Monte Carlo methods for exploring sensitivity to distributional assumptions in a Bayesian analysis of a series of 2 x 2 tables.

This paper develops Monte Carlo methods for a Bayesian analysis of a series of 2 x 2 tables under a variety of distributional assumptions. I assume that the data in each table were generated from a pair of binomial distributions and the logarithm of odds of a favourable response follows a bivariate distribution with means that are linear functions of covariates and an arbitrary covariance matrix. I use Gibbs and importance sampling methods to obtain various characteristics of the posterior distribution of the quantities of interest. I apply the method to analyse the data from a population case-control study. Given the size of the population at risk I also derive the posterior distribution of the risk difference defined as the difference in the probabilities of disease development in the exposed and unexposed groups.

[1]  Linda June Davis,et al.  Generalization of the Mantel―Haenszel estimator to nonconstant odds ratios , 1985 .

[2]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[3]  D. Rubin,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[4]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[5]  D. Spiegelhalter,et al.  Bayes Factors for Linear and Log‐Linear Models with Vague Prior Information , 1982 .

[6]  Cyrus R. Mehta,et al.  Computing an Exact Confidence Interval for the Common Odds Ratio in Several 2×2 Contingency Tables , 1985 .

[7]  G. Kneale.  Problems arising in estimating from retrospective survey data the latent periods of juvenile cancers initiated by obstetric radiography. , 1971, Biometrics.

[8]  R. Kass,et al.  Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models) , 1989 .

[9]  John J. Gart,et al.  Point and interval estimation of the common odds ratio in the combination of 2 × 2 tables with fixed marginals , 1970 .

[10]  P. McCullagh.  On the Elimination of Nuisance Parameters in the Proportional Odds Model , 1984 .

[11]  N. Breslow,et al.  Methods of estimation in log odds ratio regression models. , 1986, Biometrics.

[12]  A. Stewart,et al.  Age-distribution of cancers caused by obstetric x-rays and their relevance to cancer latent periods. , 1970, Lancet.

[13]  N. Mantel.  Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure , 1977 .

[14]  Pooling controls from different studies. , 1991, Statistics in medicine.

[15]  N Breslow,et al.  Regression analysis of the log odds ratio: a method for retrospective studies. , 1976, Biometrics.

[16]  John J. Gart,et al.  THE COMPARISON OF PROPORTIONS: A REVIEW OF SIGNIFICANCE TESTS, CONFIDENCE INTERVALS AND ADJUSTMENTS FOR STRATIFICATION' , 1971 .

[17]  S. Radhakrishna,et al.  Combination of results from several 2 X 2 contingency tables , 1965 .

[18]  W. Haenszel,et al.  Statistical aspects of the analysis of data from retrospective studies of disease. , 1959, Journal of the National Cancer Institute.

[19]  J. Gart,et al.  On the conditional moments of the k-statistics for the Poisson distribution , 1970 .

[20]  C. N. Morris.  Comment: Simulation in Hierarchical Models , 1987 .

[21]  P. Sheehe,et al.  Combination of log relative risk in retrospective studies of disease. , 1966, American journal of public health and the nation's health.

[22]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[23]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[24]  M. Tanner,et al.  Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler , 1992 .

[25]  Marvin Zelen,et al.  The analysis of several 2× 2 contingency tables , 1971 .

[26]  Small sample considerations in combining 2 X 2 tables. , 1967, Biometrics.

[27]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[28]  John J. Gart,et al.  On the Combination of Relative Risks , 1962 .

[29]  S. E. Hills,et al.  Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling , 1990 .

[30]  H. Busch.  METHODS IN CANCER RESEARCH , 1969 .

[31]  J C Wakefield,et al.  Hierarchical models for multicentre binary response studies. , 1990, Statistics in medicine.