Interpreting and Visualizing Regression Models Using Stata

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Michael Mitchell’s Interpreting and Visualizing Regression Models Using Stata is a clear treatment of how to carefully present results from model-fitting in a wide variety of settings. It is a boon to anyone who has to present the tangible meaning of a complex model in a clear fashion, regardless of the audience. As an example, many experienced researchers start to squirm when asked to give a simple explanation of the practical meaning of interactions in nonlinear models such as logistic regression. The techniques presented in Mitchell's book make answering those questions easy. The overarching theme of the book is that graphs make interpreting even the most complicated models containing interaction terms, categorical variables, and other intricacies straightforward.

 

Using a dataset based on the General Social Survey, Mitchell starts with a basic linear regression with a single independent variable and then illustrates how to tabulate and graph predicted values. Mitchell focuses on Stata’s margins and marginsplot commands, which play a central role in the book and which greatly simplify the calculation and presentation of results from regression models. In particular, through use of the marginsplot command, Mitchell shows how you can graphically visualize every model presented in the book. Gaining insight into results is much easier when you can view them in a graph rather than in a mundane table of results.

 

Mitchell then proceeds to more-complicated models where the effects of the independent variables are nonlinear. After discussing how to detect nonlinear effects, he presents examples using both standard polynomial terms (squares and cubes of variables) as well as fractional polynomial models, where independent variables can be raised to powers like −1 or 1/2. In all cases, Mitchell again uses the marginsplot command to illustrate the effect that changing an independent variable has on the dependent variable. Piecewise-linear models are presented as well; these are linear models in which the slope or intercept is allowed to change depending on the range of an independent variable. Mitchell also uses the contrast command when discussing categorical variables; as the name suggests, this command allows you to easily contrast predictions made for various levels of the categorical variable.

 

Interaction terms can be tricky to interpret, but Mitchell shows how graphs produced by marginsplot greatly clarify results. Individual chapters are devoted to two- and three-way interactions containing all continuous or all categorical variables and include many practical examples. Raw regression output including interactions of continuous and categorical variables can be nigh impossible to interpret, but again Mitchell makes this a snap through judicious use of the margins and marginsplot commands in subsequent chapters.

 

The first two-thirds of the book is devoted to cross-sectional data, while the final third considers longitudinal data and complex survey data. A significant difference between this book and most others on regression models is that Mitchell spends quite some time on fitting and visualizing discontinuous models—models where the outcome can change value suddenly at thresholds. Such models are natural in settings such as education and policy evaluation, where graduation or policy changes can make sudden changes in income or revenue.

 

This book is a worthwhile addition to the library of anyone involved in statistical consulting, teaching, or collaborative applied statistical environments. Graphs greatly aid the interpretation of regression models, and Mitchell’s book shows you how.

 

Comments from readers

 

I just received Michael Mitchell’s new book, Interpreting and Visualizing Regression Models Using Stata. Nobody can make Stata graphic capabilities as easy to use as Mitchell. This new book gives me new ways to interpret all sorts of regression models including multilevel models. I'm recommending it to all my students. The new Stata 12 features he explains in this book are compelling.

 

Alan C. Acock
Oregon State University

 

I received my copy last week and it is an amazing resource beyond the visualization aspect. As we would expect, Michael Mitchell did more than explain how the visualization can assist in the interpretation of the models and interaction effects. He al so provides great insight regarding the interpretation of a variety of interaction effects in nonlinear models as well. This is definitely a worthy addition to the library and could help save grad students a great deal of agony when it comes to interpreting and understanding the results of their analyses.

 

William R. Buchanan
Performing Arts & Creative Education Solutions (PACES) Consulting

 

Michael Mitchell is a senior statistician in disaster preparedness and response. He is the author of A Visual Guide to Stata Graphics as well as Data Management Using Stata. Previously, he worked for 12 years as a statistical consultant and manager of the UCLA ATS Statistical Consulting Group. There, he envisioned the UCLA Statistical Consulting Resources website and wrote hundreds of webpages about Stata.

List of tables
List of figures
Preface (PDF)
Acknowledgments
1 Introduction
1.1 Overview of the book 
1.2 Getting the most out of this book 
1.3 Downloading the example datasets and programs 
1.4 The GSS dataset 
1.4.1 Income 
1.4.2 Age 
1.4.3 Education 
1.4.4 Gender 
1.5 The pain datasets 
1.6 The optimism datasets 
1.7 The school datasets 
1.8 The sleep datasets 
I Continuous predictors
2 Continuous predictors: Linear
2.1 Chapter overview 
2.2 Simple linear regression 
2.2.1 Computing predicted means using the margins command 
2.2.2 Graphing predicted means using the marginsplot command 
2.3 Multiple regression 
2.3.1 Computing adjusted means using the margins command 
2.3.2 Some technical details about adjusted means 
2.3.3 Graphing adjusted means using the marginsplot command 
2.4 Checking for nonlinearity graphically 
2.4.1 Using scatterplots to check for nonlinearity 
2.4.2 Checking for nonlinearity using residuals 
2.4.3 Checking for nonlinearity using locally weighted smoother 
2.4.4 Graphing outcome mean at each level of predictor 
2.4.5 Summary 
2.5 Checking for nonlinearity analytically 
2.5.1 Adding power terms 
2.5.2 Using factor variables 
2.6 Summary 
3 Continuous predictors: Polynomials
3.1 Chapter overview 
3.2 Quadratic (squared) terms 
3.2.1 Overview 
3.2.2 Examples 
3.3 Cubic (third power) terms 
3.3.1 Overview 
3.3.2 Examples 
3.4 Fractional polynomial regression 
3.4.1 Overview 
3.4.2 Example using fractional polynomial regression 
3.5 Main effects with polynomial terms 
3.6 Summary 
4 Continuous predictors: Piecewise models
4.1 Chapter overview 
4.2 Introduction to piecewise regression models 
4.3 Piecewise with one known knot 
4.3.1 Overview 
4.3.2 Examples using the GSS 
4.4 Piecewise with two known knots 
4.4.1 Overview 
4.4.2 Examples using the GSS 
4.5 Piecewise with one knot and one jump 
4.5.1 Overview 
4.5.2 Examples using the GSS 
4.6 Piecewise with two knots and two jumps 
4.6.1 Overview 
4.6.2 Examples using the GSS 
4.7 Piecewise with an unknown knot 
4.8 Piecewise model with multiple unknown knots 
4.9 Piecewise models and the marginsplot command 
4.10 Automating graphs of piecewise models 
4.11 Summary 
5 Continuous by continuous interactions
5.1 Chapter overview 
5.2 Linear by linear interactions 
5.2.1 Overview 
5.2.2 Example using GSS data 
5.2.3 Interpreting the interaction in terms of age 
5.2.4 Interpreting the interaction in terms of education 
5.2.5 Interpreting the interaction in terms of age slope 
5.2.6 Interpreting the interaction in terms of the educ slope 
5.3 Linear by quadratic interactions 
5.3.1 Overview 
5.3.2 Example using GSS data 
5.4 Summary 
6 Continuous by continuous by continuous interactions
6.1 Chapter overview 
6.2 Overview 
6.3 Examples using the GSS data 
6.3.1 A model without a three-way interaction 
6.3.2 A three-way interaction model 
6.4 Summary 
II Categorical predictors
7 Categorical predictors
7.1 Chapter overview 
7.2 Comparing two groups using a t test 
7.3 More groups and more predictors 
7.4 Overview of contrast operators 
7.5 Compare each group against a reference group 
7.5.1 Selecting a specific contrast 
7.5.2 Selecting a different reference group 
7.5.3 Selecting a contrast and reference group 
7.6 Compare each group against the grand mean 
7.6.1 Selecting a specific contrast 
7.7 Compare adjacent means 
7.7.1 Reverse adjacent contrasts 
7.7.2 Selecting a specific contrast 
7.8 Comparing the mean of subsequent or previous levels 
7.8.1 Comparing the mean of previous levels 
7.8.2 Selecting a specific contrast 
7.9 Polynomial contrasts 
7.10 Custom contrasts 
7.11 Weighted contrasts 
7.12 Pairwise comparisons 
7.13 Interpreting confidence intervals 
7.14 Testing categorical variables using regression 
7.15 Summary 
8 Categorical by categorical interactions
8.1 Chapter overview 
8.2 Two by two models: Example 1 
8.2.1 Simple effects 
8.2.2 Estimating the size of the interaction 
8.2.3 More about interaction 
8.2.4 Summary 
8.3 Two by three models 
8.3.1 Example 2 
8.3.2 Example 3 
8.3.3 Summary 
8.4 Three by three models: Example 4 
8.4.1 Simple effects 
8.4.2 Simple contrasts 
8.4.3 Partial interaction 
8.4.4 Interaction contrasts 
8.4.5 Summary 
8.5 Unbalanced designs 
8.6 Main effects with interactions: anova versus regress 
8.7 Interpreting confidence intervals 
8.8 Summary 
9 Categorical by categorical by categorical interactions
9.1 Chapter overview 
9.2 Two by two by two models 
9.2.1 Simple interactions by season 
9.2.2 Simple interactions by depression status 
9.2.3 Simple effects 
9.3 Two by two by three models 
9.3.1 Simple interactions by depression status 
9.3.2 Simple partial interaction by depression status 
9.3.3 Simple contrasts 
9.3.4 Partial interactions 
9.4 Three by three by three models and beyond 
9.4.1 Partial interactions and interaction contrasts 
9.4.2 Simple interactions 
9.4.3 Simple effects and simple comparisons 
9.5 Summary 
III Continuous and categorical predictors
10 Linear by categorical interactions
10.1 Chapter overview 
10.2 Linear and two-level categorical: No interaction 
10.2.1 Overview 
10.2.2 Examples using the GSS 
10.3 Linear by two-level categorical interactions 
10.3.1 Overview 
10.3.2 Examples using the GSS 
10.4 Linear by three-level categorical interactions 
10.4.1 Overview 
10.4.2 Examples using the GSS 
10.5 Summary 
11 Polynomial by categorical interactions
11.1 Chapter overview 
11.2 Quadratic by categorical interactions 
11.2.1 Overview 
11.2.2 Quadratic by two-level categorical 
11.2.3 Quadratic by three-level categorical 
11.3 Cubic by categorical interactions 
11.4 Summary 
12 Piecewise by categorical interactions
12.1 Chapter overview 
12.2 One knot and one jump 
12.2.1 Comparing slopes across gender 
12.2.2 Comparing slopes across education 
12.2.3 Difference in differences of slopes 
12.2.4 Comparing changes in intercepts 
12.2.5 Computing and comparing adjusted means 
12.2.6 Graphing adjusted means 
12.3 Two knots and two jumps 
12.3.1 Comparing slopes across gender 
12.3.2 Comparing slopes across education 
12.3.3 Difference in differences of slopes 
12.3.4 Comparing changes in intercepts by gender 
12.3.5 Comparing changes in intercepts by education 
12.3.6 Computing and comparing adjusted means 
12.3.7 Graphing adjusted means 
12.4 Comparing coding schemes 
12.4.1 Coding scheme #1 
12.4.2 Coding scheme #2 
12.4.3 Coding scheme #3 
12.4.4 Coding scheme #4 
12.4.5 Choosing coding schemes 
12.5 Summary 
13 Continuous by continuous by categorical interactions
13.1 Chapter overview 
13.2 Linear by linear by categorical interactions 
13.2.1 Fitting separate models for males and females 
13.2.2 Fitting a combined model for males and females 
13.2.3 Interpreting the interaction focusing in the age slope 
13.2.4 Interpreting the interaction focusing on the educ slope 
13.2.5 Estimating and comparing adjusted means by gender 
13.3 Linear by quadratic by categorical interactions 
13.3.1 Fitting separate models for males and females 
13.3.2 Fitting a common model for males and females 
13.3.3 Interpreting the interaction 
13.3.4 Estimating and comparing adjusted means by gender 
13.4 Summary 
14 Continuous by categorical by categorical interactions
14.1 Chapter overview 
14.2 Simple effects of gender on the age slope 
14.3 Simple effects of education on the age slope 
14.4 Simple contrasts on education for the age slope 
14.5 Partial interaction on education for the age slope 
14.6 Summary 
IV Beyond ordinary linear regression
15 Multilevel models
15.1 Chapter overview 
15.2 Example 1: Continuous by continuous interaction 
15.3 Example 2: Continuous by categorical interaction 
15.4 Example 3: Categorical by continuous interaction 
15.5 Example 4: Categorical by categorical interaction 
15.6 Summary 
16 Time as a continuous predictor
16.1 Chapter overview 
16.2 Example 1: Linear effect of time 
16.3 Example 2: Linear effect of time by a categorical predictor 
16.4 Example 3: Piecewise modeling of time 
16.5 Example 4: Piecewise effects of time by a categorical predictor 
16.5.1 Baseline slopes 
16.5.2 Change in slopes: Treatment versus baseline 
16.5.3 Jump at treatment 
16.5.4 Comparisons among groups 
16.6 Summary 
17 Time as a categorical predictor
17.1 Chapter overview 
17.2 Example 1: Time treated as a categorical variable 
17.3 Example 2: Time (categorical) by two groups 
17.4 Example 3: Time (categorical) by three groups 
17.5 Comparing models with different residual covariance structures 
17.6 Summary 
18 Nonlinear models
18.1 Chapter overview 
18.2 Binary logistic regression 
18.2.1 A logistic model with one categorical predictor 
18.2.2 A logistic model with one continuous predictor 
18.2.3 A logistic model with covariates 
18.3 Multinomial logistic regression 
18.4 Ordinal logistic regression 
18.5 Poisson regression 
18.6 More applications of nonlinear models 
18.6.1 Categorical by categorical interaction 
18.6.2 Categorical by continuous interaction 
18.6.3 Piecewise modeling 
18.7 Summary 
19 Complex survey data
V Appendices
A The margins command
A.1 The predict() and expression() options 
A.2 The at() option 
A.3 Margins with factor variables 
A.4 Margins with factor variables and the at() option 
A.5 The dydx() and related options 
B The marginsplot command
C The contrast command
D The pwcompare command
References