Regression and Other Stories(Gelman et al., 2020)
exercise 13.8
Question.
Linear transformations: In the regression from the previous exercise, suppose you replaced
height in inches by height in centimeters. What would then be the intercept and slope?
and, why the intercept and slope interpretation is like that?
This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ Regression and Other Stories (corrections up to 26 Feb 2022) Please do not reproduce in any form without permission Andrew Gelman Department of Statistics and Department of Political Science Columbia University Jennifer Hill Steinhardt School of Education and Human Development New York University Aki Vehtari Department of Computer Science Aalto University ©2002–2022 by Andrew Gelman, Jennifer Hill, and Aki Vehtari Published 2020 by Cambridge University Press https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ Contents Preface xi What you should be able to do after reading and working through this book xi Fun chapter titles xii Additional material for teaching and learning xiii Part 1: Fundamentals 1 1 Overview 3 1.1 The three challenges of statistics 3 1.2 Why learn regression? 4 1.3 Some examples of regression 5 1.4 Challenges in building, understanding, and interpreting regressions 9 1.5 Classical and Bayesian inference 13 1.6 Computing least squares and Bayesian regression 16 1.7 Bibliographic note 17 1.8 Exercises 17 2 Data and measurement 21 2.1 Examining where data come from 21 2.2 Validity and reliability 23 2.3 All graphs are comparisons 25 2.4 Data and adjustment: trends in mortality rates 31 2.5 Bibliographic note 33 2.6 Exercises 34 3 Some basic methods in mathematics and probability 35 3.1 Weighted averages 35 3.2 Vectors and matrices 36 3.3 Graphing a line 37 3.4 Exponential and power-law growth and decline; logarithmic and log-log relationships 38 3.5 Probability distributions 40 3.6 Probability modeling 45 3.7 Bibliographic note 47 3.8 Exercises 47 4 Statistical inference 49 4.1 Sampling distributions and generative models 49 4.2 Estimates, standard errors, and confidence intervals 50 4.3 Bias and unmodeled uncertainty 55 4.4 Statistical significance, hypothesis testing, and statistical errors 57 4.5 Problems with the concept of statistical significance 60 4.6 Example of hypothesis testing: 55,000 residents need your help! 63 4.7 Moving beyond hypothesis testing 66 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ VI CONTENTS 4.8 Bibliographic note 67 4.9 Exercises 67 5 Simulation 69 5.1 Simulation of discrete probability models 69 5.2 Simulation of continuous and mixed discrete/continuous models 71 5.3 Summarizing a set of simulations using median and median absolute deviation 73 5.4 Bootstrapping to simulate a sampling distribution 73 5.5 Fake-data simulation as a way of life 76 5.6 Bibliographic note 76 5.7 Exercises 76 Part 2: Linear regression 79 6 Background on regression modeling 81 6.1 Regression models 81 6.2 Fitting a simple regression to fake data 82 6.3 Interpret coefficients as comparisons, not effects 84 6.4 Historical origins of regression 85 6.5 The paradox of regression to the mean 87 6.6 Bibliographic note 90 6.7 Exercises 91 7 Linear regression with a single predictor 93 7.1 Example: predicting presidential vote share from the economy 93 7.2 Checking the model-fitting procedure using fake-data simulation 97 7.3 Formulating comparisons as regression models 99 7.4 Bibliographic note 101 7.5 Exercises 101 8 Fitting regression models 103 8.1 Least squares, maximum likelihood, and Bayesian inference 103 8.2 Influence of individual points in a fitted regression 107 8.3 Least squares slope as a weighted average of slopes of pairs 108 8.4 Comparing two fitting functions: lm and stan_glm 109 8.5 Bibliographic note 111 8.6 Exercises 111 9 Prediction and Bayesian inference 113 9.1 Propagating uncertainty in inference using posterior simulations 113 9.2 Prediction and uncertainty: predict, posterior_linpred, and posterior_predict 115 9.3 Prior information and Bayesian synthesis 119 9.4 Example of Bayesian inference: beauty and sex ratio 121 9.5 Uniform, weakly informative, and informative priors in regression 123 9.6 Bibliographic note 128 9.7 Exercises 128 10 Linear regression with multiple predictors 131 10.1 Adding predictors to a model 131 10.2 Interpreting regression coefficients 133 10.3 Interactions 134 10.4 Indicator variables 136 10.5 Formulating paired or blocked designs as a regression problem 139 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ CONTENTS VII 10.6 Example: uncertainty in predicting congressional elections 140 10.7 Mathematical notation and statistical inference 144 10.8 Weighted regression 147 10.9 Fitting the same model to many datasets 148 10.10 Bibliographic note 149 10.11 Exercises 150 11 Assumptions, diagnostics, and model evaluation 153 11.1 Assumptions of regression analysis 153 11.2 Plotting the data and fitted model 156 11.3 Residual plots 161 11.4 Comparing data to replications from a fitted model 163 11.5 Example: predictive simulation to check the fit of a time-series model 166 11.6 Residual standard deviation f and explained variance '2 168 11.7 External validation: checking fitted model on new data 171 11.8 Cross validation 172 11.9 Bibliographic note 180 11.10 Exercises 180 12 Transformations and regression 183 12.1 Linear transformations 183 12.2 Centering and standardizing for models with interactions 185 12.3 Correlation and “regression to the mean” 187 12.4 Logarithmic transformations 189 12.5 Other transformations 195 12.6 Building and comparing regression models for prediction 199 12.7 Models for regression coefficients 206 12.8 Bibliographic note 210 12.9 Exercises 211 Part 3: Generalized linear models 215 13 Logistic regression 217 13.1 Logistic regression with a single predictor 217 13.2 Interpreting logistic regression coefficients and the divide-by-4 rule 220 13.3 Predictions and comparisons 222 13.4 Latent-data formulation 226 13.5 Maximum likelihood and Bayesian inference for logistic regression 228 13.6 Cross validation and log score for logistic regression 230 13.7 Building a logistic regression model: wells in Bangladesh 232 13.8 Bibliographic note 237 13.9 Exercises 237 14 Working with logistic regression 241 14.1 Graphing logistic regression and binary data 241 14.2 Logistic regression with interactions 242 14.3 Predictive simulation 247 14.4 Average predictive comparisons on the probability scale 249 14.5 Residuals for discrete-data regression 253 14.6 Identification and separation 256 14.7 Bibliographic note 259 14.8 Exercises 259 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ VIII CONTENTS 15 Other generalized linear models 263 15.1 Definition and notation 263 15.2 Poisson and negative binomial regression 264 15.3 Logistic-binomial model 270 15.4 Probit regression: normally distributed latent data 272 15.5 Ordered and unordered categorical regression 273 15.6 Robust regression using the t model 278 15.7 Constructive choice models 279 15.8 Going beyond generalized linear models 283 15.9 Bibliographic note 286 15.10 Exercises 286 Part 4: Before and after fitting a regression 289 16 Design and sample size decisions 291 16.1 The problem with statistical power 291 16.2 General principles of design, as illustrated by estimates of proportions 293 16.3 Sample size and design calculations for continuous outcomes 297 16.4 Interactions are harder to estimate than main effects 301 16.5 Design calculations after the data have been collected 304 16.6 Design analysis using fake-data simulation 306 16.7 Bibliographic note 310 16.8 Exercises 310 17 Poststratification and missing-data imputation 313 17.1 Poststratification: using regression to generalize to a new population 313 17.2 Fake-data simulation for regression and poststratification 320 17.3 Models for missingness 322 17.4 Simple approaches for handling missing data 324 17.5 Understanding multiple imputation 326 17.6 Nonignorable missing-data models 332 17.7 Bibliographic note 333 17.8 Exercises 333 Part 5: Causal inference 337 18 Causal inference and randomized experiments 339 18.1 Basics of causal inference 339 18.2 Average causal effects 342 18.3 Randomized experiments 345 18.4 Sampling distributions, randomization distributions, and bias in estimation 346 18.5 Using additional information in experimental design 347 18.6 Properties, assumptions, and limitations of randomized experiments 350 18.7 Bibliographic note 355 18.8 Exercises 356 19 Causal inference using regression on the treatment variable 363 19.1 Pre-treatment covariates, treatments, and potential outcomes 363 19.2 Example: the effect of showing children an educational television show 364 19.3 Including pre-treatment predictors 367 19.4 Varying treatment effects, interactions, and poststratification 370 19.5 Challenges of interpreting regression coefficients as treatment effects 373 19.6 Do not adjust for post-treatment variables 374 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ CONTENTS IX 19.7 Intermediate outcomes and causal paths 376 19.8 Bibliographic note 379 19.9 Exercises 380 20 Observational studies with all confounders assumed to be measured 383 20.1 The challenge of causal inference 383 20.2 Using regression to estimate a causal effect from observational data 386 20.3 Assumption of ignorable treatment assignment in an observational study 388 20.4 Imbalance and lack of complete overlap 391 20.5 Example: evaluating a child care program 394 20.6 Subclassification and average treatment effects 397 20.7 Propensity score matching for the child care example 399 20.8 Restructuring to create balanced treatment and control groups 405 20.9 Additional considerations with observational studies 413 20.10 Bibliographic note 416 20.11 Exercises 417 21 Additional topics in causal inference 421 21.1 Estimating causal effects indirectly using instrumental variables 421 21.2 Instrumental variables in a regression framework 427 21.3 Regression discontinuity: known assignment mechanism but no overlap 432 21.4 Identification using variation within or between groups 440 21.5 Causes of effects and effects of causes 445 21.6 Bibliographic note 449 21.7 Exercises 450 Part 6: What comes next? 455 22 Advanced regression and multilevel models 457 22.1 Expressing the models so far in a common framework 457 22.2 Incomplete data 458 22.3 Correlated errors and multivariate models 459 22.4 Regularization for models with many predictors 459 22.5 Multilevel or hierarchical models 460 22.6 Nonlinear models, a demonstration using Stan 460 22.7 Nonparametric regression and machine learning 464 22.8 Computational efficiency 467 22.9 Bibliographic note 471 22.10 Exercises 471 Appendixes 473 A Computing in R 475 A.1 Downloading and installing R and Stan 475 A.2 Accessing data and code for the examples in the book 476 A.3 The basics 476 A.4 Reading, writing, and looking at data 481 A.5 Making graphs 482 A.6 Working with messy data 484 A.7 Some R programming 488 A.8 Working with rstanarm fit objects 490 A.9 Bibliographic note 492 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Copyright by Andrew Gelman, Jennifer Hill, and Aki Vehtari 2020. The book web page https://avehtari.github.io/ROS-Examples/ X CONTENTS B 10 quick tips to improve your regression modeling 493 B.1 Think about variation and replication 493 B.2 Forget about statistical significance 493 B.3 Graph the relevant and not the irrelevant 493 B.4 Interpret regression coefficients as comparisons 494 B.5 Understand statistical methods using fake-data simulation 494 B.6 Fit many models 495 B.7 Set up a computational workflow 495 B.8 Use transformations 496 B.9 Do causal inference in a targeted way, not as a byproduct of a large regression 496 B.10 Learn methods through live examples 496 References 497 Author Index 515 Subject Index 520 https://avehtari.github.io/ROS-Examples/ This book has been published by Cambridge University Press as Regression and Other Stories by Andrew Gelman, Jennifer Hill, and Aki Vehtari. This PDF is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works.