# Multilevel Statistical Models: 3 rd edition, 2003 Contents

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1 Multilevel Statistical Models: 3 rd edition, 2003 Contents Preface Acknowledgements Notation Two and three level models. A general classification notation and diagram Glossary Chapter 1 An introduction to multilevel models 1.1 Hierarchically structured data 1.2 School effectiveness 1.3 Sample survey methods 1.4 Repeated measures data 1.5 Event history models 1.6 Discrete response data 1.7 Multivariate models 1.8 Nonlinear models 1.9 Measurement errors 1.10 Random cross classifications and multiple membership structures Factor analysis and structural equation models 1.12 Levels of aggregation and ecological fallacies 1.13 Causality 1.14 Other references 1.15 A caveat Chapter 2 The basic 2-level model 2.1 Introduction 2.2 The 2-level model 2.3 Parameter estimation The variance components model The general 2-level model with random coefficients 2.4 Maximum likelihood estimation using Iterative Generalised Least Squares (IGLS) 2.5 Marginal models and Generalized Estimating Equations (GEE)

2 2.6 Residuals 2.7 The adequacy of Ordinary Least Squares estimates. 2.8 A 2-level example using longitudinal educational achievement data Checking for outlying units Model checking using estimated residuals 2.9 General model diagnostics 2.10 Higher level explanatory variables and compositional effects 2.11 Transforming to Normality 2.12 Hypothesis testing and confidence intervals Fixed parameters Random parameters Hypothesis testing for non-nested models Inferences for residual estimates 2.13 Bayesian estimation using Markov Chain Monte Carlo (MCMC) Gibbs sampling Metropolis Hastings (MH) sampling Convergence of MCMC chains Making inferences An example 2.14 Data augmentation Appendix 2.1 The general structure and maximum likelihood estimation for a multilevel model Appendix 2.2 Multilevel residuals estimation Shrunken estimates Delta method estimators for the covariance matrix of residuals Appendix 2.3 The EM algorithm Appendix 2.4 MCMC sampling Gibbs sampling Metropolis Hastings (MH) sampling Hierarchical Centering Chapter 3. Three level models and more complex hierarchical structures. 3.1 Complex variance structures Partitioning the variance and intra-unit correlation Variances for subgroups defined at level Variance as a function of predicted value

3 3.1.4 Variances for subgroups defined at higher levels 3.2 A 3-level complex variation model. 3.3 Parameter Constraints 3.4 Weighting units Weighted residuals 3.5 Robust (Sandwich) Estimators and Jacknifing 3.6 The bootstrap The fully Non-parametric bootstrap The fully parametric bootstrap The iterated parametric bootstrap and bias correction The residuals bootstrap 3.7 Aggregate level analyses Inferences about residuals from aggregate level analyses 3.6 Meta analysis Aggregate and mixed level analysis Defining origin and scale An example: meta analysis of class size data Practical issues 3.7 Design issues Chapter 4. Multilevel Models for discrete response data 4.1 Generalised linear models 4.1 Proportions as responses 4.2 An example from a fertility survey 4.3 Models for multiple response categories 4.4 Models for counts 4.5 Ordered responses 4.6 Mixed discrete - continuous response models 4.7 A latent variable model for binary and ordered responses 4.8 Partitioning variation in discrete response models Model linearisation (Method A) Simulation (Method B) A binary linear model (Method C) A latent variable approach (Method D) An Example

4 Appendix 4.1. Generalised linear model estimation Approximate quasilikelihood estimates Differentials for some discrete response models The Logit - Binomial model The Logit - Multinomial (Multivariate Logit) model The Log - Poisson model The log log - Binomial model Appendix 4.2 Maximum likelihood estimation for generalised linear models Simulated maximum likelihood estimation Residuals Cross-classifications and multiple membership models Computing issues Maximum likelihood estimation via Quadrature Appendix 4.3 MCMC estimation for generalised linear models MH sampling Latent variable models for binary data Multicategory ordered responses Proportions as responses Appendix 4.4. Bootstrap estimation for generalised linear models The iterated bootstrap Chapter 5. Models for Repeated measures Data 5.1 Repeated measures data 5.2 A 2-level repeated measures model 5.3 A polynomial model example for adolescent growth and the prediction of adult height 5.4 Modelling an autocorrelation structure at level A growth model with autocorrelated residuals 5.6 Multivariate repeated measures models 5.7 Scaling across time 5.8 Cross-over designs 5.9 Missing data 5.10 Longitudinal discrete response data Chapter 6 Multivariate multilevel data 6.1 Introduction 6.2 The basic 2-level multivariate model

5 6.3 Rotation Designs 6.4 A rotation design example using Science test scores 6.5 Informative subject choice in examinations 6.5 Principal Components analysis Chapter 7 Multilevel factor analysis and structural equation models 7.1 A two stage 2 level factor model 7.2 A general multilevel factor model 7.3 MCMC estimation for the factor model A two level factor example 7.4 Structural equation models 7.5 Discrete response multilevel structural equation models Chapter 8. Nonlinear multilevel models 8.1 Introduction 8.2 Nonlinear functions of linear components 8.3 Estimating population means 8.4 Nonlinear functions for variances and covariances 8.5 Examples of nonlinear growth and nonlinear level 1 variance 8.6 Multivariate Nonlinear Models Appendix 8.1 Nonlinear model estimation Modelling variances and covariances as nonlinear functions Likelihood values Chapter 9. Multilevel modelling in sample surveys 9.1 Sample survey structures 9.2 Population structures Superpopulations Finite population inference 9.3 Small area estimation Information at domain level only Longitudinal data Multivariate responses Chapter 10 Multilevel event history models 10.1 Introduction 10.2 Censoring 10.3 Hazard and survival funtions

6 10.4 Parametric proportional hazard models 10.5 The semiparametric Cox model 10.6 Tied observations 10.7 Repeated measures proportional hazard models 10.8 Example using birth interval data 10.9 Log duration models Censored data Infinite durations Examples with birth interval data and children s activity episodes The discrete time (piecewise) proportional hazards model A 2-level repeated measures discrete time event history model Partnership data example General discrete time event history models Chapter 11. Cross classified data structures 11.1 Random cross classifications 11.2 A basic cross classified model 11.3 Examination results for a cross classification of schools 11.4 Interactions in cross classifications 11.5 Cross classifications with one unit per cell 11.6 Multivariate cross classified models 11.7 A general notation for cross classifications 11.8 MCMC estimation in cross classified models Appendix 11.1 IGLS Estimation for cross classified data An efficient IGLS algorithm Computational considerations Chapter 12 Multiple membership models 12.1 Multiple membership structures 12.2 Notation and classifications for multiple membership structures 12.3 An example of salmonella infection 12.4 A repeated measures multiple membership model 12.5 Individuals as higher level units 12.6 Spatial models 12.7 Missing identification models Chapter 13 Measurement errors in multilevel models

7 13.1 A basic measurement error model 13.2 Moment based estimators Measurement errors in level 1 variables Measurement errors in higher level variables 13.3 A 2-level example with measurement error at both levels 13.4 Multivariate responses 13.5 Nonlinear models 13.6 Measurement errors for discrete explanatory variables 13.7 MCMC estimation for measurement error models Appendix 13.1 Measurement errors Moment based estimators for a basic 2-level Model Parameter estimation Random coefficients for explanatory variables measured with error Nonlinear models 13.2 MCMC estimation for measurement error models Chapter 14. Missing data in multilevel models 14.1 A multivariate model for missing data 14.2 Creating a completed data set 14.3 Multiple imputation and error corrections 14.4 Discrete variables with missing data An example with missing data 14.6 MCMC estimation for missing data 14.7 Informatively missing data Chapter 15. Software for multilevel modelling, resources and further developments 15.1 Software packages and resources 15.2 Further developments References

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