Random effects models with non-parametric priors.

We discuss the performance of non-parametric maximum likelihood (NPML) estimators for the distribution of a univariate random effect in the analysis of longitudinal data. For continuous data, we analyse generated and real data sets, and compare the NPML method to those that assume a Gaussian random effects distribution and to ordinary least squares. For binary outcomes we use generated data to study the moderate and large-sample performance of the NPML compared with a method based on a Gaussian random effect distribution in logistic regression. We find that estimated fixed effects are compatible for all approaches, but that appropriate standard errors for the NPML require adjusting the likelihood-based standard errors. We conclude that the non-parametric approach provides an attractive alternative to Gaussian-based methods, though additional evaluations are necessary before it can be recommended for general use.

[1]  B. Lindsay.  The Geometry of Mixture Likelihoods: A General Theory , 1983 .

[2]  Diane Lambert,et al.  Identifiability of finite mixtures of logistic regression models , 1991 .

[3]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[4]  B. Lindsay.  The Geometry of Mixture Likelihoods, Part II: The Exponential Family , 1983 .

[5]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

[6]  J. Heckman,et al.  A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data , 1984 .

[7]  B. Lindsay,et al.  Semiparametric Estimation in the Rasch Model and Related Exponential Response Models, Including a Simple Latent Class Model for Item Analysis , 1991 .

[8]  Deborah Burr,et al.  On Errors-in-Variables in Binary Regression—Berkson Case , 1988 .

[9]  Martin Abba Tanner,et al.  Tools for Statistical Inference: Observed Data and Data Augmentation Methods , 1993 .

[10]  J. Ware,et al.  Random-effects models for serial observations with binary response. , 1984, Biometrics.

[11]  R. N. Kackar,et al.  Approximations for Standard Errors of Estimators of Fixed and Random Effects in Mixed Linear Models , 1984 .

[12]  Thomas A. Louis,et al.  Empirical Bayes Ranking Methods , 1989 .

[13]  Dean Follmann,et al.  Consistent estimation in the rasch model based on nonparametric margins , 1988 .

[14]  M. Gail,et al.  Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates , 1984 .

[15]  L. Zhao,et al.  Correlated binary regression using a quadratic exponential model , 1990 .

[16]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[17]  S. Fienberg,et al.  Longitudinal analysis of categorical epidemiological data: a study of Three Mile Island. , 1985, Environmental health perspectives.

[18]  T. Louis,et al.  Empirical Bayes Confidence Intervals Based on Bootstrap Samples , 1987 .

[19]  D. Ruppert,et al.  Random-Effect Models in Nonlinear Regression with Applications to Bioassay , 1989 .

[20]  Christine Waternaux,et al.  Methods for Analysis of Longitudinal Data: Blood-Lead Concentrations and Cognitive Development , 1989 .

[21]  H. Teicher.  Identifiability of Mixtures , 1961 .

[22]  N M Laird,et al.  Maximum likelihood regression methods for paired binary data. , 1990, Statistics in medicine.

[23]  N. Laird.  Nonparametric Maximum Likelihood Estimation of a Mixing Distribution , 1978 .

[24]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[25]  T. A. Louis,et al.  Using empirical Bayes methods in biopharmaceutical research. , 1991, Statistics in medicine.

[26]  Diane Lambert,et al.  Generalizing Logistic Regression by Nonparametric Mixing , 1989 .

[27]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[28]  H. Preisler.  Analysis of a Toxicological Experiment Using a Generalized Linear Model with , 1989 .

[29]  David C. Schmittlein,et al.  Surprising Inferences from Unsurprising Observations: Do Conditional Expectations Really Regress to the Mean? , 1989 .

[30]  H. Robbins.  The Empirical Bayes Approach to Statistical Decision Problems , 1964 .

[31]  N. Laird,et al.  Maximum likelihood computations with repeated measures: application of the EM algorithm , 1987 .

[32]  A S Whittemore,et al.  Methods for analyzing panel studies of acute health effects of air pollution. , 1979, Biometrics.

[33]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[34]  Bradley P. Carlin,et al.  Approaches for Empirical Bayes Confidence Intervals , 1990 .

[35]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[36]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.