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Model selection is an important part of any statistical analysis and, indeed, is central to the pursuit of science in general. Many authors have examined the question of model selection from both frequentist and Bayesian perspectives, and many tools for selecting the “best model” have been suggested in the literature. This paper considers the various proposals from a Bayesian decision–theoretic perspective.

This article examines the evidence of vote tampering in a District Justice election in Beaver County, Pennsylvania. An informal exploratory data analysis and a legal history are followed by a formal Bayesian model of the data from the vote count on election night and the recount completed 2 months later. The evidence suggests that persons unknown could have gained access to the boxes containing the paper ballots, and surprising patterns of changes in the counts support the inference that certain boxes were tampered with. Three methods are compared not only with respect to the overall matter of whether tampering occurred, but also with respect to which precincts were likely to have been tampered with, and to what extent. The results are generally consistent across methods. The Bayesian model is validated by using it on the data for a race (for Superior Court) in the same election in which vote tampering is not suspected. The results show that the model gives a predictive distribution of just a few votes un...

This article explores a reparameterization of a 2 × 2 table. This reparameterization turns out to be useful in considering the consequences of amalgamating several 2 × 2 tables. A formal theorem is derived that permits characterization of Yule's association paradox. The same theorem also permits discussion of the likely direction of error incurred when using capture–recapture methods to estimate the size of a population. This has implications for the “wily trout” argument in the census correction controversy.

Jurors who are opposed to the death penalty are excluded from serving on trials in which the defendant may be sentenced to death. The exclusion of such jurors from death-penalty trials raises some concern about the fairness of these juries in deciding the guilt or innocence of a defendant. This article uses data from a survey of behaviors and beliefs of jurors conducted at the conclusion of non–capital offense trials to study this issue. A Bayesian hierarchical model is proposed for the probabilities of voting guilty or not guilty on a particular trial given various death-penalty beliefs. The Gibbs sampler is used to fit the model. The posterior distributions of the log odds of voting guilty for jurors who would be excluded from death-penalty juries versus those who would not be excluded suggest that there may be differences between the decisions of jurors on non–capital offense trials. We believe that they are likely to differ on death-penalty trials as well. The results from fitting the hierarc...

In the usual, countably additive definition of probability, it is not possible to have a distribution giving equal probabilities to every one of the natural numbers. Yet such a distribution would be interesting and potentially useful. This article considers an approach to this problem based on finitely additive probability. We give a necessary and sufficient condition for when specifications of the probabilities of an arbitrary collection of subsets of a space Ω can be extended to define a finitely additive probability on all the subsets of Ω. This is applied to probability statements modeling the uniform distribution on the natural numbers, using relative frequencies and residue classes to make precise notions of uniformity. Tight bounds are given on the possible values of the probability of an arbitrary set under both interpretations. These bounds are applied to several sets of interest.

We consider the optimal design of experiments in which estimation and design are performed by different parties. The parties are assumed to share similar goals, as reflected by a common loss function, but they may have different prior beliefs. After presenting a few motivating examples, we examine the problem of optimal sample size selection under a normal likelihood with constant cost per observation. We also consider the problem of optimal allocation for given overall sample sizes. We present results under both squared-error loss and a logarithmic utility, paying attention to the differences between one- and two-prior optimal designs. An asymmetric discrepancy measure features repeatedly in our development, and we question the extent of its role in optimal two-prior design.

One method for evaluating the sensitivity of a Bayesian analysis is to embed the prior into a class of priors. Then bounds on prior and posterior quantities of interest must be computed. This approach to inference, often called robust Bayesian inference, has received much attention lately. Implementing robust Bayesian methods entails difficult computations, especially if the parameter space is high dimensional. In this article we develop a Monte Carlo approach to computing these bounds and also explore some interesting theoretical properties of certain classes of priors. The methods can be useful in other situations in which bounds on expectations are required.

This article deals with a novel utility function to design experiments in a Bayesian framework. This utility function is a linear combination of the gain in Shannon information and of the total outcome of the experiment, defined as the sum of observed values in the dependent variable of a linear model. Thus the expected posterior utility to be maximized is a combination of the Bayes D-optimality criterion and the posterior expectation of the total output. Earlier studies have shown that Bayesian parallels of the classical D-, A-, and E-optimal designs can be obtained by considering utility or loss functions concerned with efficient estimation of the parameters of interest. A different requirement that might be desirable in applied problems is to combine the accuracy of parameter estimation with the maximization of experimental output. The utility function considered here does this. We look at the implications of using this utility in deriving designs in the context of hierarchical linear models. ...

Federal law prohibits discrimination in employment decisions against persons 40 years old and older. This article uses data from an actual case to illustrate several methods of showing adverse impact, a legal doctrine under which only the effects of the employer's acts are at issue and not the motives with which they were done. The strengths and weaknesses of the Fisher exact test of significance, a Bayesian analysis, and a method of paired observations inspired by the Mann-Whitney-Wilcoxon statistic are assessed. One important conclusion from this analysis is that it is useful to have several different kinds of analysis bearing on the same issue. To the extent that the analyses agree, credibility is added to each.

Several axiom systems for preference among acts lead to the existence of a unique probability and a state-independent utility such that acts are ranked according to their expected utilities. These axioms have been used as a foundation for Bayesian decision theory and the subjective probability calculus. In this paper, we note that the uniqueness of the probability is relative to the choice of what counts as a constant outcome. Although it is sometimes clear what should be considered constant, there are many cases in which there are several possible choices. Each choice can lead to a different “unique” probability and utility. By focusing attention on state-dependent utilities, we determine conditions under which a truly unique probability and utility can be determined from an agent’s expressed preferences among acts. Suppose that an agent’s preference can be represented in terms of a probability P and a utility U. That is, the agent prefers one act to another if and only if the expected utility of the one act is higher than that of the other. There are many other equivalent representations in terms of probabilities Q, which are mutually absolutely continuous with P, and state-dependent utilities V, which differ from U by possibly different positive affine transformations in each state of nature. An example is described in which two different but equivalent state-independent utility representations exist for the same preference structure. What differs between the two representations is which acts count as constants. The acts involve receiving different amounts of one or the other of two currencies and the states are different exchange rates between the currencies. It is easy to see how it would not be possible for constant amounts of both currencies to simultaneously have constant values across the different states. Savage (Foundations of statistics. John Wiley, New York, 1954, sec. 5.5) discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility. Savage attributed the phenomenon to the construction of a “small world”. We show that the small world problem is just another example of two different, but equivalent, representations treating different acts as constants. Finally, we prove a theorem (similar to one of Karni, Decision making under uncertainty. Harvard University Press, Cambridge, 1985) that shows how to elicit a unique state-dependent utility and does not assume that there are prizes with constant value. To do this, we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment.

Projecting populations that have sparse or unreliable data, such as those of many developing countries, presents a challenge to demographers. The assumptions that they make to project data-poor populations frequently fall into the realm of “educated guesses,” and the resulting projections, often regarded as forecasts, are valid only to the extent that the assumptions on which they are based reasonably represent the past or future, as the case may be. These traditional projection techniques do not incorporate a demographer's assessment of uncertainty in the assumptions. Addressing the challenges of forecasting a data-poor population, we project the Iraqi Kurdish population using a Bayesian approach. This approach incorporates a demographer's uncertainty about past and future characteristics of the population in the form of elicited prior distributions.

Partial table of contents: MAJOR ISSUES Ethically Optimizing Clinical Trials (K. Schaffner) Admissibility of Treatment (N. Sedransk) TEST CASE: VERAPAMIL/NITROPRUSSIDE The Mechanics of Conducting a Clinical Trial (E. Heitmiller & T. Blanck) Issues of Statistical Design (N. Sedransk) Operational History and Procedural Feasibility (J. Kadane) Verapamil versus Nitroprusside: Results of the Clinical Trial I (J. Kadane & N. Sedransk) Verapamil versus Nitroprusside: Results of the Clinical Trial II (E. Heitmiller, et al.) OTHER ISSUES Author's Response to Commentaries I and II (D. Kairys) EPILOGUE Epilogue (J. Kadane) Indexes.

We propose a model for determining the appropriate number of peremptory challenges for each party to a jury trial to ensure fairness. We suppose that all jurors not removed for cause may be thought of as in one of three categories: biased against the prosecution/plaintiff, biased against the defense, and fair. We propose that a fair number of peremptory challenges for a side is that number sufficient to challenge all jurors biased against it, with a certain probability. Given estimates for the various parameters in the model, an exact algorithm, and an approximation are proposed and illustrated.

Tierney and Kadane (1986) presented a simple second-order approximation for posterior expectations of positive functions. They used Laplace's method for asymptotic evaluation of integrals, in which the integrand is written as f(θ)exp(-nh(θ)) and the function h is approximated by a quadratic. The form in which they applied Laplace's method, however, was fully exponential: The integrand was written instead as exp[− nh(θ) + log f(θ)]; this allowed first-order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second-order expansion for the ratio. Other second-order expansions (Hartigan 1965; Johnson 1970; Lindley 1961, 1980; Mosteller and Wallace 1964) require computation of more derivatives of the log-likelihood function. In this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions. To obtain a second-order approximation to an expectation E(g(θ)), we use the fully exponential method to approximate...

This article describes approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary (i.e., not necessarily positive) parameters. These approximations can also be used to compute approximate predictive densities. To apply the proposed method, one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima. Nevertheless, the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third-order expansions of the likelihood and requiring the evaluation of third derivatives. The approximate marginal posterior densities behave very much like saddle-point approximations for sampling distributions. The principal regularity condition required is that the likelihood times prior be unimodal.

Bayesian methods are given for finite-category sampling when some of the observations suffer missing category distinctions. Dickey's (1983) generalization of the Dirichlet family of prior distributions is found to be closed under such censored sampling. The posterior moments and predictive probabilities are proportional to ratios of B. C. Carlson's multiple hypergeometric functions. Closed-form expressions are developed for the case of nested reported sets, when Bayesian estimates can be computed easily from relative frequencies. Effective computational methods are also given in the general case. An example involving surveys of death-penalty attitudes is used throughout to illustrate the theory. A simple special case of categorical missing data is a two-way contingency table with cross-classified count data xij (i = 1, …, r; j = 1, …, c), together with supplementary trials counted only in the margin distinguishing the rows, yi (i = 1, …, r). There could also be further supplementary trials report...

The wisdom of adjusting U.S. decennial census counts and the theory and methods that might be used for that purpose have been matters of hot dispute in certain recent court cases involving expert testimony from numerous leading statisticians and demographers. This article by Ericksen and Kadane and the ensuing comments address the theoretical and practical issues from both sides of the controversy. In order to provide a context for the reader, I am including the following brief history, as it has been explained to me. Between the 1970 and the 1980 decennial censuses, a demand for greater accuracy of census counts arose because of the increased uses of census numbers in fund-allocation formulas for states and other political jurisdictions. The undercount became a particular target of concern, because estimates provided by the Bureau of the Census show that it is more concentrated among blacks than whites, among men than women, and among the young rather than the old. A lawsuit was filed on Census ...

Potential jurors in capital cases are often queried on their attitudes toward capital punishment. The extreme groups say they would never or they would always approve capital punishment, given a guilty verdict. In many jurisdictions, these two groups are routinely excluded from juries deciding whether the defendant is guilty in capital cases. This exclusion persists even when the potential jurors say they could be fair and impartial in deciding guilt or innocence. The current study shows that this exclusion creates a bias that almost certainly works against the defendant.

This article describes the mathematical theory underlying an interactive computer program for eliciting the hyperparameters of a subjective conjugate distribution for the multiple linear regression model with the usual normal error structure. Although the methods are heuristic, they are shown to produce hyperparameter estimates satisfying the constraints satisfied by the hyperparameters themselves. An application is given to the problem of predicting the time to fatigue failure of an asphalt-concrete road as a function of several design variables concerning the road.

Natural test statistics for the hypothesis that an equation is overidentified have been developed by Anderson and Rubin and by Basmann. If the disturbances are jointly normal, serially uncorrected, and small, both the above overidentification test statistics have the Fisher variance-ratio distribution asymptotically as the variance of the error terms gets small. This gives an analytic explanation of Monte Carlo results of Basmann. The results given apply to linear models in which predetermined variables are exogenous.

"In 1980, several cities and states sued the U.S. Census Bureau to correct census results. This correction would adjust for the differential undercounting of Blacks and Hispanics, especially in cities. In this article, the authors, each of whom testified for New York City and State in their joint lawsuit against the Census Bureau, describe the likely pattern of the undercount and present a method to adjust for it." The authors describe available methods for data adjustment and introduce a regression-based composite method of adjustment, which is used to estimate the undercounts for 66 areas. "As expected, we find that the highest undercount rates are in large cities, and the lowest are in states and state remainders with small percentages of Blacks and Hispanics. Next, we analyze how sensitive our estimates are to changes in data and modeling assumptions. We find that these changes do not affect the estimates very much. Our conclusion is that regardless of whether we use one of the simple methods or the composite method and regardless of how we vary the assumptions of the composite method, an adjustment reliably reduces population shares in states with few minorities and increases the shares of large cities."

As we leave our editorships, we take this opportunity to pass some thoughts on to JASA readers, authors, and reviewers. With more than 16,000 subscribers, JASA reaches far more readers than any other statistical research journal. To meet the needs of this broad audience, we have tried to publish a wide range of innovative, readable articles of immediate or potential usefulness for statistical practice. Submissions that tackle significant substantive problems, even with incomplete success, often are more desirable than the extensions and polishing, or further mathematization, of well-worked areas. When writing a manuscript to be submitted to JASA, authors should consider carefully the points just raised. In particular, the first two sections of an article should include the main results and motivating ideas, although not necessarily in the most general setting, to be accessible to a general audience. The ratio of published articles to first submissions continues to be 20%-25% and is determined largely by JASA's page limitations. Many articles have to be trimmed, or the acceptance rate would be lower still. Thus, brief, easily understood articles have the best chance for publication, and they are also likely to be reviewed faster (see Tables 1 and 2 for times until first decisions). For authors who seek information on preparing articles for JASA, we recommend the American Statistical Association Style Guide and "Writing Technical Papers or