Data Analysis Using Regression and Multilevel/Hierarchical Models

by Andrew Gelman , Jennifer Hill

• ISBN13: 9780521686891
• ISBN10: 052168689X
• Edition: 1st
• Format: Paperback
• Copyright: 2006-12-18
• Publisher: Cambridge University Press

Summary

Data Analysis Using Regression and Multilevel/Hierarchical Models is a comprehensive manual for the applied researcher who wants to perform data analysis using linear and nonlinear regression and multilevel models. The book introduces a wide variety of models, whilst at the same time instructing the reader in how to fit these models using available software packages. The book illustrates the concepts by working through scores of real data examples that have arisen from the authors' own applied research, with programming codes provided for each one. Topics covered include causal inference, including regression, poststratification, matching, regression discontinuity, and instrumental variables, as well as multilevel logistic regression and missing-data imputation. Practical tips regarding building, fitting, and understanding are provided throughout.

Author Biography

Jennifer Hill is Assistant Professor of Public Affairs in the Department of International and Public Affairs at Columbia University.

Table of Contents

List of examples	p. xvii
Preface	p. xix
Why?	p. 1
What is multilevel regression modeling?	p. 1
Some examples from our own research	p. 3
Motivations for multilevel modeling	p. 6
Distinctive features of this book	p. 8
Computing	p. 9
Concepts and methods from basic probability and statistics	p. 13
Probability distributions	p. 13
Statistical inference	p. 16
Classical confidence intervals	p. 18
Classical hypothesis testing	p. 20
Problems with statistical significance	p. 22
55,000 residents desperately need your help!	p. 23
Bibliographic note	p. 26
Exercises	p. 26
Single-level regression	p. 29
Linear regression: the basics	p. 31
One predictor	p. 31
Multiple predictors	p. 32
Interactions	p. 34
Statistical inference	p. 37
Graphical displays of data and fitted model	p. 42
Assumptions and diagnostics	p. 45
Prediction and validation	p. 47
Bibliographic note	p. 49
Exercises	p. 49
Linear regression: before and after fitting the model	p. 53
Linear transformations	p. 53
Centering and standardizing, especially for models with interactions	p. 55
Correlation and "regression to the mean"	p. 57
Logarithmic transformations	p. 59
Other transformations	p. 65
Building regression models for prediction	p. 68
Fitting a series of regressions	p. 73
Bibliographic note	p. 74
Exercises	p. 74
Logistic regression	p. 79
Logistic regression with a single predictor	p. 79
Interpreting the logistic regression coefficients	p. 81
Latent-data formulation	p. 85
Building a logistic regression model: wells in Bangladesh	p. 86
Logistic regression with interactions	p. 92
Evaluating, checking, and comparing fitted logistic regressions	p. 97
Average predictive comparisons on the probability scale	p. 101
Identifiability and separation	p. 104
Bibliographic note	p. 105
Exercises	p. 105
Generalized linear models	p. 109
Introduction	p. 109
Poisson regression, exposure, and overdispersion	p. 110
Logistic-binomial model	p. 116
Probit regression: normally distributed latent data	p. 118
Ordered and unordered categorical regression	p. 119
Robust regression using the t model	p. 124
Building more complex generalized linear models	p. 125
Constructive choice models	p. 127
Bibliographic note	p. 131
Exercises	p. 132
Working with regression inferences	p. 135
Simulation of probability models and statistical inferences	p. 137
Simulation of probability models	p. 137
Summarizing linear regressions using simulation: an informal Bayesian approach	p. 140
Simulation for nonlinear predictions: congressional elections	p. 144
Predictive simulation for generalized linear models	p. 148
Bibliographic note	p. 151
Exercises	p. 152
Simulation for checking statistical procedures and model fits	p. 155
Fake-data simulation	p. 155
Example: using fake-data simulation to understand residual plots	p. 157
Simulating from the fitted model and comparing to actual data	p. 158
Using predictive simulation to check the fit of a time-series model	p. 163
Bibliographic note	p. 165
Exercises	p. 165
Causal inference using regression on the treatment variable	p. 167
Causal inference and predictive comparisons	p. 167
The fundamental problem of causal inference	p. 170
Randomized experiments	p. 172
Treatment interactions and poststratification	p. 178
Observational studies	p. 181
Understanding causal inference in observational studies	p. 186
Do not control for post-treatment variables	p. 188
Intermediate outcomes and causal paths	p. 190
Bibliographic note	p. 194
Exercises	p. 194
Causal inference using more advanced models	p. 199
Imbalance and lack of complete overlap	p. 199
Subclassification: effects and estimates for different subpopulations	p. 204
Matching: subsetting the data to get overlapping and balanced treatment and control groups	p. 206
Lack of overlap when the assignment mechanism is known: regression discontinuity	p. 212
Estimating causal effects indirectly using instrumental variables	p. 215
Instrumental variables in a regression framework	p. 220
Identification strategies that make use of variation within or between groups	p. 226
Bibliographic note	p. 229
Exercises	p. 231
Multilevel regression	p. 235
Multilevel structures	p. 237
Varying-intercept and varying-slope models	p. 237
Clustered data: child support enforcement in cities	p. 237
Repeated measurements, time-series cross sections, and other non-nested structures	p. 241
Indicator variables and fixed or random effects	p. 244
Costs and benefits of multilevel modeling	p. 246
Bibliographic note	p. 247
Exercises	p. 248
Multilevel linear models: the basics	p. 251
Notation	p. 251
Partial pooling with no predictors	p. 252
Partial pooling with predictors	p. 254
Quickly fitting multilevel models in R	p. 259
Five ways to write the same model	p. 262
Group-level predictors	p. 265
Model building and statistical significance	p. 270
Predictions for new observations and new groups	p. 272
How many groups and how many observations per group are needed to fit a multilevel model?	p. 275
Bibliographic note	p. 276
Exercises	p. 277
Multilevel linear models: varying slopes, non-nested models, and other complexities	p. 279
Varying intercepts and slopes	p. 279
Varying slopes without varying intercepts	p. 283
Modeling multiple varying coefficients using the scaled inverse-Wishart distribution	p. 284
Understanding correlations between group-level intercepts and slopes	p. 287
Non-nested models	p. 289
Selecting, transforming, and combining regression inputs	p. 293
More complex multilevel models	p. 297
Bibliographic note	p. 297
Exercises	p. 298
Multilevel logistic regression	p. 301
State-level opinions from national polls	p. 301
Red states and blue states: what's the matter with Connecticut?	p. 310
Item-response and ideal-point models	p. 314
Non-nested overdispersed model for death sentence reversals	p. 320
Bibliographic note	p. 321
Exercises	p. 322
Multilevel generalized linear models	p. 325
Overdispersed Poisson regression: police stops and ethnicity	p. 325
Ordered categorical regression: storable votes	p. 331
Non-nested negative-binomial model of structure in social networks	p. 332
Bibliographic note	p. 342
Exercises	p. 342
Fitting multilevel models	p. 343
Multilevel modeling in Bugs and R: the basics	p. 345
Why you should learn Bugs	p. 345
Bayesian inference and prior distributions	p. 345
Fitting and understanding a varying-intercept multilevel model using R and Bugs	p. 348
Step by step through a Bugs model, as called from R	p. 353
Adding individual- and group-level predictors	p. 359
Predictions for new observations and new groups	p. 361
Fake-data simulation	p. 363
The principles of modeling in Bugs	p. 366
Practical issues of implementation	p. 369
Open-ended modeling in Bugs	p. 370
Bibliographic note	p. 373
Exercises	p. 373
Fitting multilevel linear and generalized linear models in Bugs and R	p. 375
Varying-intercept, varying-slope models	p. 375
Varying intercepts and slopes with group-level predictors	p. 379
Non-nested models	p. 380
Multilevel logistic regression	p. 381
Multilevel Poisson regression	p. 382
Multilevel ordered categorical regression	p. 383
Latent-data parameterizations of generalized linear models	p. 384
Bibliographic note	p. 385
Exercises	p. 385
Likelihood and Bayesian inference and computation	p. 387
Least squares and maximum likelihood estimation	p. 387
Uncertainty estimates using the likelihood surface	p. 390
Bayesian inference for classical and multilevel regression	p. 392
Gibbs sampler for multilevel linear models	p. 397
Likelihood inference, Bayesian inference, and the Gibbs sampler: the case of censored data	p. 402
Metropolis algorithm for more general Bayesian computation	p. 408
Specifying a log posterior density, Gibbs sampler, and Metropolis algorithm in R	p. 409
Bibliographic note	p. 413
Exercises	p. 413
Debugging and speeding convergence	p. 415
Debugging and confidence building	p. 415
General methods for reducing computational requirements	p. 418
Simple linear transformations	p. 419
Redundant parameters and intentionally nonidentifiable models	p. 419
Parameter expansion: multiplicative redundant parameters	p. 424
Using redundant parameters to create an informative prior distribution for multilevel variance parameters	p. 427
Bibliographic note	p. 434
Exercises	p. 434
Prom data collection to model understanding to model checking	p. 435
Sample size and power calculations	p. 437
Choices in the design of data collection	p. 437
Classical power calculations: general principles, as illustrated by estimates of proportions	p. 439
Classical power calculations for continuous outcomes	p. 443
Multilevel power calculation for cluster sampling	p. 447
Multilevel power calculation using fake-data simulation	p. 449
Bibliographic note	p. 454
Exercises	p. 454
Understanding and summarizing the fitted models	p. 457
Uncertainty and variability	p. 457
Superpopulation and finite-population variances	p. 459
Contrasts and comparisons of multilevel coefficients	p. 462
Average predictive comparisons	p. 466
R[superscript 2] and explained variance	p. 473
Summarizing the amount of partial pooling	p. 477
Adding a predictor can increase the residual variance!	p. 480
Multiple comparisons and statistical significance	p. 481
Bibliographic note	p. 484
Exercises	p. 485
Analysis of variance	p. 487
Classical analysis of variance	p. 487
ANOVA and multilevel linear and generalized linear models	p. 490
Summarizing multilevel models using ANOVA	p. 492
Doing ANOVA using multilevel models	p. 494
Adding predictors: analysis of covariance and contrast analysis	p. 496
Modeling the variance parameters: a split-plot latin square	p. 498
Bibliographic note	p. 501
Exercises	p. 501
Causal inference using multilevel models	p. 503
Multilevel aspects of data collection	p. 503
Estimating treatment effects in a multilevel observational study	p. 506
Treatments applied at different levels	p. 507
Instrumental variables and multilevel modeling	p. 509
Bibliographic note	p. 512
Exercises	p. 512
Model checking and comparison	p. 513
Principles of predictive checking	p. 513
Example: a behavioral learning experiment	p. 515
Model comparison and deviance	p. 524
Bibliographic note	p. 526
Exercises	p. 527
Missing-data imputation	p. 529
Missing-data mechanisms	p. 530
Missing-data methods that discard data	p. 531
Simple missing-data approaches that retain all the data	p. 532
Random imputation of a single variable	p. 533
Imputation of several missing variables	p. 539
Model-based imputation	p. 540
Combining inferences from multiple imputations	p. 542
Bibliographic note	p. 542
Exercises	p. 543
Appendixes	p. 545
Six quick tips to improve your regression modeling	p. 547
Fit many models	p. 547
Do a little work to make your computations faster and more reliable	p. 547
Graphing the relevant and not the irrelevant	p. 548
Transformations	p. 548
Consider all coefficients as potentially varying	p. 549
Estimate causal inferences in a targeted way, not as a byproduct of a large regression	p. 549
Statistical graphics for research and presentation	p. 551
Reformulating a graph by focusing on comparisons	p. 552
Scatterplots	p. 553
Miscellaneous tips	p. 559
Bibliographic note	p. 562
Exercises	p. 563
Software	p. 565
Getting started with R, Bugs, and a text editor	p. 565
Fitting classical and multilevel regressions in R	p. 565
Fitting models in Bugs and R	p. 567
Fitting multilevel models using R, Stata, SAS, and other software	p. 568
Bibliographic note	p. 573
References	p. 575
Author index	p. 601
Subject index	p. 607
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