Multilevel Statistical Models

Harvey Goldstein, Professor of social sciences, University of Bristol and Associate Editor for the Statistical Modelling Journal, and previous Editor of the Royal Statistical Society's Journal, Series A.

"This book is suitable as a comprehensive text for postgraduate courses, as well as a general reference guide. Applied statisticians in the social sciences, economics, biological and medical disciplines will find this book beneficial. See the review of the third edition." (Zentralblatt MATH, 1 December 2013)"This book would also serve as an outstanding general reference on multilevel models, since it offers concise and easy to follow descriptions of the various multilevel models and their applications, in addition to the references on which this work is based. I really enjoyed reading this book, and am sure that others will have a similar pleasurable experience." (Journal of Biopharmaceutical Statistics (JBS), 2012)

• ContentsDedicationPrefaceAcknowledgementsNotationA general classification notation and diagramGlossaryChapter 1 An introduction to multilevel models1.1 Hierarchically structured data1.2 School effectiveness1.3 Sample survey methods1.4 Repeated measures data1.5 Event history and survival models1.6 Discrete response data1.7 Multivariate models1.8 Nonlinear models1.9 Measurement errors1.10 Cross classifications and multiple membership structures.1.11 Factor analysis and structural equation models1.12 Levels of aggregation and ecological fallacies1.13 Causality1.14 The latent normal transformation and missing data1.15 Other texts1.16 A caveat Chapter 2 The 2-level model2.1 Introduction2.2 The 2-level model2.3 Parameter estimation2.4 Maximum likelihood estimation using Iterative Generalised Least Squares (IGLS)2.5 Marginal models and Generalized Estimating Equations (GEE)2.6 Residuals2.7 The adequacy of Ordinary Least Squares estimates.2.8 A 2-level example using longitudinal educational achievement data2.9 General model diagnostics2.10 Higher level explanatory variables and compositional effects2.11 Transforming to normality2.12 Hypothesis testing and confidence intervals2.13 Bayesian estimation using Markov Chain Monte Carlo (MCMC)2.14 Data augmentationAppendix 2.1 The general structure and maximum likelihood estimation for a multilevel modelAppendix 2.2 Multilevel residuals estimationAppendix 2.3 Estimation using profile and extended likelihoodAppendix 2.4 The EM algorithmAppendix 2.5 MCMC samplingChapter 3. Three level models and more complex hierarchical structures.3.1 Complex variance structures3.2 A 3-level complex variation model example.3.3 Parameter Constraints3.4 Weighting units3.5 Robust (Sandwich) Estimators and Jacknifing3.6 The bootstrap3.7 Aggregate level analyses3.8 Meta analysis3.9 Design issuesChapter 4. Multilevel Models for discrete response data4.1 Generalised linear models4.2 Proportions as responses4.3 Examples4.4 Models for multiple response categories4.5 Models for counts4.6 Mixed discrete - continuous response models4.7 A latent normal model for binary responses4.8 Partitioning variation in discrete response modelsAppendix 4.1. Generalised linear model estimationAppendix 4.2 Maximum likelihood estimation for generalised linear modelsAppendix 4.3 MCMC estimation for generalised linear modelsAppendix 4.4. Bootstrap estimation for generalised linear modelsChapter 5. Models for repeated measures data5.1 Repeated measures data5.2 A 2-level repeated measures model5.3 A polynomial model example for adolescent growth and the prediction of adult height5.4 Modelling an autocorrelation structure at level 1.5.5 A growth model with autocorrelated residuals5.6 Multivariate repeated measures models5.7 Scaling across time5.8 Cross-over designs5.9 Missing data5.10 Longitudinal discrete response dataChapter 6. Multivariate multilevel data6.1 Introduction6.2 The basic 2-level multivariate model6.3 Rotation Designs6.4 A rotation design example using Science test scores6.5 Informative response selection: subject choice in examinations6.6 Multivariate structures at higher levels and future predictions6.7 Multivariate responses at several levels6.8 Principal Components analysisAppendix 6.1 MCMC algorithm for a multivariate normal response model with constraintsChapter 7. Latent normal models for multivariate data7.1 The normal multilevel multivariate model7.2 Sampling binary responses7.3 Sampling ordered categorical responses7.4 Sampling unordered categorical responses7.5 Sampling count data7.6 Sampling continuous non-normal data7.7 Sampling the level 1 and level 2 covariance matrices7.8 Model fit7.9 Partially ordered data7.10 Hybrid normal/ordered variables7.11 DiscussionChapter 8. Multilevel factor analysis, structural equation and mixture models8.1 A 2-stage 2-level factor model8.2 A general multilevel factor model8.3 MCMC estimation for the factor model8.4 Structural equation models8.5 Discrete response multilevel structural equation models8.6 More complex hierarchical latent variable models8.7 Multilevel mixture models Chapter 9. Nonlinear multilevel models9.1 Introduction9.2 Nonlinear functions of linear components9.3 Estimating population means9.4 Nonlinear functions for variances and covariances9.5 Examples of nonlinear growth and nonlinear level 1 varianceAppendix 9.1 Nonlinear model estimationChapter 10. Multilevel modelling in sample surveys10.1 Sample survey structures10.2 Population structures10.3 Small area estimation Chapter 11 Multilevel event history and survival models11.1 Introduction11.2 Censoring11.3 Hazard and survival funtions11.4 Parametric proportional hazard models11.5 The semiparametric Cox model11.6 Tied observations11.7 Repeated events proportional hazard models11.8 Example using birth interval data11.9 Log duration models11.10 Examples with birth interval data and children’s activity episodes11.11 The grouped discrete time hazards model11.12 Discrete time latent normal event history modelsChapter 12. Cross classified data structures12.1 Random cross classifications12.2 A basic cross classified model12.3 Examination results for a cross classification of schools12.4 Interactions in cross classifications12.5 Cross classifications with one unit per cell12.6 Multivariate cross classified models12.7 A general notation for cross classifications12.8 MCMC estimation in cross classified modelsAppendix 12.1 IGLS Estimation for cross classified data.Chapter 13 Multiple membership models13.1 Multiple membership structures13.2 Notation and classifications for multiple membership structures13.3 An example of salmonella infection13.4 A repeated measures multiple membership model13.5 Individuals as higher level units13.5.1 Example of research grant awards13.6 Spatial models13.7 Missing identification modelsAppendix 13.1 MCMC estimation for multiple membership models.Chapter 14 Measurement errors in multilevel models14.1 A basic measurement error model14.2 Moment based estimators14.3 A 2-level example with measurement error at both levels.14.4 Multivariate responses14.5 Nonlinear models14.6 Measurement errors for discrete explanatory variables14.7 MCMC estimation for measurement error modelsAppendix 14.1 Measurement error estimation14.2 MCMC estimation for measurement error modelsChapter 15. Smoothing models for multilevel data.15.1 Introduction15.2. Smoothing estimators15.3 Smoothing splines15.4 Semi parametric smoothing models15.5 Multilevel smoothing models15.6 General multilevel semi-parametric smoothing models15.7 Generalised linear models15.8 An exampleFixedRandom15.9 ConclusionsChapter 16. Missing data, partially observed data and multiple imputation16.1 Creating a completed data set16.2 Joint modelling for missing data16.3 A two level model with responses of different types at both levels.16.4 Multiple imputation16.5 A simulation example of multiple imputation for missing data16.6 Longitudinal data with attrition16.7 Partially known data values16.8 ConclusionsChapter 17 Multilevel models with correlated random effects17.1 Non-independence of level 2 residuals17.2 MCMC estimation for non-independent level 2 residuals17.3 Adaptive proposal distributions in MCMC estimation17.4 MCMC estimation for non-independent level 1 residuals17.5 Modelling the level 1 variance as a function of explanatory variables with random effects17.6 Discrete responses with correlated random effects17.7 Calculating the DIC statistic17.8 A growth data set17.9 ConclusionsChapter 18. Software for multilevel modellingReferencesAuthor indexSubject index