Empirical Bayes Confidence Intervals Research Articles

Robust Empirical Bayes Confidence Intervals

We construct robust empirical Bayes confidence intervals (EBCIs) in a normal means problem. The intervals are centered at the usual linear empirical Bayes estimator, but use a critical value accounting for shrinkage. Parametric EBCIs that assume a normal distribution for the means (Morris (1983b)) may substantially undercover when this assumption is violated. In contrast, our EBCIs control coverage regardless of the means distribution, while remaining close in length to the parametric EBCIs when the means are indeed Gaussian. If the means are treated as fixed, our EBCIs have an average coverage guarantee: the coverage probability is at least 1 − α on average across the n EBCIs for each of the means. Our empirical application considers the effects of U.S. neighborhoods on intergenerational mobility.

Bias corrected empirical Bayes confidence intervals for the selected mean

ABSTRACTEmpirical Bayes (EB) methods are very useful for post selection inference. Following Datta et al. (2002), we construct EB confidence intervals for the selected population mean. The EB intervals are adjusted to achieve the target coverage probabilities asymptotically up to the second order. Both unconditional coverage probabilities of EB intervals and corresponding probabilities conditional on ancillary statistics are found.

A second-order efficient empirical Bayes confidence interval

We introduce a new adjusted residual maximum likelihood method (REML) in the context of producing an empirical Bayes (EB) confidence interval for a normal mean, a problem of great interest in different small area applications. Like other rival empirical Bayes confidence intervals such as the well-known parametric bootstrap empirical Bayes method, the proposed interval is second-order correct, that is, the proposed interval has a coverage error of order $O(m^{-{3}/{2}})$. Moreover, the proposed interval is carefully constructed so that it always produces an interval shorter than the corresponding direct confidence interval, a property not analytically proved for other competing methods that have the same coverage error of order $O(m^{-{3}/{2}})$. The proposed method is not simulation-based and requires only a fraction of computing time needed for the corresponding parametric bootstrap empirical Bayes confidence interval. A Monte Carlo simulation study demonstrates the superiority of the proposed method over other competing methods.

Empirical Bayes Confidence Intervals for Selected Parameters in High-Dimensional Data

Modern statistical problems often involve a large number of populations and hence a large number of parameters that characterize these populations. It is common for scientists to use data to select the most significant populations, such as those with the largest t statistics. The scientific interest often lies in studying and making inferences regarding these parameters, called the selected parameters, corresponding to the selected populations. The current statistical practices either apply a traditional procedure assuming there were no selection—a practice that is not valid—or they use the Bonferroni-type procedure that is valid but very conservative and often noninformative. In this article, we propose valid and sharp confidence intervals that allow scientists to select parameters and to make inferences for the selected parameters based on the same data. This type of confidence interval allows the users to zero in on the most interesting selected parameters without collecting more data. The validity of confidence intervals is defined as the controlling of Bayes coverage probability so that it is no less than a nominal level uniformly over a class of prior distributions for the parameter. When a mixed model is assumed and the random effects are the key parameters, this validity criterion is exactly the frequentist criterion, since the Bayes coverage probability is identical to the frequentist coverage probability. Assuming that the observations are normally distributed with unequal and unknown variances, we select parameters with the largest t statistics. We then construct sharp empirical Bayes confidence intervals for these selected parameters, which have either a large Bayes coverage probability or a small Bayes false coverage rate uniformly for a class of priors. Our intervals, applicable to any high-dimensional data, are applied to microarray data and are shown to be better than all the alternatives. It is also anticipated that the same intervals would be valid for any selection rule. Supplementary materials for this article are available online.

Corrected empirical Bayes confidence intervals in nested error regression models

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized to be useful because it gives a stable and reliable estimate for a mean of a small area. In practical situations where EBLUP is applied to real data, it is important to evaluate how much EBLUP is reliable. One method for the purpose is to construct a confidence interval based on EBLUP. In this paper, we obtain an asymptotically corrected empirical Bayes confidence interval in a nested error regression model with unbalanced sample sizes and unknown components of variance. The coverage probability is shown to satisfy the confidence level in the second-order asymptotics. It is numerically revealed that the corrected confidence interval is superior to the conventional confidence interval based on the sample mean in terms of the coverage probability and the expected width of the interval. Finally, it is applied to the posted land price data in Tokyo and the neighboring prefecture.

Empirical Bayes Confidence Intervals Shrinking Both Means and Variances

SummaryWe construct empirical Bayes intervals for a large number p of means. The existing intervals in the literature assume that variances σi2 are either equal or unequal but known. When the variances are unequal and unknown, the suggestion is typically to replace them by unbiased estimators Si2. However, when p is large, there would be advantage in ‘borrowing strength’ from each other. We derive double-shrinkage intervals for means on the basis of our empirical Bayes estimators that shrink both the means and the variances. Analytical and simulation studies and application to a real data set show that, compared with the t-intervals, our intervals have higher coverage probabilities while yielding shorter lengths on average. The double-shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone. Also, the intervals are explicitly defined and can be computed immediately.

Empirical Bayes Confidence Intervals for Means of Natural Exponential Family-Quadratic Variance Function Distributions with Application to Small Area Estimation

The paper develops empirical Bayes (EB) confidence intervals for population means with distributions belonging to the natural exponential family-quadratic variance function (NEF-QVF) family when the sample size for a particular population is moderate or large. The basis for such development is to find an interval centred around the posterior mean which meets the target coverage probability asymptotically, and then show that the difference between the coverage probabilities of the Bayes and EB intervals is negligible up to a certain order. The approach taken is Edgeworth expansion so that the sample sizes from the different populations need not be significantly large. The proposed intervals meet the target coverage probabilities asymptotically, and are easy to construct. We illustrate use of these intervals in the context of small area estimation both through real and simulated data. The proposed intervals are different from the bootstrap intervals. The latter can be applied quite generally, but the order of accuracy of these intervals in meeting the desired coverage probability is unknown.

Empirical Bayes and Fully Bayes procedures to detect high-risk areas in disease mapping

Disease mapping studies have experienced an enormous development in the last twenty years. Both an Empirical Bayes (EB) and a Fully Bayes (FB) approach have been used for smoothing purposes. However, an excess of smoothing might hinder the detection of true high-risk areas. Identifying these extreme regions minimizing the misclassification of background or normal areas, and then, avoiding false alarms is crucial in epidemiology. Bayesian decision rules, based on the posterior distribution of the relative risks, have been investigated for this task, but no similar studies have been conducted under the EB approach. Within this framework, second order correct estimators of the MSE of the log-relative risk predictor can be used to build appropriate confidence intervals for the relative risks. Their ability to detect high-risk areas is investigated through a simulation study using the geographical structure of the well-known Scottish lip cancer data. Bayesian credibility intervals and decision rules, based on the posterior distribution of the relative risks, are also investigated to check if any of the approaches outperforms the others when classifying high-risk regions. The conclusion is that Bayesian decision rules, exploiting the posterior distribution of the relative risks, are more powerful to detect high-risk areas than EB confidence intervals, but no general rules can be defined as a global criterion to be routinely applied in every real setting.

Estimation in Bayesian Disease Mapping

Recent work on Bayesian inference of disease mapping models discusses the advantages of the fully Bayesian (FB) approach over its empirical Bayes (EB) counterpart, suggesting that FB posterior standard deviations of small-area relative risks are more reflective of the uncertainty associated with the relative risk estimation than counterparts based on EB inference, since the latter fail to account for the variability in the estimation of the hyperparameters. In this article, an EB bootstrap methodology for relative risk inference with accurate parametric EB confidence intervals is developed, illustrated, and contrasted with the hyperprior Bayes. We elucidate the close connection between the EB bootstrap methodology and hyperprior Bayes, present a comparison between FB inference via hybrid Markov chain Monte Carlo and EB inference via penalized quasi-likelihood, and illustrate the ability of parametric bootstrap procedures to adjust for the undercoverage in the "naive" EB interval estimates. We discuss the important roles that FB and EB methods play in risk inference, map interpretation, and real-life applications. The work is motivated by a recent analysis of small-area infant mortality rates in the province of British Columbia in Canada.

On an Asymptotic Theory of Conditional and Unconditional Coverage Probabilities of Empirical Bayes Confidence Intervals

Empirical Bayes (EB) methodology is now widely used in statistics. However, construction of EB confidence intervals is still very limited. Following Cox (1975), Hill (1990) and Carlin & Gelfand (1990, 1991), we consider EB confidence intervals, which are adjusted so that the actual coverage probabilities asymptotically meet the target coverage probabilities up to the second order. We consider both unconditional and conditional coverage, conditioning being done with respect to an ancillary statistic.

Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are pth degree piecewise polynomials with p− 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally‐adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot‐selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global‐penalty parameter. The method is developed first for univariate models and then extended to additive models.

Empirical Bayes estimation and inference for the random effects model with binary response.

Waclawiw and Liang introduced an estimating function-based approach for estimating the parameters of the classical two-stage random effects model for longitudinal data. In the present paper, the authors conduct a case study of the general approach for the binary response setting, where the fully specified parametric two-stage model with fixed and univariate random effects has an analytically intractable likelihood. With successful convergence of the algorithm, the authors propose a fully parametric bootstrapping method for deriving empirical Bayes confidence intervals for all model parameters. The bootstrapping approach is a blend of the estimating function technique with the developments of Laird and Louis. The estimating function approach to estimation and inference provides a general framework for the analysis of a wide variety of medical data, including the setting of small and varying numbers of discrete repeated observations. An application of the methodology to the analysis of binary responses in a crossover clinical trial is presented.

Approaches for Empirical Bayes Confidence Intervals

Approaches for Empirical Bayes Confidence Intervals

Approaches for Empirical Bayes Confidence Intervals

Abstract Parametric empirical Bayes (EB) methods of point estimation date to the landmark paper by James and Stein (1961). Interval estimation through parametric empirical Bayes techniques has a somewhat shorter history, which was summarized by Laird and Louis (1987). In the exchangeable case, one obtains a “naive” EB confidence interval by simply taking appropriate percentiles of the estimated posterior distribution of the parameter, where the estimation of the prior parameters (“hyperparameters”) is accomplished through the marginal distribution of the data. Unfortunately, these “naive” intervals tend to be too short, since they fail to account for the variability in the estimation of the hyperparameters. That is, they do not attain the desired coverage probability in the EB sense defined by Morris (1983a, b). They also provide no statement of conditional calibration (Rubin 1984). In this article we propose a conditional bias correction method for developing EM intervals that corrects these deficiencies...

Sensitivity of bayes and empirical bayes estimates

The main objective is to investigate the behaviour of Bayes and empirical Bayes confidence intervals for a mean to changes from normality in the specification of either the sampling or prior distributions. To do this the posterior mean and variance are calculated when the prior and sampling distributions are defined by Edgeworth expansions, corrective terms being obtained as functions of the standardized higher cumulants. Comparison with the empirical Bayes estimates, calculated under normal assumptions, gives, in particular, the sensitivity to distributional shape. A correction for errors incurred by the estimation of the prior is briefly considered. The results are illustrated by an application to real data.

Empirical Bayes Confidence Intervals for a Series of Related Experiments

When data from different studies are analyzed, often one objective of the analysis is to obtain improved parameter estimates and confidence intervals for each individual study. James-Stein or empirical Bayes estimation is well suited to producing estimates. This paper applies recent results on the construction of confidence intervals for the parameters of the individual studies. A naive empirical Bayes approach to confidence intervals assumes that the estimated posterior for each study can be used directly for inferences. The naive approach generally leads to confidence intervals with less than the desired coverage. Morris (1983, Journal of the American Statistical Association 78, 47-59) suggested one approach to the construction of empirical Bayes confidence intervals based on the use of a Bayes hyperprior distribution. Laird and Louis (1987, Journal of the American Statistical Association 82, 739-757) define the empirical Bayes bootstrap for drawing bootstrap samples from the compound (empirical Bayes) model and present a procedure for obtaining empirical Bayes confidence intervals which uses the bootstrap samples to adjust for uncertainty in the estimate of the prior. This paper compares the empirical performance of classical, Morris's, and bootstrap intervals on a random sample of bioassays from the National Cancer Institute data base on potential chemical carcinogens.

Approximated Bayes and empirical Bayes confidence intervals—The known variance case

In this paper hierarchical Bayes and empirical Bayes results are used to obtain confidence intervals of the population means in the case of real problems. This is achieved by approximating the posterior distribution with a Pearson distribution. In the first example hierarchical Bayes confidence intervals for the Efron and Morris (1975, J. Amer. Statist. Assoc., 70, 311–319) baseball data are obtained. The same methods are used in the second example to obtain confidence intervals of treatment effects as well as the difference between treatment effects in an analysis of variance experiment. In the third example hierarchical Bayes intervals of treatment effects are obtained and compared with normal approximations in the unequal variance case.

Bayes and Empirical Bayes Confidence Intervals in Applied Research

Bayes and Empirical Bayes Confidence Intervals in Applied Research

Field conditions for the use of certain empirical bayes estimators

Established here are the minimal field conditions for obtaining large sample, normal empirical Bayes limits for mean estimation. For an adverse (uniform) prior distribution, the minimum sample size, the sample number, and relative sample error conditions are established through Monte Carlo results. For as little as two samples of size 25, even where sampling error is half that of the prior, new, easily computed empirical Bayes confidence intervals are shown to improve classical estimation.