Preface
Acknowledgments
About the Authors
1 Basic Concepts for Statistical Modeling
1.1 Introduction
1.2 Parameter Versus Statistic
1.3 Probability Definition
1.4 Conditional Probability
1.5 Concepts of Prevalence and Incidence
1.6 Random Variables
1.7 Probability Distributions
1.8 Centrality and Dispersion Parameters of a Random Variable
1.9 Independence and Dependence of Random Variables
1.10 Special Probability Distributions
1.10.1 Binomial Distribution
1.10.2 Poisson Distribution
1.10.3 Normal Distribution
1.11 Hypothesis Testing
1.12 Confidence Intervals
1.13 Clinical Significance Versus Statistical Significance
1.14 Data Management
1.14.1 Study Design
1.14.2 Data Collection
1.14.3 Data Entry
1.14.4 Data Screening
1.14.5 What to Do When Detecting a Data Issue
1.14.6 Impact of Data Issues and How to Proceed
1.15 Concept of Causality
References
2 Introduction to Simple Linear Regression Models
2.1 Introduction
2.2 Specific Objectives
2.3 Model Definition
2.4 Model Assumptions
2.5 Graphic Representation
2.6 Geometry of the Simple Regression Model
2.7 Estimation of Parameters
2.8 Variance of Estimators
2.9 Hypothesis Testing About the Slope of the Regression Line
2.9.1 Using the Student's t-Distribution
2.9.2 Using ANOVA
2.10 Coefficient of Determination
R2
2.11 Pearson Correlation Coefficient
2.12 Estimation of Regression Line Values and Prediction
2.12.1 Confidence Interval for the Regression Line
2.12.2 Prediction Interval of Actual Values of the Response
2.13 Example
2.14 Predictions
2.14.1 Predictions with the Database Used by the Model
2.14.2 Predictions with Data Not Used to Create the Model
2.14.3 Residual Analysis
2.15 Conclusions
Practice Exercise
References
3 Matrix Representation of the Linear Regression Model
3.1 Introduction
3.2 Specific Objectives
3.3 Definition
3.3.1 Matrix
3.4 Matrix Representation of a SLRM
3.5 Matrix Arithmetic
3.5.1 Addition and Subtraction of Matrices
3.6 Matrix Multiplication
3.7 Special Matrices
3.8 Linear Dependence
3.9 Rank of a Matrix
3.10 Inverse Matrix [A
–1]
3.11 Application of an Inverse Matrix in a SLRM
3.12 Estimation of
β Parameters in a SLRM
3.13 Multiple Linear Regression Model (MLRM)
3.14 Interpretation of the Coefficients in a MLRM
3.15 ANOVA in a MLRM
3.16 Using Indicator Variables
(Dummy Variables)
3.17 Polynomial Regression Models
3.18 Centering
3.19 Multicollinearity
3.20 Interaction Terms
3.21 Conclusion
Practice Exercise
References
4 Evaluation of Partial Tests of Hypotheses in a MLRM
4.1 Introduction
4.2 Specific Objectives
4.3 Definition of Partial Hypothesis
4.4 Evaluation Process of Partial Hypotheses
4.5 Special Cases
4.6 Examples
4.7 Conclusion
Practice Exercise
References
5 Selection of Variables in a Multiple Linear Regression Model
5.1 Introduction
5.2 Specific Objectives
5.3 Selection of Variables According to the Study Objectives
5.4 Criteria for Selecting the Best Regression Model
5.4.1 Coefficient of Determination, R2
5.4.2 Adjusted Coefficient of Determination, R2A
5.4.3 Mean Square Error (MSE)
5.4.4 Mallows's Cp
5.4.5 Akaike Information Criterion
5.4.6 Bayesian Information Criterion
5.4.7 All Possible Models
5.5
Stepwise Method in Regression
5.5.1 Forward Selection
5.5.2 Backward Elimination
5.5.3 Stepwise Selection
5.6 Limitations of
Stepwise Methods
5.7 Conclusion
Practice Exercise
References
6 Correlation Analysis
6.1 Introduction
6.2 Specific Objectives
6.3 Main Correlation Coefficients Based on SLRM
6.3.1 Pearson Correlation Coefficient ρ
6.3.2 Relationship Between r and β1
6.4 Major Correlation Coefficients Based on MLRM
6.4.1 Pearson Correlation Coefficient of Zero Order
6.4.2 Multiple Correlation Coefficient
6.5 Partial Correlation Coefficient
6.5.1 Partial Correlation Coefficient of the First Order
6.5.2 Partial Correlation Coefficient of the Second Order
6.5.3 Semipartial Correlation Coefficient
6.6 Significance Tests
6.7 Suggested Correlations
6.8 Example
6.9 Conclusion
Practice Exercise
References
7 Strategies for Assessing the Adequacy of the Linear Regression Model
7.1 Introduction
7.2 Specific Objectives
7.3 Residual Definition
7.4 Initial Exploration
7.5 Initial Considerations
7.6 Standardized Residual
7.7 Jackknife Residuals (R-Student Residuals)
7.8 Normality of the Errors
7.9 Correlation of Errors
7.10 Criteria for Detecting Outliers, Leverage, and Influential Points
7.11 Leverage Values
7.12 Cook's Distance
7.13 COV RATIO
7.14 DFBETAS
7.15 DFFITS
7.16 Summary of the Results
7.17 Multicollinearity
7.18 Transformation of Variables
7.19 Conclusion
Practice Exercise
References
8 Weighted Least-Squares Linear Regression
8.1 Introduction
8.2 Specific Objectives
8.3 Regression Model with Transformation into the Original Scale of
Y
8.4 Matrix Notation of the Weighted Linear Regression Model
8.5 Application of the WLS Model with Unequal Number of Subjects
8.5.1 Design without Intercept
8.5.2 Model with Intercept and Weighting Factor
8.6 Applications of the WLS Model When Variance Increases
8.6.1 First Alternative
8.6.2 Second Alternative
8.7 Conclusions
Practice Exercise
References
9 Generalized Linear Models
9.1 Introduction
9.2 Specific Objectives
9.3 Exponential Family of Probability Distributions
9.3.1 Binomial Distribution
9.3.2 Poisson Distribution
9.4 Exponential Family of Probability Distributions with Dispersion
9.5 Mean and Variance in EF and EDF
9.6 Definition of a Generalized Linear Model
9.7 Estimation Methods
9.8
Deviance Calculation
9.9 Hypothesis Evaluation
9.10 Analysis of Residuals
9.11 Model Selection
9.12 Bayesian Models
9.13 Conclusions
References
10 Poisson Regression Models for Cohort Studies
10.1 Introduction
10.2 Specific Objectives
10.3 Incidence Measures
10.3.1 Incidence Density
10.3.2 Cumulative Incidence
10.4 Confounding Variable
10.5 Stratified Analysis
10.6 Poisson Regression Model
10.7 Definition of Adjusted Relative Risk
10.8 Interaction Assessment
10.9 Relative Risk Estimation
10.10 Implementation of the Poisson Regression Model
10.11 Conclusion
Practice Exercise
References
11 Logistic Regression in Case–Control Studies
11.1 Introduction
11.2 Specific Objectives
11.3 Graphical Representation
11.4 Definition of the Odds Ratio
11.5 Confounding Assessment
11.6 Effect Modification
11.7 Stratified Analysis
11.8 Unconditional Logistic Regression Model
11.9 Types of Logistic Regression Models
11.9.1 Binary Case
11.9.2 Binomial Case
11.10 Computing the OR
crude
11.11 Computing the Adjusted OR
11.12 Inference on
OR
11.13 Example of the Application of ULR Model: Binomial Case
11.14 Conditional Logistic Regression Model
11.15 Conclusions
Practice Exercise
References
12 Regression Models in a Cross-Sectional Study
12.1 Introduction
12.2 Specific Objectives
12.3 Prevalence Estimation Using the Normal Approach
12.4 Definition of the Magnitude of the Association
12.5 POR Estimation
12.5.1 Woolf's Method
12.5.2 Exact Method
12.6 Prevalence Ratio
12.7 Stratified Analysis
12.8 Logistic Regression Model
12.8.1 Modeling Prevalence Odds Ratio
12.8.2 Modeling Prevalence Ratio
12.9 Conclusions
Practice Exercise
References
13 Solutions to Practice Exercises
Chapter 2 Practice Exercise
Chapter 3 Practice Exercise
Chapter 4 Practice Exercise
Chapter 5 Practice Exercise
Chapter 6 Practice Exercise
Chapter 7 Practice Exercise
Chapter 8 Practice Exercise
Chapter 10 Practice Exercise
Chapter 11 Practice Exercise
Chapter 12 Practice Exercise
Index
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