Generalized least squares (GLS) estimates of the level-2 coefficients,

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1 Contents 1 Conceptual and Statistical Background for Two-Level Models The general two-level model Level-1 model Level-2 model Parameter estimation Empirical Bayes ("EB") estimates of randomly varying level-1 coefficients, q j Generalized least squares (GLS) estimates of the level-2 coefficients, qs Maximum likelihood estimates of variance and covariance components, at level 1, and T at level Some other useful statistics Hypothesis testing Restricted versus full maximum likelihood Generalized Estimating Equations Working with HLM Constructing the MDM file from raw data Executing analyses based on the MDM file Model checking based on the residual file Windows, interactive, and batch execution An example using HLM2 in Window mode Constructing the MDM file from raw data Executing analyses based on the MDM file Annotated HLM2 output Model checking based on the residual file Handling of missing data The Basic Model Specifications - HLM2 dialog box Other analytic options Controlling the iterative procedure Estimation control Constraints on the fixed effects To put constraints on fixed effects Modeling heterogeneity of level-1 variances Specifying level-1 deletion variables Using design weights Hypothesis testing Output options Models without a level-1 intercept Coefficients having a random effect with no corresponding fixed effect Exploratory analysis of potential level-2 predictors

2 3 Conceptual and Statistical Background for Three-Level Models The general three-level model Level-1 model Level-2 model Level-3 model Parameter estimation Hypothesis testing Working with HLM An example using HLM3 in Windows mode Constructing the MDM file from raw data Executing analyses based on the MDM file An annotated example of HLM3 output Model checking based on the residual files Specification of a conditional model Other program features Basic specifications Iteration control Estimation settings Hypothesis testing Output settings Conceptual and Statistical Background for Hierarchical Generalized Linear Models (HGLM) The two-level HLM as a special case of HGLM Level-1 sampling model Level-1 link function Level-1 structural model Two- and three-level models for binary outcomes Level-1 sampling model Level-1 link function Level-1 structural model Level-2 and Level-3 models The model for count data Level-1 sampling model Level-1 link function Level-1 structural model Level-2 model The model for multinomial data Level-1 sampling model Level-1 link function Level-1 structural model Level-2 model The model for ordinal data Level-1 sampling model Level-1 structural model Parameter estimation Estimation via PQL Properties of the estimators Parameter estimation: A high-order Laplace approximation of maximum likelihood Unit-specific and population-average models

3 5.8 Over-dispersion and under-dispersion Restricted versus full PQL versus full ML Hypothesis testing Fitting HGLMs (Nonlinear Models) Executing nonlinear analyses based on the MDM file Case 1: a Bernoulli model Case 2: a binomial model (number of trials, m i j 1) Case 3: Poisson model with equal exposure Case 4: Poisson model with variable exposure Case 5: Multinomial model Case 6: Ordinal model Additional features Over-dispersion Laplace approximation for binary outcome models Printing variance-covariance matrices for fixed effects Fitting HGLMs with three levels Conceptual and Statistical Background for Hierarchical Multivariate Linear Models (HMLM) Unrestricted model Level-1 model Level-2 model Combined model HLM with homogenous level-1 variance Level-1 model Level-2 model Combined model HLM with varying level-1 variance HLM with a log-linear model for the level-1 variance First-order auto-regressive model for the level-1 residuals HMLM2: A multilevel, multivariate model Level-1 model Level-2 model Level-3 model The combined model Working with HMLM/HMLM An analysis using HMLM via Windows mode Constructing the MDM from raw data Executing analyses based on the MDM file An annotated example of HMLM An analysis using HMLM2 via Windows mode Executing analyses based on the MDM file

4 9 Special Features Latent variable analysis A latent variable analysis using HMLM: Example A latent variable analysis using HMLM: Example Applying HLM to multiply-imputed data Data with multiply-imputed values for the outcome or one covariate Calculations performed Working with plausible values in HLM Data with multiply-imputed values for the outcome and covariates "V-Known" models for HLM Data input format Creating the MDM file Estimating a V-known model V-known analyses where Q = Conceptual and Statistical Background for Cross-classified Random Effect Models (HCM2) The general cross-classified random effects models Level-1 or "within-cell" model Level-2 or "between-cell" model Parameter estimation Hypothesis testing Working with HCM An example using HCM2 in Windows mode Constructing the MDM file from raw data SPSS input Level-2 row-factor file Level-2 column-factor file Executing analyses based on the MDM file Specification of a conditional model with the effect associated with a row-specific predictor fixed Other program features Graphing Data and Models Data based graphs two level analyses Box and whisker plots Scatter plots Line plots two-level analyses Model-based graphs two level Model graphs Level-1 equation modeling Level-1 residual box-and-whisker plots Level-1 residual vs predicted value Level-1 EB/OLS coefficient confidence intervals Graphing categorical predictors Three-level applications

5 A Using HLM2 in interactive and batch mode A.1 Using HLM2 in interactive mode A.1.1 Example: constructing an MDM file for the HS&B data using SPSS file input A.1.2 Example: constructing an MDM file for the HS&B data using ASCII file input A.2 Rules for format statements A.2.1 Example: Executing an analysis using HSB.MDM A.3 Using HLM in batch and/or interactive mode A.4 Using HLM2 in batch mode A.5 Printing of variance and covariance matrices for fixed effects and level-2 variances 257 A.6 Preliminary exploratory analysis with HLM B Using HLM3 in Interactive and Batch Mode B.1 Using HLM3 in interactive mode B.1.1 Example: constructing an MDM file for the public school data using SPSS file input B.1.2 Example: constructing an MDM file for the HS&B data using ASCII file input B.1.3 Example: Executing an analysis using EG.MDM B.2 Using HLM3 in batch mode B.2.1 Table of keywords and options B.3 Printing of variance and covariance matrices C Using HGLM in Interactive and Batch Mode C.1 Example: Executing an analysis using THAIUGRP.MDM D Using HMLM in Interactive and Batch Mode D.1 Constructing an MDM file D.2 Executing analyses based on MDM files D.2.1 Table of keywords and options D.2.2 Table of HMLM2 keywords and options E Using Special Features in Interactive and Batch Mode E.1 Example: Latent variable analysis using the National Youth Study data sets E.2 A latent variable analysis to run regression with missing data E.3 Commands to apply HLM to multiply-imputed data F Using HCM2 in Interactive and Batch Mode F.1 Using HCM2 in interactive mode F.1.1 Example: constructing an MDM file for the educational attainment data as described in Chapter 11 using SPSS file input F.1.2 Example: Executing an unconditional model analysis using ATTAIN.MDM F.1.3 Example: Executing a conditional model analysis using ATTAIN.MDM F.2 Using HCM2 in batch mode References