Quantitative Methods in Population Health: Extensions of Ordinary Regression (Wiley Series in Probability and Statistics)

Quantitative Methods in Population Health......Page 3
List of Figures......Page 7
List of Tables......Page 9
Contents......Page 11
Preface......Page 17
Acknowledgments......Page 19
Acronyms......Page 21
I.1 Newborn Lung Project......Page 23
I.2 Wisconsin Diabetes Registry......Page 24
I.3 Wisconsin Sleep Cohort Study......Page 25
Suggested Reading......Page 27
1.1 The Ordinary Linear Regression Equation and Its Assumptions......Page 29
1.1.1 Straight-Line Relationship......Page 30
1.1.2 Equal Variance Assumption......Page 33
1.1.3 Normality Assumption......Page 34
1.1.4 Independence Assumption......Page 35
1.2 A Note on How the Least-Squares Estimators are Obtained......Page 36
Output Packet I: Examples of Ordinary Regression Analyses......Page 37
2.1 Maximum Likelihood Estimation......Page 49
2.2 Example......Page 52
2.3 Properties of Maximum Likelihood Estimators......Page 53
Output Packet II: Using PROC MIXED and Comparisons to PROC REG......Page 54
3 Reformulating Ordinary Regression Analysis in Matrix Notation......Page 58
3.1 Writing the Ordinary Regression Equation in Matrix Notation......Page 59
3.1.1 Example......Page 60
3.2 Obtaining the Least-Squares Estimator b in Matrix Notation......Page 61
3.2.1 Example: Matrices in Regression Analysis......Page 62
3.3 List of Matrix Operations to Know......Page 64
4.1 Variance and Correlation Matrices......Page 66
4.2.1 Two Variables......Page 68
4.2.2 Many Variables......Page 70
5.1 Usual Standard Error of Least-Squares Estimator of Regression Slope in Nonmatrix Formulation......Page 79
5.2 Standard Errors of Least-Squares Regression Estimators in Matrix Notation......Page 80
5.2.1 Example......Page 81
5.3 The Large Sample Variance Matrix of Maximum Likelihood Estimators......Page 82
5.4 Tests and Confidence Intervals......Page 84
5.4.1 Example-Comparing PROC REG and PROC MIXED......Page 85
6.1 Ordinary Least Squares with Unequal Variance......Page 90
6.1.1 Examples......Page 92
6.2.1 The Functional Transformation Approach......Page 94
6.2.2 The Linear Transformation Approach......Page 96
6.2.3 Standard Errors of Weighted Regression Estimators......Page 101
Output Packet III: Applying the Empirical Option to Adjust Standard Errors......Page 103
Output Packet IV: Analyses with Transformation of the Outcome Variable to Equalize Residual Variance......Page 111
Output Packet V: Weighted Regression Analyses of GHb Data on Age......Page 121
7 Application of Weighting with Probability Sampling and Nonresponse......Page 125
7.1 Sample Surveys with Unequal Probability Sampling......Page 126
7.1.1 Example......Page 129
7.2 Examining the Impact of Nonresponse......Page 130
7.2.1 Example (of Reweighting as Well as Some SAS Manipulations)......Page 132
7.2.2 A Few Comments on Weighting by a Variable Versus Including it in the Regression Model......Page 135
Output Packet VI: Survey and Missing Data Weights......Page 137
8 Principles in Dealing with Correlated Data......Page 147
8.1 Analysis of Correlated Data by Ordinary Unweighted Least-Squares Estimation......Page 148
8.1.1 Example......Page 149
8.1.2 Deriving the Variance Estimator......Page 150
8.2 Specifying Correlation and Variance Matrices......Page 152
8.3 The Least-Squares Equation Incorporating Correlation......Page 154
8.3.1 Another Application of the Spectral Theorem......Page 155
8.4 Applying the Spectral Theorem to the Regression Analysis of Correlated Data......Page 156
8.5 Analysis of Correlated Data by Maximum Likelihood......Page 157
8.5.1 Non equal Variance......Page 158
8.5.2 Correlated Errors......Page 159
8.5.3 Example......Page 160
Output Packet VII: Analysis of Longitudinal Data in Wisconsin Sleep Cohort......Page 163
9 A Further Study of How the Transformation Works with Correlated Data......Page 173
9.1 Why Would b(W) and b(B) Differ?......Page 175
9.2 How the Between- and Within-Individual Estimators are Combined......Page 177
9.3 How to Proceed in Practice......Page 179
9.3.1 Example......Page 180
Output Packet VIII: Investigating and Fitting Within- and Between-Individual Effects......Page 182
10.1 Random Intercept......Page 184
10.1.1 Example......Page 187
10.2 Random Slopes......Page 189
10.2.1 Example......Page 193
10.3.1 Example......Page 195
Output Packet IX: Fitting Random Effects Models......Page 197
11 The Normal Distribution and Likelihood Revisited......Page 209
11.1 PROC GENMOD......Page 210
11.1.1 Example......Page 211
Output Packet X: Introducing PROC GENMOD......Page 212
12.1 The Exponential Family......Page 218
12.1.1 The Binomial Distribution......Page 220
12.1.2 The Poisson Distribution......Page 221
12.2 Score Equations for the Exponential Family and the Canonical Link......Page 222
12.3 Other Link Functions......Page 224
12.3.1 Example......Page 225
13.1 A Brief Review of Logistic Regression......Page 227
13.1.1 Example: Review of the Output from PROC LOGIST......Page 228
13.2 Analysis of Binomial Data in the Generalized Linear Models Framework......Page 230
13.2.1 Example of Logistic Regression with Binary Outcome......Page 234
13.2.2 Example with Binomial Outcome......Page 235
13.3 Other Links for Binary and Binomial Data......Page 237
13.3.1 Example......Page 239
Output Packet XI: Logistic Regression Analysis with PROC LOGIST and PROC GENMOD......Page 240
Output Packet XII: Analysis of Grouped Binomial Data......Page 249
Output Packet XIII: Some Goodness-of-Fit Tests for Binomial Outcome......Page 251
Output Packet XIV: Three Link Functions for Binary Outcome......Page 257
Output Packet XV: Poisson Regression......Page 275
Output Packet XVI: Dealing with Overdispersion in Rates......Page 282
14.1 Review of Rates......Page 264
14.1.1 Relationship Between Rate and Risk......Page 266
14.2 Regression Analysis......Page 267
14.3 Example with Cancer Mortality Rates......Page 269
14.3.1 Example with Hospitalization of Infants......Page 270
14.4 Overdispersion......Page 271
14.4.1 Fitting a Dispersion Parameter......Page 272
14.4.3 Using Robust Standard Errors......Page 273
14.4.4 Applying Adjustments for Over Dispersion to the Examples......Page 274
15.1 A Brief Review and Reformulation of the Normal Distribution, Least Squares and Likelihood......Page 291
15.2 Further Developments for the Exponential Family......Page 292
15.3 How are the Generalized Estimating Equations Justified?......Page 294
15.3.1 Analysis of Longitudinal Systolic Blood Pressure by PROC MIXED and GENMOD......Page 295
15.3.2 Analysis of Longitudinal Hypertension Data by PROC GENMOD......Page 297
15.3.3 Analysis of Hospitalizations Among VLBW Children Up to Age 5......Page 299
15.4 Another Way to Deal with Correlated Binary Data......Page 301
Output Packet XVII: Mixed Versus GENMOD for Longitudinal SBP and Hypertension Data......Page 302
Output Packet XVIII: Longitudinal Analysis of Rates......Page 313
Output Packet XIX: Conditional Logistic Regression of Hypertension Data......Page 316
References......Page 318
Appendix: Matrix Operations......Page 323
A.1 Adding Matrices......Page 324
A.3 Multiplying Matrices by Each Other......Page 325
A.4 The Inverse of a Matrix......Page 327
Index......Page 331