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Parametric Empirical Bayes Inference: Theory and Applications
Abstract
This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some successful applications of major importance are considered. Recent results concerning estimates of error and confidence intervals are described and illustrated with data.
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“The highly parallel nature of gene expression experiments lends itself to a particular class of statistical methods, called parametric empirical Bayes, that borrow information between genes in a dynamic way (14,15).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [15]
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Chan, D. C. (2016) · Journal of Political Economy · doi:10.1086/685910 · Cited by 95
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Ghosh, M., & Meeden, G. D. (1986) · Journal of the American Statistical Association · doi:10.1080/01621459.1986.10478373 · Cited by 94
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“Combining Equations (1) and (2), the underlying hospital-speci®c rate, mkh, may be generically written as E mkhjyk; nk ; gk; s2k ykh nkh 1ÿ Skh logitÿ1 gkZTh Skh; 3 where Skh 2 0; 1 is the shrinkage factor (Morris, 1983) associated with the hth provider and kth measure.”
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“…for u is given in two stages: u b; σ2π ∼ Np yb; σ2πI ; b; σ2π ∼ π1 σ2π π2 b : Here b = β1; : : : ; βk t and σ2π are the unknown “hyperparameters,” and y is a p × k matrix of known covariates which has rank k. [See Morris (1983) and Berger (1985) for background and discussion of this model.]”
Huang, I., Frangakis, C., Dominici, F., Diette, G. B., & Wu, A. W. (2005) · Health Services Research · doi:10.1111/j.1475-6773.2005.00352.x · Cited by 89
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Calkins, D. G., Mallister, D. C., Pitcher, K. W., & Pendleton, G. W. (1999) · Marine Mammal Science · doi:10.1111/j.1748-7692.1999.tb00813.x · Cited by 88
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MethodMortality Effects and Choice Across Private Health Insurance Plans
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Knapp, G., Biggerstaff, B. J., & Hartung, J. G. (2006) · Biometrical Journal · doi:10.1002/bimj.200510175 · Cited by 84
“Morris, C. N. (1983).”
Hidden Markov Models for Longitudinal Comparisons
Scott, S. L., James, G. M., & Sugar, C. A. (2005) · Journal of the American Statistical Association · doi:10.1198/016214504000001592 · Cited by 83
“Figure 5 between 0 and 1 (Morris, 1983).”
Greenland, S. (1992) · Statistics in Medicine · doi:10.1002/sim.4780110208 · Cited by 85
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BackgroundBayesian data analysis in population ecology: motivations, methods, and benefits
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“…frequentist solution to these problems, often referred to as ''best unbiased prediction'' (Laird and Ware 1982) or ''empirical Bayes estimation'' (Morris 1983; Laird and Louis 1987), does not provide an automatic procedure that can account fully for the uncertainty involved in estimating a…”
BackgroundReceptive Field Inference with Localized Priors
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“likelihood, and maximum marginal likelihood [11,34,39–41].”
Estimating Hospital Quality with Quasi-Experimental DataPreprint
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“Another quick solution could be to just use the small subset of collected sample and run a multivariate polynomial regression on the configuration parameters vs pc (or rc) to predict power (performance) in all other configurations.”
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SELTZER, M. H., Wong, W. H., & Bryk, A. S. (1996) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986021002131 · Cited by 79
“7 Morris & Normand (1992) point out that in HM settings where the random regression parameters are scalars assumed normally distributed at Level 2, and where the sampling variances of the p7 are known and equivalent across units (i.”
BackgroundVeroniki, A. A., Mavridis, D., Higgins, J. P. T., & Salanti, G. (2014) · BMC Medical Research Methodology · doi:10.1186/1471-2288-14-106 · Cited by 79
“In the present study we also included the restricted maximum likelihood [34] and the empirical Bayes [35] estimators in conjunction with the inverse variance approach.”
MethodEmpirical Bayes estimation smoothing of relative risks in disease mapping
Meza, J. L. (2002) · Journal of Statistical Planning and Inference · doi:10.1016/s0378-3758(02)00322-1 · Cited by 78
“Morris (1983), Prasad and Rao (1990), Singh et al. (1998) and Butar and Lahiri (2003) all examine normal linear models.”
BackgroundEnriching the Analysis of Genomewide Association Studies with Hierarchical Modeling
Chen, G. K., & Witte, J. S. (2007) · The American Journal of Human Genetics · doi:10.1086/519794 · Cited by 79
“5,7,9,19,20 Thus, Wald statistics testing can be used to provide GWA rankings on the basis of ˜ b information from both maximum-likelihood estimatesand the additional information contained in the second-stage covariates.”
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Coluccia, A., & Ricciato, F. (2014) · IEEE Communications Letters · doi:10.1109/lcomm.2014.040214.132781 · Cited by 78
“To overcome these limitations we resort to an Empirical Bayes approach [15], [5] which incorporates the advantages of Bayesian models but requires a less stringent specification of prior knowledge.”
BackgroundMethodEmpirical Bayes methods for stabilizing incidence rates before mapping.
Devine, O., Louis, T. A., & Halloran, M. (1994) · Epidemiology · doi:10.1097/00001648-199411000-00010 · Cited by 87
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MethodMultilevel Analysis of Repeated Measures Data
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“The estimator O j is also called an empirical Bayes estimator( Morris, 1983 ).”
Robust Empirical Bayes Analyses of Event Rates
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“, Morris, 1983). Because this estimation problem is not featured in the Bryk-Raudenbush methods, it is not part of the treatment here. Some general discussion of empirical Bayes estimates for measurement of change problems is given in Rogosa et. al. (1982).”
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“Morris [ 21 ]). That is, we first generate { hi} (i = 1 .....”
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“We shall work in the parametric empirical Bayes (EB) setting (Morris 1983a) and let rr index the members of the family n, although rr could be viewed as indexing all distributions, producing the nonparametric empirical Bayes of Robbins (1983). By construction, the ¥i = (¥il" .”
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“Previous work has shown that hierarchical modeling can considerably improve conventional estimation (Morris 1983; Greenland 1993).”
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“More generally, Fay and Herriot (1979) and Morris (1983) use”
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Allison, D. B., Fernandez, J. R., Heo, M., Zhu, S., Etzel, C. J., Beasley, T. M., & Amos, C. I. (2002) · The American Journal of Human Genetics · doi:10.1086/339273 · Cited by 67
“A second alternative is the use of empirical Bayes approaches (Morris 1983; Carlin and Louis 2000a, 2000b).”
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Devine, O. J., & Louis, T. A. (1994) · Statistics in Medicine · doi:10.1002/sim.4780131104 · Cited by 67
Reinsel, G. C. (1985) · Journal of the American Statistical Association · doi:10.1080/01621459.1985.10478164 · Cited by 67
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Ericksen, E. P., & Kadane, J. B. (1985) · Journal of the American Statistical Association · doi:10.1080/01621459.1985.10477140 · Cited by 66
Li, Z., & Begg, C. (1994) · Journal of the American Statistical Association · doi:10.1080/01621459.1994.10476892 · Cited by 65
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Green, E. J., & Strawderman, W. E. (1991) · Journal of the American Statistical Association · doi:10.1080/01621459.1991.10475144 · Cited by 64
Market Failure in Kidney Exchange
Agarwal, N., Itai Ashlagi, A., Azevedo, E. M., Featherstone, C. R., & Karaduman, Ö. (2019) · American Economic Review · doi:10.1257/aer.20180771 · Cited by 64
Correlates of Care Quality in Long-Term Care Facilities: A Multilevel Analysis
Bravo, G., Wals, P. D., Dubois, M., & Charpentier, M. (1999) · The Journals of Gerontology Series B · doi:10.1093/geronb/54b.3.p180 · Cited by 64
“Their estimates of the |3s are empirical Bayes estimates (Morris, 1983), also referred to as shrunken estimates (James & Stein, 1961).”
BackgroundA Model for Incorporating Historical Controls into a Meta-Analysis
Begg, C. B., & Pilote, L. (1991) · Biometrics · doi:10.2307/2532647 · Cited by 64
“If we further assume that these random effects are normally distributed, then we can greatly reduce the number of nuisance parameters (Berger, 1985, Chap. 4; Morris, 1983).”
BackgroundDusen, B. V., & Nissen, J. (2019) · Physical Review Physics Education Research · doi:10.1103/physrevphyseducres.15.020108 · Cited by 62
“Of the factors, the sample size within the course is the most important [29].”
BackgroundMethods for estimating between‐study variance and overall effect in meta‐analysis of odds ratios
Bakbergenuly, I., Hoaglin, D. C., & Kulinskaya, E. (2020) · Research Synthesis Methods · doi:10.1002/jrsm.1404 · Cited by 63
Learning About and From a Distribution of Program Impacts Using Multisite Trials
Raudenbush, S. W., & Bloom, H. S. (2015) · American Journal of Evaluation · doi:10.1177/1098214015600515 · Cited by 63
Sensitivity Analysis for Fixed Effects in the Hierarchical Model: A Gibbs Sampling Approach
SELTZER, M. H. (1993) · Journal of Educational Statistics · doi:10.3102/10769986018003207 · Cited by 63
Two Modeling Strategies for Empirical Bayes EstimationPreprint
Efron, B. (2016) · Cited by 63
Kliem, S., & Kröger, C. (2013) · Behaviour Research and Therapy · doi:10.1016/j.brat.2013.08.005 · Cited by 62
“HLM use information from all of the studies to obtain an empirical Bayes estimate of each study’s effects (Morris, 1983).”
MethodEfficiently Simulating the Coverage Properties of Interval Estimates
Rubin, D. B., & Nathaniel Schenker, A. (1986) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2347266 · Cited by 62
Embracing the disharmony in medical imaging: A Simple and effective framework for domain adaptation
Wang, R., Chaudhari, P., & Davatzikos, C. A. (2021) · Medical Image Analysis · doi:10.1016/j.media.2021.102309 · Cited by 69
“Methods based on parametric empirical Bayes (Morris, 1983) such as ComBat methods (Johnson et al., 2007; Pomponio et al., 2020) remove sources of variability, specifically site differences, by performing location and scale adjustments to the data.”
MethodShrinkage estimators of time series seasonal factors and their effect on forecasting accuracy
Miller, D. M., & Williams, D. W. (2002) · International Journal of Forecasting · doi:10.1016/s0169-2070(02)00077-8 · Cited by 62
Interindustry Variation in the Costs of Job Displacement
Carrington, W. J., & Zaman, A. (1994) · Journal of Labor Economics · doi:10.1086/298357 · Cited by 62
Metadata
- Authors
- Morris, C. N.
- Affiliations
- The University of Texas at Austin (US)
- Corresponding author
- Morris, Carl N.
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 1983-03
- Volume
- 78
- Issue
- 381
- Pages
- 47-55
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W1983607152
- Cited by
- 1,448
- References
- 36
- Citations by year
- 2026: 32 · 2025: 30 · 2024: 52 · 2023: 42 · 2022: 40 · 2021: 51 · 2020: 62 · 2019: 44 · 2018: 49 · 2017: 43 · 2016: 54 · 2015: 63 · 2014: 66 · 2013: 49 · 2012: 48
- FWCI
- 24.44
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the American Statistical Association (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- parametric empirical Bayes, confidence intervals, shrinkage estimation
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