Bayesian Analysis of Marginal Log-Linear Graphical Models for Three Way Contingency Tables (original) (raw)

Bayesian Analysis of Graphical Models of Marginal Independence for Three Way Contingency Tables

2012

This paper deals with the Bayesian analysis of graphical models of marginal independence for three way contingency tables. Each marginal independence model corresponds to a particular factorization of the cell probabilities and a conjugate analysis based on Dirichlet prior can be performed. We illustrate a comprehensive Bayesian analysis of such models, involving suitable choices of prior parameters, estimation, model determination, as well as the allied computational issues. The posterior distributions of the marginal log-linear parameters is indirectly obtained using simple Monte Carlo schemes. The methodology is illustrated using two real data sets.

Bayesian Inference and Model Selection for Association Models in Contingency Tables

In this work we present Bayesian hypothesis tests for the independence between two categorical variables in contingency tables. Initially we implement conjugate analysis based on the Multinomial-Dirichlet setup. We compute the Bayes factor and assess the sensitivity of the results to the prior distribution. Then we focus on log-linear models. We compare the saturated and the model of independence with the utilization of Bayes factor compared with several model selection criteria. Finally, we focus on Bayesian inference in association models, a class of models describing the structure of association between two categorical variables. MCMC procedures are facilitated with the help of WinBUGS in order to estimate the posterior distributions of model parameters. Laplace approximation methods are applied so as to approximately compute the marginal likelihood. We illustrate all methods using real datasets.

Objective Bayesian Analysis of Contingency Tables

2002

The statistical analysis of contingency tables is typically carried out with a hypothesis test. In the Bayesian paradigm, default priors for hypothesis tests are typically improper, and cannot be used. Although such priors are available, and proper, for testing contingency tables, we show that for testing independence they can be greatly improved on by so-called intrinsic priors. We also argue that because there is no realistic situation that corresponds to the case of conditioning on both margins of a contingency table, the proper analysis of an a × b contingency table should only condition on either the table total or on only one of the margins. The posterior probabilities from the intrinsic priors provide reasonable answers in these cases. Examples using simulated and real data are given.

Bayesian inference for contingency tables with given marginals

Journal of the Italian Statistical Society, 1995

In this paper we introduce a class of prior distributions for contingency tables with given marginals. We are interested in the structure of concordance/discordance of such tables. There is actually a minor limitation in that the marginals are required to assume only rational values. We do argue, though, that this is not a serious drawback for all applicatory purposes. The posterior and predictive distributions given an Msample are computed. Examples of Bayesian estimates of some classical indices of concordance are also given. Moreover, we show how to use simulation in order to overcome some difficulties which arise in the computation of the posterior distribution.

Bayesian estimation of unrestricted and order-restricted association models for a two-way contingency table

Computational Statistics & Data Analysis, 2007

In two-way contingency tables analysis, a popular class of models for describing the structure of the association between the two categorical variables are the so-called "association" models. Such models assign scores to the classification variables which can be either fixed and prespecified or unknown parameters to be estimated. Under the row-column (RC) association model, both row and column scores are unknown parameters without any restriction concerning their ordinality. It is natural to impose order restrictions on the scores when the classification variables are ordinal. The Bayesian approach for the RC (unrestricted and restricted) model is adopted. MCMC methods are facilitated in order the parameters to be estimated. Furthermore, an alternative parametrization of the association models is proposed. This new parametrization simplifies computation in the MCMC procedure and leads to a natural parameter space for the order constrained model. The proposed methodology is illustrated via a popular dataset.

Analysis of Contingency Tables Bayesian Analysis of Contingency Tables

2000

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we

Bayesian estimation of the orthogonal decomposition of a contingency table

Austrian Journal of Statistics, 2016

In a multinomial sampling, contingency tables can be parametrized by probabilities of each cell. These probabilities constitute the joint probability function of two or more discrete random variables. These probability tables have been previously studied from a compositional point of view. The compositional analysis of probability tables ensures coherence when analysing sub-tables. The main results are:(1) given a probability table, the closest independent probability table is the product of their geometric marginals;(2) the probability table can be orthogonally decomposed into an independent table and an interaction table;(3) the departure of independence can be measured using simplicial deviance, which is the Aitchison square norm of the interaction table.In previous works, the analysis has been performed from a frequentist point of view. This contribution is aimed at providing a Bayesian assessment of the decomposition. The resulting model is a log-linear one, which parameters ar...

Bayesian Analysis of Contingency Tables

Communications in Statistics - Theory and Methods, 2005

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.

Bayesian methods for contingency tables using Gibbs sampling

Statistical Papers, 2004

Cell counts in contingency tables can be smoothed using loglinear models. Recently, sampling-based methods such as Markov chain Monte Carlo (MCMC) have been introduced, making it possible to sample from posterior distributions. The novelty of the approach presented here is that all conditional distributions can be specified directly, so that straightforward Gibbs sampling is possible. Thus, the model is constructed in a way that makes burn-in and checking convergence a relatively minor issue. The emphasis of this paper is on smoothing cell counts in contingency tables, and not so much on estimation of regression parameters. Therefore, the * 34 prior distribution consists of two stages. We rely on a normal nonconjugate prior at the first stage, and a vague prior for hyperparameters at the second stage. The smoothed counts tend to compromise between the observed data and a log-linear model. The methods are demonstrated with a sparse data table taken from a multi-center clinical trial.

Bayesian Testing and Estimation of Association in a Two-Way Contingency Table

Journal of the American Statistical Association, 1997

In a two-way contingency table, one is interested in checking the goodness of t of simple models such as independence, quasi-independence, symmetry, or constant association, and estimating parameters which describe the association structure of the table. In a large table, one may be interested in detecting a few outlying cells which deviate from the main association pattern in the table. Bayesian tests of the above hypotheses are described using a prior de ned on the set of interaction terms of the log-linear model. These tests and associated estimation procedures have several advantages over classical tting/estimation procedures First, the tests above can give measures of evidence in support of simple hypotheses. Second, the Bayes factors can be used to give estimates of association parameters of the table which allow for uncertainty that the hypothesized model is true. These methods are illustrated for a number of tables.