Generalized Linear Mixed Models

This article provides an overview of generalized linear mixed models (GLMMs), how they are fit to data, and the inferences possible when using them. GLMMs are a class of statistical models that handle a wide variety of distributions for the outcome, accommodate nonlinear models, and model correlated data. As regression methods, they are not only capable of estimation and testing of covariate effects but also can be used to draw inferences about correlation structures in the data and are able to calculate predicted values that take into account not only covariates but also observed outcomes. We briefly describe software available for fitting GLMMs.

[1]  C. Eisenhart.  The assumptions underlying the analysis of variance. , 1947, Biometrics.

[2]  Melvin R. Novick,et al.  Some latent train models and their use in inferring an examinee's ability , 1966 .

[3]  R. Lyman Ott.,et al.  An introduction to statistical methods and data analysis , 1977 .

[4]  Martin Crowder,et al.  Beta-binomial Anova for Proportions , 1978 .

[5]  P. McCullagh.  Regression Models for Ordinal Data , 1980 .

[6]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[7]  D. Bystrak,et al.  The north american breeding bird survey. , 1981 .

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

[9]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

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

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

[12]  K Y Liang,et al.  Longitudinal data analysis for discrete and continuous outcomes. , 1986, Biometrics.

[13]  K. Liang,et al.  Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions , 1987 .

[14]  Robert D. Gibbons,et al.  Trend in correlated proportions , 1987 .

[15]  H. Goldstein.  Multilevel Statistical Models , 2006 .

[16]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .

[17]  N M Laird,et al.  Missing data in longitudinal studies. , 1988, Statistics in medicine.

[18]  P. Albert,et al.  Models for longitudinal data: a generalized estimating equation approach. , 1988, Biometrics.

[19]  M. Conaway.  Analysis of Repeated Categorical Measurements with Conditional Likelihood Methods , 1989 .

[20]  D. Cox,et al.  Analysis of Binary Data (2nd ed.). , 1990 .

[21]  Mark R. Conaway,et al.  A random effects model for binary data , 1990 .

[22]  B W Turnbull,et al.  Analysis of multi-type recurrent events in longitudinal studies; application to a skin cancer prevention trial. , 1990, Biometrics.

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

[24]  G. Robinson.  That BLUP is a Good Thing: The Estimation of Random Effects , 1991 .

[25]  R. Schall.  Estimation in generalized linear models with random effects , 1991 .

[26]  S. Zeger,et al.  Multivariate Regression Analyses for Categorical Data , 1992 .

[27]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[28]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[29]  Anthony S. Bryk,et al.  Hierarchical Linear Models: Applications and Data Analysis Methods , 1992 .

[30]  Jim Albert,et al.  A Bayesian Analysis of a Poisson Random Effects Model for Home Run Hitters , 1992 .

[31]  W. Gilks,et al.  Random-effects models, for longitudinal data using Gibbs sampling. , 1993, Biometrics.

[32]  T. Stukel,et al.  Comparison of methods for the analysis of longitudinal interval count data. , 1993, Statistics in medicine.

[33]  John B. Willett,et al.  It’s About Time: Using Discrete-Time Survival Analysis to Study Duration and the Timing of Events , 1993 .

[34]  Stephen W. Raudenbush,et al.  A Crossed Random Effects Model for Unbalanced Data With Applications in Cross-Sectional and Longitudinal Research , 1993 .

[35]  John A. Nelder,et al.  Generalized linear models. 2nd ed. , 1993 .

[36]  Russell D. Wolfinger,et al.  Laplace's approximation for nonlinear mixed models. , 1993 .

[37]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[38]  D. Hedeker,et al.  Effects of social support and relapse prevention training as adjuncts to a televised smoking-cessation intervention. , 1993, Journal of consulting and clinical psychology.

[39]  S. K. Van Den Eeden,et al.  A conditional analysis for two-treatment multiple-period crossover designs with binomial or poisson outcomes and subjects who drop out. , 1993, Statistics in medicine.

[40]  David H. Uttal,et al.  Subject-specific and Population-averaged Continuation Ratio Logit Models for Multiple Discrete Time Survival Profiles , 1994 .

[41]  D. Hedeker,et al.  A random-effects ordinal regression model for multilevel analysis. , 1994, Biometrics.

[42]  C. Mcgilchrist.  Estimation in Generalized Mixed Models , 1994 .

[43]  S. Lipsitz,et al.  Performance of generalized estimating equations in practical situations. , 1994, Biometrics.

[44]  W. Link,et al.  Observer differences in the North American Breeding Bird Survey , 1994 .

[45]  C. Geyer.  Estimating Normalizing Constants and Reweighting Mixtures , 1994 .

[46]  A. Keen,et al.  A simple approach for the analysis of generalizea linear mixed models , 1994 .

[47]  George Casella,et al.  Estimation with Selected Binomial Information or do you Really Believe that Dave Winfield is Batting .471 , 1994 .

[48]  C. McCulloch.  Maximum Likelihood Variance Components Estimation for Binary Data , 1994 .

[49]  D. Stram,et al.  Variance components testing in the longitudinal mixed effects model. , 1994, Biometrics.

[50]  A. Welsh,et al.  ASYMPTOTIC PROPERTIES OF RESTRICTED MAXIMUM LIKELIHOOD (REML) ESTIMATES FOR HIERARCHICAL MIXED LINEAR MODELS , 1994 .

[51]  A. Kuk.  Asymptotically Unbiased Estimation in Generalized Linear Models with Random Effects , 1995 .

[52]  G. Fitzmaurice,et al.  A caveat concerning independence estimating equations with multivariate binary data. , 1995, Biometrics.

[53]  Noreen Goldman,et al.  An assessment of estimation procedures for multilevel models with binary responses , 1995 .

[54]  N. Breslow,et al.  Bias correction in generalised linear mixed models with a single component of dispersion , 1995 .

[55]  C. McCulloch,et al.  A Note on the Existence of the Posterior Distribution for a Class of Mixed Models for Binomial Responses , 1995 .

[56]  P. Diggle.  Analysis of Longitudinal Data , 1995 .

[57]  T R Ten Have,et al.  A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses. , 1996, Biometrics.

[58]  Harvey Goldstein,et al.  Improved Approximations for Multilevel Models with Binary Responses , 1996 .

[59]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[60]  M. Lindstrom,et al.  A survey of methods for analyzing clustered binary response data , 1996 .

[61]  P. Cortesi,et al.  Distribution and diversity of vegetative compatibility types in subpopulations of Cryphonectria parasitica in Italy , 1996 .

[62]  N. Breslow,et al.  Bias Correction in Generalized Linear Mixed Models with Multiple Components of Dispersion , 1996 .

[63]  T. T. Have.  A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses. , 1996 .

[64]  David G. Clayton,et al.  Generalized linear mixed models , 1996 .

[65]  Jeroen K. Vermunt,et al.  Log-Linear Models for Event Histories , 1997 .

[66]  D. Hedeker,et al.  Random effects probit and logistic regression models for three-level data. , 1997, Biometrics.

[67]  M. Segal,et al.  An Assessment of Approximate Maximum Likelihood Estimators in Generalized Linear Mixed Models , 1997 .

[68]  C. McCulloch.  Maximum Likelihood Algorithms for Generalized Linear Mixed Models , 1997 .

[69]  P. Diggle,et al.  Analysis of Longitudinal Data. , 1997 .

[70]  J. Kalbfleisch,et al.  Between- and within-cluster covariate effects in the analysis of clustered data. , 1998, Biometrics.

[71]  P. Cortesi,et al.  Genetics of Vegetative Incompatibility inCryphonectria parasitica , 1998, Applied and Environmental Microbiology.

[72]  J K Lindsey,et al.  On the appropriateness of marginal models for repeated measurements in clinical trials. , 1998, Statistics in medicine.

[73]  D. Hedeker,et al.  A Multilevel Thresholds of Change Model for Analysis of Stages of Change Data , 1998 .

[74]  J. Booth,et al.  Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm , 1999 .

[75]  Jun S. Liu,et al.  Monte Carlo EM with importance reweighting and its applications in random effects models 1 1 This wo , 1999 .

[76]  D. Hedeker.  MIXNO: a computer program for mixed-effects nominal logistic regression , 1999 .

[77]  Roel Bosker,et al.  Multilevel analysis : an introduction to basic and advanced multilevel modeling , 1999 .

[78]  J. Booth,et al.  2. Random-Effects Modeling of Categorical Response Data , 2000 .

[79]  D. Hall.  Zero‐Inflated Poisson and Binomial Regression with Random Effects: A Case Study , 2000, Biometrics.

[80]  S. Raudenbush,et al.  Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation , 2000 .

[81]  A. Agresti,et al.  Multinomial logit random effects models , 2001 .

[82]  Ulf Böckenholt,et al.  Mixed-effects analyses of rank-ordered data , 2001 .

[83]  Joseph L Schafer,et al.  A Two-Part Random-Effects Model for Semicontinuous Longitudinal Data , 2001 .

[84]  S. R. Searle,et al.  Generalized, Linear, and Mixed Models , 2005 .

[85]  P. Heagerty,et al.  Misspecified maximum likelihood estimates and generalised linear mixed models , 2001 .

[86]  Stan Lipovetsky,et al.  Latent Variable Models and Factor Analysis , 2001, Technometrics.

[87]  A. Agresti,et al.  Modeling Clustered Ordered Categorical Data: A Survey , 2001 .

[88]  S. Rabe-Hesketh,et al.  Reliable Estimation of Generalized Linear Mixed Models using Adaptive Quadrature , 2002 .

[89]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[90]  Eric R. Ziegel,et al.  Multivariate Statistical Modelling Based on Generalized Linear Models , 2002, Technometrics.

[91]  Thomas A. Louis,et al.  Matching conditional and marginal shapes in binary random intercept models using a bridge distribution function , 2003 .

[92]  Debashis Ghosh,et al.  Impact of diabetes on cognitive function among older Latinos: a population-based cohort study. , 2003, Journal of clinical epidemiology.

[93]  Francis Tuerlinckx,et al.  A nonlinear mixed model framework for item response theory. , 2003, Psychological methods.

[94]  Sophia Rabe-Hesketh,et al.  Multilevel logistic regression for polytomous data and rankings , 2003 .

[95]  Herwig Friedl,et al.  Negative binomial loglinear mixed models , 2003 .

[96]  D. Hedeker.  A mixed‐effects multinomial logistic regression model , 2003, Statistics in medicine.

[97]  Mark Von Tress,et al.  Generalized, Linear, and Mixed Models , 2003, Technometrics.

[98]  S. Rabe-Hesketh,et al.  Generalized multilevel structural equation modeling , 2004 .

[99]  Sreenivasan Ravi,et al.  Multilevel Statistical Models, 3rd edn , 2005 .

[100]  Risto Lehtonen,et al.  Multilevel Statistical Models , 2005 .

[101]  Charles E. McCulloch,et al.  Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models , 2005 .

[102]  S. Rabe-Hesketh,et al.  Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects , 2005 .

[103]  Sophia Rabe-Hesketh,et al.  Multilevel and Longitudinal Modeling Using Stata , 2005 .

[104]  Stan Lipovetsky,et al.  Generalized Latent Variable Modeling: Multilevel,Longitudinal, and Structural Equation Models , 2005, Technometrics.

[105]  J. Pinheiro,et al.  Efficient Laplacian and Adaptive Gaussian Quadrature Algorithms for Multilevel Generalized Linear Mixed Models , 2006 .

[106]  Donald Hedeker,et al.  Longitudinal Data Analysis , 2006 .

[107]  Sophia Rabe-Hesketh,et al.  Redundant Overdispersion Parameters in Multilevel Models for Categorical Responses , 2007 .

[108]  U L F BCKENHOLT,et al.  MIXED-EFFECTS ANALYSES OF RANK-ORDERED DATA , 2022 .