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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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Ranking Hospitals on Surgical Mortality: The Importance of Reliability Adjustment
Dimick, J. B., Staiger, D. O., & Birkmeyer, J. D. (2010) · Health Services Research · doi:10.1111/j.1475-6773.2010.01158.x · Cited by 221
“In empirical Bayes methods, the mortality is ''shrunk'' back toward the average mortality, with the degree of shrinkage related to the reliability, or precision, with which mortality is measured (Morris 1983).”
Value added of teachers in high-poverty schools and lower poverty schools
Sass, T. R., Hannaway, J., Xu, Z., Figlio, D. N., & Feng, L. (2012) · Journal of Urban Economics · doi:10.1016/j.jue.2012.04.004 · Cited by 217
“In addition to placing a restriction on the minimum number of students per teacher, it is also possible to mitigate dispersion in estimated teacher effects through the use of “Empirical Bayes” or “shrinkage” estimators (see Morris (1983), Jacob and Lefgren (2005)).”
MethodCommon Structure of Smoothing Techniques in Statistics
Titterington, D. M. (1985) · International Statistical Review · doi:10.2307/1402932 · Cited by 219
What to Do When K-Means Clustering Fails: A Simple yet Principled Alternative Algorithm
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“This approach is referred to as parametric EB point estimation [48].”
Improving the Statistical Approach to Health Care Provider Profiling
Christiansen, C. L., & Morris, C. N. (1997) · Annals of Internal Medicine · doi:10.7326/0003-4819-127-8_part_2-199710151-00065 · Cited by 216
“One can estimate this distribution by using a hierarchical model (13) and the data for all hospitals.”
MethodExploratory Data Analysis for Complex Models
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A Factor-Analytic Probit Model for Representing the Market Structure in Panel Data
Elrod, T., & Keane, M. P. (1995) · Journal of Marketing Research · doi:10.1177/002224379503200103 · Cited by 213
Modelling clustered survival data from multicentre clinical trials
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“This superior performance exempli es the well-known result of James and Stein [34,48,49] for empirical Bayes or shrinkage estimators.”
Volatility in School Test Scores: Implications for Test-Based Accountability Systems
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Regression Discontinuity Inference with Specification Error
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Combining single-case experimental data using hierarchical linear models.
Noortgate, W. V. D., & Onghena, P. (2003) · School Psychology Quarterly · doi:10.1521/scpq.18.3.325.22577 · Cited by 195
Large‐Scale Perturbations: Opportunities for Innovation
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Estimation of the predictive power of the model in mixed‐effects meta‐regression: A simulation study
López-López, J. A., Marín‐Martínez, F., Sánchez‐Meca, J., Noortgate, W. V. D., & Viechtbauer, W. (2013) · British Journal of Mathematical and Statistical Psychology · doi:10.1111/bmsp.12002 · Cited by 196
“Several alternatives have been proposed in the literature to estimate the total heterogeneity variance, τ2, in random-effects models (DerSimonian & Laird, 1986; Morris, 1983; S anchez-Meca&Mar ın-Mart ınez, 2008; Sidik& Jonkman, 2005b, 2007; Viechtbauer, 2005).”
BackgroundHierarchical linear models for the quantitative integration of effect sizes in single-case research
Noortgate, W. V. D., & Onghena, P. (2003) · Behavior Research Methods, Instruments, & Computers · doi:10.3758/bf03195492 · Cited by 190
“Optimal estimates of the case-specific effects are obtained by using empirical Bayes techniques (Morris, 1983).”
MethodTHE PROBLEM OF MULTIPLE INFERENCE IN STUDIES DESIGNED TO GENERATE HYPOTHESES
Thomas, D. C., Siemiatycki, J., Dewar, R. A. D., Robins, J. M., Goldberg, M. S., & Armstrong, B. (1985) · American Journal of Epidemiology · doi:10.1093/oxfordjournals.aje.a114189 · Cited by 190
“Empirical-Bayes (EB) methods (14) were developed to provide less arbitrary "best estimates" for ensembles of related parameters.”
MethodGeneral maximum likelihood empirical Bayes estimation of normal means
Jiang, W., & Zhang, C. (2009) · The Annals of Statistics · doi:10.1214/08-aos638 · Cited by 188
“This greedy approach, also called nonparametric EB [26], was proposed the earliest among the three, but it is also the least understood, in spite of [28, 29, 30, 36, 37, 38].”
BackgroundEvaluation of some random effects methodology applicable to bird ringing data
Burnham, K. P., & White, G. C. (2002) · Journal of Applied Statistics · doi:10.1080/02664760120108755 · Cited by 186
“This coherent relationship does not hold for the usual shrinkage estimator found in the statistical literature (Morris, 1983; Louis, 1984).”
BackgroundMultilevel Logistic Regression for Polytomous Data and Rankings
Skrondal, A., & Rabe‐Hesketh, S. (2003) · Psychometrika · doi:10.1007/bf02294801 · Cited by 186
“We suggest using parametric empirical Bayes methods (e.g., Morris, 1983) for predicting values of the latent variables for the individual units.”
MethodO’Brien, S. M., Jacobs, J. P., Pasquali, S. K., Gaynor, J. W., Karamlou, T., Welke, K. F., Filardo, G., Han, J. M., Kim, S., Shahian, D. M., et al. (2015) · The Annals of Thoracic Surgery · doi:10.1016/j.athoracsur.2015.07.014 · Cited by 185
“Heuristically, the empirical Bayes estimate for a given procedure is a weighted average of the procedure’s actual observed risk and a model-based estimate of the procedure’s risk derived by borrowing data from other procedures.”
MethodInfluentialPeer effects on worker output in the laboratory generalize to the field
Herbst, D., & Mas, A. (2015) · Science · doi:10.1126/science.aac9555 · Cited by 185
Mean Squared Error of Estimation or Prediction Under a General Linear Model
Harville, D. A., & Jeske, D. R. (1992) · Journal of the American Statistical Association · doi:10.1080/01621459.1992.10475274 · Cited by 182
UNCERTAINTY AND VARIABILITY IN DEMOGRAPHY AND POPULATION GROWTH: A HIERARCHICAL APPROACH
Clark, J. S. (2003) · Ecology · doi:10.1890/0012-9658(2003)084[1370:uavida]2.0.co;2 · Cited by 183
Bingenheimer, J. B., & Raudenbush, S. W. (2004) · Annual Review of Public Health · doi:10.1146/annurev.publhealth.25.050503.153925 · Cited by 186
“Empirical Bayes estimators, or adaptations thereof, have been used to characterize toxoplasmosis rates in cities in El Salvador (39, 104); to obtain age- and sex-specific stomach, bladder, and lung cancer rates in Missouri counties (147, 148); to derive age-standardized lip cancer rates for areas in Scotland (11); to…”
MethodMukherjee, B., & Chatterjee, N. (2007) · Biometrics · doi:10.1111/j.1541-0420.2007.00953.x · Cited by 180
“The specific form of the “shrinkage” weights is motivated by the EB perspective (Morris, 1983; Greenland, 1993).”
InfluentialLavy, V., & Sand, E. (2018) · Journal of Public Economics · doi:10.1016/j.jpubeco.2018.09.007 · Cited by 180
“The GEMS records were created and 17 Following Morris (1983) and the teacher value added literature (for example, Kane and Staiger 2008) we construct the EB shrinkage factor for teacher i by the ratio of signal variance to signal variance plus noise variance of teacher i. Similarly to the teacher value added literatur…”
MethodInfluentialTHE ROLE OF RISK IN EXPLAINING DIFFERENCES IN PROFITABILITY.
Aaker, D. A., & Jacobson, R. (1987) · Academy of Management Journal · doi:10.2307/256274 · Cited by 179
Exploratory disease mapping: kriging the spatial risk function from regional count data.
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Jacob, B. A., & Lefgren, L. J. (2007) · The Quarterly Journal of Economics · doi:10.1162/qjec.2007.122.4.1603 · Cited by 172
On measuring the variability of small area estimators under a basic area level model
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Effect of Statins on Kidney Disease Outcomes: A Systematic Review and Meta-analysis
Su, X., Zhang, L., Lv, J., Wang, J., Hou, W., Xie, X., & Zhang, H. (2016) · American Journal of Kidney Diseases · doi:10.1053/j.ajkd.2016.01.016 · Cited by 164
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Hierarchical Poisson Regression Modeling
Christiansen, C. L., & Morris, C. N. (1997) · Journal of the American Statistical Association · doi:10.1080/01621459.1997.10474013 · Cited by 165
A unified jackknife theory for empirical best prediction with M-estimation
Jiang, J., Lahiri, P., & Wan, S. (2002) · The Annals of Statistics · doi:10.1214/aos/1043351257 · Cited by 166
“Note that the cut-off point (m − 3)/(m − 1) was suggested by Morris (1983).”
BackgroundMethodBuild, Compute, Critique, Repeat: Data Analysis with Latent Variable Models
Blei, D. (2014) · Annual Review of Statistics and Its Application · doi:10.1146/annurev-statistics-022513-115657 · Cited by 311
Gatsonis, C., Epstein, A., Newhouse, J., Normand, S., & Mcneil, B. (1995) · Medical Care · doi:10.1097/00005650-199506000-00005 · Cited by 174
Estimating a Population of Parameter Values Using Bayes and Empirical Bayes Methods
Louis, T. A. (1984) · Journal of the American Statistical Association · doi:10.1080/01621459.1984.10478062 · Cited by 218
Robust Estimation of Mean Squared Error of Small Area Estimators
Lahiri, P., & Rao, J. N. K. (1995) · Journal of the American Statistical Association · doi:10.1080/01621459.1995.10476570 · Cited by 159
Faust, W. C., Díaz, M., & Pohl, H. G. (2008) · The Journal of Urology · doi:10.1016/j.juro.2008.09.039 · Cited by 158
Leveraging Lotteries for School Value-Added: Testing and Estimation*
Angrist, J. D., Hull, P., Pathak, P. A., & Walters, C. (2017) · The Quarterly Journal of Economics · doi:10.1093/qje/qjx001 · Cited by 158
An Empirical Comparison of Statistical Models for Value-Added Assessment of School Performance
Tekwe, C. D., Carter, R. L., Ma, C., Algina, J. J., Lucas, M. E., Roth, J., Ariet, M., Fisher, T. J., & Resnick, M. B. (2004) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986029001011 · Cited by 156
“2 We say “perceived theoretical deficiency” because the shrinkage estimators available for use in practice are only approximations of theoretically superior Stein-type shrinkage estimators (Efron & Morris, 1975; Morris, 1983) and the related best linear unbiased predictors (BLUPs), which are Empirical Bayes estimators…”
BackgroundA multilevel meta-analysis of single-subject experimental design studies
Van den Noortgate, W., & Onghena, P. (2008) · Evidence-Based Communication Assessment and Intervention · doi:10.1080/17489530802505362 · Cited by 192
“It can be shown that these estimates are typically more accurate than the estimates based on the data from that participant or study only (Morris, 1983).”
BackgroundEmpirical Bayes Gibbs sampling.
Casella, G. (2001) · Biostatistics · doi:10.1093/biostatistics/2.4.485 · Cited by 175
Applied Bayesian Modeling And Causal Inference From Incomplete-Data Perspectives
Gelman, A., & Xiaoli, M. (2014) · Cited by 152
Langan, D., Higgins, J. P. T., & Simmonds, M. (2017) · Research Synthesis Methods · doi:10.1002/jrsm.1198 · Cited by 149
“Approximate restricted maximum likelihood (Morris, 1983) ARML ^τ 2¼ Pk i¼1a 2 i k k 1 ^ θ i ^θð Þ2 ^σ 2i Xk i¼1a 2 i , where ai ¼ 1= σ̂2i þτ̂2 .”
Empirical Bayes Procedures for Stabilizing Maps of U.S. Cancer Mortality Rates
Mantón, K. G., Woodbury, M. A., Stallard, E., Riggan, W. B., Creason, J. P., & Pellom, A. C. (1989) · Journal of the American Statistical Association · doi:10.1080/01621459.1989.10478816 · Cited by 148
Dynamic criteria and the measurement of change.
Hofmann, D. A., Jacobs, R. R., & Baratta, J. E. (1993) · Journal of Applied Psychology · doi:10.1037/0021-9010.78.2.194 · Cited by 148
Robust variance estimation for random effects meta-analysis
Sidik, K., & Jonkman, J. N. (2006) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2005.07.019 · Cited by 138
Long-Term Health Outcomes in Women With Silicone Gel Breast Implants
Balk, E. M., Earley, A., Avendano, E. E., & Raman, G. (2015) · Annals of Internal Medicine · doi:10.7326/m15-1169 · Cited by 144
Methods to increase reproducibility in differential gene expression via meta-analysis
Sweeney, T. E., Haynes, W. A., Vallania, F., Ioannidis, J. P. A., & Khatri, P. (2016) · Nucleic Acids Research · doi:10.1093/nar/gkw797 · Cited by 137
Greenland, S. (1993) · Statistics in Medicine · doi:10.1002/sim.4780120802 · Cited by 140
Triple-goal Estimates in Two-stage Hierarchical Models
Shen, W., & Louis, T. A. (1998) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/1467-9868.00135 · Cited by 142
Composite Measures For Predicting Surgical Mortality In The Hospital
Dimick, J. B., Staiger, D. O., Baser, O., & Birkmeyer, J. D. (2009) · Health Affairs · doi:10.1377/hlthaff.28.4.1189 · Cited by 140
Empirical Bayes: Past, Present and Future
Carlin, B. P., & Louis, T. A. (2000) · Journal of the American Statistical Association · doi:10.1080/01621459.2000.10474331 · Cited by 138
“Further work by Morris (1983) and Hill (1990) 1288 Journal of the American Statistical Association, December 2000 encouraged a more unifying role for EB, in which evaluations of procedures could be made by averaging over both the data space Y and the parameter space 0, with the EB position emerging…”
BackgroundInfluentialEmpirical-Bayes Adjustments for Multiple Comparisons Are Sometimes Useful
Greenland, S., & Robins, J. M. (1991) · Epidemiology · doi:10.1097/00001648-199107000-00002 · Cited by 137
Standard Errors of Prediction in Generalized Linear Mixed Models
Booth, J. G., & Hobert, J. P. (1998) · Journal of the American Statistical Association · doi:10.1080/01621459.1998.10474107 · Cited by 133
“From a Bayesian perspective, the plug-in estimate can be interpreted as a first-order approximation to the Bayesian posterior variance based on vague prior information (Morris 1983).”
MethodHypothesis tests for population heterogeneity in meta‐analysis
Viechtbauer, W. (2006) · British Journal of Mathematical and Statistical Psychology · doi:10.1348/000711005x64042 · Cited by 136
“…(e.g. Berkey, Hoaglin, Mosteller, & Colditz, 1995; Normand, 1999; Thompson & Sharp, 1999) is given by ŝ 2ðREMLÞ u < Pk i¼1w 2 1 kðk2 1Þ21 ESi 2 m̂ðMLÞu 2 2s 21iPk i¼1w 2 i : ð12Þ It appears that (12) originated with C. N. Morris (1983), who suggested it as an ‘approximate’ REML estimator (p. 53).”
Empirical Bayes Methods for Combining Likelihoods
Efron, B. (1996) · Journal of the American Statistical Association · doi:10.1080/01621459.1996.10476919 · Cited by 134
“Data from Sacks et al. (1990). Carl Morris, in Part 8 of his Commentary following this article, points out discrepancies between Table 1 and the original data for cities 4, 24, 32, and 33.”
BackgroundRandom effects models with non‐parametric priors
Butler, S. M., & Louis, T. A. (1992) · Statistics in Medicine · doi:10.1002/sim.4780111416 · Cited by 132
EFD: A Hybrid Knowledge/Statistical‐Based System for the Detection of Fraud
Major, J. A., & Riedinger, D. R. (1992) · International Journal of Intelligent Systems · doi:10.1002/int.4550070709 · Cited by 130
“With the W statistic from above, use an empirical Bayes technique to re-estimate the provider-specific effect based on both the observed effect and the systematic variation among provider effects (Morris, 1983; Morrison and Schmittlein, 1981).”
Remaining Useful Life Prognosis Based on Ensemble Long Short-Term Memory Neural Network
Cheng, Y., Wu, J., Zhu, H., Or, S. W., & Shao, X. (2020) · IEEE Transactions on Instrumentation and Measurement · doi:10.1109/tim.2020.3031113 · Cited by 133
“Then, the marginal distribution of ˆ R ( m ) t is written as [32] MD ˆ R ( m t (cid:17) Using the obtained sample points ˆ R ( m ) t , the likelihood function L is defined as [33] (cid:4) where .”
BackgroundMethodRaballo, A., Poletti, M., Preti, A., & Parnas, J. (2020) · Schizophrenia Bulletin · doi:10.1093/schbul/sbaa201 · Cited by 136
SCHWARTZ, C. C., HAROLDSON, M. A., WHITE, G. C., HARRIS, R. B., CHERRY, S., KEATING, K. A., MOODY, D., & SERVHEEN, C. (2006) · Wildlife Monographs · doi:10.2193/0084-0173(2006)161[1:tsaeio]2.0.co;2 · Cited by 277
Rehospitalizations Following Sepsis
Chang, D., Tseng, C., & Shapiro, M. F. (2015) · Critical Care Medicine · doi:10.1097/ccm.0000000000001159 · Cited by 127
Shrinkage Estimation of Price and Promotional Elasticities: Seemingly Unrelated Equations
Blattberg, R. C., & George, E. I. (1991) · Journal of the American Statistical Association · doi:10.1080/01621459.1991.10475034 · Cited by 127
Fischer, R. D., & Karl, J. A. (2021) · Social Psychological and Personality Science · doi:10.1177/19485506211019844 · Cited by 150
Minimax Multiple Shrinkage Estimation
George, E. I. (1986) · The Annals of Statistics · doi:10.1214/aos/1176349849 · Cited by 120
Kindig, D. A., & Cheng, E. R. (2013) · Health Affairs · doi:10.1377/hlthaff.2011.0892 · Cited by 119
Chang, D., & Shapiro, M. F. (2016) · JAMA Internal Medicine · doi:10.1001/jamainternmed.2016.4298 · Cited by 119
Yan, B., Su, X., Xu, B., Qiao, X., & Wang, L. (2018) · PLoS ONE · doi:10.1371/journal.pone.0206134 · Cited by 134
Constrained Bayes Estimation With Applications
Ghosh, M. (1992) · Journal of the American Statistical Association · doi:10.1080/01621459.1992.10475236 · Cited by 118
Chatterjee, S., Lahiri, P., & Li, H. (2008) · The Annals of Statistics · doi:10.1214/07-aos512 · Cited by 118
“However, Basu, Ghosh and Mukerjee (2003) showed that the resulting empirical Bayes interval proposed by Morris (1983a) still has coverage error of O ( n − 1 ).”
BackgroundMethodInfluentialEmpirical Bayes Estimation of Rates in Longitudinal Studies
Hui, S. L., & Berger, J. O. (1983) · Journal of the American Statistical Association · doi:10.1080/01621459.1983.10477015 · Cited by 118
Principals as Agents: Subjective Performance Measurement in Education
Jacob, B., & Lefgren, L. (2005) · doi:10.3386/w11463 · Cited by 118
Pastor, D. A., & Lazowski, R. A. (2017) · Multivariate Behavioral Research · doi:10.1080/00273171.2017.1365684 · Cited by 124
Hierarchical Regression Analysis Applied to a Study of Multiple Dietary Exposures and Breast Cancer
Witte, J. S., Greenland, S., Haile, R. W., & Bird, C. L. (1994) · Epidemiology · doi:10.1097/00001648-199411000-00009 · Cited by 115
The statistical analysis of treatment effects in 24‐hour ambulatory blood pressure recordings
Marler, M., Jacob, R. G., Lehoczky, J. P., & Shapiro, A. P. (1988) · Statistics in Medicine · doi:10.1002/sim.4780070608 · Cited by 115
Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction
Robert, C. P. (2012) · CHANCE · doi:10.1080/09332480.2012.726568 · Cited by 115
“…the vignette about this “sin” in the previous issue) • The prior relies on an estimator, whose variability is not accounted for in the subsequent analysis (Morris, 1983) The original motivation for the approach (Robbins, 1955) The core of Large-Scale Inference is multiple testing and the…”
BackgroundShifts of Reference Points for Framing of Strategic Decisions and Changing Risk-Return Associations
Lehner, J. M. (2000) · Management Science · doi:10.1287/mnsc.46.1.63.15130 · Cited by 111
Hospital‐Level Variation in the Use of Intensive Care
Seymour, C. W., Iwashyna, T. J., Ehlenbach, W. J., Wunsch, H., & Cooke, C. R. (2012) · Health Services Research · doi:10.1111/j.1475-6773.2012.01402.x · Cited by 112
“Xi,h = vector of patient-level factors for patient i in hospital h Using empirical Bayes methods, we determined the adjusted predicted ICU use rate and 95 percent confidence intervals at the hospital level for an average patient (Morris 1983; Normand, Glickman, and Gatsonis 1997).”
Fast Adaptive Penalized Splines
Krivobokova, T., Crainiceanu, C. M., & Kauermann, G. (2008) · Journal of Computational and Graphical Statistics · doi:10.1198/106186008x287328 · Cited by 109
“For further discussion we refer to Morris (1983), Laird and Louis (1987), Kass and Steffey (1989), or Ruppert and Carroll (2000).”
BackgroundMethodInfluentialMethods for Analysis of Longitudinal Data: Blood-Lead Concentrations and Cognitive Development
WATERNAUX, C. M., Laird, N. M., & Ware, J. H. (1989) · Journal of the American Statistical Association · doi:10.1080/01621459.1989.10478736 · Cited by 111
“Thus Vi probably underestimates the true variance, but better methods (Kackar and Harville 1984; Morris 1983) have not been where Wi = Ii-I, and”
MethodAvoiding zero between‐study variance estimates in random‐effects meta‐analysis
Chung, Y., Rabe‐Hesketh, S., & Choi, I. (2013) · Statistics in Medicine · doi:10.1002/sim.5821 · Cited by 109
“Morris [31] suggests an approximate REML estimator in the empirical Bayes framework, which requires an iterative method.”
MethodEmpirical Bayes Ranking Methods
Laird, N. M., & Louis, T. A. (1989) · Journal of Educational Statistics · doi:10.2307/1164724 · Cited by 110
On measures of uncertainty of empirical Bayes small-area estimators
Butar, F. B., & Lahiri, P. (2002) · Journal of Statistical Planning and Inference · doi:10.1016/s0378-3758(02)00323-3 · Cited by 109
“Morris (1983) and Laird and Louis (1987) are among handful of researchers who addressed this important issue.”
BackgroundMethodInfluentialMixed Model for Analyzing Geographic Variability in Mortality Rates
Tsutakawa, R. K. (1988) · Journal of the American Statistical Association · doi:10.1080/01621459.1988.10478562 · Cited by 109
“Modeling the variability between and within units borrows strength from many sources and improves estimates for all units, as frequently demonstrated by empirical Bayes methods (Morris 1983).”
BackgroundInference for Multivariate Normal Hierarchical Models
Everson, P. J., & Morris, C. N. (2000) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/1467-9868.00239 · Cited by 107
Self-citationEconometrics and decision theory
Chamberlain, G. (2000) · Journal of Econometrics · doi:10.1016/s0304-4076(99)00039-1 · Cited by 106
“See Morris (1983b).”
Two‐stage methods for the analysis of pooled data
Stukel, T. A., Demidenko, E., Dykes, J., & Karagas, M. R. (2001) · Statistics in Medicine · doi:10.1002/sim.852 · Cited by 105
“In this situation, Morris [52] derives a Stein-type estimator using a”
MethodThe Measurement of Student Ability in Modern Assessment Systems
Jacob, B. A., & Rothstein, J. (2016) · The Journal of Economic Perspectives · doi:10.1257/jep.30.3.85 · Cited by 108
“In this concluding section, we discuss their implications for several of the running examples discussed throughout this article.”
Worker Beliefs About Outside Options
Jäger, S., Roth, C., Roussille, N., & Schoefer, B. (2024) · The Quarterly Journal of Economics · doi:10.1093/qje/qjae001 · Cited by 104
Methodological Advances in Analyzing the Effects of Schools and Classrooms on Student Learning
Raudenbush, S. W., & Bryk, A. S. (1988) · Review of Research in Education · doi:10.2307/1167369 · Cited by 103
Robust Empirical Bayes Analyses of Event Rates
Gaver, D. P., & O’Muircheartaigh, I. G. (1987) · Technometrics · doi:10.2307/1269878 · Cited by 102
Skills, productivity and the evaluation of teacher performance
Harris, D. N., & Sass, T. R. (2014) · Economics of Education Review · doi:10.1016/j.econedurev.2014.03.002 · Cited by 108
The 1988 Neyman Memorial Lecture: A Galtonian Perspective on Shrinkage Estimators
Stigler, S. M. (1990) · Statistical Science · doi:10.1214/ss/1177012274 · Cited by 100
Combining Generative Models for Multifocal Glioma Segmentation and Registration
Kwon, D., Shinohara, R. T., Akbari, H., & Davatzikos, C. A. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-10404-1_95 · Cited by 100
“…the image likelihood defined as the multivariate Gaussian for Φ . p ( q , h | η ∗ , x ) is the prior function for { q , h } to be used as part of an empirical Bayes approach [11], and we assume the probability distribution of hyperparam-eters η given Y , i.e. p ( η | Y ), is sharply peaked at η ∗ .”
MethodChanges in affect interrelations as a function of stressful events
Zautra, A., Berkhof, J. H., & Nicolson, N. A. (2002) · Cognition & Emotion · doi:10.1080/02699930143000257 · Cited by 99
“The adjusted scores are defined as: PAij * = PAij 7 g (1)i * 7 d (1)i * Eventij; NAij * = NAij 7 g (2)i * 7 d (2)i * Eventij, where g (1)i *, g (2)i *, d (1)i *, and d (2)i * are the empirical Bayes estimates (Bryk & Raudenbush , 1992, p. 40; Morris, 1983).”
The Hospital Compare Mortality Model and the Volume–Outcome Relationship
Silber, J. H., Rosenbaum, P. R., Brachet, T., Ross, R. N., Bressler, L. J., Even‐Shoshan, O., Lorch, S. A., & Volpp, K. G. M. (2010) · Health Services Research · doi:10.1111/j.1475-6773.2010.01130.x · Cited by 99
“In thinking about shrinkage, the notation of Morris (1983) and more recently Dimick et al. (2009) and Mukamel et al. (2010) is helpful.”
MethodBerger, J. O., & Deely, J. (1988) · Journal of the American Statistical Association · doi:10.1080/01621459.1988.10478606 · Cited by 99
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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