Part 2: Linear regression

6. Background on regression modeling

6.1 Regression models

6.2 Fitting a simple regression to fake data

6.3 Interpret coefficients as comparisons, not effects

6.4 Historical origins of regression

6.5 The paradox of regression to the mean

6.6 Bibliographic note

6.7 Exercises

7. Linear regression with a single predictor

7.1 Example: predicting presidential vote share from the economy

7.2 Checking the model-fitting procedure using fake-data simulation

7.3 Formulating comparisons as regression models

7.4 Bibliographic note

7.5 Exercises

8. Fitting regression models

8.1 Least squares, maximum likelihood, and Bayesian inference

8.2 Influence of individual points in a fitted regression

8.3 Least squares slope as a weighted average of slopes of pairs

8.4 Comparing two fitting functions: lm and stan_glm

8.5 Bibliographic note

8.6 Exercises

9. Prediction and Bayesian inference

9.1 Propagating uncertainty in inference using posterior simulations

9.2 Prediction and uncertainty: predict, posterior_linpred, and posterior_predict

9.3 Prior information and Bayesian synthesis

9.4 Example of Bayesian inference: beauty and sex ratio

9.5 Uniform, weakly informative, and informative priors in regression

9.6 Bibliographic note

9.7 Exercises

10. Linear regression with multiple predictors

10.1 Adding predictors to a model

10.2 Interpreting regression coefficients

10.3 Interactions

10.4 Indicator variables

10.5 Formulating paired or blocked designs as a regression problem

10.6 Example: uncertainty in predicting congressional elections

10.7 Mathematical notation and statistical inference

10.8 Weighted regression

10.9 Fitting the same model to many datasets

10.10 Bibliographic note

10.11 Exercises

11. Assumptions, diagnostics, and model evaluation

11.1 Assumptions of regression analysis

11.2 Plotting the data and fitted model

11.3 Residual plots

11.4 Comparing data to replications from a fitted model

11.5 Example: predictive simulation to check the fit of a time-series model

11.6 Residual standard deviation σ and explained variance R²

11.7 External validation: checking fitted model on new data

11.8 Cross validation

11.9 Bibliographic note

11.10 Exercises

12. Transformations and regression

12.1 Linear transformations

12.2 Centering and standardizing for models with interactions

12.3 Correlation and 'regression to the mean'

12.4 Logarithmic transformations

12.5 Other transformations

12.6 Building and comparing regression models for prediction

12.7 Models for regression coefficients

12.8 Bibliographic note

12.9 Exercises

Part 3: Generalized linear models

13. Logistic regression

13.1 Logistic regression with a single predictor

13.2 Interpreting logistic regression coefficients and the divide-by-4 rule

13.3 Predictions and comparisons

13.4 Latent-data formulation

13.5 Maximum likelihood and Bayesian inference for logistic regression

13.6 Cross validation and log score for logistic regression

13.7 Building a logistic regression model: wells in Bangladesh

13.8 Bibliographic note

13.9 Exercises

14. Working with logistic regression

14.1 Graphing logistic regression and binary data

14.2 Logistic regression with interactions

14.3 Predictive simulation

14.4 Average predictive comparisons on the probability scale

14.5 Residuals for discrete-data regression

14.6 Identification and separation

14.7 Bibliographic note

14.8 Exercises

15. Other generalized linear models

15.1 Definition and notation

15.2 Poisson and negative binomial regression

15.3 Logistic-binomial model

15.4 Probit regression: normally distributed latent data

15.5 Ordered and unordered categorical regression

15.6 Robust regression using the t model

15.7 Constructive choice models

15.8 Going beyond generalized linear models

15.9 Bibliographic note

15.10 Exercises

Part 4: Before and after fitting a regression

16. Design and sample size decisions

16.1 The problem with statistical power

16.2 General principles of design, as illustrated by estimates of proportions

16.3 Sample size and design calculations for continuous outcomes

16.4 Interactions are harder to estimate than main effects

16.5 Design calculations after the data have been collected

16.6 Design analysis using fake-data simulation

16.7 Bibliographic note

16.8 Exercises

17. Poststratification and missing-data imputation

17.1 Poststratification: using regression to generalize to a new population

17.2 Fake-data simulation for regression and poststratification

17.3 Models for missingness

17.4 Simple approaches for handling missing data

17.5 Understanding multiple imputation

17.6 Nonignorable missing-data models

17.7 Bibliographic note

17.8 Exercises

Part 5: Causal inference

18. Causal inference and randomized experiments

18.1 Basics of causal inference

18.2 Average causal effects

18.3 Randomized experiments

18.4 Sampling distributions, randomization distributions, and bias in estimation

18.5 Using additional information in experimental design

18.6 Properties, assumptions, and limitations of randomized experiments

18.7 Bibliographic note

18.8 Exercises

19. Causal inference using regression on the treatment variable

19.1 Pre-treatment covariates, treatments, and potential outcomes

19.2 Example: the effect of showing children an educational television show

19.3 Including pre-treatment predictors

19.4 Varying treatment effects, interactions, and poststratification

19.5 Challenges of interpreting regression coefficients as treatment effects

19.6 Do not adjust for post-treatment variables

19.7 Intermediate outcomes and causal paths

19.8 Bibliographic note

19.9 Exercises

20. Observational studies with all confounders assumed to be measured

20.1 The challenge of causal inference

20.2 Using regression to estimate a causal effect from observational data

20.3 Assumption of ignorable treatment assignment in an observational study

20.4 Imbalance and lack of complete overlap

20.5 Example: evaluating a child care program

20.6 Subclassification and average treatment effects

20.7 Propensity score matching for the child care example

20.8 Restructuring to create balanced treatment and control groups

20.9 Additional considerations with observational studies

20.10 Bibliographic note

20.11 Exercises

21. Additional topics in causal inference

21.1 Estimating causal effects indirectly using instrumental variables

21.2 Instrumental variables in a regression framework

21.3 Regression discontinuity: known assignment mechanism but no overlap

21.4 Identification using variation within or between groups

21.5 Causes of effects and effects of causes

21.6 Bibliographic note

21.7 Exercises

Part 6: What comes next?

22. Advanced regression and multilevel models

22.1 Expressing the models so far in a common framework

22.2 Incomplete data

22.3 Correlated errors and multivariate models

22.4 Regularization for models with many predictors

22.5 Multilevel or hierarchical models

22.6 Nonlinear models, a demonstration using Stan

22.7 Nonparametric regression and machine learning

22.8 Computational efficiency

22.9 Bibliographic note

22.10 Exercises