CHAP T. LE
Distinguished Professor of Biostatistics
and Director of Biostatistics
Comprehensive Cancer Center
University of Minnesota
CONTENTS
Preface xiii
1 Descriptive Methods for Categorical Data 1
1.1 Proportions, 1
1.1.1 Comparative Studies, 2
1.1.2 Screening Tests, 5
1.1.3 Displaying Proportions, 8
1.2 Rates, 11
1.2.1 Changes, 11
1.2.2 Measures of Morbidity and Mortality, 13
1.2.3 Standardization of Rates, 16
1.3 Ratios, 18
1.3.1 Relative Risk, 19
1.3.2 Odds and Odds Ratio, 19
1.3.3 Generalized Odds for Ordered 2 k Tables, 22
1.3.4 Mantel–Haenszel Method, 26
1.3.5 Standardized Mortality Ratio, 30
1.4 Notes on Computations, 31
Exercises, 34
2 Descriptive Methods for Continuous Data 57
2.1 Tabular and Graphical Methods, 57
2.1.1 One-Way Scatter Plots, 57
2.1.2 Frequency Distribution, 58
vii
2.1.3 Histogram and the Frequency Polygon, 62
2.1.4 Cumulative Frequency Graph and Percentiles, 67
2.1.5 Stem-and-Leaf Diagrams, 70
2.2 Numerical Methods, 72
2.2.1 Mean, 73
2.2.2 Other Measures of Location, 76
2.2.3 Measures of Dispersion, 77
2.2.4 Box Plots, 80
2.3 Special Case of Binary Data, 81
2.4 Coe‰cients of Correlation, 83
2.4.1 Pearson’s Correlation Coe‰cient, 85
2.4.2 Nonparametric Correlation Coe‰cients, 88
2.5 Notes on Computations, 90
Exercises, 92
3 Probability and Probability Models 108
3.1 Probability, 108
3.1.1 Certainty of Uncertainty, 109
3.1.2 Probability, 109
3.1.3 Statistical Relationship, 111
3.1.4 Using Screening Tests, 115
3.1.5 Measuring Agreement, 118
3.2 Normal Distribution, 120
3.2.1 Shape of the Normal Curve, 120
3.2.2 Areas under the Standard Normal Curve, 123
3.2.3 Normal Distribution as a Probability Model, 128
3.3 Probability Models for Continuous Data, 131
3.4 Probability Models for Discrete Data, 132
3.4.1 Binomial Distribution, 133
3.4.2 Poisson Distribution, 136
3.5 Brief Notes on the Fundamentals, 137
3.5.1 Mean and Variance, 137
3.5.2 Pair-Matched Case–Control Study, 138
3.6 Notes on Computations, 140
Exercises, 141
4 Estimation of Parameters 147
4.1 Basic Concepts, 148
4.1.1 Statistics as Variables, 149
4.1.2 Sampling Distributions, 149
4.1.3 Introduction to Confidence Estimation, 152
4.2 Estimation of Means, 152
4.2.1 Confidence Intervals for a Mean, 154
4.2.2 Uses of Small Samples, 156
viii CONTENTS
4.2.3 Evaluation of Interventions, 158
4.3 Estimation of Proportions, 160
4.4 Estimation of Odds Ratios, 165
4.5 Estimation of Correlation Coe‰cients, 168
4.6 Brief Notes on the Fundamentals, 171
4.7 Notes on Computations, 173
Exercises, 173
5 Introduction to Statistical Tests of Significance 188
5.1 Basic Concepts, 190
5.1.1 Hypothesis Tests, 190
5.1.2 Statistical Evidence, 191
5.1.3 Errors, 192
5.2 Analogies, 194
5.2.1 Trials by Jury, 194
5.2.2 Medical Screening Tests, 195
5.2.3 Common Expectations, 195
5.3 Summaries and Conclusions, 196
5.3.1 Rejection Region, 197
5.3.2 p Values, 198
5.3.3 Relationship to Confidence Intervals, 201
5.4 Brief Notes on the Fundamentals, 203
5.4.1 Type I and Type II Errors, 203
5.4.2 More about Errors and p Values, 203
Exercises, 204
6 Comparison of Population Proportions 208
6.1 One-Sample Problem with Binary Data, 208
6.2 Analysis of Pair-Matched Data, 210
6.3 Comparison of Two Proportions, 213
6.4 Mantel–Haenszel Method, 218
6.5 Inferences for General Two-Way Tables, 223
6.6 Fisher’s Exact Test, 229
6.7 Ordered 2 k Contingency Tables, 230
6.8 Notes on Computations, 234
Exercises, 234
7 Comparison of Population Means 246
7.1 One-Sample Problem with Continuous Data, 246
7.2 Analysis of Pair-Matched Data, 248
7.3 Comparison of Two Means, 253
7.4 Nonparametric Methods, 257
7.4.1 Wilcoxon Rank-Sum Test, 257
7.4.2 Wilcoxon Signed-Rank Test, 261
CONTENTS ix
7.5 One-Way Analysis of Variance, 263
7.6 Brief Notes on the Fundamentals, 269
7.7 Notes on Computations, 270
Exercises, 270
8 Correlation and Regression 282
8.1 Simple Regression Analysis, 283
8.1.1 Simple Linear Regression Model, 283
8.1.2 Scatter Diagram, 283
8.1.3 Meaning of Regression Parameters, 284
8.1.4 Estimation of Parameters, 285
8.1.5 Testing for Independence, 289
8.1.6 Analysis-of-Variance Approach, 292
8.2 Multiple Regression Analysis, 294
8.2.1 Regression Model with Several Independent
Variables, 294
8.2.2 Meaning of Regression Parameters, 295
8.2.3 E¤ect Modifications, 295
8.2.4 Polynomial Regression, 296
8.2.5 Estimation of Parameters, 296
8.2.6 Analysis-of-Variance Approach, 297
8.2.7 Testing Hypotheses in Multiple Linear Regression, 298
8.3 Notes on Computations, 305
Exercises, 306
9 Logistic Regression 314
9.1 Simple Regression Analysis, 316
9.1.1 Simple Logistic Regression Model, 317
9.1.2 Measure of Association, 318
9.1.3 E¤ect of Measurement Scale, 320
9.1.4 Tests of Association, 321
9.1.5 Use of the Logistic Model for Di¤erent Designs, 322
9.1.6 Overdispersion, 323
9.2 Multiple Regression Analysis, 325
9.2.1 Logistic Regression Model with Several Covariates, 326
9.2.2 E¤ect Modifications, 327
9.2.3 Polynomial Regression, 328
9.2.4 Testing Hypotheses in Multiple Logistic
Regression, 329
9.2.5 Receiver Operating Characteristic Curve, 336
9.2.6 ROC Curve and Logistic Regression, 337
9.3 Brief Notes on the Fundamentals, 339
Exercise, 341
x CONTENTS
10 Methods for Count Data 350
10.1 Poisson Distribution, 350
10.2 Testing Goodness of Fit, 354
10.3 Poisson Regression Model, 356
10.3.1 Simple Regression Analysis, 357
10.3.2 Multiple Regression Analysis, 360
10.3.3 Overdispersion, 368
10.3.4 Stepwise Regression, 370
Exercise, 372
11 Analysis of Survival Data and Data from Matched Studies 379
11.1 Survival Data, 381
11.2 Introductory Survival Analyses, 384
11.2.1 Kaplan–Meier Curve, 384
11.2.2 Comparison of Survival Distributions, 386
11.3 Simple Regression and Correlation, 390
11.3.1 Model and Approach, 391
11.3.2 Measures of Association, 392
11.3.3 Tests of Association, 395
11.4 Multiple Regression and Correlation, 395
11.4.1 Proportional Hazards Model with Several
Covariates, 396
11.4.2 Testing Hypotheses in Multiple Regression, 397
11.4.3 Time-Dependent Covariates and Applications, 401
11.5 Pair-Matched Case–Control Studies, 405
11.5.1 Model, 406
11.5.2 Analysis, 407
11.6 Multiple Matching, 409
11.6.1 Conditional Approach, 409
11.6.2 Estimation of the Odds Ratio, 410
11.6.3 Testing for Exposure E¤ect, 411
11.7 Conditional Logistic Regression, 413
11.7.1 Simple Regression Analysis, 414
11.7.2 Multiple Regression Analysis, 418
Exercises, 426
12 Study Designs 445
12.1 Types of Study Designs, 446
12.2 Classification of Clinical Trials, 447
12.3 Designing Phase I Cancer Trials, 448
12.4 Sample Size Determination for Phase II Trials and
Surveys, 451
12.5 Sample Sizes for Other Phase II Trials, 453
CONTENTS xi
12.5.1 Continuous Endpoints, 454
12.5.2 Correlation Endpoints, 454
12.6 About Simon’s Two-Stage Phase II Design, 456
12.7 Phase II Designs for Selection, 457
12.7.1 Continuous Endpoints, 457
12.7.2 Binary Endpoints, 458
12.8 Toxicity Monitoring in Phase II Trials, 459
12.9 Sample Size Determination for Phase III Trials, 461
12.9.1 Comparison of Two Means, 462
12.9.2 Comparison of Two Proportions, 464
12.9.3 Survival Time as the Endpoint, 466
12.10 Sample Size Determination for Case–Control Studies, 469
12.10.1 Unmatched Designs for a Binary Exposure, 469
12.10.2 Matched Designs for a Binary Exposure, 471
12.10.3 Unmatched Designs for a Continuous
Exposure, 473
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Distinguished Professor of Biostatistics
and Director of Biostatistics
Comprehensive Cancer Center
University of Minnesota
CONTENTS
Preface xiii
1 Descriptive Methods for Categorical Data 1
1.1 Proportions, 1
1.1.1 Comparative Studies, 2
1.1.2 Screening Tests, 5
1.1.3 Displaying Proportions, 8
1.2 Rates, 11
1.2.1 Changes, 11
1.2.2 Measures of Morbidity and Mortality, 13
1.2.3 Standardization of Rates, 16
1.3 Ratios, 18
1.3.1 Relative Risk, 19
1.3.2 Odds and Odds Ratio, 19
1.3.3 Generalized Odds for Ordered 2 k Tables, 22
1.3.4 Mantel–Haenszel Method, 26
1.3.5 Standardized Mortality Ratio, 30
1.4 Notes on Computations, 31
Exercises, 34
2 Descriptive Methods for Continuous Data 57
2.1 Tabular and Graphical Methods, 57
2.1.1 One-Way Scatter Plots, 57
2.1.2 Frequency Distribution, 58
vii
2.1.3 Histogram and the Frequency Polygon, 62
2.1.4 Cumulative Frequency Graph and Percentiles, 67
2.1.5 Stem-and-Leaf Diagrams, 70
2.2 Numerical Methods, 72
2.2.1 Mean, 73
2.2.2 Other Measures of Location, 76
2.2.3 Measures of Dispersion, 77
2.2.4 Box Plots, 80
2.3 Special Case of Binary Data, 81
2.4 Coe‰cients of Correlation, 83
2.4.1 Pearson’s Correlation Coe‰cient, 85
2.4.2 Nonparametric Correlation Coe‰cients, 88
2.5 Notes on Computations, 90
Exercises, 92
3 Probability and Probability Models 108
3.1 Probability, 108
3.1.1 Certainty of Uncertainty, 109
3.1.2 Probability, 109
3.1.3 Statistical Relationship, 111
3.1.4 Using Screening Tests, 115
3.1.5 Measuring Agreement, 118
3.2 Normal Distribution, 120
3.2.1 Shape of the Normal Curve, 120
3.2.2 Areas under the Standard Normal Curve, 123
3.2.3 Normal Distribution as a Probability Model, 128
3.3 Probability Models for Continuous Data, 131
3.4 Probability Models for Discrete Data, 132
3.4.1 Binomial Distribution, 133
3.4.2 Poisson Distribution, 136
3.5 Brief Notes on the Fundamentals, 137
3.5.1 Mean and Variance, 137
3.5.2 Pair-Matched Case–Control Study, 138
3.6 Notes on Computations, 140
Exercises, 141
4 Estimation of Parameters 147
4.1 Basic Concepts, 148
4.1.1 Statistics as Variables, 149
4.1.2 Sampling Distributions, 149
4.1.3 Introduction to Confidence Estimation, 152
4.2 Estimation of Means, 152
4.2.1 Confidence Intervals for a Mean, 154
4.2.2 Uses of Small Samples, 156
viii CONTENTS
4.2.3 Evaluation of Interventions, 158
4.3 Estimation of Proportions, 160
4.4 Estimation of Odds Ratios, 165
4.5 Estimation of Correlation Coe‰cients, 168
4.6 Brief Notes on the Fundamentals, 171
4.7 Notes on Computations, 173
Exercises, 173
5 Introduction to Statistical Tests of Significance 188
5.1 Basic Concepts, 190
5.1.1 Hypothesis Tests, 190
5.1.2 Statistical Evidence, 191
5.1.3 Errors, 192
5.2 Analogies, 194
5.2.1 Trials by Jury, 194
5.2.2 Medical Screening Tests, 195
5.2.3 Common Expectations, 195
5.3 Summaries and Conclusions, 196
5.3.1 Rejection Region, 197
5.3.2 p Values, 198
5.3.3 Relationship to Confidence Intervals, 201
5.4 Brief Notes on the Fundamentals, 203
5.4.1 Type I and Type II Errors, 203
5.4.2 More about Errors and p Values, 203
Exercises, 204
6 Comparison of Population Proportions 208
6.1 One-Sample Problem with Binary Data, 208
6.2 Analysis of Pair-Matched Data, 210
6.3 Comparison of Two Proportions, 213
6.4 Mantel–Haenszel Method, 218
6.5 Inferences for General Two-Way Tables, 223
6.6 Fisher’s Exact Test, 229
6.7 Ordered 2 k Contingency Tables, 230
6.8 Notes on Computations, 234
Exercises, 234
7 Comparison of Population Means 246
7.1 One-Sample Problem with Continuous Data, 246
7.2 Analysis of Pair-Matched Data, 248
7.3 Comparison of Two Means, 253
7.4 Nonparametric Methods, 257
7.4.1 Wilcoxon Rank-Sum Test, 257
7.4.2 Wilcoxon Signed-Rank Test, 261
CONTENTS ix
7.5 One-Way Analysis of Variance, 263
7.6 Brief Notes on the Fundamentals, 269
7.7 Notes on Computations, 270
Exercises, 270
8 Correlation and Regression 282
8.1 Simple Regression Analysis, 283
8.1.1 Simple Linear Regression Model, 283
8.1.2 Scatter Diagram, 283
8.1.3 Meaning of Regression Parameters, 284
8.1.4 Estimation of Parameters, 285
8.1.5 Testing for Independence, 289
8.1.6 Analysis-of-Variance Approach, 292
8.2 Multiple Regression Analysis, 294
8.2.1 Regression Model with Several Independent
Variables, 294
8.2.2 Meaning of Regression Parameters, 295
8.2.3 E¤ect Modifications, 295
8.2.4 Polynomial Regression, 296
8.2.5 Estimation of Parameters, 296
8.2.6 Analysis-of-Variance Approach, 297
8.2.7 Testing Hypotheses in Multiple Linear Regression, 298
8.3 Notes on Computations, 305
Exercises, 306
9 Logistic Regression 314
9.1 Simple Regression Analysis, 316
9.1.1 Simple Logistic Regression Model, 317
9.1.2 Measure of Association, 318
9.1.3 E¤ect of Measurement Scale, 320
9.1.4 Tests of Association, 321
9.1.5 Use of the Logistic Model for Di¤erent Designs, 322
9.1.6 Overdispersion, 323
9.2 Multiple Regression Analysis, 325
9.2.1 Logistic Regression Model with Several Covariates, 326
9.2.2 E¤ect Modifications, 327
9.2.3 Polynomial Regression, 328
9.2.4 Testing Hypotheses in Multiple Logistic
Regression, 329
9.2.5 Receiver Operating Characteristic Curve, 336
9.2.6 ROC Curve and Logistic Regression, 337
9.3 Brief Notes on the Fundamentals, 339
Exercise, 341
x CONTENTS
10 Methods for Count Data 350
10.1 Poisson Distribution, 350
10.2 Testing Goodness of Fit, 354
10.3 Poisson Regression Model, 356
10.3.1 Simple Regression Analysis, 357
10.3.2 Multiple Regression Analysis, 360
10.3.3 Overdispersion, 368
10.3.4 Stepwise Regression, 370
Exercise, 372
11 Analysis of Survival Data and Data from Matched Studies 379
11.1 Survival Data, 381
11.2 Introductory Survival Analyses, 384
11.2.1 Kaplan–Meier Curve, 384
11.2.2 Comparison of Survival Distributions, 386
11.3 Simple Regression and Correlation, 390
11.3.1 Model and Approach, 391
11.3.2 Measures of Association, 392
11.3.3 Tests of Association, 395
11.4 Multiple Regression and Correlation, 395
11.4.1 Proportional Hazards Model with Several
Covariates, 396
11.4.2 Testing Hypotheses in Multiple Regression, 397
11.4.3 Time-Dependent Covariates and Applications, 401
11.5 Pair-Matched Case–Control Studies, 405
11.5.1 Model, 406
11.5.2 Analysis, 407
11.6 Multiple Matching, 409
11.6.1 Conditional Approach, 409
11.6.2 Estimation of the Odds Ratio, 410
11.6.3 Testing for Exposure E¤ect, 411
11.7 Conditional Logistic Regression, 413
11.7.1 Simple Regression Analysis, 414
11.7.2 Multiple Regression Analysis, 418
Exercises, 426
12 Study Designs 445
12.1 Types of Study Designs, 446
12.2 Classification of Clinical Trials, 447
12.3 Designing Phase I Cancer Trials, 448
12.4 Sample Size Determination for Phase II Trials and
Surveys, 451
12.5 Sample Sizes for Other Phase II Trials, 453
CONTENTS xi
12.5.1 Continuous Endpoints, 454
12.5.2 Correlation Endpoints, 454
12.6 About Simon’s Two-Stage Phase II Design, 456
12.7 Phase II Designs for Selection, 457
12.7.1 Continuous Endpoints, 457
12.7.2 Binary Endpoints, 458
12.8 Toxicity Monitoring in Phase II Trials, 459
12.9 Sample Size Determination for Phase III Trials, 461
12.9.1 Comparison of Two Means, 462
12.9.2 Comparison of Two Proportions, 464
12.9.3 Survival Time as the Endpoint, 466
12.10 Sample Size Determination for Case–Control Studies, 469
12.10.1 Unmatched Designs for a Binary Exposure, 469
12.10.2 Matched Designs for a Binary Exposure, 471
12.10.3 Unmatched Designs for a Continuous
Exposure, 473
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