Data Analysis Using Regression and Multilevel/Hierarchical Models

Transcription

1 Data Analysis Using Regression and Multilevel/Hierarchical Models ANDREW GELMAN Columbia University JENNIFER HILL Columbia University CAMBRIDGE UNIVERSITY PRESS

2 Contents List of examples V a 9 e xv " Preface < xix 1 Why? What is multilevel regression modeling? Some examples from our own research Motivations for multilevel modeling Distinctive features of this book Computing 9 2 Concepts and methods from basic probability and statistics Probability distributions Statistical inference Classical confidence intervals Classical hypothesis testing Problems with Statistical significance ,000 residents desperately need your help! Bibliographie note Exercises 26 Part 1A: Single-level regression 29 3 Linear regression: the basics One predictor Multiple predictors Interactions Statistical inference Graphical displays of data and fitted model Assumptions and diagnostics Prediction and Validation Bibliographie note Exercises 49 4 Linear regression: before and after fltting the model Linear transformations Centering and standardizing, especially for modeis with interactions Correlation and "regression to the mean" Logarithmic transformations Other transformations Building regression modeis for prediction Fitting a series of regressions 73

3 X CONTENTS 4.8 Bibliographie note Exercises 74 5 Logistic regression Logistic regression with a Single predictor Interpreting the logistic regression coefficients Latent-data formulation Building a logistic regression model: wells in Bangladesh Logistic regression with interactions Evaluating, checking, and comparing fitted logistic regressions Average predictive comparisons on the probability scale Identifiability and Separation Bibliographie note Exercises Generalized linear modeis Introduction Poisson regression, exposure, and overdispersion Logistic-binomial model Probit regression: normally distributed latent data Ordered and unordered categorical regression Robust regression using the t model Building more complex generalized linear modeis Constructive choiee modeis Bibliographie note Exercises 132 Part 1B: Working with regression inferences Simulation of probability modeis and Statistical inferences Simulation of probability modeis Summarizing linear regressions using Simulation: an informal Bayesian approach Simulation for nonlinear predictions: congressional elections Predictive Simulation for generalized linear modeis Bibliographie note Exercises Simulation for checking Statistical procedures ahd model fits Fake-data Simulation Example: using fake-data Simulation to understand residual plots Simulating from the fitted model and comparing to actual data Using predictive Simulation to check the fit of a time-series model Bibliographie note Exercises Causal inference using regression on the treatment variable Causal inference and predictive comparisons The fundamental problem of causal inference Randomized experiments Treatment interactions and poststratification 178

4 CONTENTS 9.5 Observational studies Understanding causal inference in observational studies Do not control for post-treatment variables Intermediate outcomes and causal paths Bibliographie note Exercises Causal inference using more advanced modeis Imbalance and lack of complete overlap Subclassification: effects and estimates for different subpopulations Matching: subsetting the data to get overlapping and balanced treatment and control groups Lack of overlap when the assignment mechanism is known: regression discontinuity Estimating causal effects indirectly using instrumental variables Instrumental variables in a regression Framework Identification strategies that make use of Variation within or between groups Bibliographie note Exercises 231 Part 2A: Multilevel regression Multilevel struetures Varying-intercept and varying-slope modeis Clustered data: child support enforcement in cities Repeated measurements, time-series cross sections, and other non-nested struetures Indicator variables and fixed or random effects Costs and benefits of multilevel modeling Bibliographie note, Exercises Multilevel linear modeis: the basics Notation Partial pooling with no predictors Partial pooling with predictors Quickly fitting multilevel modeis in R Five ways to write the same model Group-level predictors Model building and Statistical significance Predictions for new observations and new groups How many groups and how many observations per group are needed to fit a multilevel model? Bibliographie note n Exercises Multilevel linear modeis: varying slopes, non-nested modeis, and other complexities Varying intereepts and slopes Varying slopes without varying intereepts 283 XI

5 Xll CONTENTS 13.3 Modeling multiple varying coefficients using the scaled inverse- Wishart distribution Understanding correlations between group-level intercepts and slopes Non-nested models Selecting, transforming, and combining regression inputs More complex multilevel models Bibliographie note Exercises Multilevel logistic regression State-level opinions from national polls Red states and blue states: what's the matter with Connecticut? Item-response and ideal-point models ~ Non-nested overdispersed model for death sentence reversals Bibliographie note Exercises Multilevel generalized linear models Overdispersed Poisson regression: police stops and ethnicity Ordered categorical regression: storable votes Non-nested negative-binomial model of structure in social networks Bibliographie note Exercises 342 Part 2B: Fitting multilevel models Multilevel modeling in Bugs and R: the basics Why you should learn Bugs Bayesian inference and prior distributions Fitting and understanding a varying-intercept multilevel model using R and Bugs Step by step through a Bugs model, as called from R Adding individual- and group-level predictors Predictions for new observations and new groups Fake-data Simulation The principles of modeling in Bugs Practical issues of implementation Open-ended modeling in Bugs Bibliographie note Exercises Fitting multilevel linear and generalized linear models in Bugs and R Varying-intercept, varying-slope models Varying intercepts and slopes with group-level predictors Non-nested models Multilevel logistic regression Multilevel Poisson regression Multilevel ordered categorical regression Latent-data parameterizations of generalized linear models 384

6 CONTENTS 17.8 Bibliographie note Exercises Likelihood and Bayesian inference and computation Least Squares and maximum likelihood estimation Uncertainty estimates using the likelihood surface Bayesian inference for classical and multilevel regression Gibbs sampler for multilevel linear modeis Likelihood inference, Bayesian inference, and the Gibbs sampler: the case of censored data Metropolis algorithm for more general Bayesian computation Specifying a log posterior density, Gibbs sampler, and Metropolis algorithm in R Bibliographie note Exercises Debugging and speeding convergence Debugging and confidence building General methods for reducing computational requirements Simple linear transformations Redundant parameters and intentionally nonidentifiable modeis Parameter expansion: multiplicative redundant parameters Using redundant parameters to create an informative prior distribution for multilevel variance parameters Bibliographie note Exercises 434 Part 3: Prom data collection to model understanding to model checking Sample size and power calculations Choices in the design of data collection Classical power calculations: general principles, as illustrated by estimates of proportions Classical power calculations for continuous outcomes Multilevel power calculation for Cluster sampling Multilevel power calculation using fake-data Simulation Bibliographie note Exercises Understanding and summarizing the fitted modeis Uncertainty and variability Superpopulation and finite-population variances Contrasts and comparisons of multilevel coefficients Average predictive comparisons R 2 and explained variance Summarizing the amount of partial pooling Adding a predictor can increase the residual variance! Multiple comparisons and Statistical significance Bibliographie note Exercises 485 xiii

7 XIV CONTENTS 22 Analysis of variance Classical analysis of variance ANOVA and multilevel linear and generalized linear modeis Summarizing multilevel modeis using ANOVA Doing ANOVA using multilevel modeis Adding predictors: analysis of covariance and contrast analysis Modeling the variance parameters: a split-plot latin Square Bibliographie note Exercises Causal inference using multilevel modeis Multilevel aspects of data collection " Estimating treatment effects in a multilevel observational study Treatments applied at different levels Instrumental variables and multilevel modeling Bibliographie note Exercises Model checking and comparison Principles of predictive checking Example: a behavioral learning experiment Model comparison and deviance Bibliographie note Exercises Missing-data imputation Missing-data mechanisms Missing-data methods that discard data Simple missing-data approaches that retain all the data Random imputation of a Single variable Imputation of several missing variables Model-based imputation Combining inferences from multiple imputations Bibliographie note Exercises 543 Appendixes 545 A Six quick tips to improve your regression modeling 547 A.l Fit many modeis 547 A.2 Do a little work to make your computations faster and more reliable 547 A.3 Graphing the relevant and not the irrelevant 548 A.4 Transformations 548 A.5 Consider all coefficients as potentially varying 549 A.6 Estimate causal inferences in a targeted way, not as a byproduct of a large regression 549 B Statistical graphics for research and presentation 551 B.l Reformulating a graph by focusing on comparisons 552 B.2 Scatterplots 553 B.3 Miscellaneous tips 559

8 CONTENTS B.4 Bibliographie note 562 B.5 Exercises 563 C Software 565 C.l Getting started with R, Bugs, and a text editor 565 C.2 Fitting classical and multilevel regressions in R 565 C.3 Fitting modeis in Bugs and R 567 C.4 Fitting multilevel modeis using R, Stata, SAS, and other Software 568 C.5 Bibliographie note 573 References 575 Author index 601 Subject index 607 xv