Preface

1. Introduction

1.1 Example: Treatment of Back Pain

1.2 The Family of Multipredictor Regression Methods

1.3 Motivation for Multipredictor Regression

1.3.1 Prediction

1.3.2 Isolating the Effect of a Single Predictor

1.3.3 Understanding Multiple Predictors

1.4 Guide to the Book

2. Exploratory and Descriptive Methods

2.1 Data Checking

2.2 Types of Data

2.3 One-Variable Descriptions

2.3.1 Numerical Variables

2.3.2 Categorical Variables

2.4 Two-Variable Descriptions

2.4.1 Outcome Versus Predictor Variables

2.4.2 Continuous Outcome Variable

2.4.3 Categorical Outcome Variable

2.5 Multivariable Descriptions

2.6 Summary

2.7 Problems

3. Basic Statistical Methods

3.1

*t*-Test and Analysis of Variance

3.1.1 *t*-Test

3.1.2 One- and Two-Sided Hypothesis Test

3.1.3 Paired *t*-Test

3.1.4 One-Way Analysis of Variance

3.1.5 Pairwise Comparisons in ANOVA

3.1.6 Multi-way ANOVA and ANCOVA

3.1.7 Robustness to Violations of Normality Assumption

3.1.8 Nonparametric Alternatives

3.1.9 Equal Variance Assumption

3.2 Correlation Coefficient

3.2.1 Spearman Rank Correlation Coefficient

3.2.2 Kendall's *τ*

3.3 Simple Linear Regression Model

3.3.1 Systematic Part of the Model

3.3.2 Random Part of the Model

3.3.3 Assumptions About the Predictor

3.3.4 Ordinary Least Squares Estimation

3.3.5 Fitted Values and Residuals

3.3.6 Sums of Squares

3.3.7 Standard Errors of the Regression Coefficients

3.3.8 Hypothesis Tests and Confidence Intervals

3.3.9 Slope, Correlation Coefficient, and *R*^{2}

3.4 Contingency Table Methods for Binary Outcomes

3.4.1 Measures of Risk and Association for Binary Outcomes

3.4.2 Tests of Association in Contingency Tables

3.4.3 Predictors with Multiple Categories

3.4.4 Analyses Involving Multiple Categorical Predictors

3.4.5 Collapsibility of Standard Measures of Association

3.5 Basic Methods for Survival Analysis

3.5.1 Right Censoring

3.5.2 Kaplan–Meier Estimator of the Survival Function

3.5.3 Interpretation of Kaplan–Meier Curves

3.5.4 Median Survival

3.5.5 Cumulative Event Function

3.5.6 Comparing Groups Using the Logrank Test

3.6 Bootstrap Confidence Intervals

3.7 Interpretation of Negative Findings

3.8 Further Notes and References

3.9 Problems

3.10 Learning objectives

4. Linear Regression

4.1 Example: Exercise and Glucose

4.2 Multiple Linear Regression Model

4.2.1 Systematic Part of the Model

4.2.2 Random Part of the Model

4.2.3 Generalization of *R*^{2} and *r*

4.2.4 Standardized Regression Coefficients

4.3 Categorical Predictors

4.3.1 Binary Predictors

4.3.2 Multilevel Categorical Predictors

4.3.3 The *F*-Test

4.3.4 Multiple Pairwise Comparisons Between Categories

4.3.5 Testing for Trend Across Categories

4.4 Confounding

4.4.1 Range of Confounding Patterns

4.4.2 Confounding Is Difficult to Rule Out

4.4.3 Adjusted Versus Unadjusted *β*s

4.4.4 Example: BMI and LDL

4.5 Mediation

4.5.1 Indirect Effects via the Mediator

4.5.2 Overall and Direct Effects

4.5.3 Percent Explained

4.5.4 Example: BMI, Exercise, and Glucose

4.5.5 Pitfalls in Evaluating Mediation

4.6 Interaction

4.6.1 Example: Hormone Therapy and Statin Use

4.6.2 Example: BMI and Statin Use

4.6.3 Interaction and Scale

4.6.4 Example: Hormone Therapy and Baseline LDL

4.6.5 Details

4.7 Checking Model Assumptions and Fit

4.7.1 Linearity

4.7.2 Normality

4.7.3 Constant Variance

4.7.4 Outlying, High Leverage, and Influential Points

4.7.5 Interpretation of Results for Log Transformed Variables

4.7.6 When to Use Transformations

4.8 Sample Size, Power, and Detectable Effects

4.8.1 Calculations Using Standard Errors Based on Published
Data

4.9 Summary

4.10 Further Notes and References

4.10.1 Generalized Additive Models

4.11 Problems

4.12 Learning Objectives

5. Logistic Regression

5.1 Single Predictor Models

5.1.1 Interpretation of Regression Coefficients

5.1.2 Categorical Predictors

5.2 Multipredictor Models

5.2.1 Likelihood Ratio Tests

5.2.2 Confounding

5.2.3 Mediation

5.2.4 Interaction

5.2.5 Prediction

5.2.6 Prediction Accuracy

5.3 Case–Control Studies

5.3.1 Matched Case–Control Studies

5.4 Checking Model Assumptions and Fit

5.4.1 Linearity

5.4.2 Outlying and Influential Points

5.4.3 Model Adequacy

5.4.4 Technical Issues in Logistic Model Fitting

5.5 Alternative Strategies for Binary Outcomes

5.5.1 Infectious Disease Transmission Models

5.5.2 Pooled Logistic Regression

5.5.3 Regression Models Based on Risk
Differences and Relative Risks

5.5.4 Exact Logistic Regression

5.5.5 Nonparametric Binary Regression

5.5.6 More Than Two Outcome Levels

5.6 Likelihood

5.7 Sample Size, Power, and Detectable Effects

5.8 Summary

5.9 Further Notes and References

5.10 Problems

5.11 Learning Objectives

6. Survival Analysis

6.1 Survival Data

6.1.1 Why Linear and Logistic Regression Would not Work

6.1.2 Hazard Function

6.1.3 Hazard Ratio

6.1.4 Proportional Hazards Assumption

6.2 Cox Proportional Hazards Models

6.2.1 Proportional Hazards Models

6.2.2 Parametric Versus Semi-Parametric Models

6.2.3 Hazard Ratios, Risk, and Survival Times

6.2.4 Hypothesis Tests and Confidence Intervals

6.2.5 Binary Predictors

6.2.6 Multilevel Categorical Predictors

6.2.7 Continuous Predictors

6.2.8 Confounding

6.2.9 Mediation

6.2.10 Interaction

6.2.11 Model Building

6.2.12 Adjusted Survival Curves for Comparing Groups

6.2.13 Predicted Survival for Specific Covariate Patterns

6.3 Extensions to the Cox Model

6.3.1 Time-Dependent Covariates

6.3.2 Stratified Cox Model

6.4 Checking Model Assumptions and Fit

6.4.1 Log-Linearity of the Hazard Function

6.4.2 Proportional Hazards

6.5 Competing Risks Data

6.5.1 What Are Competing Risks Data?

6.5.2 Notation for Competing Risks Data

6.5.3 Summaries for Competing Risks Data

6.6 Some Details

6.6.1 Bootstrap Confidence Intervals

6.6.2 Prediction

6.6.3 Adjusting for Nonconfounding Covariates

6.6.4 Independent Censoring

6.6.5 Interval Censoring

6.6.6 Left-Truncation

6.7 Sample Size, Power, and Detectable Effects

6.8 Summary

6.9 Further Notes and References

6.10 Problems

6.11 Learning Objectives

7. Repeated Measures and Longitudinal Data Analysis

7.1 A Simple Repeated Measures Example: Fecal Fat

7.1.1 Model Equations for the Fecal Fat Example

7.1.2 Correlations Within Subjects

7.1.3 Estimates of the Effects of Pill Type

7.2 Hierarchical Data

7.2.1 Example: Treatment of Back Pain

7.2.2 Example: Physician Profiling

7.2.3 Analysis Strategies for Hierarchical Data

7.3 Longitudinal Data

7.3.1 Analysis Strategies for Longitudinal Data

7.3.2 Analyzing Change Scores

7.4 Generalized Estimating Equations

7.4.1 Example: Birthweight and Birth Order Revisited

7.4.2 Correlation Structures

7.4.3 Working Correlation and Robust Standard Errors

7.4.4 Tests and Confidence Intervals

7.4.5 Use of **xtgee** for Clustered Logistic Regression

7.5 Random Effects Models

7.6 Re-Analysis of the Georgia Babies Data Set

7.7 Analysis of the SOF BMD Data

7.7.1 Time Varying Predictors

7.7.2 Separating Between- and Within-Cluster Information

7.7.3 Prediction

7.7.4 A Logistic Analysis

7.8 Marginal Versus Conditional Models

7.9 Example: Cardiac Injury Following Brain Hemorrhage

7.9.1 Bootstrap Analysis

7.10 Power and Sample Size for Repeated Measures Designs

7.10.1 Between-Cluster Predictor

7.10.2 Within-Cluster Predictor

7.11 Summary

7.12 Further Notes and References

7.12.1 Missing Data

7.12.2 Computing

7.13 Problems

7.14 Learning Objectives

8. Generalized Linear Models

8.1 Example: Treatment for Depression

8.1.1 Statistical Issues

8.1.2 Model for the Mean Response

8.1.3 Choice of Distribution

8.1.4 Interpreting the Parameters

8.1.5 Further Notes

8.2 Example: Costs of Phototherapy

8.2.1 Model for the Mean Response

8.2.2 Choice of Distribution

8.2.3 Interpreting the Parameters

8.3 Generalized Linear Models

8.3.1 Example: Risky Drug Use Behavior

8.3.2 Modeling Data with Many Zeros

8.3.3 Example: A Randomized Trial to Reduce Risk of Fracture

8.3.4 Relationship of Mean to Variance

8.3.5 Non-Linear Models

8.4 Sample Size for the Poisson Model

8.5 Summary

8.6 Further Notes and References

8.7 Problems

8.8 Learning Objectives

9. Strengthening Causal Inference

9.1 Potential Outcomes and Causal Effects

9.1.1 Average Causal Effects

9.1.2 Marginal Structural Model

9.1.3 Fundamental Problem of Causal Inference

9.1.4 Randomization Assumption

9.1.5 Conditional Independence

9.1.6 Marginal and Conditional Means

9.1.7 Potential Outcomes Estimation

9.1.8 Inverse Probability Weighting

9.2 Regression as a Basis for Causal Inference

9.2.1 No Unmeasured Confounders

9.2.2 Correct Model Specification

9.2.3 Overlap and the Positivity Assumption

9.2.4 Lack of Overlap and Model Misspecification

9.2.5 Adequate Sample Size and Number of Events

9.2.6 Example: Phototherapy for Neonatal Jaundice

9.3 Marginal Effects and Potential Outcomes Estimation

9.3.1 Marginal and Conditional Effects

9.3.2 Contrasting Conditional and Marginal Effects

9.3.3 When Marginal and Conditional Odds-Ratios Differ

9.3.4 Potential Outcomes Estimation

9.3.5 Marginal Effects in Longitudinal Data

9.4 Propensity Scores

9.4.1 Estimation of Propensity Scores

9.4.2 Effect Estimation Using Propensity Scores

9.4.3 Inverse Probability Weights

9.4.4 Checking for Propensity Score/Exposure Interaction

9.4.5 Addressing Positivity Violations Using Restriction

9.4.6 Average Treatment Effect in the Treated (ATT)

9.4.7 Recommendations for Using Propensity Scores

9.5 Time-Dependent Treatments

9.5.1 Models Using Time-Dependent IP Weights

9.5.2 Implementation

9.5.3 Drawbacks and Difficulties

9.5.4 Focusing of New Users

9.5.5 Nested New-User Cohorts

9.6 Mediation

9.7 Instrumental Variables

9.7.1 Vulnerabilities

9.7.2 Structural Equations and Instrumental Variables

9.7.3 Checking IV Assumptions

9.7.4 Example: Effect of Hormone Therapy on Change in LDL

9.7.5 Extension to Binary Exposures and Outcomes

9.7.6 Example: Phototherapy for Neonatal Jaundice

9.7.7 Interpretation of IV Estimates

9.8 Trials with Incomplete Adherence to Treatment

9.8.1 Intention-to-Treat

9.8.2 As-Treated Comparisons by Treatment Received

9.8.3 Instrumental Variables

9.8.4 Principal Stratification

9.9 Summary

9.10 Further Notes and References

9.11 Problems

9.12 Learning Objectives

10. Predictor Selection

10.1 Prediction

10.1.1 Bias–Variance Trade-off and Overfitting

10.1.2 Measures of Prediction Error

10.1.3 Optimism-Corrected Estimates of Prediction Error

10.1.4 Minimizing Prediction Error Without Overfitting

10.1.5 Point Scores

10.1.6 Example: Risk Stratification of Patients with Heart
Disease

10.2 Evaluating a Predictor of Primary Interest

10.2.1 Including Predictors for Face Validity

10.2.2 Selecting Predictors on Statistical Grounds

10.2.3 Interactions With the Predictor of Primary Interest

10.2.4 Example: Incontinence as a Risk Factor for Falling

10.2.5 Directed Acyclic Graphs

10.2.6 Randomized Experiments

10.3 Identifying Multiple Important Predictors

10.3.1 Ruling Out Confounding Is Still Central

10.3.2 Cautious Interpretation Is Also Key

10.3.3 Example: Risk Factors for Coronary Heart Disease

10.3.4 Allen–Cady Modified Backward Selection

10.4 Some Details

10.4.1 Collinearity

10.4.2 Number of Predictors

10.4.3 Alternatives to Backward Selection

10.4.4 Model Selection and Checking

10.4.5 Model Selection Complicates Inference

10.5 Summary

10.6 Further Notes and References

10.7 Problems

10.8 Learning Objectives

11. Missing Data

11.1 Why Missing Data Can Be a Problem

11.1.1 Missing Predictor in Linear Regression

11.1.2 Missing Outcome in Longitudinal Data

11.2 Classifications of Missing Data

11.2.1 Mechanisms for Missing Data

11.3 Simple Approaches to Handling Missing Data

11.3.1 Include a Missing Data Category

11.3.2 Last Observation or Baseline Carried Forward

11.4 Methods for Handling Missing Data

11.5 Missing Data in the Predictors and Multiple Imputation

11.5.1 Remarks About Using Multiple Imputation

11.5.2 Approaches to Multiple Imputation

11.5.3 Multiple Imputation for HERS

11.6 Deciding Which Missing Data Mechanism May Be Applicable

11.7 Missing Outcomes, Missing Completely at Random

11.8 Missing Outcomes, Covariate-Dependent Missing Completely at Random

11.9 Missing Outcomes for Longitudinal Studies, Missing at Random

11.9.1 ML and MAR

11.9.2 Multiple Imputation

11.9.3 Inverse Probability Weighting

11.10 Technical Details About Maximum Likelihood and Data Which Are
Missing at Random

11.10.1 An Example of the EM Algorithm

10.10.2 The EM Algorithm Imputes the Missing Data

10.10.3 ML Versus MI with Missing Outcomes

11.11 Methods for Data that Are Missing Not at Random

11.11.1 Pattern Mixture Models

11.11.2 Multiple Imputation Under MNAR

11.11.3 Joint Modeling of Outcomes and the Dropout Process

11.12 Summary

11.13 Further Notes and References

11.14 Problems

11.15 Learning Objectives

12. Complex Surveys

12.1 Overview of Complex Survey Designs

12.2 Inverse Probability Weighting

12.2.1 Accounting for Inverse Probability Weights in the
Analysis

12.2.2 Inverse Probability Weights and Missing Data

12.3 Clustering and Stratification

12.3.1 Design Effects

12.4 Example: Diabetes in NHANES

12.5 Some Details

12.5.1 Ignoring Secondary Levels of Clustering

12.5.2 Other Methods of Variance Estimation

12.5.3 Model Checking

12.5.4 Postestimation Capabilities in Stata

12.5.5 Other Statistical Packages for Complex Surveys

12.6 Summary

12.7 Further Notes and References

12.8 Problems

12.9 Learning Objectives

13. Summary

13.1 Introduction

13.2 Selecting Appropriate Statistical Methods

13.3 Planning and Executing a Data Analysis

13.3.1 Analysis Plans

13.3.2 Choice of Software

13.3.3 Data Preparation

13.3.4 Record Keeping and Reproducibility of Results

13.3.5 Data Security

13.3.6 Consulting a Statistician

13.3.7 Use of Internet Resources

13.4 Further Notes and References

13.4.1 Multiple Hypothesis Tests

13.4.2 Statistical Learning

References

Index