this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the U.S. Space Shuttle.

e wrong, which is even worse. Please let us know of any successes or failures. Beware - Gibbs sampling can be dangerous!. BUGS c flcopyright MRC Biostatistics Unit 1995. ALL RIGHTS RESERVED. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The work was funded in part by ESRC (UK) Award Number H519 25 5023. 1 2 Contents 1 Introduction 5 1.1 What is BUGS? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 For what kind of problems is BUGS best suited? . . . . . . . . . . . . . . . . . . . . . 5 1.3 Markov Chain Monte Carlo (MCMC) techniques . . . . . . . . . . . . . . . . . . . . 5 1.4 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Hardware platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Software . . .

`Publication bias' is a relatively new statistical phenomenon that only arises when one attempts through a meta-analysis to review all studies, significant or insignificant, in order to provide a total perspective on a particular issue. This has recently received some notoriety as an issue in the evaluation of the relative risk of lung cancer associated with passive smoking, following legal challenges to a 1992 EPA analysis which concluded that such exposure is associated with significant excess risk of lung cancer. We introduce a Bayesian approach which estimates and adjusts for publication bias. Estimation is based on a data augmentation principle within a hierarchical model, and the number and outcomes of unobserved studies are simulated using Gibbs sampling methods. This technique yields a quantitative adjustment for the passive smoking meta-analysis. We estimate that there may be both negative and positive but insignificant studies omitted, and that failing to allow for these woul...

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When combining results from separate investigations in a meta-analysis, random effects methods enable the modeling of differences between studies by incorporating a heterogeneity parameter ø 2 that accounts explicitly for across-study variation. We develop a simple form for the variance of Cochran's homogeneity statistic Q, leading to interval estimation of ø 2 utilizing an approximating distribution for Q; this enables us to extend the point estimation of DerSimonian and Laird. We also develop asymptotic likelihood methods and compared them with this method. We then use these approximating distributions to give a new method of calculating the weight given to the individual studies' results when estimating the overall mean which takes into account variation in these point estimates of ø 2 . Two examples illustrate the methods presented, where we show that the new weighting scheme is between the standard fixed and random effects models in down-weighting the results of large studie...

Hierarchically exchangeable data are characterized by the exchangeability of a population of units and the exchangeability of observations from each individual unit. A flexible model for such data is the hierarchical logistic-normal model, which provides unconstrained sampling distributions at the within-unit level and an unconstrained covariance structure at the betweenunit level. Also, the sampling distribution at the between-unit level is unimodal in a weak sense. Parameter estimation and inference for the hierarchical logistic-normal model is relatively straightforward via Markov chain Monte Carlo or an approximate EM algorithm. These and other features of the hierarchical logistic normal model are explored, and the model is applied to the analysis of tumor locations in a mammalian population. A comparison is made to a similar data analysis based on Dirichlet distributions.

This paper addresses the use of weight functions to model publication bias in meta-analysis. Since this bias is hard to gauge, we introduce a non-parametric "-contamination class of weight functions. We then illustrate how to explore sensitivity of conclusions to the specification of the weight function by examining the range of results for the entire class. We find lower bounds on the coverage of confidence intervals. If no publication bias is present, results are robust even when considered over the entire "-contamination class. However, if publication bias is present, then the coverage provided by the usual interval estimator is not robust. In this case, an alternative interval estimator is suggested. We also illustrate how both upper and lower bounds on posterior quantities of interest may be found for the case in which prior information is available. Some key words: Weight functions; Selection bias; Meta-analysis. 1 Introduction This paper addresses the use of weight functions t...

There are currently several classical and Bayesian methods of meta-analysis available for combining epidemiological results. We describe and compare these in a consistent framework, and apply them to published studies of the relative risk of lung cancer associated with exposure to environmental tobacco smoke in the workplace. We find that although all methods give reasonably similar combined estimates of relative risk of lung cancer associated with this exposure (none of which is significantly raised above unity, in either a frequentist or a Bayesian sense), the approximations arising from classical methods appear to be non-conservative and should be used with caution. The Bayesian methods, which account more explicitly for possible inhomogeneity in studies, give slightly lower estimates again of relative risk and wider posterior credible intervals, indicating that inference from the nonBayesian approaches might be optimistic. Keywords: Passive smoking, meta-analysis, environmental tob...

Meta-analysis is well known to be susceptible to publication bias (PB), caused by lack of publication of all works on the area in question. This paper applies a recently developed Bayesian method for assessing PB to a collection of studies of the possible effects of oral contraceptive use on incidence of cervical cancer. We build on a meta-analysis developed in Delgado-Rodriguez et al. [2]. We apply a more formal approach to evaluating and adjusting for PB than the ad hoc funnel plot procedure used in that paper. We conclude that there is support for the existence of publication bias in the main data set analyzed in [2], and we show it is probably caused by the explicit disregard of `low quality' studies in [2]. Overall we conclude that there is rather weak support for a positive association between oral contraceptive use and incidence of cervical cancer. 1 Introduction Meta-analysis is a widely applied technique for statistically combining analyses from individual studies into a sing...

We introduce a Bayesian approach which estimates and adjusts for selection bias in a set of studies used in a meta-analysis. We use a hierarchical model for study outcome, and propose an additional model component to account for publication bias, which is the possibility that studies of interest are not equally likely to be published and hence observed studies are not a random sample. Estimation is based on the data augmentation principle and the number and outcomes of unobserved studies are simulated using Gibbs sampling methods. After examining simulation performance, we apply our techniques to a meta-analysis of 35 studies of the relationship between lung cancer and spousal exposure to environmental tobacco smoke. We find that the 95% posterior probability interval for relative risk is shifted downward after allowing for this. These results are consistent with earlier, ad hoc, approaches to this problem. Keywords and phrases: Meta-analysis, publication bias, missing studies, Markov...