Description
Book SynopsisA one-stop guide for public health students and practitioners learning the applications of classical regression models in epidemiology This book is written for public health professionals and students interested in applying regression models in the field of epidemiology.
Table of ContentsPreface xv
Acknowledgments xvii
About the Authors xix
1 Basic Concepts for Statistical Modeling 1
1.1 Introduction 1
1.2 Parameter Versus Statistic 2
1.3 Probability Definition 3
1.4 Conditional Probability 3
1.5 Concepts of Prevalence and Incidence 4
1.6 Random Variables 4
1.7 Probability Distributions 4
1.8 Centrality and Dispersion Parameters of a Random Variable 6
1.9 Independence and Dependence of Random Variables 7
1.10 Special Probability Distributions 7
1.10.1 Binomial Distribution 7
1.10.2 Poisson Distribution 8
1.10.3 Normal Distribution 9
1.11 Hypothesis Testing 11
1.12 Confidence Intervals 14
1.13 Clinical Significance Versus Statistical Significance 14
1.14 Data Management 15
1.14.1 Study Design 15
1.14.2 Data Collection 16
1.14.3 Data Entry 17
1.14.4 Data Screening 18
1.14.5 What to Do When Detecting a Data Issue 19
1.14.6 Impact of Data Issues and How to Proceed 20
1.15 Concept of Causality 21
References 22
2 Introduction to Simple Linear Regression Models 25
2.1 Introduction 25
2.2 Specific Objectives 26
2.3 Model Definition 26
2.4 Model Assumptions 28
2.5 Graphic Representation 29
2.6 Geometry of the Simple Regression Model 29
2.7 Estimation of Parameters 30
2.8 Variance of Estimators 31
2.9 Hypothesis Testing About the Slope of the Regression Line 32
2.9.1 Using the Student’s t-Distribution 32
2.9.2 Using ANOVA 32
2.10 Coefficient of Determination R2 34
2.11 Pearson Correlation Coefficient 34
2.12 Estimation of Regression Line Values and Prediction 35
2.12.1 Confidence Interval for the Regression Line 35
2.12.2 Prediction Interval of Actual Values of the Response 36
2.13 Example 36
2.14 Predictions 39
2.14.1 Predictions with the Database Used by the Model 40
2.14.2 Predictions with Data Not Used to Create the Model 42
2.14.3 Residual Analysis 44
2.15 Conclusions 46
Practice Exercise 47
References 48
3 Matrix Representation of the Linear Regression Model 49
3.1 Introduction 49
3.2 Specific Objectives 49
3.3 Definition 50
3.3.1 Matrix 50
3.4 Matrix Representation of a SLRM 50
3.5 Matrix Arithmetic 51
3.5.1 Addition and Subtraction of Matrices 51
3.6 Matrix Multiplication 52
3.7 Special Matrices 53
3.8 Linear Dependence 54
3.9 Rank of a Matrix 54
3.10 Inverse Matrix [A−1] 54
3.11 Application of an Inverse Matrix in a SLRM 56
3.12 Estimation of β Parameters in a SLRM 56
3.13 Multiple Linear Regression Model (MLRM) 57
3.14 Interpretation of the Coefficients in a MLRM 58
3.15 ANOVA in a MLRM 58
3.16 Using Indicator Variables (Dummy Variables) 60
3.17 Polynomial Regression Models 63
3.18 Centering 64
3.19 Multicollinearity 65
3.20 Interaction Terms 65
3.21 Conclusion 66
Practice Exercise 66
References 67
4 Evaluation of Partial Tests of Hypotheses in a MLRM 69
4.1 Introduction 69
4.2 Specific Objectives 69
4.3 Definition of Partial Hypothesis 70
4.4 Evaluation Process of Partial Hypotheses 71
4.5 Special Cases 71
4.6 Examples 72
4.7 Conclusion 75
Practice Exercise 75
References 75
5 Selection of Variables in a Multiple Linear Regression Model 77
5.1 Introduction 77
5.2 Specific Objectives 77
5.3 Selection of Variables According to the Study Objectives 77
5.4 Criteria for Selecting the Best Regression Model 78
5.4.1 Coefficient of Determination, R2 78
5.4.2 Adjusted Coefficient of Determination, R2A 78
5.4.3 Mean Square Error (MSE) 79
5.4.4 Mallows’s Cp 79
5.4.5 Akaike Information Criterion 79
5.4.6 Bayesian Information Criterion 80
5.4.7 All Possible Models 80
5.5 Stepwise Method in Regression 80
5.5.1 Forward Selection 81
5.5.2 Backward Elimination 82
5.5.3 Stepwise Selection 82
5.6 Limitations of Stepwise Methods 83
5.7 Conclusion 83
Practice Exercise 84
References 85
6 Correlation Analysis 87
6.1 Introduction 87
6.2 Specific Objectives 87
6.3 Main Correlation Coefficients Based on SLRM 87
6.3.1 Pearson Correlation Coefficient ρ 88
6.3.2 Relationship Between r and ^β1 89
6.4 Major Correlation Coefficients Based on MLRM 89
6.4.1 Pearson Correlation Coefficient of Zero Order 89
6.4.2 Multiple Correlation Coefficient 90
6.5 Partial Correlation Coefficient 90
6.5.1 Partial Correlation Coefficient of the First Order 91
6.5.2 Partial Correlation Coefficient of the Second Order 91
6.5.3 Semipartial Correlation Coefficient 91
6.6 Significance Tests 92
6.7 Suggested Correlations 92
6.8 Example 92
6.9 Conclusion 94
Practice Exercise 95
References 95
7 Strategies for Assessing the Adequacy of the Linear Regression Model 97
7.1 Introduction 97
7.2 Specific Objectives 98
7.3 Residual Definition 98
7.4 Initial Exploration 98
7.5 Initial Considerations 102
7.6 Standardized Residual 102
7.7 Jackknife Residuals (R-Student Residuals) 104
7.8 Normality of the Errors 105
7.9 Correlation of Errors 106
7.10 Criteria for Detecting Outliers, Leverage, and Influential Points 107
7.11 Leverage Values 108
7.12 Cook’s Distance 108
7.13 COV Ratio 109
7.14 DFBETAS 110
7.15 DFFITS 110
7.16 Summary of the Results 111
7.17 Multicollinearity 111
7.18 Transformation of Variables 114
7.19 Conclusion 114
Practice Exercise 115
References 116
8 Weighted Least-Squares Linear Regression 117
8.1 Introduction 117
8.2 Specific Objectives 117
8.3 Regression Model with Transformation into the Original Scale of Y 117
8.4 Matrix Notation of the Weighted Linear Regression Model 119
8.5 Application of the WLS Model with Unequal Number of Subjects 120
8.5.1 Design without Intercept 121
8.5.2 Model with Intercept and Weighting Factor 122
8.6 Applications of the WLS Model When Variance Increases 123
8.6.1 First Alternative 123
8.6.2 Second Alternative 124
8.7 Conclusions 125
Practice Exercise 126
References 127
9 Generalized Linear Models 129
9.1 Introduction 129
9.2 Specific Objectives 129
9.3 Exponential Family of Probability Distributions 130
9.3.1 Binomial Distribution 130
9.3.2 Poisson Distribution 131
9.4 Exponential Family of Probability Distributions with Dispersion 131
9.5 Mean and Variance in EF and EDF 132
9.6 Definition of a Generalized Linear Model 133
9.7 Estimation Methods 134
9.8 Deviance Calculation 135
9.9 Hypothesis Evaluation 136
9.10 Analysis of Residuals 138
9.11 Model Selection 139
9.12 Bayesian Models 139
9.13 Conclusions 140
References 140
10 Poisson Regression Models for Cohort Studies 141
10.1 Introduction 141
10.2 Specific Objectives 142
10.3 Incidence Measures 142
10.3.1 Incidence Density 142
10.3.2 Cumulative Incidence 145
10.4 Confounding Variable 146
10.5 Stratified Analysis 147
10.6 Poisson Regression Model 148
10.7 Definition of Adjusted Relative Risk 149
10.8 Interaction Assessment 150
10.9 Relative Risk Estimation 151
10.10 Implementation of the Poisson Regression Model 152
10.11 Conclusion 161
Practice Exercise 162
References 162
11 Logistic Regression in Case–Control Studies 165
11.1 Introduction 165
11.2 Specific Objectives 166
11.3 Graphical Representation 166
11.4 Definition of the Odds Ratio 167
11.5 Confounding Assessment 168
11.6 Effect Modification 168
11.7 Stratified Analysis 169
11.8 Unconditional Logistic Regression Model 170
11.9 Types of Logistic Regression Models 171
11.9.1 Binary Case 172
11.9.2 Binomial Case 172
11.10 Computing the ORcrude 173
11.11 Computing the Adjusted OR 173
11.12 Inference on OR 174
11.13 Example of the Application of ULR Model: Binomial Case 175
11.14 Conditional Logistic Regression Model 178
11.15 Conclusions 183
Practice Exercise 183
References 188
12 Regression Models in a Cross-Sectional Study 191
12.1 Introduction 191
12.2 Specific Objectives 192
12.3 Prevalence Estimation Using the Normal Approach 192
12.4 Definition of the Magnitude of the Association 198
12.5 POR Estimation 200
12.5.1 Woolf’s Method 200
12.5.2 Exact Method 202
12.6 Prevalence Ratio 204
12.7 Stratified Analysis 204
12.8 Logistic Regression Model 207
12.8.1 Modeling Prevalence Odds Ratio 207
12.8.2 Modeling Prevalence Ratio 209
12.9 Conclusions 210
Practice Exercise 210
References 211
13 Solutions to Practice Exercises 213
Chapter 2 Practice Exercise 213
Chapter 3 Practice Exercise 216
Chapter 4 Practice Exercise 220
Chapter 5 Practice Exercise 221
Chapter 6 Practice Exercise 223
Chapter 7 Practice Exercise 225
Chapter 8 Practice Exercise 228
Chapter 10 Practice Exercise 230
Chapter 11 Practice Exercise 233
Chapter 12 Practice Exercise 240
Index 245
