Innovations in bayes and empirical bayes methods: estimating parameters, populations and ranks. 1999

T A Louis, and W Shen
Division of Biostatistics, The University of Minnesota, School of Public Health, 420 Delaware Street, Box 303, Minneapolis, MN 55455, USA. tom@biostat.umn.edu

By formalizing the relation among components and 'borrowing information' among them, Bayes and empirical Bayes methods can produce more valid, efficient and informative statistical evaluations than those based on traditional methods. In addition, Bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty. Computing innovations enable implementation of complex and relevant models, thereby substantially increasing the role of Bayes/empirical Bayes methods in important statistical assessments. Policy-relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics, geographic regions or research studies. Typical assessments include inference for individual parameters, synthesis over the collection of components (for example, the parameter histogram) and comparisons among parameters (for example, ranks). The relative importance of these goals depends on the context. Bayesian structuring provides a guide to valid inference. For example, while posterior means are the 'obvious' and optimal estimates for individual components under squared error loss, their empirical distribution function (EDF) is underdispersed and never valid for estimating the EDF of the true, underlying parameters. Effective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate. Similarly, ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate. Effective estimates should be based on a loss function that caters directly to ranks. Using examples of 'borrowing information', shrinkage and the variance/bias trade-off we motivate Bayes and empirical Bayes analysis. Then, we outline the formal approach and discuss 'triple-goal' estimates with values that when ranked produce optimal ranks, for which the EDF is an optimal estimate of the parameter EDF and such that the values themselves are effective estimates of co-ordinate-specific parameters. We use basic models and data analysis examples to highlight the conceptual and structural issues.

UI MeSH Term Description Entries
D002273 Carcinogens Substances that increase the risk of NEOPLASMS in humans or animals. Both genotoxic chemicals, which affect DNA directly, and nongenotoxic chemicals, which induce neoplasms by other mechanism, are included. Carcinogen,Oncogen,Oncogens,Tumor Initiator,Tumor Initiators,Tumor Promoter,Tumor Promoters,Initiator, Tumor,Initiators, Tumor,Promoter, Tumor,Promoters, Tumor
D003627 Data Interpretation, Statistical Application of statistical procedures to analyze specific observed or assumed facts from a particular study. Data Analysis, Statistical,Data Interpretations, Statistical,Interpretation, Statistical Data,Statistical Data Analysis,Statistical Data Interpretation,Analyses, Statistical Data,Analysis, Statistical Data,Data Analyses, Statistical,Interpretations, Statistical Data,Statistical Data Analyses,Statistical Data Interpretations
D006166 Guatemala A country in CENTRAL AMERICA bordering the North Pacific Ocean, between EL SALVADOR and MEXICO, and bordering the Gulf of Honduras (Caribbean Sea) between HONDURAS and BELIZE.
D006721 Honduras A country in CENTRAL AMERICA, bordering the Caribbean Sea, between GUATEMALA and NICARAGUA and bordering the Gulf of Fonseca (North Pacific Ocean), between EL SALVADOR and NICARAGUA.
D006801 Humans Members of the species Homo sapiens. Homo sapiens,Man (Taxonomy),Human,Man, Modern,Modern Man
D000818 Animals Unicellular or multicellular, heterotrophic organisms, that have sensation and the power of voluntary movement. Under the older five kingdom paradigm, Animalia was one of the kingdoms. Under the modern three domain model, Animalia represents one of the many groups in the domain EUKARYOTA. Animal,Metazoa,Animalia
D001026 Coronary Artery Bypass Surgical therapy of ischemic coronary artery disease achieved by grafting a section of saphenous vein, internal mammary artery, or other substitute between the aorta and the obstructed coronary artery distal to the obstructive lesion. Aortocoronary Bypass,Bypass, Coronary Artery,Bypass Surgery, Coronary Artery,Coronary Artery Bypass Grafting,Coronary Artery Bypass Surgery,Aortocoronary Bypasses,Artery Bypass, Coronary,Artery Bypasses, Coronary,Bypass, Aortocoronary,Bypasses, Aortocoronary,Bypasses, Coronary Artery,Coronary Artery Bypasses
D001499 Bayes Theorem A theorem in probability theory named for Thomas Bayes (1702-1761). In epidemiology, it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the likelihood of that characteristic in healthy and diseased individuals. The most familiar application is in clinical decision analysis where it is used for estimating the probability of a particular diagnosis given the appearance of some symptoms or test result. Bayesian Analysis,Bayesian Estimation,Bayesian Forecast,Bayesian Method,Bayesian Prediction,Analysis, Bayesian,Bayesian Approach,Approach, Bayesian,Approachs, Bayesian,Bayesian Approachs,Estimation, Bayesian,Forecast, Bayesian,Method, Bayesian,Prediction, Bayesian,Theorem, Bayes
D001681 Biological Assay A method of measuring the effects of a biologically active substance using an intermediate in vivo or in vitro tissue or cell model under controlled conditions. It includes virulence studies in animal fetuses in utero, mouse convulsion bioassay of insulin, quantitation of tumor-initiator systems in mouse skin, calculation of potentiating effects of a hormonal factor in an isolated strip of contracting stomach muscle, etc. Bioassay,Assay, Biological,Assays, Biological,Biologic Assay,Biologic Assays,Assay, Biologic,Assays, Biologic,Bioassays,Biological Assays
D012377 Rodentia A mammalian order which consists of 29 families and many genera. Beavers,Capybaras,Castor Beaver,Dipodidae,Hydrochaeris,Jerboas,Rodents,Beaver,Capybara,Hydrochaeri,Jerboa,Rodent,Rodentias

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