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1 Why? 1
1.1 What is multilevel regression modeling? 1
1.2 Some examples from our own research 3
1.3 Motivations for multilevel modeling 6
2 Concepts and methods from basic probability and statistics 13
2.1 Probability distributions 13
2.2 Statistical inference 16
2.3 Classical confidence intervals 18
2.4 Classical hypothesis testing 20
2.5 Problems with statistical significance 22
2.6 55,000 residents desperately need your help! 23
2.7 Bibliographic note 26
Part 1A: Single-level regression 29
3 Linear regression: the basics 31
3.1 One predictor 31
3.2 Multiple predictors 32
3.3 Interactions 34
3.4 Statistical inference 37
3.5 Graphical displays of data and fitted model 42
3.6 Assumptions and diagnostics 45
3.7 Prediction and validation 47
4 Linear regression: before and after fitting the model 53
4.1 Linear transformations 53
4.2 Centering and standardizing, especially for models with interactions 55
4.3 Correlation and “regression to the mean” 57
4.4 Logarithmic transformations 59
4.5 Other transformations 65
4.6 Building regression models for prediction 68
4.7 Fitting a series of regressions 73
5 Logistic regression 79
5.1 Logistic regression with a single predictor 79
5.2 Interpreting the logistic regression coefficients 81
5.3 Latent-data formulation 85
5.4 Building a logistic regression model: wells in Bangladesh 86
5.5 Logistic regression with interactions 92
5.6 Evaluating, checking, and comparing fitted logistic regressions 97
5.7 Average predictive comparisons on the probability scale 101
5.8 Identifiability and separation 104
6 Generalized linear models 109
6.1 Introduction 109
6.2 Poisson regression, exposure, and overdispersion 110
6.3 Logistic-binomial model 116
6.4 Probit regression: normally distributed latent data 118
6.5 Multinomial regression 119
6.6 Robust regression using the t model 124
6.7 Building more complex generalized linear models 125
6.8 Constructive choice models 127
7 Simulation of probability models and statistical inferences 137
7.1 Simulation of probability models 137
7.2 Summarizing linear regressions using simulation: an informal
Bayesian approach 140
7.3 Simulation for nonlinear predictions: congressional elections 144
7.4 Predictive simulation for generalized linear models 148
8 Simulation for checking statistical procedures and model fits 155
8.1 Fake-data simulation 155
8.2 Example: using fake-data simulation to understand residual plots 157
8.3 Simulating from the fitted model and comparing to actual data 158
8.4 using predictive simulation to check the fit of a time-series model 163
9 Causal inference using regression on the treatment variable 167
9.1 Causal inference and predictive comparisons 167
9.2 The fundamental problem of causal inference 170
9.3 Randomized experiments 172
9.4 Treatment interactions and poststratification 178
9.5 Observational studies 181
9.6 Understanding causal inference in observational studies 186
9.7 Do not control for post-treatment variables 188
9.8 Intermediate outcomes and causal paths 190
10 Causal inference using more advanced models 199
10.1 Imbalance and lack of complete overlap 199
10.2 Subclassification: effects and estimates for different subpopulations 204
10.3 Matching: subsetting the data to get overlapping and balanced
treatment and control groups 206
10.4 Lack of overlap when the assignment mechanism is known:
regression discontinuity 212
10.5 Estimating causal effects indirectly using instrumental variables 215
10.6 Instrumental variables in a regression framework 220
10.7 Identification strategies that make use of variation within or between
Part 2A: Multilevel regression 235
11 Multilevel structures 237
11.1 Varying-intercept and varying-slope models 237
11.2 Clustered data: child support enforcement in cities 237
11.3 Repeated measurements, time-series cross sections, and other
non-nested structures 241
11.4 Indicator variables and fixed or random effects 244
11.5 Costs and benefits of multilevel modeling 246
12 Multilevel linear models: the basics 251
12.1 Notation 251
12.2 Partial pooling with no predictors 252
12.3 Partial pooling with predictors 254
12.4 Quickly fitting multilevel models in R 259
12.5 Five ways to write the same model 262
12.6 Group-level predictors 265
12.7 Model building and statistical significance 270
12.8 Predictions for new observations and new groups 272
12.9 How many groups and how many observations per group are
needed to fit a multilevel model? 275
13 Multilevel linear models: varying slopes, non-nested models, and
other complexities 279
13.1 Varying intercepts and slopes 279
13.2 Varying slopes without varying intercepts 283
13.3 Modeling multiple varying coefficients using the scaled inverse-
Wishart distribution 284
13.4 Understanding correlations between group-level intercepts and
slopes 287
13.5 Non-nested models 289
13.6 Selecting, transforming, and combining regression inputs 293
13.7 More complex multilevel models 297
14 Multilevel logistic regression 301
14.1 State-level opinions from national polls 301
14.2 Red states and blue states: what’s the matter with Connecticut? 310
14.3 Item-response and ideal-point models 314
14.4 Non-nested overdispersed model for death sentence reversals 320
15 Multilevel generalized linear models 325
15.1 Overdispersed Poisson regression: police stops and ethnicity 325
15.2 Ordered categorical regression: storable votes 331
15.3 Non-nested negative-binomial model of structure in social networks 332
Part 2B: Fitting multilevel models 343
16 Multilevel modeling in Bugs and R: the basics 345
16.1 Why you should learn Bugs 345
16.2 Bayesian inference and prior distributions 345
16.3 Fitting and understanding a varying-intercept multilevel model
using R and Bugs 348
16.4 Step by step through a Bugs model, as called from R 353
16.5 Adding individual- and group-level predictors 359
16.6 Predictions for new observations and new groups 361
16.7 Fake-data simulation 363
16.8 The principles of modeling in Bugs 366
16.9 Practical issues of implementation 369
16.10 Open-ended modeling in Bugs 370
17 Fitting multilevel linear and generalized linear models in Bugs
and R 375
17.1 Varying-intercept, varying-slope models 375
17.2 Varying intercepts and slopes with group-level predictors 379
17.3 Non-nested models 380
17.4 Multilevel logistic regression 381
17.5 Multilevel Poisson regression 382
17.6 Multilevel ordered categorical regression 383
17.7 Latent-data parameterizations of generalized linear models 384
18 Likelihood and Bayesian inference and computation 387
18.1 Least squares and maximum likelihood estimation 387
18.2 Uncertainty estimates using the likelihood surface 390
18.3 Bayesian inference for classical and multilevel regression 392
18.4 Gibbs sampler for multilevel linear models 397
18.5 Likelihood inference, Bayesian inference, and the Gibbs sampler:
the case of censored data 402
18.6 Metropolis algorithm for more general Bayesian computation 408
18.7 Specifying a log posterior density, Gibbs sampler, and Metropolis
algorithm in R 409
19 Debugging and speeding convergence 415
19.1 Debugging and confidence building 415
19.2 General methods for reducing computational requirements 418
19.3 Simple linear transformations 419
19.4 Redundant parameters and intentionally nonidentifiable models 419
19.5 Parameter expansion: multiplicative redundant parameters 424
19.6 Using redundant parameters to create an informative prior
distribution for multilevel variance parameters 427
Part 3: From data collection to model understanding to model
checking 435
20 Sample size and power calculations 437
20.1 Choices in the design of data collection 437
20.2 Classical power calculations: general principles, as illustrated by
estimates of proportions 439
20.3 Classical power calculations for continuous outcomes 443
20.4 Multilevel power calculation for cluster sampling 447
20.5 Multilevel power calculation using fake-data simulation 449
21 Understanding and summarizing the fitted models 457
21.1 Uncertainty and variability 457
21.2 Superpopulation and finite-population variances 459
21.3 Contrasts and comparisons of multilevel coefficients 462
21.4 Average predictive comparisons 466
21.5 R2 and explained variance 473
21.6 Summarizing the amount of partial pooling 477
21.7 Adding a predictor can increase the residual variance! 480
21.8 Multiple comparisons and statistical significance 481
22 Analysis of variance 487
22.1 Classical analysis of variance 487
22.2 ANOVA and multilevel linear and generalized linear models 490
22.3 Summarizing multilevel models using ANOVA 492
22.4 Doing ANOVA using multilevel models 494
22.5 Adding predictors: analysis of covariance and contrast analysis 496
22.6 Modeling the variance parameters: a split-plot latin square 498
23 Causal inference using multilevel models 503
23.1 Multilevel aspects of data collection 503
23.2 Estimating treatment effects in a multilevel observational study 506
23.3 Treatments applied at different levels 507
23.4 Instrumental variables and multilevel modeling 509
24 Model checking and comparison 513
24.1 Principles of predictive checking 513
24.2 Example: a behavioral learning experiment 515
24.3 Model comparison and deviance 524
25 Missing-data imputation 529
25.1 Missing-data mechanisms 530
25.2 Missing-data methods that discard data 531
25.3 Simple missing-data approaches that retain all the data 532
25.4 Random imputation of a single variable 533
25.5 Imputation of several missing variables 539
25.6 Model-based imputation 540
25.7 Combining inferences from multiple imputations 542
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