Contents

1. Introduction 1

1.1 Probability, 1

1.2 Parameter Estimation, 21

1.3 Random Sampling, 27

2. Measurement Issues in Epidemiology 31

2.1 Systematic and Random Error, 31

2.2 Measures of Effect, 33

2.3 Confounding, 40

2.4 Collapsibility Approach to Confounding, 46

2.5 Counterfactual Approach to Confounding, 55

2.6 Methods to Control Confounding, 67

2.7 Bias Due to an Unknown Confounder, 69

2.8 Misclassification, 72

2.9 Scope of this Book, 75

3. Binomial Methods for Single Sample Closed Cohort Data 77

3.1 Exact Methods, 77

3.2 Asymptotic Methods, 82

4. Odds Ratio Methods for Unstratified Closed Cohort Data 89

4.1 Asymptotic Unconditional Methods for a Single 2 × 2 Table, 90

4.2 Exact Conditional Methods for a Single 2 × 2 Table, 101

4.3 Asymptotic Conditional Methods for a Single 2 × 2 Table, 106

4.4 Cornfield’s Approximation, 109

4.5 Summary of Examples and Recommendations, 112

4.6 Asymptotic Methods for a Single 2 × I Table, 112

5. Odds Ratio Methods for Stratified Closed Cohort Data 119

5.1 Asymptotic Unconditional Methods for J (2 × 2) Tables, 119

5.2 Asymptotic Conditional Methods for J (2 × 2) Tables, 129

5.3 Mantel–Haenszel Estimate of the Odds Ratio, 132

5.4 Weighted Least Squares Methods for J (2 × 2) Tables, 134

5.5 Interpretation Under Heterogeneity, 136

5.6 Summary of 2 × 2 Examples and Recommendations, 137

5.7 Asymptotic Methods for J (2 × I ) Tables, 138

6. Risk Ratio Methods for Closed Cohort Data 143

6.1 Asymptotic Unconditional Methods for a Single 2 × 2 Table, 143

6.2 Asymptotic Unconditional Methods for J (2 × 2) Tables, 145

6.3 Mantel–Haenszel Estimate of the Risk Ratio, 148

6.4 Weighted Least Squares Methods for J (2 × 2) Tables, 149

6.5 Summary of Examples and Recommendations, 150

7. Risk Difference Methods for Closed Cohort Data 151

7.1 Asymptotic Unconditional Methods for a Single 2 × 2 Table, 151

7.2 Asymptotic Unconditional Methods for J (2 × 2) Tables, 152

7.3 Mantel–Haenszel Estimate of the Risk Difference, 155

7.4 Weighted Least Squares Methods for J (2 × 2) Tables, 157

7.5 Summary of Examples and Recommendations, 157

8. Survival Analysis 159

8.1 Open Cohort Studies and Censoring, 159

8.2 Survival Functions and Hazard Functions, 163

8.3 Hazard Ratio, 166

8.4 Competing Risks, 167

9. Kaplan–Meier and Actuarial Methods for Censored Survival Data 171

9.1 Kaplan–Meier Survival Curve, 171

9.2 Odds Ratio Methods for Censored Survival Data, 178

9.3 Actuarial Method, 189

10. Poisson Methods for Censored Survival Data 193

10.1 Poisson Methods for Single Sample Survival Data, 193

10.2 Poisson Methods for Unstratified Survival Data, 206

10.3 Poisson Methods for Stratified Survival Data, 218

11. Odds Ratio Methods for Case-Control Data 229

11.1 Justification of the Odds Ratio Approach, 229

11.2 Odds Ratio Methods for Matched-Pairs Case-Control Data, 236

11.3 Odds Ratio Methods for (1 : M) Matched Case-Control Data, 244

12. Standardized Rates and Age–Period–Cohort Analysis 249

12.1 Population Rates, 249

12.2 Directly Standardized Death Rate, 251

12.3 Standardized Mortality Ratio, 255

12.4 Age–Period–Cohort Analysis, 258

13. Life Tables 263

13.1 Ordinary Life Table, 264

13.2 Multiple Decrement Life Table, 270

13.3 Cause-Deleted Life Table, 274

13.4 Analysis of Morbidity Using Life Tables, 276

14. Sample Size and Power 281

14.1 Sample Size for a Prevalence Study, 281

14.2 Sample Size for a Closed Cohort Study, 283

14.3 Sample Size for an Open Cohort Study, 285

14.4 Sample Size for an Incidence Case-Control Study, 287

14.5 Controlling for Confounding, 291

14.6 Power, 292

15. Logistic Regression and Cox Regression 295

15.1 Logistic Regression, 296

15.2 Cox Regression, 305

Appendix A Odds Ratio Inequality 307

Appendix B Maximum Likelihood Theory 311

B.1 Unconditional Maximum Likelihood, 311

B.2 Binomial Distribution, 313

B.3 Poisson Distribution, 320

B.4 Matrix Inversion, 323

Appendix C Hypergeometric and Conditional Poisson Distributions 325

C.1 Hypergeometric, 325

C.2 Conditional Poisson, 326

C.3 Hypergeometric Variance Estimate, 327

C.4 Conditional Poisson Variance Estimate, 328

Appendix D Quadratic Equation for the Odds Ratio 329

Appendix E Matrix Identities and Inequalities 331

E.1 Identities and Inequalities for J (1 × I ) and J (2 × I ) Tables, 331

E.2 Identities and Inequalities for a Single Table, 336

E.3 Hypergeometric Distribution, 336

E.4 Conditional Poisson Distribution, 337

Appendix F Survival Analysis and Life Tables 339

F.1 Single Cohort, 339

F.2 Comparison of Cohorts, 340

F.3 Life Tables, 341

Appendix G Confounding in Open Cohort and Case-Control Studies 343

G.1 Open Cohort Studies, 343

G.2 Case-Control Studies, 350

Appendix H Odds Ratio Estimate in a Matched Case-Control Study 353

H.1 Asymptotic Unconditional Estimate of Matched-Pairs Odds

Ratio, 353

H.2 Asymptotic Conditional Analysis of (1 : M) Matched

Case-Control Data, 354

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