Models For Contingency Tables Research Articles

Markov bases for typical block effect models of two-way contingency tables

Markov basis for statistical model of contingency tables gives a useful tool for performing the conditional test of the model via the Markov chain Monte Carlo method. In this paper, we derive explicit forms of Markov bases for change point models and block diagonal effect models, which are typical block-wise effect models of two-way contingency tables, and perform conditional tests with some real data sets.

Split variable selection for tree modeling on rank data

A variable selection method for constructing decision trees with rank data is proposed. It utilizes conditional independence tests based on loglinear models for contingency tables. Compared with other selection methods, our method is computationally more efficient. Moreover, our method is relatively unbiased and powerful in selecting the correct split variables. Simulation results and a real data study are given to demonstrate the strength of our method.

Conditional symmetry models for three-way contingency tables

We generalize the classical conditional or triangular symmetry model for I×I contingency tables to three-way I×I×I tables with commensurable ordinal classification variables. The construction of the new family of models is such that the desirable property that connects conditional symmetry to complete symmetry and marginal homogeneity models in two-way tables is retained in three-way tables. Furthermore, connections between our proposed models obey a coherent structure. We provide maximum likelihood estimation for the new models which is illustrated with a real data example.

On partitioning conditional independence model for three-way contingency tables

For three-way contingency tables with two ordinal and one nominal variables, this paper shows that (i) the conditional independence model holds if and only if conditional covariance between ordinal variables at each level of the nominal variable equals zero and the heterogeneous linear-by-linear association model holds, (ii) the conditional independence model holds if and only if an average conditional covariance between ordinal variables across levels of the nominal variable equals zero and the homogeneous linear-by-linear association model holds, and (iii) the likelihood ratio test statistic for the conditional independence model is asymptotically equivalent to the sum of the likelihood ratio test statistic for the decomposed models.

Relational models for contingency tables

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions for the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models.

Hierarchical subspace models for contingency tables

For the statistical analysis of multiway contingency tables, we propose modeling interaction terms in each maximal compact component of a hierarchical model. By this approach we can search for parsimonious models with smaller degrees of freedom than the usual hierarchical model, while preserving the localization property of the inference in the hierarchical model. This approach also enables us to evaluate the localization property of a given log-affine model. We discuss estimation and exact tests of the proposed model and illustrate the advantage of the proposed modeling with some data sets.

Factors Influencing the Mixture Index of Model Fit in Contingency Tables Showing Independence

This study: (a) evaluated several competing techniques for dealing with sampling zeros for the two-point mixture model index, * π , in contingency tables when the independence assumption holds; and (b) investigated the performance of the estimate, * π , in various combinations of conditions, as a function of different sizes of tables, different marginal distributions and different sample sizes. Furthermore, the standard error of π ∗ estimated by use of a method proposed by Rudas, Clogg and Lindsay (1994) and the “true” standard error based on empirical simulations in various scenarios especially when encountering small sample sizes and π ∗ close to 0 were compared. These goals were achieved by Monte Carlo methods that simulate a variety of scenarios and then were applied to two real data examples. BACKGROUND AND DEVELOPMENT Traditional methods for evaluating contingency table models based on chi square statistics or quantities derived from them are not appealing in many applied research settings. According to Rudas (1998), “First, when the model is not true, a comparison of the data to what could only be expected if it were is of very little meaning; second, the actual distribution of the statistic may be very different from the reference distribution if some of the underlying assumptions are violated.” In addition, conventional methods are sensitive to sample size; often a model is rejected when fitted to a large data set even though the model may represent a reasonable summary of the data for practical purposes. In sharp contrast to chi-squared tests of fit methods, which rely heavily on size of the table, sample size and actual true probabilities, the mixture index of fit proposed by Rudas,

Non-HLA-DRB1 RA-associated risk alleles associate with anti-CCP and specific ACPA levels

Background and objectivesAnti-citrullinated protein antibodies (ACPA) with different fine specificities are exclusively found in sera and synovial fluid of rheumatoid arthritis (RA) patients. The presence and the levels of all...

Probability matrices, non-negative rank, and parameterization of mixture models

In this paper, we parameterize non-negative matrices of sum one and rank at most two using the least possible number of parameters. We also show how this parameterization relates to a class of statistical models, known in Probability and Statistics as mixture models for contingency tables. In particular, we show how to use this parameterization to make some optimization problems computationally easier.

Minimal and minimal invariant Markov bases of decomposable models for contingency tables

We study Markov bases of decomposable graphical models consisting of primitive moves (i.e., square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models.

Latent Block Model for Contingency Table

Although many clustering procedures aim to construct an optimal partition of objects or, sometimes, variables, there are other methods, called block clustering methods, which simultaneously consider the two sets and organize the data into homogeneous blocks. This kind of method has practical importance in a wide variety of applications such as text and market basket data analysis. Typically, the data that arise in these applications are arranged as a two-way contingency table. Using Poisson distributions, a latent block model for these data is proposed and, setting it under the maximum likelihood approach and the classification maximum likelihood approach, various algorithms are provided. Their performances are evaluated and compared to a simple use of EM or CEM applied separately on the rows and columns of the contingency table.

Markov bases of binary graph models of [formula omitted]-minor free graphs

The Markov width of a graph is a graph invariant defined as the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We show that a graph has Markov width at most four if and only if it contains no K 4 as a minor, answering a question of Develin and Sullivant. We also present a lower bound of order Ω ( n 2 − ε ) on the Markov width of K n .

The association of snus and smoking behaviour: a cohort analysis of Swedish males in the 1990s

The European Union has banned sales of moist snuff (snus) in all member states, with the exception of Sweden. The ban is motivated by the potential adverse health effects of snus, but snus may also help people to avoid smoking or stop tobacco use. The purpose of this study is to investigate the association between snus and smoking behaviour. The Swedish Survey of Living Conditions (ULF) health interview panel running from 1988/9 to 1996/7 was used to examine the gross and net flows between smoking and snus among Swedish males. Females were excluded from the analysis due to low snus prevalence. Contingency table models were used to investigate several hypotheses about the relationships between snus and smoking behaviour. We found clear associations between the two habits. For the younger cohort (age 16-44 years), snus use contributed to approximately six smoking quitters per smoking starter attributable to snus. For the older cohort (age 45-84) there were slightly more than two quitters per starter. In terms of odds ratios, in the younger group smoking cessation attributable to snus was twice as common as smoking initiation, but in the older group the odds of starting smoking attributable to snus was 2.5 times higher than for quitting. Snus contributed to the reduction of smoking among Swedish males in the 1990s. Snus had different effects among non-smokers and smokers in different age groups.

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 classifications, this article proposes a new model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate t-distribution with equal marginal variances having any degrees of freedom. The proposed model indicates that the difference between the symmetric two probabilities raised to the power is proportional to the distance from the main diagonal of the table. The model is an extension of the symmetry model. An example is given. The simulation studies based on bivariate t-distribution are given.

A class of statistical models to weaken independence in two-way contingency tables

In this paper we study a new class of statistical models for contingency tables. We define this class of models through a subset of the binomial equations of the classical independence model. We prove that they are log-linear and we use some notions from Algebraic Statistics to compute their sufficient statistic and their parametric representation. Moreover, we show how to compute maximum likelihood estimates and to perform exact inference through the Diaconis-Sturmfels algorithm. Examples show that these models can be useful in a wide range of applications.

Categorical time series models for contingency tables

Models for time-dependent contingency tables are presented. Multinomial-logit, conditional exponential family, Markov chain and multinomial-Dirichlet models are discussed for bivariate binary time series. The models are applied to two real data sets.

A Markov basis for conditional test of common diagonal effect in quasi-independence model for square contingency tables

In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger (or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by the Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.

Iterative proportional scaling via decomposable submodels for contingency tables

We propose iterative proportional scaling (IPS) via decomposable submodels for maximizing the likelihood function of a hierarchical model for contingency tables. In ordinary IPS the proportional scaling is performed by cycling through the members of the generating class of a hierarchical model. We propose the adjustment of more marginals at each step. This is accomplished by expressing the generating class as a union of decomposable submodels and cycling through the decomposable models. We prove the convergence of our proposed procedure, if the amount of scaling is adjusted properly at each step. We also analyze the proposed algorithms around the maximum likelihood estimate (MLE) in detail. The faster convergence of our proposed procedure is illustrated by numerical examples.

An Introduction to Categorical Data Analysis (second edition) Alan Agresti (2007) ISBN: 9780471226185; 372 pages; £50.30, $94.9, €66.40 Wiley; http://www.wiley.com/

This second edition is pitched at a level suitable for those taking an introductory course in categorical data analysis. The presentation has a low technical level and does not require familiarity with advanced mathematics. However, the author comments that as well as students it is also written for applied statisticians and practising scientists involved in data analysis. The high quality of the book can be judged from the fact that all of these are likely to find the book irreplaceable as a standard text on the subject. The author expresses a preference for logistic regression over log linear models because most applications have a single binomial or multinomial response variable. Therefore, this new edition places greater emphasis on logistic regression and less on log linear models compared with the first edition. From a pharmaceutical perspective this looks to be a good change in emphasis. The other main change from the first edition is the addition of two chapters. The new Chapter 9 covers the analysis of clustered correlated categorical data introducing marginal models and generalized estimating equations. Chapter 10 focuses on random effects and generalized linear mixed models. The content is arranged in 11 chapters as follows: Introduction; Contingency Tables; Generalized Linear Models; Logistic Regression; Building and Applying Logistic Regression Models; Multicategory Logit Models; Loglinear Models for Contingency Tables; Models for Matched Pairs; Modelling Correlated, Clustered Responses; Random Effects: Generalized Linear Mixed Models; A Historical Tour of Categorical Data Analysis. Each chapter ends with a problem section that will enable the reader to work on further examples and idiosyncratically some of the odd-numbered examples have solutions at the back of the book. An Appendix giving the SAS code for some of the examples is also included as well as a link to other software packages if SAS is not your software of choice. There are so many excellent features of this book that it is hard to know where to start. First of all the range of techniques covered is impressive. The practising statistician or scientist is likely to find all he or she needs to know day to day about categorical data analysis in this one book. Next the clarity of the exposition is superb; particularly so because the book avoids heavy mathematical development in favour of well-written English. The slow transition from basic concepts in the early chapters to the later more complex methods is handled very well and it encourages the reader to explore less familiar analysis techniques. The author also takes care to point out when methods may not be optimum due to small sample sizes or sparse cells. It is good to see the consistent focus on estimation, interpretation and understanding alongside statistical inference. Some useful guidance is given on the use of statistical software to implement the methods. The range of examples is very broad, covering data sets such as collisions involving British trains, width of female crab shells, AZT and Aids, teratology in rats and many, many more. An index of examples is provided, which covers more than three pages – showing how strongly the text is based on real data sets. All are well chosen and interesting and do an excellent job illustrating the range of analysis techniques discussed. This book has to be an essential reference for any practising pharmaceutical statistician. Is it needed alongside Alan Agresti's other superb book Categorical Data Analysis (Second Edition 2002)? I think so. The latter is more complete in terms of technical development and covers a broader range of topics. However, for me An Introduction to Categorical Data Analysis is the first volume to refer to when guidance is needed. It will also be a useful source of ideas to explain the essence of categorical data analyses to non-statisticians. Non-statisticians may well find the whole book too challenging but they will learn much from the early chapters, which are easy to follow and cover many of the important ideas in categorical data analysis.

Clustering of contingency table and mixture model

Basing cluster analysis on mixture models has become a classical and powerful approach. It enables some classical criteria such as the well-known k-means criterion to be explained. To classify the rows or the columns of a contingency table, an adapted version of k-means known as Mndki2, which uses the chi-square distance, can be used. Unfortunately, this simple, effective method which can be used jointly with correspondence analysis based on the same representation of the data, cannot be associated with a mixture model in the same way as the classical k-means algorithm. In this paper we show that the Mndki2 algorithm can be viewed as an approximation of a classifying version of the EM algorithm for a mixture of multinomial distributions. A comparison of the algorithms belonging in this context are experimentally investigated using Monte Carlo simulations.