Three nonlinear count models, Poisson R.egression (PR), Negative Binomial Regression (NBR), and Generalized Poisson Regression (GPR) are used for assessing the effects of risk factors on agricultural injuries from farm injury data. A sample of 1,322 respondents who participated in the farm safety/injury baseline survey in nine rural counties in Alabama and Mississippi, aged 18 years and older are considered for analysis. The dispersion parameter estimates and their standard errors for GPR models were consistently smaller than that of NBR models. Estimated dispersion parameters in the NBR and GPR models were positive and significantly different from zero. Estimated goodness-of-fit measures showed that GPR models outperformed the NBR and PR models.

Regression analysis of count data

Preface. List of examples. 1. Introduction. 1.1 What are Bayesian methods? 1.2 What do we mean by 'health--care evaluation'? 1.3 A Bayesian approach to evaluation. 1.4 The aim of this book and the intended audience. 1.5 Structure of the book. 2. Basic Concepts from Traditional Statistical Analysis. 2.1 Probability. 2.1.1 What is probability? 2.1.2 Odds and log--odds. 2.1.3 Bayes theorem for simple events. 2.2 Random variables, parameters and likelihood. 2.2.1 Random variables and their distributions. 2.2.2 Expectation, variance, covariance and correlation. 2.2.3 Parametric distributions and conditional independence. 2.2.4 Likelihoods. 2.3 The normal distribution. 2.4 Normal likelihoods. 2.4.1 Normal approximations for binary data. 2.4.2 Normal likelihoods for survival data. 2.4.3 Normal likelihoods for count responses. 2.4.4 Normal likelihoods for continuous responses. 2.5 Classical inference. 2.6 A catalogue of useful distributionsaeo. 2.6.1 Binomial and Bernoulli. 2.6.2 Poisson. 2.6.3 Beta. 2.6.4 Uniform. 2.6.5 Gamma. 2.6.6 Root--inverse--gamma. 2.6.7 Half--normal. 2.6.8 Log--normal. 2.6.9 Student's t. 2.6.10 Bivariate normal. 2.7 Key points. Exercises. 3. An Overview of the Bayesian Approach. 3.1 Subjectivity and context. 3.2 Bayes theorem for two hypotheses. 3.3 Comparing simple hypotheses: likelihood ratios and Bayes factors. 3.4 Exchangeability and parametric modellingaeo. 3.5 Bayes theorem for general quantities. 3.6 Bayesian analysis with binary data. 3.6.1 Binary data with a discrete prior distribution. 3.6.2 Conjugate analysis for binary data. 3.7 Bayesian analysis with normal distributions. 3.8 Point estimation, interval estimation and interval hypotheses. 3.9 The prior distribution. 3.10 How to use Bayes theorem to interpret trial results. 3.11 The 'credibility' of significant trial resultsaeo. 3.12 Sequential use of Bayes theoremaeo. 3.13 Predictions. 3.13.1 Predictions in the Bayesian framework. 3.13.2 Predictions for binary dataaeo. 3.13.3 Predictions for normal data. 3.14 Decision--making. 3.15 Design. 3.16 Use of historical data. 3.17 Multiplicity, exchangeability and hierarchical models. 3.18 Dealing with nuisance parametersaeo. 3.18.1 Alternative methods for eliminating nuisance parametersaeo. 3.18.2 Profile likelihood in a hierarchical modelaeo. 3.19 Computational issues. 3.19.1 Monte Carlo methods. 3.19.2 Markov chain Monte Carlo methods. 3.19.3 WinBUGS. 3.20 Schools of Bayesians. 3.21 A Bayesian checklist. 3.22 Further reading. 3.23 Key points. Exercises. 4. Comparison of Alternative Approaches to Inference. 4.1 A structure for alternative approaches. 4.2 Conventional statistical methods used in health--care evaluation. 4.3 The likelihood principle, sequential analysis and types of error. 4.3.1 The likelihood principle. 4.3.2 Sequential analysis. 4.3.3 Type I and Type II error. 4.4 P--values and Bayes factorsaeo. 4.4.1 Criticism of P--values. 4.4.2 Bayes factors as an alternative to P--values: simple hypotheses. 4.4.3 Bayes factors as an alternative to P--values: composite hypotheses. 4.4.4 Bayes factors in preference studies. 4.4.5 Lindley's paradox. 4.5 Key points. Exercises. 5. Prior Distributions. 5.1 Introduction. 5.2 Elicitation of opinion: a brief review. 5.2.1 Background to elicitation. 5.2.2 Elicitation techniques. 5.2.3 Elicitation from multiple experts. 5.3 Critique of prior elicitation. 5.4 Summary of external evidenceaeo. 5.5 Default priors. 5.5.1 'Non--informative' or 'reference' priors: 5.5.2 'Sceptical' priors. 5.5.3 'Enthusiastic' priors. 5.5.4 Priors with a point mass at the null hypothesis ('lump--and--smear' priors)aeo. 5.6 Sensitivity analysis and 'robust' priors. 5.7 Hierarchical priors. 5.7.1 The judgement of exchangeability. 5.7.2 The form for the random--effects distribution. 5.7.3 The prior for the standard deviation of the random effectsaeo. 5.8 Empirical criticism of priors. 5.9 Key points. Exercises. 6. Randomised Controlled Trials. 6.1 Introduction. 6.2 Use of a loss function: is a clinical trial for inference or decision? 6.3 Specification of null hypotheses. 6.4 Ethics and randomisation: a brief review. 6.4.1 Is randomisation necessary? 6.4.2 When is it ethical to randomise? 6.5 Sample size of non--sequential trials. 6.5.1 Alternative approaches to sample--size assessment. 6.5.2 'Classical power': hybrid classical--Bayesian methods assuming normality. 6.5.3 'Bayesian power'. 6.5.4 Adjusting formulae for different hypotheses. 6.5.5 Predictive distribution of power and necessary sample size. 6.6 Monitoring of sequential trials. 6.6.1 Introduction. 6.6.2 Monitoring using the posterior distribution. 6.6.3 Monitoring using predictions: 'interim power'. 6.6.4 Monitoring using a formal loss function. 6.6.5 Frequentist properties of sequential Bayesian methods. 6.6.6 Bayesian methods and data monitoring committees. 6.7 The role of 'scepticism' in confirmatory studies. 6.8 Multiplicity in randomised trials. 6.8.1 Subset analysis. 6.8.2 Multi--centre analysis. 6.8.3 Cluster randomization. 6.8.4 Multiple endpoints and treatments. 6.9 Using historical controlsaeo. 6.10 Data--dependent allocation. 6.11 Trial designs other than two parallel groups. 6.12 Other aspects of drug development. 6.13 Further reading. 6.14 Key points. Exercises. 7. Observational Studies. 7.1 Introduction. 7.2 Alternative study designs. 7.3 Explicit modelling of biases. 7.4 Institutional comparisons. 7.5 Key points. Exercises. 8. Evidence Synthesis. 8.1 Introduction. 8.2 'Standard' meta--analysis. 8.2.1 A Bayesian perspective. 8.2.2 Some delicate issues in Bayesian meta--analysis. 8.2.3 The relationship between treatment effect and underlying risk. 8.3 Indirect comparison studies. 8.4 Generalised evidence synthesis. 8.5 Further reading. 8.6 Key points. Exercises. 9. Cost--effectiveness, Policy--Making and Regulation. 9.1 Introduction. 9.2 Contexts. 9.3 'Standard' cost--effectiveness analysis without uncertainty. 9.4 'Two--stage' and integrated approaches to uncertainty in cost--effectiveness modeling. 9.5 Probabilistic analysis of sensitivity to uncertainty about parameters: two--stage approach. 9.6 Cost--effectiveness analyses of a single study: integrated approach. 9.7 Levels of uncertainty in cost--effectiveness models. 9.8 Complex cost--effectiveness models. 9.8.1 Discrete--time, discrete--state Markov models. 9.8.2 Micro--simulation in cost--effectiveness models. 9.8.3 Micro--simulation and probabilistic sensitivity analysis. 9.8.4 Comprehensive decision modeling. 9.9 Simultaneous evidence synthesis and complex cost--effectiveness modeling. 9.9.1 Generalised meta--analysis of evidence. 9.9.2 Comparison of integrated Bayesian and two--stage approach. 9.10 Cost--effectiveness of carrying out research: payback models. 9.10.1 Research planning in the public sector. 9.10.2 Research planning in the pharmaceutical industry. 9.10.3 Value of information. 9.11 Decision theory in cost--effectiveness analysis, regulation and policy. 9.12 Regulation and health policy. 9.12.1 The regulatory context. 9.12.2 Regulation of pharmaceuticals. 9.12.3 Regulation of medical devices. 9.13 Conclusions. 9.14 Key points. Exercises. 10. Conclusions and Implications for Future Research. 10.1 Introduction. 10.2 General advantages and problems of a Bayesian approach. 10.3 Future research and development. Appendix: Websites and Software. A.1 The site for this book. A.2 Bayesian methods in health--care evaluation. A.3 Bayesian software. A.4 General Bayesian sites. References. Index.

Bayesian approaches to clinical trials and health-care evaluation

: Scientific learning is an iterative process employing Criticism and Estimation. Correspondingly the formulated model factors into two complimentary parts - a predictive part allowing model criticism, and a Bayes posterior part allowing estimation. Implications for significance tests, the theory of precise measurement, and for ridge estimates are considered. Predictive checking functions for transformation, serial correlation, bad values, and their relation with Bayesian options are considered. Robustness is seen from a Bayesian viewpoint and examples are given. For the bad value problem a comparison with M estimators is made. (Author)

Sampling and Bayes' inference in scientific modelling and robustness

(1976). Survival Distributions: Reliability Applications in the Biomedical Sciences. Technometrics: Vol. 18, No. 4, pp. 501-501.

Survival Distributions: Reliability Applications in the Biomedical Sciences

The Bayesian approach involves synthesising data and judgement in order to reach conclusions about unknown quantities and make predictions. Bayesian methods have become increasingly popular in recent years, notably in medical research, and although there are a number of books on Bayesian analysis, few cover clinical trials and biostatistical applications in any detail. Bayesian Approaches to Clinical Trials and Health-Care Evaluation provides a valuable overview of this rapidly evolving field, including basic Bayesian ideas, prior distributions, clinical trials, observational studies, evidence synthesis and cost-effectiveness analysis. Covers a broad array of essential topics, building from the basics to more advanced techniques.

Bayesian Approaches to Clinical Trials and Health-Care Evaluation: Spiegelhalter/Clinical Trials and Health-Care Evaluation

In fitting a generalized linear model, many authors have noticed that data sets can show greater residual variability than predicted under the exponential family. Two main approaches have been used to model this overdispersion. The first approach uses a sampling density which is a conjugate mixture of exponential family distributions. The second uses a quasilikelihood which adds a new scale parameter to the exponential likelihood. The approaches are compared by means of a Bayesian analysis using noninformative priors. In examples, it is indicated that the posterior analysis can be significantly different using the two approaches.



Plusieurs auteurs ont remarque que lors de ľajustement ďun modele lineaire generalise, les donnees peuvent presenter plus de variabilite residuelle que celle predite sous un modele exponentiel. Deux approches sont utilisees pour representer cette dispersion excedentaire. La premiere utilise une densite echantillonnale qui est un melange conjuge de densites de la famille exponentielle. La deuxieme approche utilise une fonction de quasi-vraisemblance qui ajoute un nouveau parametre de dispersion a la fonction de vraisemblance exponentielle. Les deux approches sont comparees en faisant une analyse bayesienne avec des lois a priori non informatives. Des exemples illustrent que les deux approches peuvent mener a des analyses a posteriori differentes.

http://www.jstor.org/stable/pdfplus/3315529.pdf

A Bayesian approach to some overdispersion models

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses

Bayesian testing of an exponential point null hypothesis

Consider the situation in which there are several different therapeutic agents. It is desired to select the best agent and to examine its efficacy relative to the control. Too often clinical trials terminate with negative outcomes in part due to inadequate phase II studies. A two-stage phase II based on a Bayesian approach is considered in order to reduce such likelihood. The first stage consists of selecting the best agent and the second stage consists of examining the relative efficacy of the selected agent compared to the control. A formal phase III clinical trial can be initiated when the particular agent is shown to be promising on the basis of the proposed phase II study. The Bayesian approach employed uses an ad hoc likelihood due to the fact that the exact likelihood is complex and intractable. In this sense the proposed approach is thus an approximation. A simulation study is conducted to investigate the performance of the proposed Bayesian approach and compared to two fixed-sample-size approaches. Due to the fact that the procedure is approximate, the simulation study is essential to assess the usefulness of the procedure. The study suggests that the Bayesian approach is an attractive alternative to fixed-sample-size approaches.

Bayesian approach to two-stage phase II trial.

The problem of simultaneously estimating p normal variances is investigated when the parameters are believed a priori to be similar in size. A hierarchical Bayes approach is employed and the resulting estimator is compared to common estimators used including one proposed by Box and Tiao (1973) using a Bayesian approach with a noninformative prior. The technique is then applied to estimate components of variance in the one way layout random effect model of the analysis of variance.

Hierarchical bayes estimation of normal variances with application to a random effect model

Consider the problem of simultaneously estimating p Gamma means when the means are believed a priori to be exchangeable. A Bayesian hierarchical model is employed to model this exchangeability assumption. Several approximation methods are discussed to compute the hierarchical Bayes estimate of the Gamma means. The accuracy of the different approaches is investigated.

Patricia A. Pepple

Papers

A Bayesian approach to some overdispersion models

Bayesian testing of an exponential point null hypothesis

Bayesian approach to two-stage phase II trial.

Hierarchical bayes estimation of normal variances with application to a random effect model

Approximation methods for estimating gamma means using a hierarchical exchangeable model