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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Parametric empirical Bayes borrows information between genes in a dynamic way.
“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]
Cites
Cross validatory choice and assessment of statistical predictions
Stone, M. (1974) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1974.tb00994.x · Cited by 10,881 · see citing works
Journal of the Royal Statistical Society.
(1923) · The Economic Journal · doi:10.1093/ej/33.130.269 · Cited by 3,289 · see citing works
The Predictive Sample Reuse Method with Applications
Geisser, S. (1975) · Journal of the American Statistical Association · doi:10.1080/01621459.1975.10479865 · Cited by 2,748 · see citing works
Estimation with Quadratic Loss
James, W., & Stein, C. M. (1992) · Springer series in statistics · doi:10.1007/978-1-4612-0919-5_30 · Cited by 2,319 · see citing works
Bayes Estimates for the Linear Model
Lindley, D. V., & Smith, A. F. M. (1972) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1972.tb00885.x · Cited by 2,025 · see citing works
Estimates of Income for Small Places: An Application of James-Stein Procedures to Census Data
Fay, R. E., & Herriot, R. A. (1979) · Journal of the American Statistical Association · doi:10.1080/01621459.1979.10482505 · Cited by 1,283 · see citing works
Stein's Estimation Rule and its Competitors—An Empirical Bayes Approach
Efron, B., & Morris, C. (1973) · Journal of the American Statistical Association · doi:10.1080/01621459.1973.10481350 · Cited by 981 · see citing works
Sampling and Bayes' Inference in Scientific Modelling and Robustness
Box, G. E. P. (1980) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2982063 · Cited by 905 · see citing works
Data Analysis Using Stein's Estimator and its Generalizations
Efron, B., & Morris, C. N. (1975) · Journal of the American Statistical Association · doi:10.1080/01621459.1975.10479864 · Cited by 863 · see citing works
An Empirical Bayes Approach to Statistics
Robbins, H. E. (1992) · Springer series in statistics · doi:10.1007/978-1-4612-0919-5_26 · Cited by 858 · see citing works
Mathematical Methods in Risk Theory
Bühlmann, H. (1970) · Springer eBooks · doi:10.1007/978-3-540-30711-2 · Cited by 733 · see citing works
Cross-Validatory Choice and Assessment of Statistical Predictions (With Discussion)
Stone, M. (1976) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1976.tb01573.x · Cited by 696 · see citing works
Natural Exponential Families with Quadratic Variance Functions
Morris, C. N. (1982) · The Annals of Statistics · doi:10.1214/aos/1176345690 · Cited by 617 · see citing works
The Empirical Bayes Approach to Statistical Decision Problems
Robbins, H. E. (1964) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177703729 · Cited by 617 · see citing works
The Estimation of Probabilities: An Essay on Modern Bayesian Methods
Sparks, J. N., & Good, I. J. (1967) · American Educational Research Journal · doi:10.2307/1162122 · Cited by 533 · see citing works
Mathematical Methods in Risk Theory.
Beard, R. E., & Buhlmann, H. (1971) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2344252 · Cited by 509 · see citing works
AN EMPIRICAL BAYES APPROACH TO STATISTICS
Robbins, H. E. (1956) · University of California Press eBooks · doi:10.1525/9780520313880-015 · Cited by 458 · see citing works
The Estimation of Probabilities: An Essay on Modern Bayesian Methods
Hill, B. M., & Good, I. J. (1965) · Journal of the American Statistical Association · doi:10.2307/2283422 · Cited by 339 · see citing works
A Simulation Study of Alternatives to Ordinary Least Squares
Dempster, A. P., Schatzoff, M., & Wermuth, N. (1977) · Journal of the American Statistical Association · doi:10.1080/01621459.1977.10479910 · Cited by 302 · see citing works
Natural Exponential Families with Quadratic Variance Functions: Statistical Theory
Morris, C. N. (1983) · The Annals of Statistics · doi:10.1214/aos/1176346158 · Cited by 253 · see citing works
Asymptotically Subminimax Solutions of Compound Statistical Decision Problems
Robbins, H. E. (1985) · Springer eBooks · doi:10.1007/978-1-4612-5110-1_1 · Cited by 234 · see citing works
A Robust Generalized Bayes Estimator and Confidence Region for a Multivariate Normal Mean
Berger, J. O. (1980) · The Annals of Statistics · doi:10.1214/aos/1176345068 · Cited by 178 · see citing works
Using Empirical Bayes Techniques in the Law School Validity Studies
Rubin, D. B. (1980) · Journal of the American Statistical Association · doi:10.1080/01621459.1980.10477553 · Cited by 157 · see citing works
Asymptotically Subminimax Solutions of Compound Statistical Decision Problems
Robbins, H. E. (1951) · University of California Press eBooks · doi:10.1525/9780520411586-011 · Cited by 148 · see citing works
Biased versus unbiased estimation
Efron, B. (1975) · Advances in Mathematics · doi:10.1016/0001-8708(75)90114-0 · Cited by 115 · see citing works
Empirical Bayes Methods Applied to Estimating Fire Alarm Probabilities
Carter, G. M., & Rolph, J. E. (1974) · Journal of the American Statistical Association · doi:10.1080/01621459.1974.10480222 · Cited by 112 · see citing works
The Quality Measurement Plan (QMP)
Hoadley, B. (1981) · Bell System Technical Journal · doi:10.1002/j.1538-7305.1981.tb00239.x · Cited by 86 · see citing works
Predictions for National Football League Games via Linear-Model Methodology
Harville, D. A. (1980) · Journal of the American Statistical Association · doi:10.1080/01621459.1980.10477504 · Cited by 86 · see citing works
Parametric Empirical Bayes Confidence Intervals
Morris, C. N. (1983) · Elsevier eBooks · doi:10.1016/b978-0-12-121160-8.50008-9 · Cited by 83 · see citing works
Compound Decisions and Empirical Bayes
Copas, J. B. (1969) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1969.tb00800.x · Cited by 66 · see citing works
g2-minimax estimators in the exponential family
JACKSON, D. A., O'Donovan, T. M., Zimmer, W. J., & Deely, J. (1970) · Biometrika · doi:10.1093/biomet/57.2.439 · Cited by 33 · see citing works
An empirical Bayes estimation problem
Robbins, H. E. (1980) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.77.12.6988 · Cited by 31 · see citing works
Gini's Contributions to the Theory of Inference
Forcina, A. (1982) · International Statistical Review · doi:10.2307/1402459 · Cited by 21 · see citing works
Interval estimation for empirical Bayes generalizations of Stein's estimator
Morris, C. N. (1977) · Medical Entomology and Zoology · Cited by 18 · see citing works
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limma powers differential expression analyses for RNA-sequencing and microarray studies
Ritchie, M. E., Phipson, B., Wu, D., Hu, Y., Law, C. W., Shi, W., & Smyth, G. K. (2015) · Nucleic Acids Research · doi:10.1093/nar/gkv007 · Cited by 40,458 · see citing works
“These methods all help improve inference at both the gene and gene set level in small experiments. genes in a dynamic way (14,15).”
MethodConducting Meta-Analyses in <i>R</i> with the <b>metafor</b> Package
Viechtbauer, W. (2010) · Journal of Statistical Software · doi:10.18637/jss.v036.i03 · Cited by 20,464 · see citing works
“…estimator (DerSimonian and Laird 1986; Raudenbush 2009), the Sidik-Jonkman estimator (Sidik and Jonkman 2005a,b), the maximum-likelihood or restricted maximum-likelihood estimator (Viechtbauer 2005; Raudenbush 2009), or the empirical Bayes estimator (Morris 1983; Berkey et al. 1995).”
BackgroundBlei, D. M., Ng, A. Y., & Jordan, M. I. (2002) · The MIT Press eBooks · doi:10.7551/mitpress/1120.003.0082 · Cited by 12,649 · see citing works
Sampling-based approaches to calculating marginal densities
Gelfand, A. E., & Smith, A. F. M. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10476213 · Cited by 6,606 · see citing works
The Handbook of Research Synthesis and Meta-Analysis
(2019) · doi:10.7758/9781610448864 · Cited by 4,553 · see citing works
Gelman, A. (2006) · Bayesian Analysis · doi:10.1214/06-ba117a · Cited by 4,050 · see citing works
“On the theoretical side, hierarchical models allow a more “objective” approach to inference by estimating the parameters of prior distributions from data rather than requiring them to be specified using subjective information (see James and Stein, 1960, Efron and Morris, 1975, and Morris, 1983).”
A Re-Evaluation of Random-Effects Meta-Analysis
Higgins, J. P. T., Thompson, S. G., & Spiegelhalter, D. J. (2009) · Journal of the Royal Statistical Society Series A: Statistics in Society · doi:10.1111/j.1467-985x.2008.00552.x · Cited by 2,569 · see citing works
Graphical tools for network meta-analysis in STATA
Chaimani, A., Higgins, J. P. T., Mavridis, D., Spyridonos, P., & Salanti, G. (2013) · PLoS ONE · doi:10.1371/journal.pone.0076654 · Cited by 2,473 · see citing works
“Method of moments and restricted maximum likelihood are those most commonly used and have been implemented in STATA’s metan command along with an empirical Bayes method [32,33].”
MethodBayes Empirical Bayes Inference of Amino Acid Sites Under Positive Selection
Yang, Z. (2005) · Molecular Biology and Evolution · doi:10.1093/molbev/msi097 · Cited by 2,325 · see citing works
“Several procedures have been suggested in the statistics literature to correct for the bias in the NEB approach to achieve approximately correct frequentist coverage probabilities (e.g., Morris 1983; Laird and Louis 1987; Carlin and Gelfand 1990).”
MethodRao, J. N. K., & Molina, I. (2003) · Encyclopedia of Statistical Sciences · doi:10.1002/0471667196.ess0731 · Cited by 2,137 · see citing works
Improved tests for a random effects meta‐regression with a single covariate
Knapp, G., & Hartung, J. (2003) · Statistics in Medicine · doi:10.1002/sim.1482 · Cited by 2,075 · see citing works
“[1] uses the empirical Bayes (EB) estimator [13] of the between-trial variance which is also iteratively given as the solution of the equation”
BackgroundMethodEfficacy of BCG Vaccine in the Prevention of Tuberculosis
Colditz, G. A. (1994) · JAMA · doi:10.1001/jama.1994.03510330076038 · Cited by 1,944 · see citing works
Bias and Efficiency of Meta-Analytic Variance Estimators in the Random-Effects Model
Viechtbauer, W. (2005) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986030003261 · Cited by 1,699 · see citing works
“It appears that Equation 19 originated with Morris (1983), who suggested it as an “approximate” REML estimate (p. 53).”
BackgroundThat BLUP is a Good Thing: The Estimation of Random Effects
Robinson, G. K. (1991) · Statistical Science · doi:10.1214/ss/1177011926 · Cited by 1,694 · see citing works
Methods to estimate the between‐study variance and its uncertainty in meta‐analysis
Veroniki, A. A., Jackson, D., Viechtbauer, W., Bender, R., Bowden, J., Knapp, G., Kuss, O., Higgins, J. P., Langan, D., & Salanti, G. (2016) · Research Synthesis Methods · doi:10.1002/jrsm.1164 · Cited by 1,685 · see citing works
“However, several investigators claim that in practice the differences between frequentist and Bayesian approaches appear to be small (Morris, 1983; Thompson and Sharp, 1999).”
An Overview of the Logic and Rationale of Hierarchical Linear Models
Hofmann, D. A. (1997) · Journal of Management · doi:10.1177/014920639702300602 · Cited by 1,584 · see citing works
“Bryk and Raudenbush (1992) drawing on the OLS regression rule of thumb of 10 observations per predictor, mentioned that analogous rules for hierarchical linear models can be developed.”
BackgroundMethodInfluentialSchäfer, J., & Strimmer, K. (2005) · Statistical Applications in Genetics and Molecular Biology · doi:10.2202/1544-6115.1175 · Cited by 1,556 · see citing works
Prior distributions for variance parameters in hierarchical modelsPreprint
Gelman, A. (2004) · RePEc: Research Papers in Economics · Cited by 1,558 · see citing works
Application of Hierarchical Linear Models to Assessing Change
Bryk, A., & Raudenbush, S. (1987) · Psychological Bulletin · doi:10.1037/0033-2909.101.1.147 · Cited by 1,613 · see citing works
Modeling Multilevel Data Structures
Steenbergen, M. R., & Jones, B. (2002) · American Journal of Political Science · doi:10.2307/3088424 · Cited by 1,508 · see citing works
“The first, is to simply adjust the GLS and EB estimators by incorporating estimates of the sampling variance of the (co)variance components (Morris 1983; Kackar and Harvill e 1984), possibly via bootstrapping (Laird and Louis 1987).”
MethodGroup representations in probability and statistics
Diaconis, P. (1988) · doi:10.1214/lnms/1215467407 · Cited by 1,381 · see citing works
Machine Learning Methods That Economists Should Know About
Athey, S., & Imbens, G. W. (2019) · Annual Review of Economics · doi:10.1146/annurev-economics-080217-053433 · Cited by 1,163 · see citing works
“, Morris [1983]).”
Bayesian Statistics and Marketing
Rossi, P. E., & Allenby, G. M. (2003) · Marketing Science · doi:10.1287/mksc.22.3.304.17739 · Cited by 1,116 · see citing works
A comparison of heterogeneity variance estimators in simulated random‐effects meta‐analyses
Langan, D., Higgins, J. P. T., Jackson, D., Bowden, J., Veroniki, A. A., Kontopantelis, E., Viechtbauer, W., & Simmonds, M. (2019) · Research Synthesis Methods · doi:10.1002/jrsm.1316 · Cited by 1,074 · see citing works
Blei, D. M., & Jordan, M. I. (2003) · International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR) · doi:10.1145/860435.860460 · Cited by 1,078 · see citing works
“With K draws from the prior distribution, however, we are in a good position to take the empirical Bayes perspective [9] and compute a maximum likelihood estimate of η.”
MethodPhipson, B., Lee, S., Majewski, I. J., Alexander, W. S., & Smyth, G. K. (2016) · The Annals of Applied Statistics · doi:10.1214/16-aoas920 · Cited by 966 · see citing works
“Morris, C. N. (1983). Parametric empirical Bayes inference: theory and applications. Journal of the American Statistical Association 78, 47–55. Murie, C., Woody, O., Lee, A. Y. and Nadon, R. (2009). Comparison of small n statistical tests of differential expression applied to microarrays. BMC Bioinformatics 10, 45. Pi…”
BackgroundInfluentialIllustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling
Gelfand, A. E., Hills, S. E., Racine-Poon, A., & Smith, A. F. M. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10474968 · Cited by 970 · see citing works
Bayes-Stein Estimation for Portfolio Analysis
Jorion, P. (1986) · Journal of Financial and Quantitative Analysis · doi:10.2307/2331042 · Cited by 961 · see citing works
The Handbook of Research Synthesis
Ziegel, E. R., Cooper, H., & Hedges, L. V. (1995) · Technometrics · doi:10.2307/1269651 · Cited by 936 · see citing works
Harbord, R. M., & Higgins, J. P. T. (2008) · The Stata Journal: Promoting communications on statistics and Stata · doi:10.1177/1536867x0800800403 · Cited by 897 · see citing works
“It is found by iterating the following equation (Morris 1983; Berkey et al. 1995):”
InfluentialA random‐effects regression model for meta‐analysis
Berkey, C. S., Hoaglin, D. C., Mosteller, F. C., & Colditz, G. A. (1995) · Statistics in Medicine · doi:10.1002/sim.4780140406 · Cited by 882 · see citing works
Regression discontinuity inference with specification error
Lee, D. S., & Card, D. (2007) · Journal of Econometrics · doi:10.1016/j.jeconom.2007.05.003 · Cited by 855 · see citing works
A Hierarchical Model for Studying School Effects
Raudenbush, S. W., & Bryk, A. S. (1986) · Sociology of Education · doi:10.2307/2112482 · Cited by 818 · see citing works
Statistical Modelling Issues in School Effectiveness Studies
Aitkin, M. A., & Longford, N. T. (1986) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2981882 · Cited by 789 · see citing works
Estimating Teacher Impacts on Student Achievement: An Experimental Evaluation
Kane, T. J., & Staiger, D. O. (2008) · National Bureau of Economic Research · doi:10.3386/w14607 · Cited by 784 · see citing works
The Estimation of the Mean Squared Error of Small-Area Estimators
Prasad, N., & Rao, J. N. K. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10475320 · Cited by 768 · see citing works
Small Area Estimation: An Appraisal
Ghosh, M., & Rao, J.-N.-K. (1994) · Statistical Science · doi:10.1214/ss/1177010647 · Cited by 871 · see citing works
Effects of omega-3 fatty acids on serum markers of cardiovascular disease risk: a systematic review
Balk, E. M., Lichtenstein, A. H., Chung, M., Kupelnick, B., Chew, P., & Lau, J. (2006) · Atherosclerosis · doi:10.1016/j.atherosclerosis.2006.02.012 · Cited by 732 · see citing works
League Tables and Their Limitations: Statistical Issues in Comparisons of Institutional Performance
Goldstein, H., & Spiegelhalter, D. J. (1996) · Journal of the Royal Statistical Society. Series A (Statistics in Society) · doi:10.2307/2983325 · Cited by 1,069 · see citing works
International Portfolio Diversification with Estimation Risk
Jorion, P. (1985) · The Journal of Business · doi:10.1086/296296 · Cited by 672 · see citing works
Guidelines for reporting mortality and morbidity after cardiac valve interventions
Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · Journal of Thoracic and Cardiovascular Surgery · doi:10.1016/j.jtcvs.2007.12.002 · Cited by 664 · see citing works
Counting Your Customers: Who-Are They and What Will They Do Next?
Schmittlein, D. C., Morrison, D. G., & Colombo, R. (1987) · Management Science · doi:10.1287/mnsc.33.1.1 · Cited by 662 · see citing works
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Xu, R. (2003) · Statistics in Medicine · doi:10.1002/sim.1572 · Cited by 542 · see citing works
Cressie, N., Calder, C. A., Clark, J. S., Hoef, J. M. V., & Wikle, C. K. (2009) · Ecological Applications · doi:10.1890/07-0744.1 · Cited by 541 · see citing works
“One could try to account for the variability in P̂, but formal inference is usually approximate (e.g., Morris 1983a, b); one commonly used approximation is based on multivariable differential calculus and Taylor-series expansions of estimators (e.g., Prasad and Rao 1990, Cressie 1992, Rao…”
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“Frequentist reasoning was intimately involved in Robbins' original formulation of empirical Bayes, and in significant implementations of the paradigm, such as Morris (1983) for hierarchical models .”
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Casella, G. (1985) · The American Statistician · doi:10.1080/00031305.1985.10479400 · Cited by 524 · see citing works
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Duncan, C., Jones, K., & Moon, G. (1998) · Social Science & Medicine · doi:10.1016/s0277-9536(97)00148-2 · Cited by 689 · see citing works
“…contexts identified are treated as coming from a population distribution the estimation procedure can "pool" all the information in the data thus allowing the predictions of place-specific relationships to be based on precision-weighted estimators (Morris, 1983) which take account of sample sizes.”
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Kontopantelis, E., Springate, D. A., & Reeves, D. (2013) · PLoS ONE · doi:10.1371/journal.pone.0069930 · Cited by 508 · see citing works
“Note that we do not discuss the empirical Bayes estimator proposed by Morris [28], due to its practical similarity to the model error variance estimators [18].”
BackgroundGuidelines for Reporting Mortality and Morbidity After Cardiac Valve Interventions
Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · The Annals of Thoracic Surgery · doi:10.1016/j.athoracsur.2007.12.082 · Cited by 482 · see citing works
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“In addition, all data were analyzed in a random-effects model meta-regression—a meta-analytic technique of multivariable linear regression across studies—according to the method of Morris (21) as described by Berkey and colleagues (22).”
MethodBoyle, P., Gould, A. L., & Roehrborn, C. G. (1996) · Urology · doi:10.1016/s0090-4295(96)00353-6 · Cited by 467 · see citing works
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“An EB estimate of 2 is obtained as an iterative solution to the equation [19, 20]”
MethodInfluentialTweedie’s Formula and Selection Bias
Efron, B. (2011) · Journal of the American Statistical Association · doi:10.1198/jasa.2011.tm11181 · Cited by 414 · see citing works
“(4.11) [Morris (1983) presented a hierarchical Bayes analysis of James–Stein estimation accuracy; see Remark I of Section 7.]”
BackgroundMethodInfluentialConfidence intervals for the amount of heterogeneity in meta‐analysis
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Fox, J., & Glas, C. A. W. (2001) · Psychometrika · doi:10.1007/bf02294839 · Cited by 398 · see citing works
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Koedel, C., Mihály, K., & Rockoff, J. E. (2015) · Economics of Education Review · doi:10.1016/j.econedurev.2015.01.006 · Cited by 368 · see citing works
“6 A detailed discussion of shrinkage estimators can be found in Morris (1983).”
BackgroundMeasuring Extreme Response Style
Greenleaf, E. A. (1992) · Public Opinion Quarterly · doi:10.1086/269326 · Cited by 368 · see citing works
Kass, R. E., & Steffey, D. (1989) · Journal of the American Statistical Association · doi:10.1080/01621459.1989.10478825 · Cited by 474 · see citing works
A mixture model approach for the analysis of microarray gene expression data
Allison, D. B., Gadbury, G. L., Heo, M., Fernández, J. R., Lee, C., Prolla, T. A., & Weindruch, R. H. (2002) · Computational Statistics & Data Analysis · doi:10.1016/s0167-9473(01)00046-9 · Cited by 358 · see citing works
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Clark, J. S., & Bjørnstad, O. N. (2004) · Ecology · doi:10.1890/03-0520 · Cited by 348 · see citing works
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Gelman, A., & Pardoe, I. (2006) · Technometrics · doi:10.1198/004017005000000517 · Cited by 342 · see citing works
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The Intertemporal Variability of Teacher Effect Estimates
McCaffrey, D., Sass, T. R., Lockwood, J. R., & Mihaly, K. (2009) · Education Finance and Policy · doi:10.1162/edfp.2009.4.4.572 · Cited by 406 · see citing works
Mixed model prediction and small area estimation
Jiang, J., & Lahiri, P. (2006) · Test · doi:10.1007/bf02595419 · Cited by 328 · see citing works
“This data set has been analyzed by several authors in the past, including Efron and Morris (1975), Efron (1975), Morris (1983b), Gelman et al. (1995), Datta and Lahiri (2000), Rao (2003), Lahiri and Li (2005), among others.”
BackgroundMethodInfluentialCharter school quality and parental decision making with school choice
Hanushek, E. A., Kain, J. F., Rivkin, S. G., & Branch, G. F. (2007) · Journal of Public Economics · doi:10.1016/j.jpubeco.2006.09.014 · Cited by 326 · see citing works
Education and Economic Growth: A Meta-Regression Analysis
Benos, N., & Zotou, S. (2014) · World Development · doi:10.1016/j.worlddev.2014.06.034 · Cited by 334 · see citing works
“As suggested by Thompson and Sharp (1999), the unknown variance of the random-effect model can be computed by an iterative residual (restricted) maximum likelihood process (REML), the Empirical Bayes (EB) method (see also Morris, 1983), or a momentestimator (MM).”
MethodVisualizing Assumptions and Results in Network Meta-analysis: The Network Graphs Package
Chaimani, A., & Salanti, G. (2015) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x1501500402 · Cited by 323 · see citing works
“…2005) be used for the estimation of the loop-specific heterogeneity (when tau2(loop) has been specified). eb specifies that the empirical Bayes method (Morris 1983) be used for the estimation of the loop-specific heterogeneity (when tau2(loop) has been specified). random specifies that the random-effects…”
Testing for Convergence Clubs in Income Per Capita: A Predictive Density Approach*
Canova, F. (2004) · International Economic Review · doi:10.1111/j.1468-2354.2004.00117.x · Cited by 322 · see citing works
“…a unknown break point in time series (see e.g., Ploberger et al., 1989; Bai, 1997; Polasek and Ren, 1997) and to the EB tradition of constructing posterior estimates of the coefficients of a model by plugging in estimates of the parameters of the prior (see Morris, 1983; Berger, 1985; Efron, 1996).”
BackgroundMethodInfluentialEmpirical bayes versus fully bayesian analysis of geographical variation in disease risk
Bernardinelli, L., & Montomoli, C. (1992) · Statistics in Medicine · doi:10.1002/sim.4780110802 · Cited by 302 · see citing works
Educational Applications of Hierarchical Linear Models: A Review
Raudenbush, S. W. (1988) · Journal of Educational Statistics · doi:10.3102/10769986013002085 · Cited by 301 · see citing works
Heagerty, P. J., & Zeger, S. L. (2000) · Statistical Science · doi:10.1214/ss/1009212671 · Cited by 298 · see citing works
Santesso, N., Akl, E. A., Bianchi, M., Mente, A., Mustafa, R., Heels-Ansdell, D., & Schünemann, H. J. (2012) · European Journal of Clinical Nutrition · doi:10.1038/ejcn.2012.37 · Cited by 292 · see citing works
Guidelines for reporting mortality and morbidity after cardiac valve interventions
Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · European Journal of Cardio-Thoracic Surgery · doi:10.1016/j.ejcts.2007.12.055 · Cited by 289 · see citing works
“These methods include mixed models, random and fixed effects models, generalized estimating equation approach, and hierarchic models (such as currently used for The Society of Thoracic Surgeons National Database risk assessment) [31—33].”
MethodJacob, B. A., & Lefgren, L. (2008) · Journal of Labor Economics · doi:10.1086/522974 · Cited by 494 · see citing works
Grossman, P., Loeb, S., Cohen, J., & Wyckoff, J. (2013) · American Journal of Education · doi:10.1086/669901 · Cited by 277 · see citing works
Empirical bayes meta-analysis.
Raudenbush, S. W., & Bryk, A. S. (1985) · Journal of Educational Statistics · doi:10.3102/10769986010002075 · Cited by 299 · see citing works
Estimating the Effect of Leaders on Public Sector Productivity: The Case of School Principals
Branch, G. F., Hanushek, E. A., & Rivkin, S. G. (2012) · National Bureau of Economic Research · doi:10.3386/w17803 · Cited by 264 · see citing works
A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series
Kitagawa, G., & Gersch, W. (1985) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1985.1103788 · Cited by 263 · see citing works
Health Care Exceptionalism? Performance and Allocation in the US Health Care Sector
Chandra, A., Finkelstein, A. N., Sacarny, A., & Syverson, C. (2016) · American Economic Review · doi:10.1257/aer.20151080 · Cited by 258 · see citing works
“Second, we restrict our sample to hospitals with a minimum number of patients per condition over the three-year measurement period; the cutoff threshold varies across our quality measures as described in Appendix B.”
Simple Heterogeneity Variance Estimation for Meta-Analysis
Sidik, K., & Jonkman, J. N. (2005) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1111/j.1467-9876.2005.00489.x · Cited by 254 · see citing works
“…of Hedges (1983) and Hedges and Olkin (1985), the maximum likelihood estimator of Hardy and Thompson (1996), the restricted maximum likelihood (REML) estimator of Raudenbush and Bryk (1985) and the empirical Bayes estimator of Morris (1983), which was used for meta-analysis by Berkey et al. (1995).”
MethodNatural Exponential Families with Quadratic Variance Functions: Statistical Theory
Morris, C. N. (1983) · The Annals of Statistics · doi:10.1214/aos/1176346158 · Cited by 253 · see citing works
An introduction to hierarchical linear modelling
SULLIVAN, L. M., Dukes, K. A., & Losina, E. (1999) · Statistics in Medicine · doi:10.1002/(sici)1097-0258(19990415)18:7 · Cited by 252 · see citing works
Robust Bayes and Empirical Bayes Analysis with $_\epsilon$-Contaminated Priors
Berger, J. O., & Berliner, L. M. (1986) · The Annals of Statistics · doi:10.1214/aos/1176349933 · Cited by 246 · see citing works
Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting
Ruppert, D., & Carroll, R. (2000) · Australian & New Zealand Journal of Statistics · doi:10.1111/1467-842x.00119 · Cited by 286 · see citing works
“…but then pretend that the hyperparameters were known so they do not account for extra variability in the posterior distribution caused by the estimation of hyperparameters in the prior; for discussion see, e.g., Morris (1983), Laird & Louis (1987), Kass & Steffey (1989) or Carlin & Louis (1996).”
A comparison of procedures to test for moderators in mixed-effects meta-regression models
Viechtbauer, W., López-López, J. A., Sánchez-Meca, J., & Marín-Martínez, F. (2015) · Psychological Methods · doi:10.1037/met0000023 · Cited by 241 · see citing works
“This estimator can be derived based on EB methods (Morris, 1983) and will therefore be denoted by u0007̂EB 2 .”
MethodInfluentialRaudenbush, S. W. (1993) · Journal of Educational Statistics · doi:10.2307/1165158 · Cited by 239 · see citing works
Empirical Bayes Confidence Intervals Based on Bootstrap Samples
Laird, N. M., & Louis, T. A. (1987) · Journal of the American Statistical Association · doi:10.1080/01621459.1987.10478490 · Cited by 233 · see citing works
Hobbs, B. P., Sargent, D. J., & Carlin, B. P. (2012) · Bayesian Analysis · doi:10.1214/12-ba722 · Cited by 232 · see citing works
“Empirical Bayesian In this subsection we consider parametric empirical Bayesian (EB) estimation (see e.g. Morris 1983; Kass and Steffey 1989; Carlin and Louis 2009) in the context of our proposed commensurate prior models.”
MethodWho Leaves? Teacher Attrition and Student Achievement
Boyd, D. J., Grossman, P., Lankford, H., Loeb, S., & Wyckoff, J. (2008) · National Bureau of Economic Research · doi:10.3386/w14022 · Cited by 231 · see citing works
“6 See Jacob and Lefgren (2005) for a clear summary of the approach we use as well as Kane and Staiger (2002) and Morris (1983) for earlier references.”
MethodDo Parents Value School Effectiveness?
Abdulkadiroğlu, A., Pathak, P. A., Schellenberg, J., & Walters, C. (2020) · American Economic Review · doi:10.1257/aer.20172040 · Cited by 226 · see citing works
“Relative to the unbiased but noisy estimate θ̂j , this EB shrinkage estimator uses the prior distribution to reduce sampling variance at the cost of increased bias, yielding a minimum mean squared error (MSE) prediction of θj (Robbins, 1956; Morris, 1983).”
MethodPrediction in Multilevel Generalized Linear Models
Skrondal, A., & Rabe‐Hesketh, S. (2009) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/j.1467-985x.2009.00587.x · Cited by 220 · see citing works
“Similarly, empirical Bayesians evaluate inferences with respect to joint sampling of ζ j and y j (e.g. Morris (1983)).”
BackgroundMethodA multivariate mixed linear model for meta-analysis.
Kalaian, H. A., & Raudenbush, S. W. (1996) · Psychological Methods · doi:10.1037/1082-989x.1.3.227 · Cited by 229 · see citing works
- 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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