) and the model with both main and all interactive effects of time variables provides best prediction performance. Also, the findings showed that Hour 18, weekdays 1, 6, 7 (Friday and Weekends), and month 8 (August) have the largest number of crash occurrences. It is suggested that both main and interactive effects of time variables should be included for model development, which otherwise might yield misleading information. It is anticipated that research results will benefit the safety professionals with better understanding of the temporal patterns of crashes with different time periods and allow the safety administrators to allocate the safety resources.

Asymmetry models based on ordered score and separations of symmetry model for square contingency tables

Summary This study proposes two original asymmetry models based on ordered scores for square contingency tables with the same row and column ordinal classifications. The proposed models can be applied to cases in which the scores of all categories are known or unknown. In the proposed models, the log odds for an observation falling in the (i, j)th cell instead of the (j, i)th cell are inversely proportional to the difference of the ordered scores corresponding to categories i and j. The asymmetry parameter of the proposed model can be useful for inferring whether the row variable is stochastically greater than the column variable or vice versa. The proposed models constantly hold when the symmetry model holds, but the converse is not necessarily true. This study also examines what is necessary for a model, in addition to the proposed models, to satisfy the symmetry model, and gives separations of the symmetry model using the proposed and marginal mean equality models. We apply real data to show the utility of the proposed models. The proposed models provide a better fit than that of the existing models.

A new general class of RC association models: Estimation and main properties

The paper introduces a new class of row-column (RC) association models for contingency tables by allowing the user to select both the scale on which interactions are measured as in Kateri and Papaioannou (1994) and the type of logit (local, global, continuation) suitable for the row and column classification variables as in Bartolucci and Forcina (2002). These choices determine the matrix of interactions which is subjected to rank constraints. Examples are provided where the new models fit substantially better than traditional ones; an intuitive explanation for this behavior is outlined and supported by numerical investigations. An extension of the optimality property in Kateri and Papaioannou (1994) is derived, leading to a general representation theorem and reconstruction formulas for the joint probabilities. These results are the key to show that, given marginal logits, the generalized interactions introduced in this paper determine uniquely the bivariate distribution; in addition, for each pair of logit types, we establish which kind of positive association is implied by our extended interactions being non negative. Quick model selection within this wide class can be performed by an efficient algorithm for computing maximum likelihood estimates which allows for additional linear constraints both on marginal logits and interactions.

Extended diagonal uniform association symmetry model for square contingency tables with ordinal categories

This study proposes original symmetry models with a structure wherein the local odds ratios increase exponentially as the row number of the square contingency table increases in each sub-table. The proposed models provide a better fit than the existing models.

Marginal Continuation odds Ratio Model and Decomposition of Marginal Homogeneity Model for Multi-way Contingency Tables

For square contingency tables with ordered categories, the marginal homogeneity model is represented by various expressions, and some extensions of the marginal homogeneity model were proposed. Herein we consider the marginal continuation-ratio to examine a new expression of the marginal homogeneity model. We also propose an extension of the marginal homogeneity model using the ratio of marginal continuation-ratios; namely, the marginal continuation odds ratio. The proposed model can be interpreted in various ways. Additionally, we propose a generalization of it, and decompose the marginal homogeneity model using the generalized model. Furthermore, we extend the models and decompositions into multi-way contingency tables.

New statistics to test log-linear modeling hypothesis with no distributional specifications and clusters with homogeneous correlation

Traditionally, the Dirichlet-multinomial distribution has been recognized as a key model for contingency tables generated by cluster sampling schemes. There are, however, other possible distributions appropriate for these contingency tables. This paper introduces new statistics capable of testing log-linear modeling hypotheses with distributional unspecification, when the individuals of the clusters are possibly homogeneously correlated. An estimator for the intracluster correlation coefficient, valid for different cluster sizes, plays a crucial role in the construction of the goodness-of-fit test-statistics.

Population size estimation with interval censored counts and external information: Prevalence of multiple sclerosis in Rome.

We discuss Bayesian log-linear models for incomplete contingency tables with both missing and interval censored cells, with the aim of obtaining reliable population size estimates. We also discuss use of external information on the censoring probability, which may substantially reduce uncertainty. We show in simulation that information on lower bounds and external information can each improve the mean squared error of population size estimates, even when the external information is not completely accurate. We conclude with an original example on estimation of prevalence of multiple sclerosis in the metropolitan area of Rome, where five out of six lists have interval censored counts. External information comes from mortality rates of multiple sclerosispatients.

Assessment of SNP-SNP interactions by using square contingency table analysis.

The evolution of SNP-SNP interactions has become an interesting field in genetic epidemiology. Most of the studies, aimed to analyze the relationship between genetic factors and disease of interest, are focused on single SNP associations. However, for quantitative traits, influenced by the interplay of environmental and more than one genetic factors, interaction between the multi factors should be taken into consideration. In this study, symmetry models for square contingency tables are applied to the cross-classified SNP-SNP interactions data. Results from a genome-wide association analysis of blood pressure are used as a prior evidence for the interacted SNPs.

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

ABSTRACTThis 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.

Objective Bayesian comparison of order-constrained models in contingency tables

In social and biomedical sciences, testing in contingency tables often involves order restrictions on cell probabilities parameters. We develop objective Bayes methods for order-constrained testing and model comparison when observations arise under product binomial or multinomial sampling. Specifically, we consider tests for monotone order of the parameters against equality of all parameters. Our strategy combines in a unified way both the intrinsic prior methodology and the encompassing prior approach in order to compute Bayes factors and posterior model probabilities. Performance of our method is evaluated on several simulation studies and real datasets.

Asymptotic cumulants of the minimum phi-divergence estimator for categorical data under possible model misspecification

The asymptotic cumulants of the minimum phi-divergence estimators of the parameters in a model for categorical data are obtained up to the fourth order with the higher-order asymptotic variance under possible model misspecification. The corresponding asymptotic cumulants up to the third order for the studentized minimum phi-divergence estimator are also derived. These asymptotic cumulants, when a model is misspecified, depend on the form of the phi-divergence. Numerical illustrations with simulations are given for typical cases of the phi-divergence, where the maximum likelihood estimator does not necessarily give best results. Real data examples are shown using log-linear models for contingency tables.

Lasso Regularization for Selection of Log-linear Models: An Application to Educational Assortative Mating

Log-linear models for contingency tables are a key tool for the study of categorical inequalities in sociology. However, the conventional approach to model selection and specification suffers from at least two limitations: reliance on oftentimes equivocal diagnostics yielded by fit statistics, and the inability to identify patterns of association not covered by model candidates. In this article, we propose an application of Lasso regularization that addresses the aforementioned limitations. We evaluate our method through a Monte Carlo experiment and an empirical study of educational assortative mating in Chile, 1990–2015. Results demonstrate that our approach has the virtue, relative to ad hoc specification searches, of offering a principled statistical criterion to inductively select a model. Importantly, we show that in situations where conventional fit statistics provide conflicting diagnostics, our Lasso-based approach is consistent in its model choice, yielding solutions that are both predictive and parsimonious.

Graphics for Uncertainty

SummaryGraphical methods such as colour shading and animation, which are widely available, can be very effective in communicating uncertainty. In particular, the idea of a ‘density strip’ provides a conceptually simple representation of a distribution and this is explored in a variety of settings, including a comparison of means, regression and models for contingency tables. Animation is also a very useful device for exploring uncertainty and this is explored particularly in the context of flexible models, expressed in curves and surfaces whose structure is of particular interest. Animation can further provide a helpful mechanism for exploring data in several dimensions. This is explored in the simple but very important setting of spatiotemporal data.

Decomposition of diamond model for square contingency tables with ordered categories

For square contingency tables with the same row and column ordinal classifications, this paper shows that the diamond model holds if and only if the weighted covariance for the difference between the row and column classifications and the sum of them equals zero and the uniform association diamond model holds. An example is given.

Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories

For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.

ϕ-Divergence in Contingency Table Analysis.

The -divergence association models for two-way contingency tables is a family of models that includes the association and correlation models as special cases. We present this family of models, discussing its features and demonstrating the role of -divergence in building this family. The most parsimonious member of this family, the model of -scaled uniform local association, is considered in detail. It is implemented and representative examples are commented on.