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Biostat. Methods Table of Contents
Biostatistical Methods: The Assessment of Relative Risks

John M. Lachin
John Wiley and Sons, 2000
ISBN: 0-471-36996-9

TABLE OF CONTENTS

1 Biostatistics and Biomedical Science, 1

1.1 Statistics and the Scientific Method, 1
1.2 Biostatistics, 2
1.3 Natural History of Disease Progression, 3
1.4 Types of Biomedical Studies, 5
1.5 Studies of Diabetic Nephropathy, 7

2 Relative Risk Estimates and Tests for Two Independent Groups, 13

2.1 Probability As a Measure of Risk, 14
2.1.1 Prevalence and Incidence, 14
2.1.2 Binomial Distribution and Large Sample Approximations, 14
2.1.3 Asymmetric Confidence Limits, 15
2.1.3.1 Exact Confidence Limits, 15
2.1.3.2 Logit Confidence Limits, 16
2.1.3.3 Complimentary log-log Confidence Limits, 17
2.1.3.4 Test Inverted Confidence Limits, 18
2.1.4 Case of Zero Events, 19
2.2 Measures of Relative Risk, 19
2.3 Large Sample Distribution, 23
2.3.1 Risk Difference, 23
2.3.2 Relative Risk, 24
2.3.3 Odds Ratio, 26
2.4 Sampling Models: - Likelihoods, 28
2.4.1 Unconditional Product Binomial Likelihood, 28
2.4.2 Conditional Hypergeometric Likelihood, 28
2.4.3 Maximum Likelihood Estimates, 29
2.4.4 Asymptotically Unbiased Estimates, 30
2.5 Exact Inference, 32
2.5.1 Confidence Limits, 32
2.5.2 Fisher-Irwin Exact Test, 33
2.6 Large Sample Tests, 36
2.6.1 General Considerations, 36
2.6.2 Unconditional Test, 39
2.6.3 Conditional Mantel-Haenszel Test, 40
2.6.4 Cochran's Test, 40
2.6.5 Likelihood Ratio Test, 42
2.6.6 Test-Based Confidence Limits, 43
2.6.7 Continuity Correction, 44
2.7 SAS PROC FREQ, 45
2.8 Other Measures of Differential Risk, 50
2.8.1 Attributable Risk Fraction, 50
2.8.2 Population Attributable Risk, 50
2.8.3 Number Needed to Treat, 53
2.9 Problems, 54

3 Sample Size, Power, and Efficiency, 61

3.1 Estimation Precision, 62
3.2 Power of Z -Tests, 63
3.2.1 Type I and II Errors and Power, 63
3.2.2 Power and Sample Size, 67
3.3 Test for Two Proportions, 68
3.3.1 Power of the Z -Test, 69
3.3.2 Relative Risk and Odds Ratio, 71
3.4 Power of Chi-Square Tests, 73
3.5 Efficiency, 75
3.5.1 Pitman Efficiency, 75
3.5.2 Asymptotic Relative Efficiency, 78
3.5.3 Estimation Efficiency, 79
3.5.4 Stratified Versus Unstratified Analysis of Risk Differences, 80
3.6 Problems, 83

4 Stratified-Adjusted Analysis for Two Independent Groups, 87

4.1 Introduction, 87
4.2 Mantel-Haenszel Test and Cochran's Test, 89
4.2.1 Conditional Within-Strata Analysis, 89
4.2.2 Marginal Unadjusted Analysis, 90
4.2.3 Mantel-Haenszel Test, 92
4.2.4 Cochran's Test, 93
4.3 Stratified-Adjusted Estimators, 95
4.3.1 Mantel-Haenszel Estimators, 95
4.3.2 Test-Based Confidence Limits, 96
4.3.3 Large Sample Variance of Log Odds Ratio, 96
4.3.4 Maximum Likelihood Estimators of the Common Odds Ratio, 99
4.3.5 Minimum Variance Linear Estimators (MVLE), 99
4.3.6 MVLE versus Mantel Haenszel Estimators, 101
4.3.7 SAS PROC FREQ, 103
4.4 Nature of Covariate Adjustment, 105
4.4.1 Confounding and Effect Modification, 105
4.4.2 Stratification Adjustment and Regression Adjustment, 107
4.4.3 When Does Adjustment Matter?, 108
4.5 Multivariate Tests of Hypotheses, 114
4.5.1 Multivariate Null Hypothesis, 114
4.5.2 Omnibus Test, 115
4.5.3 Bonferroni Inequality, 117
4.5.4 Partitioning of the Omnibus Alternative Hypothesis, 118
4.6 Tests of Homogeneity, 120
4.6.1 Contrast Test of Homogeneity, 120
4.6.2 Cochran's Test of Homogeneity, 122
4.6.3 Zelen's Test, 124
4.6.4 Breslow-Day Test for Odds Ratios, 124
4.7 Efficient Tests of No Partial Association, 126
4.7.1 Restricted Alternative Hypothesis of Association, 126
4.7.2 Radhakrishna Family of Efficient Tests of Association, 128
4.8 Asymptotic Relative Efficiency of Competing Tests, 133
4.8.1 Family of Tests, 133
4.8.2 Asymptotic Relative Efficiency, 135
4.9 Maximin Efficient Robust Tests, 139
4.9.1 Maximin Efficiency, 139
4.9.2 Gastwirth Scale Robust Test, 140
4.9.3 Wei-Lachin Test of Stochastic Ordering, 142
4.9.4 Comparison of Weighted Tests, 145
4.10 Random Effects Model, 145
4.10.1 Measurement Error Model, 146
4.10.2 Stratified-Adjusted Estimates from Multiple, 2x2 Tables, 147
4.11 Power and Sample Size for Tests of Association, 155
4.11.1 Power Function of the Radhakrishna Family, 155
4.11.2 Power and Sample Size for Cochran's Test, 157
4.12 Problems, 159

5 Case-Control and Matched Studies, 169

5.1 Unmatched Case-Control (Retrospective) Sampling, 169
5.1.1 Odds Ratio, 170
5.1.2 Relative Risk, 172
5.1.3 Attributable Risk, 173
5.2 Matching, 175
5.2.1 Frequency Matching, 175
5.2.2 Matched Pairs Design: Cross-Sectional or Prospective, 176
5.3 Tests of Association for Matched Pairs, 179
5.3.1 Exact Test, 179
5.3.2 McNemar's Large Sample Test, 180
5.3.3 SAS PROC FREQ, 182
5.4 Measures of Association for Matched Pairs, 183
5.4.1 Conditional Odds Ratio, 183
5.4.2 Confidence Limits for the Odds Ratio, 184
5.4.2.1 Exact Limits, 184
5.4.2.2 Large Sample Limits, 185
5.4.3 Conditional Large Sample Test and Confidence Limits, 185
5.4.4 Mantel-Haenszel Analysis, 186
5.4.5 Relative Risk for Matched Pairs, 187
5.4.6 Attributable Risk for Matched Pairs, 188
5.5 Pair-Matched Retrospective Study, 189
5.5.1 Conditional Odds Ratio, 190
5.5.2 Relative Risks from Matched Retrospective Studies, 191
5.6 Power Function of McNemar's Test, 192
5.6.1 Unconditional Power Function, 192
5.6.2 Conditional Power Function, 192
5.6.3 Other Approaches, 194
5.6.4 Matching Efficiency, 195
5.7 Stratified Analysis of Pair-Matched Tables, 195
5.7.1 Pair and Member Stratification, 196
5.7.2 Stratified Mantel-Haenszel Analysis, 197
5.7.3 MVLE, 197
5.7.4 Tests of Homogeneity and Association, 198
5.7.4.1 Conditional Odds Ratio, 198
5.7.4.2 Marginal Relative Risk, 199
5.7.5 Random Effects Model Analysis, 201
5.8 Problems, 201

6 Applications of Maximum Likelihood and Efficient Scores, 209

6.1 Binomial, 209
6.2 2x2 Table: Product Binomial (Unconditionally), 211
6.2.1 MLEs and Their Asymptotic Distribution, 211
6.2.2 Logit Model, 212
6.2.3 Tests of Significance, 217
6.2.3.1 Wald Test, 217
6.2.3.2 Likelihood Ratio Test, 217
6.2.3.3 Efficient Score Test, 218
6.3 2x2 Table, Conditionally, 219
6.4 Score-Based Estimate, 220
6.5 Stratified Score Analysis of Independent, 2x2 Tables, 222
6.5.1 Conditional Mantel-Haenszel Test and the Score Estimate, 223
6.5.2 Unconditional Cochran Test as a C(alpha) Test, 224
6.6 Matched Pairs, 226
6.6.1 Unconditional Logit Model, 226
6.6.2 Conditional Logit Model, 228
6.6.3 Conditional Likelihood Ratio Test, 230
6.6.4 Conditional Score Test, 230
6.6.5 Matched Case-Control Study, 231
6.7 Iterative Maximum Likelihood, 231
6.7.1 Newton-Raphson (or Newton's Method), 232
6.7.2 Fisher Scoring (Method of Scoring), 233
6.8 Problems, 238

7 Logistic Regression Models, 247

7.1 Unconditional Logistic Regression Model, 247
7.1.1 General Logistic Regression Model, 247
7.1.2 Logistic Regression and Binomial Logit Regression, 250
7.1.3 SAS PROCEDURES, 253
7.1.4 Stratified, 2x2 Tables, 255
7.1.5 Family of Binomial Regression Models, 257
7.2 Interpretation of the Logistic Regression Model, 259
7.2.1 Model Coefficients and Odds Ratios, 259
7.2.2 Partial Regression Coefficients, 263
7.2.3 Model Building: Stepwise Procedures, 267
7.2.4 Disproportionate Sampling, 270
7.2.5 Unmatched Case Control Study, 271
7.3 Tests of Significance, 272
7.3.1 Likelihood Ratio Tests, 272
7.3.1.1 Model Test, 272
7.3.1.2 Test of Model Components, 272
7.3.2 Efficient Scores Test, 273
7.3.2.1 Model Test, 273
7.3.2.2 Test of Model Components, 275
7.3.3 Wald Tests, 275
7.3.4 Type III Tests in SAS PROC GENMOD, 277
7.3.5 Robust Inferences, 278
7.3.6 Power and Sample Size, 283
7.4 Interactions, 285
7.4.1 Qualitative-Qualitative Covariate Interaction, 286
7.4.2 Interactions with a Quantitative Covariate, 290
7.5 Measures of the Strength of Association, 292
7.5.1 Squared Error Loss, 292
7.5.2 Entropy Loss, 293
7.6 Conditional Logistic Regression Model for Matched Studies, 296
7.6.1 Conditional Logistic Model, 296
7.6.2 Special Case: 1:1 Matching, 300
7.6.3 Matched Retrospective Study, 300
7.6.4 Fitting the General Conditional Logistic Regression Model: The Conditional PH Model, 301
7.6.5 Robust Inference, 303
7.6.6 Explained Variation, 303
7.7 Problems, 305

8 Analysis of Count Data, 317

8.1 Event Rates and the Homogeneous Poisson Model, 317
8.1.1 Poisson Process, 317
8.1.2 Doubly Homogeneous Poisson Model, 318
8.1.3 Relative Risks, 320
8.1.4 Violations of the Homogeneous Poisson Assumptions, 323
8.2 Over-Dispersed Poisson Model, 323
8.2.1 Two-Stage Random Effects Model, 324
8.2.2 Relative Risks, 327
8.2.3 Stratified-Adjusted Analyses, 329
8.3 Poisson Regression Model, 330
8.3.1 Homogeneous Poisson Regression Model, 330
8.3.2 Explained Variation, 337
8.3.3 Applications of Poisson Regression, 338
8.4 Over-Dispersed and Robust Poisson Regression, 338
8.4.1 Quasi-Likelihood Over-Dispersed Poisson Regression, 338
8.4.2 Robust Inference Using the Information Sandwich, 340
8.5 Power and Sample Size for Poisson Models, 343
8.6 Conditional Poisson Regression for Matched Sets, 344
8.7 Problems, 345

9 Analysis of Event-Time Data, 353

9.1 Introduction to Survival Analysis, 354
9.1.1 Hazard and Survival Function, 354
9.1.2 Censoring at Random, 355
9.1.3 Kaplan-Meier Estimator, 356
9.1.4 Estimation of the Hazard Function, 359
9.1.5 Comparison of Survival Probabilities for Two Groups, 361
9.2 Lifetable Construction, 368
9.2.1 Discrete Distributions: Actuarial Lifetable, 368
9.2.2 Modified Kaplan-Meier Estimator, 369
9.2.3 Competing Risks, 370
9.2.4 SAS PROC LIFETEST: Survival Estimation, 375
9.3 Family of Weighted Mantel-Haenszel Tests, 377
9.3.1 Weighted Mantel-Haenszel Test, 377
9.3.2 Mantel-logrank Test, 378
9.3.3 Modified Wilcoxon Test, 379
9.3.4 G-rho Family of Tests, 380
9.3.5 Measures of Association, 381
9.3.6 SAS PROC LIFETEST: Tests of Significance, 383
9.4 Proportional Hazards Models, 384
9.4.1 Cox's Proportional Hazards Models, 385
9.4.2 Stratified Models, 388
9.4.3 Time-Dependent Covariates, 389
9.4.4 Fitting the Model, 390
9.4.5 Robust Inference, 391
9.4.6 Adjustments for Tied Observations, 393
9.4.6.1 Discrete and Grouped Failure Time Data, 394
9.4.6.2 Cox's Adjustment for Ties, 395
9.4.6.3 Kalbfleisch-Prentice Marginal Model, 396
9.4.6.4 Peto-Breslow Adjustment for Ties, 397
9.4.6.5 Interval Censoring, 397
9.4.7 Model Assumptions, 397
9.4.8 Explained Variation, 399
9.4.9 SAS PROC PHREG, 401
9.5 Evaluation of Sample Size and Power, 409
9.5.1 Exponential Survival, 409
9.5.2 Cox's Proportional Hazards Model, 412
9.6 Analysis of Recurrent Events: The Multiplicative Intensity Model, 414
9.6.1 Counting Process Formulation, 415
9.6.2 Nelson-Aalen Estimator, 417
9.6.3 Aalen-Gill Test Statistics, 419
9.6.4 Multiplicative Intensity Model, 422
9.7 Problems, 426

Appendix Statistical Theory, 449

A.1 Introduction, 449
A.1.1 Notation, 449
A.1.2 Matrices, 450
A.1.3 Partition of Variation, 451
A.2 Central Limit Theorem and the Law of Large Numbers, 451
A.2.1 Univariate Case, 451
A.2.2 Multivariate Case, 453
A.3 Delta Method, 455
A.3.1 Univariate Case, 455
A.3.2 Multivariate Case, 456
A.4 Slutsky's Convergence Theorem, 457
A.4.1 Convergence in Distribution, 457
A.4.2 Convergence in Probability, 458
A.4.3 Convergence in Distribution of Transformations, 458
A.5 Least Squares Estimation, 460
A.5.1 Ordinary Least Squares (OLS), 460
A.5.2 Gauss-Markov Theorem, 462
A.5.3 Weighted Least Squares (WLS), 463
A.5.4 Iteratively Reweighted Least Squares (IRLS), 465
A.6 Maximum Likelihood Estimation and Efficient Scores, 465
A.6.1 Estimating Equation, 465
A.6.2 Efficient Score, 466
A.6.3 Fisher's Information Function, 467
A.6.4 Cramer-Rao Inequality: Efficient Estimators, 470
A.6.5 Asymptotic Distribution of the Efficient Score and the MLE, 471
A.6.6 Consistency and Asymptotic Efficiency of the MLE, 472
A.6.7 Estimated Information, 472
A.6.8 Invariance Under Transformations, 473
A.6.9 Independent But Not Identically Distributed Observations, 474
A.7 Likelihood Based Tests of Significance, 476
A.7.1 Wald Tests, 476
A.7.1.1 Element-wise Tests, 476
A.7.1.2 Composite Test, 476
A.7.1.3 Test of a Linear Hypothesis, 477
A.7.2 Likelihood Ratio Tests, 478
A.7.2.1 Composite Test, 478
A.7.2.2 Test of a Sub-Hypothesis, 478
A.7.3 Efficient Scores Test, 479
A.7.3.1 Composite Test, 479
A.7.3.2 Test of a Sub-Hypothesis: C(alpha) Tests, 480
A.7.3.3 Relative Efficiency Versus the Likelihood Ratio Test, 482
A.8 Explained Variation, 483
A.8.1 Squared Error Loss, 484
A.8.2 Residual Variation, 486
A.8.3 Negative Log-Likelihood Loss, 487
A.8.4 Madalla's R_LR, 2, 487
A.9 Robust Inference, 488
A.9.1 Information Sandwich, 488
A.9.1.1 Correct Model Specification, 489
A.9.1.2 Incorrect Model Specification, 490
A.9.2 Robust Confidence Limits and Tests, 493
A.10 Generalized Linear Models and Quasi-Likelihood, 494
A.10.1 Generalized Linear Models, 494
A.10.2 Exponential Family of Models, 495
A.10.3 Deviance and the Chi-Square Goodness of Fit, 498
A.10.4 Quasi-Likelihood, 500
A.10.5 Conditional GLMs, 502
A.10.6 Generalized Estimating Equations (GEE), 503

References, 505

Author Index, 525

Index, 531