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Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models
Comment from the Stata technical groupThis book is intended for use as a teaching text for a one-semester or two-quarter secondary statistics course in biostatistics and focuses on multipredictor regression models in modern medical research. It lists as a prerequisite an introductory course in statistics or biostatistics, but the first three chapters provide sufficient review material to make this requirement not critical. The authors take a unified approach to regression models. They begin with linear regression and then discuss issues such as model statement and assumptions, types of regressors (e.g., categorical vs. continuous), interactions, causation and confounding, inference and testing, diagnostics, and alternative models for when assumptions are violated. Then they discuss these same issues in the contexts of other multipredictor regression models, namely logistic regression, the Cox model, and generalized linear models (GLMs). Chapters follow covering generalized estimating equations (GEE) and the analysis of survey data. Almost all analyses are performed using Stata. Table of contentsPreface
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
1.4 Guide to the book1.3.2 Isolating the effect of a single predictor 1.3.3 Understanding multiple predictors 2. Exploratory and descriptive methods
2.1 Data checking
2.2 Types of data 2.3 One-variable descriptions
2.3.1 Numerical variables
2.4 Two-variable descriptions2.3.2 Categorical variables
2.4.1 Outcome versus predictor variables
2.5 Multivariable descriptions2.4.2 Continuous outcome variable 2.4.3 Categorical outcome variable 2.6 Problems 3. Basic statistical methods
3.1 t-test and analysis of variance
3.1.1 t-test
3.2 Correlation coefficient3.1.2 One- and two-sided hypothesis test 3.1.3 Paired t-test 3.1.4 One-way analysis of variance (ANOVA) 3.1.5 Pairwise comparisons in ANOVA 3.1.6 Multi-way ANOVA and ANCOVA 3.1.7 Robustness to violations of assumptions 3.3 Simple linear regression model
3.3.1 Systematic part of the model
3.4 Contingency table methods for binary outcomes3.3.2 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 R2
3.4.1 Measures of risk and association for binary outcomes
3.5 Basic methods for survival analysis3.4.2 Tests of association in contingency tables 3.4.3 Predictors with multiple categories 3.4.4 Analysis involving multiple categorical predictors
3.5.1 Right censoring
3.6 Bootstrap confidence intervals3.5.2 Kaplan–Meier estimator of the survival function 3.5.3 Interpretation of Kaplan–Meier curves 3.5.4 Median survival 3.5.5 Cumulative incidence function 3.5.6 Comparing groups using the logrank test 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
4.3 Categorical predictors4.2.2 Random part of the model 4.2.3 Generalization of R2 and r 4.2.4 Standardized regression coefficients
4.3.1 Binary predictors
4.4 Confounding4.3.2 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
4.4.1 Causal effects and counterfactuals
4.5 Mediation4.4.2 A linear model for the counterfactual experiment 4.4.3 Confounding of causal effects 4.4.4 Randomization assumption 4.4.5 Conditions for confounding of causal effects 4.4.6 Control of confounding 4.4.7 Range of confounding patterns 4.4.8 Diagnostics for confounding in a sampl e 4.4.9 Confounding is difficult to rule out 4.4.10 Adjusted vs. unadjusted βs 4.4.11 Example: BMI and LDL
4.5.1 Modeling mediation
4.6 Interaction4.5.2 Confidence intervals for measures of mediation 4.5.3 Example: BMI, exercise, and glucose
4.6.1 Causal effects and interaction
4.7 Checking model assumptions and fit4.6.2 Modeling interaction 4.6.3 Overall causal effect in the presence of interaction 4.6.4 Example: Hormone therapy and statin use 4.6.5 Example: BMI and statin use 4.6.7 Interaction and scale 4.6.8 Details
4.7.1 Linearity
4.8 Summary4.7.2 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 4.9 Further notes and references 4.10 Problems 4.11 Learning objectives 5. Predictor selection
5.1 Diagramming and the hypothesized causal model
5.2 Prediction
5.2.1 Bias-variance trade-off
5.3 Evaluating a predictor of primary interest5.2.2 Estimating prediction error 5.2.3 Screening candidate models 5.2.4 Classification and regression trees (CART)
5.3.1 Including predictors for face validity
5.4 Identifying multiple important predictors5.3.2 Selecting predictors on statistical grounds 5.3.3 Interactions with the predictor of primary interest 5.3.4 Example: Incontinence as a risk factor for falling 5.3.5 Randomized experiments
5.4.1 Ruling out confounding is still central
5.5 Some details5.4.2 Cautious interpretation is also key 5.4.3 Example: Risk factors for coronary heart disease 5.4.4 Allen–Cady modified backward selection
5.5.1 Collinearity
5.6 Summary5.5.2 Number of predictors 5.5.3 Alternatives to backward selection 5.5.4 Model selection and checking 5.5.5 Model selection complicates inference 5.7 Further notes and references 5.8 Problems 5.9 Learning objectives 6. Logistic regression
6.1 Single predictor models
6.1.1 Interpretation of regression coefficients
6.2 Multipredictor models6.1.2 Categorical predictors
6.2.1 Likelihood ratio tests
6.3 Case–control studies6.2.2 Confounding 6.2.3 Interaction 6.2.4 Prediction 6.2.5 Prediction accuracy
6.3.1 Matched case–control studies
6.4 Checking models assumptions and fit
6.4.1 Outlying and influential points
6.5 Alternative strategies for binary outcomes6.4.2 Linearity 6.4.3 Model adequacy 6.4.4 Technical issues in logistic model fitting
6.5.1 Infectious disease transmission models
6.6 Likelihood6.5.2 Regression models based on excess and relative risks 6.5.3 Nonparametric binary regression 6.5.4 More than two outcome levels 6.7 Summary 6.8 Further notes and references 6.9 Problems 6.10 Learning objectives 7. Survival analysis
7.1 Survival data
7.1.1 Why linear and logistic regression won't work
7.2 Cox proportional hazards models7.1.2 Hazard function 7.1.3 Hazard ratio 7.1.4 Proportional hazards assumption
7.2.1 Proportional hazards models
7.3 Extensions to the Cox model7.2.2 Parametric vs. semi-parametric models 7.2.3 Hazard ratios, risk, and survival times 7.2.4 Hypothesis tests and confidence intervals 7.2.5 Binary predictors 7.2.6 Multilevel categorical predictors 7.2.7 Continuous predictors 7.2.8 Confounding 7.2.9 Mediation 7.2.10 Interaction 7.2.11 Adjusted survival curves for comparing groups 7.2.12 Predicted survival for specific covariate patterns
7.3.1 Time-dependent covariates
7.4 Checking model assumptions and fit7.3.2 Stratified Cox model
7.4.1 Log-linearity
7.5 Some details7.4.2 Proportional hazards
7.5.1 Bootstrap confidence intervals
7.6 Summary7.5.2 Prediction 7.5.3 Adjusting for non-confounding covariates 7.5.4 Independent censoring 7.5.5 Interval censoring 7.5.6 Left truncation 7.7 Further notes and references 7.8 Problems 7.9 Learning objectives 8. Repeated measures analysis
8.1 A simple repeated measures example: fecal fat
8.1.1 Model equations for the fecal fat example
8.2 Hierarchical data8.1.2 Correlations within subjects 8.1.3 Estimates of the effects of pill type
8.2.1 Analysis strategies for hierarchical data
8.3 Longitudinal data
8.3.1 Analysis strategies for longitudinal data
8.4 Generalized Estimating Equations8.3.2 Example: Birthweight and birth order 8.3.3 When to use repeated measures analyses
8.4.1 Birthweight and birth order revisited
8.5 Random effects models8.4.2 Correlation structures 8.4.3 Working correlation and robust standard errors 8.4.4 Hypothesis tests and confidence intervals 8.4.5 use of xtgee for clustered logistic regression
8.5.1 Re-analysis of birthweight and birth order
8.6 Example: Cardiac injury following brain hemorrhage8.5.2 Prediction 8.5.3 Logistic model for low birthweight 8.5.4 Marginal versus conditional models
8.6.1 Bootstrap confidence intervals
8.7 Summary8.8 Further notes and references 8.9 Problems 8.10 Learning objectives 9. Generalized linear models
9.1 Example: Treatment for depression
9.1.1 Statistical issues
9.2 Example: Costs of Phototherapy9.1.2 Model for the mean response 9.1.3 Choice of distribution 9.1.4 Interpreting the parameters 9.1.5 Further notes
9.2.1 Model for the mean response
9.3 Generalized linear models9.2.2 Choice of distribution 9.2.3 Interpreting the parameters
9.3.1 Example: Risky drug use behavior
9.4 Summary9.3.2 Relationship of mean to variance 9.3.3 Nonlinear models 9.5 Further notes and references 9.6 Problems 9.7 Learning objectives 10. Complex surveys
10.1 Example: NHANES
10.2 Probability weights 10.3 Variance estimation
10.3.1 Design effects
10.4 Summary10.3.2 Simplification of correlation structure 10.3.3 Other methods of variance estimation 10.5 Further notes and references 10.6 Problems 10.7 Learning objectives 11. Summary
11.1 Introduction
11.2 Selecting appropriate statistical methods 11.3 Planning and executing a data analysis
11.3.1 Analysis plans
11.4 Further notes and references11.3.2 Choice of software 11.3.3 Record keeping and organization 11.3.4 Data security 11.3.5 Consulting a statisticia n 11.3.6 Use of internet resources References
Index
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