Frequentist Perspective Research Articles

Hierarchical Poisson Regression Modeling

The Poisson model and analyses here feature nonexchangeable gamma distributions (although exchangeable following a scale transformation) for individual parameters, with standard deviations proportional to means. A relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components. When tested in simulation studies, the resulting procedure provides better coverage probabilities and smaller risk than several other published rules, and thus works well from Bayesian and frequentist perspectives alike. The computations provide fast, accurate density approximations to individual parameters and to structural regression coefficients. The computer program is publicly available through Statlib.

Comparing Bayesian and frequentist estimators in the exchangeable case

Consider a sample of observations ( X 1,… X n ) with density f Q(X 1,…,X n)= ∫ ϴП i=1 nfϴ(X i)dQ(ϴ) , where f θ ( x) is a known model, θ ∈ Θ, a finite-dimensional space, and Q is an unknown distribution ranging over a suitable set of probability distributions over Θ. This is the most interesting case of exchangeable observations. The problem of estimating h( Q), a real-valued function of Q of interest, is considered, both from a Bayesian and a frequentist perspective. In particular, it is proved that the uniformly minimum variance unbiased estimator (UMVUE) for h( θ) is the UMVUE for h( Q) = E Q [ h( θ)]. Finally, following the idea in the paper by Samaniego and Reneau (1994), Bayesian and frequentist estimators of h( Q) are compared.

Selection of the reference priors for a balanced random effects model

Reference priors for a balanced random effects model with its natural parameter setting were derived by Berger and Bernado (1992a). There are many different priors corresponding to different ordered groupings. The behavior of the posterior inferences for the between variance component in the model is studied for various reference priors. Two frequentist approaches are used to justify appropriate reference priors. It is found that, for a given parameter of interest, different frequentist perspectives may result in selecting different reference priors.

Bayesian Analysis of a Protocol for Sampling Preserved Tumor Specimens

Abstract Intratumor DNA heterogeneity—as assessed by flow-cytometric cellular DNA analysis—is predictive of prognosis in many types of cancer. But some truly heterogeneous tumors apparently have a substantial chance of being misclassified as normal. One way to enhance the assay's sensitivity is to test multiple samples from each tumor specimen. This article describes an analysis of data from a study of endometrial cancer, with a view to establishing the number of samples needed to test to reliably assess heterogeneity. The analysis involves selection of a model for the data and estimation of test sensitivity and predictive value. We analyze the data from both Bayesian and frequentist perspectives. The data suggest that the degree of within-tumor correlation is relatively high; consequently, there is little to be gained from taking multiple samples per tumor.

Shortest prediction intervals for the birnbaum-saunders distribution

Shortest prediction intervals for a future observation from the Birnbaum-Saunders distribution are obtained from both frequentist and Bayesian perspectives. Comparisons are made with alternative intervals obtained via inversion. Monte Carlo simulations are performed to assess the approximate intervals.

Multiple-Imputation Inferences with Uncongenial Sources of Input

Conducting sample surveys, imputing incomplete observations, and analyzing the resulting data are three indispensable phases of modern practice with public-use data files and with many other statistical applications. Each phase inherits different input, including the information preceding it and the intellectual assessments available, and aims to provide output that is one step closer to arriving at statistical inferences with scientific relevance. However, the role of the imputation phase has often been viewed as merely providing computational convenience for users of data. Although facilitating computation is very important, such a viewpoint ignores the imputer's assessments and information inaccessible to the users. This view underlies the recent controversy over the validity of multiple-imputation inference when a procedure for analyzing multiply imputed data sets cannot be derived from (is "uncongenial" to) the model adopted for multiple imputation. Given sensible imputations and complete-data analysis procedures, inferences from standard multiple-imputation combining rules are typically superior to, and thus different from, users' incomplete-data analyses. The latter may suffer from serious nonresponse biases because such analyses often must rely on convenient but unrealistic assumptions about the nonresponse mechanism. When it is desirable to conduct inferences under models for nonresponse other than the original imputation model, a possible alternative to recreating imputations is to incorporate appropriate importance weights into the standard combining rules. These points are reviewed and explored by simple examples and general theory, from both Bayesian and frequentist perspectives, particularly from the randomization perspective. Some convenient terms are suggested for facilitating communication among researchers from different perspectives when evaluating multiple-imputation inferences with uncongenial sources of input.

Estimating the Probability of Toxicity at the Recommended Dose Following a Phase I Clinical Trial in Cancer

The problem of point and interval estimation following a Phase I trial, carried out according to the scheme outlined by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48), is investigated. A reparametrization of the model suggested in this earlier work can be seen to be advantageous in some circumstances. Maximum likelihood estimators, Bayesian estimators, and one-step estimators are considered. The continual reassessment method imposes restrictions on the sample space such that it is not possible for confidence intervals to achieve exact coverage properties, however large a sample is taken. Nonetheless, our simulations, based on a small finite sample of 20, not atypical in studies of this type, indicate that the calculated intervals are useful in most practical cases and achieve coverage very close to nominal levels in a very wide range of situations. The relative merits of the different estimators and their associated confidence intervals, viewed from a frequentist perspective, are discussed.

Some Bayesian and Non-Bayesian Procedures for the Analysis of Comparative Experiments and for Small-Area Estimation: Computational Aspects, Frequentist Properties, and Relationships

Abstract The estimation of a treatment contrast from experimental data and the estimation of a small-area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two-part mixed linear model. In this article a Bayesian approach to point and interval prediction is presented and its computational requirements are examined. Differences between the Bayesian approach and the traditional (classical) approach are discussed in general terms and, in addition, in terms of two examples taken from the literature: (1) the comparison of drug formulations in a bioavailability trial (Westlake) and (2) the estimation of corn-crop areas using satellite data (Battese, Harter, and Fuller). Some deficiencies in the classical approach are pointed out, and the Bayesian approach is considered from a frequentist perspective. It is shown, via a Monte Carlo study, that, for certain (noninformative) choices of the prior distribution, the frequentist prop...

Tolerance Intervals: Alternatives to Credibility Intervals in Validity Generalization Research

In validity generalization research, the estimated mean and variance of the true validity distribution are often used to construct a credibility interval, an inter val containing a specified proportion of the true valid ity distribution. The statistical interpretation of this in terval in the literature has varied between Bayesian and classical (frequentist) viewpoints. Credibility in tervals are here discussed from the frequentist perspec tive. These are known as "tolerance intervals" in the statistical literature. Two new methods for construct ing a credibility interval are presented. Unlike the cur rent method of constructing the credibility interval, tolerance intervals have known performance character istics across repeated applications, justifying confi dence statements. The new methods may be useful in validity generalization research involving a small or moderate number of validation studies. Index terms: Bayesian statistics, Credibility intervals, Meta- analysis, Tolerance intervals, True validity distribu tion, Validity generalization.