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Bayesian Decision Theory in Biostatistics: the Utility of Utility

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Bayesian Decision Theory in Biostatistics: the Utility of Utility
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  Bayesian Decision Theoryin Biostatistics:the Utility of Utility David Draper (joint work with Dimitris Fouskakis and Ioannis Ntzoufras ) Department of Applied Mathematics and StatisticsUniversity of California, Santa Cruz, USA draper@ams.ucsc.eduwww.ams.ucsc.edu/ ∼ draper Bayesian Biostatistics ConferenceMD Anderson Cancer Center, Houston TX  28 Jan 2009 Bayesian decision theory in biostatistics 1  What Biostatisticians Do The practice of  statistics in general (and biostatistics in particular) can beroughly divided into four activities : • Description of  available information (e.g., one or more data sets )relevant to answering a question of interest, without an attempt to generalizeoutward from the available data; • Inference about aspects of the underlying process that gave rise to the data; • Prediction of  future data values under interesting scenarios ; and • Decision-making (choosing an action from among the availablepossibilities , in spite of the current uncertainty about relevantunknowns ), e.g., experimental or sampling design .Description is largely non-probabilistic and relatively uncontroversial . Two probability paradigms are in widespread use today in biostatistics: • Frequentist probability: Restrict attention to phenomena that are inherently repeatable under (essentially) identical conditions ; Bayesian decision theory in biostatistics 2  Use of Frequentist and Bayesian Probability in Biostatistics then, for an event A of interest, P  F  ( A ) is the limiting relative frequency withwhich A occurs in the n (hypothetical) repetitions, as n → ∞ . • Bayesian probability: numerical weight of evidence in favor of anuncertain proposition, obeying a series of  reasonable axioms to ensure thatBayesian probabilities are coherent (internally logically consistent) . Two facts about these paradigms: • With the frequentist approach, inference is much easier than (good) prediction and decision-making . • For several reasons (e.g., computing technology ), the frequentistparadigm dominated work in biostatistics in the 20th century .An unpleasant by-product of these two facts is thatIn biostatistical work it’s a common practice to use frequentistinferential tools , such as hypothesis testing and benefit-onlyvariable selection methods , for decision-theoretic purposes forwhich they may not be optimal . Bayesian decision theory in biostatistics 3  Three Examples In this talk I’ll give three examples of how thinkingdecision-theoretically can lead to better results . • Variable selection in generalized linear models is a familiar task that isusually accomplished in what may be termed a benefit-only manner: we try, using inferential tools , (e.g.) to find a subset of the available predictorsthat maximizes predictive accuracy on future data .This ignores the cost of data collection of the predictors , which may varyconsiderably from one variable to another; Bayesian decision theory withan appropriate utility structure can improve on this. References: — Fouskakis D, Draper D (2008). Comparing stochastic optimization methods forvariable selection in binary outcome prediction, with application to health policy. Journal of the American Statistical Association  , forthcoming.— Fouskakis D, Ntzoufras I, Draper D (2009). Bayesian variable selection usingcost-adjusted BIC, with application to cost-effective measurement of quality of healthcare. Annals of Applied Statistics , forthcoming Bayesian decision theory in biostatistics 4  Three Examples (continued) — Fouskakis D, Ntzoufras I, Draper D (2009). Population-based reversible jumpMCMC for Bayesian variable selection and evaluation under cost constraints. Journal of the Royal Statistical Society, Series C  , forthcoming. • When a clinical trial has been adequately planned ( “appropriately powered” ), as far as sample size is concerned, tobring the notions of  clinical and statistical significance into goodagreement with respect to its primary objectives , it may well still betrue that it is “underpowered” for secondary subgroup analyses .The use of  frequentistmultiple comparisons(inferential) methods in such situations — e.g., to make choices about whether to run newtrials on the promising subgroups — is a bad idea that cannevertheless be seen in the literature (e.g., in a published trial I’m nowreanalyzing, assessing the efficacy of an HIV vaccine ). Bayesian decision theory in biostatistics 5

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