Introductory Biostatistics

Chap T. Le, PhD, is Distinguished Professor of Biostatistics and Director of Biostatistics and Bioinformatics at the University of Minnesota Masonic Cancer Center. He has provided statistical consulting for a variety of biomedical research projects, and he has worked on collaborations focusing on the analyses of survival and categorical data and, currently, in the areas of cancer and tobacco research. Dr. Le is the author of Health and Numbers: A Problems-Based Introduction to Biostatistics, Third Edition; Applied Categorical Data Analysis and Translational Research, Second Edition; and Applied Survival Analysis, all published by Wiley. Lynn E. Eberly, PhD, is Associate Professor in the Division of Biostatistics at the University of Minnesota. The author of more than 100 journal articles, Dr. Eberly has been a statistical collaborator in biomedical and public health research for more than 18 years. Her current research interests include methods for and applications to correlated data in neurodegenerative conditions, endocrinology, psychiatry/psychology, and cancer research.

Preface to the Second Edition xiii

Preface to the First Edition xv

About the Companion Website xix

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 7

1.2 Rates 10

1.2.1 Changes 11

1.2.2 Measures of Morbidity and Mortality 13

1.2.3 Standardization of Rates 15

1.3 Ratios 18

1.3.1 Relative Risk 18

1.3.2 Odds and Odds Ratio 18

1.3.3 Generalized Odds for Ordered 2 × k Tables 21

1.3.4 Mantel–Haenszel Method 25

1.3.5 Standardized Mortality Ratio 28

1.4 Notes on Computations 30

Exercises 32

2 Descriptive Methods for Continuous Data 55

2.1 Tabular and Graphical Methods 55

2.1.1 One‐Way Scatter Plots 55

2.1.2 Frequency Distribution 56

2.1.3 Histogram and Frequency Polygon 60

2.1.4 Cumulative Frequency Graph and Percentiles 64

2.1.5 Stem and Leaf Diagrams 68

2.2 Numerical Methods 69

2.2.1 Mean 69

2.2.2 Other Measures of Location 72

2.2.3 Measures of Dispersion 73

2.2.4 Box Plots 76

2.3 Special Case of Binary Data 77

2.4 Coefficients of Correlation 78

2.4.1 Pearson’s Correlation Coefficient 80

2.4.2 Nonparametric Correlation Coefficients 83

2.5 Notes on Computations 85

Exercises 87

3 Probability and Probability Models 103

3.1 Probability 103

3.1.1 Certainty of Uncertainty 104

3.1.2 Probability 104

3.1.3 Statistical Relationship 106

3.1.4 Using Screening Tests 109

3.1.5 Measuring Agreement 112

3.2 Normal Distribution 114

3.2.1 Shape of the Normal Curve 114

3.2.2 Areas Under the Standard Normal Curve 116

3.2.3 Normal Distribution as a Probability Model 122

3.3 Probability Models for Continuous Data 124

3.4 Probability Models for Discrete Data 125

3.4.1 Binomial Distribution 126

3.4.2 Poisson Distribution 128

3.5 Brief Notes on the Fundamentals 130

3.5.1 Mean and Variance 130

3.5.2 Pair‐Matched Case–Control Study 130

3.6 Notes on Computations 132

Exercises 134

4 Estimation of Parameters 141

4.1 Basic Concepts 142

4.1.1 Statistics as Variables 143

4.1.2 Sampling Distributions 143

4.1.3 Introduction to Confidence Estimation 145

4.2 Estimation of Means 146

4.2.1 Confidence Intervals for a Mean 147

4.2.2 Uses of Small Samples 149

4.2.3 Evaluation of Interventions 151

4.3 Estimation of Proportions 153

4.4 Estimation of Odds Ratios 157

4.5 Estimation of Correlation Coefficients 160

4.6 Brief Notes on the Fundamentals 163

4.7 Notes on Computations 165

Exercises 166

5 Introduction to Statistical Tests of Significance 179

5.1 Basic Concepts 180

5.1.1 Hypothesis Tests 181

5.1.2 Statistical Evidence 182

5.1.3 Errors 182

5.2 Analogies 185

5.2.1 Trials by Jury 185

5.2.2 Medical Screening Tests 186

5.2.3 Common Expectations 186

5.3 Summaries and Conclusions 187

5.3.1 Rejection Region 187

5.3.2 p Values 189

5.3.3 Relationship to Confidence Intervals 191

5.4 Brief Notes on the Fundamentals 193

5.4.1 Type I and Type II Errors 193

5.4.2 More about Errors and p Values 194

Exercises 194

6 Comparison of Population Proportions 197

6.1 One‐Sample Problem with Binary Data 197

6.2 Analysis of Pair‐Matched Data 199

6.3 Comparison of Two Proportions 202

6.4 Mantel–Haenszel Method 206

6.5 Inferences for General Two‐Way Tables 211

6.6 Fisher’s Exact Test 217

6.7 Ordered 2 × K Contingency Tables 219

6.8 Notes on Computations 222

Exercises 222

7 Comparison of Population Means 235

7.1 One‐Sample Problem with Continuous Data 235

7.2 Analysis of Pair‐Matched Data 237

7.3 Comparison of Two Means 242

7.4 Nonparametric Methods 246

7.4.1 Wilcoxon Rank‐Sum Test 246

7.4.2 Wilcoxon Signed‐Rank Test 250

7.5 One‐Way Analysis of Variance 252

7.5.1 One‐Way Analysis of Variance Model 253

7.5.2 Group Comparisons 258

7.6 Brief Notes on the Fundamentals 259

7.7 Notes on Computations 260

Exercises 260

8 Analysis of Variance 273

8.1 Factorial Studies 273

8.1.1 Two Crossed Factors 273

8.1.2 Extensions to More Than Two Factors 278

8.2 Block Designs 280

8.2.1 Purpose 280

8.2.2 Fixed Block Designs 281

8.2.3 Random Block Designs 284

8.3 Diagnostics 287

Exercises 291

9 Regression Analysis 297

9.1 Simple Regression Analysis 298

9.1.1 Correlation and Regression 298

9.1.2 Simple Linear Regression Model 301

9.1.3 Scatter Diagram 302

9.1.4 Meaning of Regression Parameters 302

9.1.5 Estimation of Parameters and Prediction 303

9.1.6 Testing for Independence 307

9.1.7 Analysis of Variance Approach 309

9.1.8 Some Biomedical Applications 311

9.2 Multiple Regression Analysis 317

9.2.1 Regression Model with Several Independent Variables 318

9.2.2 Meaning of Regression Parameters 318

9.2.3 Effect Modifications 319

9.2.4 Polynomial Regression 319

9.2.5 Estimation of Parameters and Prediction 320

9.2.6 Analysis of Variance Approach 321

9.2.7 Testing Hypotheses in Multiple Linear Regression 322

9.2.8 Some Biomedical Applications 330

9.3 Graphical and Computational Aids 334

Exercises 336

10 Logistic Regression 351

10.1 Simple Regression Analysis 353

10.1.1 Simple Logistic Regression Model 353

10.1.2 Measure of Association 355

10.1.3 Effect of Measurement Scale 356

10.1.4 Tests of Association 358

10.1.5 Use of the Logistic Model for Different Designs 358

10.1.6 Overdispersion 359

10.2 Multiple Regression Analysis 362

10.2.1 Logistic Regression Model with Several Covariates 363

10.2.2 Effect Modifications 364

10.2.3 Polynomial Regression 365

10.2.4 Testing Hypotheses in Multiple Logistic Regression 365

10.2.5 Receiver Operating Characteristic Curve 372

10.2.6 ROC Curve and Logistic Regression 374

10.3 Brief Notes on the Fundamentals 375

10.4 Notes on Computing 377

Exercises 377

11 Methods for Count Data 383

11.1 Poisson Distribution 383

11.2 Testing Goodness of Fit 387

11.3 Poisson Regression Model 389

11.3.1 Simple Regression Analysis 389

11.3.2 Multiple Regression Analysis 393

11.3.3 Overdispersion 402

11.3.4 Stepwise Regression 404

Exercises 406

12 Methods for Repeatedly Measured Responses 409

12.1 Extending Regression Methods Beyond Independent Data 409

12.2 Continuous Responses 410

12.2.1 Extending Regression using the Linear Mixed Model 410

12.2.2 Testing and Inference 414

12.2.3 Comparing Models 417

12.2.4 Special Cases: Random Block Designs and Multi‐level Sampling 418

12.3 Binary Responses 423

12.3.1 Extending Logistic Regression using Generalized Estimating Equations 423

12.3.2 Testing and Inference 425

12.4 Count Responses 427

12.4.1 Extending Poisson Regression using Generalized Estimating Equations 427

12.4.2 Testing and Inference 428

12.5 Computational Notes 431

Exercises 432

13 Analysis of Survival Data and Data from Matched Studies 439

13.1 Survival Data 440

13.2 Introductory Survival Analyses 443

13.2.1 Kaplan–Meier Curve 444

13.2.2 Comparison of Survival Distributions 446

13.3 Simple Regression and Correlation 450

13.3.1 Model and Approach 451

13.3.2 Measures of Association 452

13.3.3 Tests of Association 455

13.4 Multiple Regression and Correlation 456

13.4.1 Proportional Hazards Model with Several Covariates 456

13.4.2 Testing Hypotheses in Multiple Regression 457

13.4.3 Time‐Dependent Covariates and Applications 461

13.5 Pair‐Matched Case–Control Studies 464

13.5.1 Model 465

13.5.2 Analysis 466

13.6 Multiple Matching 468

13.6.1 Conditional Approach 469

13.6.2 Estimation of the Odds Ratio 469

13.6.3 Testing for Exposure Effect 470

13.7 Conditional Logistic Regression 472

13.7.1 Simple Regression Analysis 473

13.7.2 Multiple Regression Analysis 478

Exercises 484

14 Study Designs 493

14.1 Types of Study Designs 494

14.2 Classification of Clinical Trials 495

14.3 Designing Phase I Cancer Trials 497

14.4 Sample Size Determination for Phase II Trials and Surveys 499

14.5 Sample Sizes for Other Phase II Trials 501

14.5.1 Continuous Endpoints 501

14.5.2 Correlation Endpoints 502

14.6 About Simon’s Two‐Stage Phase II Design 503

14.7 Phase II Designs for Selection 504

14.7.1 Continuous Endpoints 505

14.7.2 Binary Endpoints 505

14.8 Toxicity Monitoring in Phase II Trials 506

14.9 Sample Size Determination for Phase III Trials 508

14.9.1 Comparison of Two Means 509

14.9.2 Comparison of Two Proportions 511

14.9.3 Survival Time as the Endpoint 513

14.10 Sample Size Determination for Case–Control Studies 515

14.10.1 Unmatched Designs for a Binary Exposure 516

14.10.2 Matched Designs for a Binary Exposure 518

14.10.3 Unmatched Designs for a Continuous Exposure 520

Exercises 522

References 529

Appendices 535

Answers to Selected Exercises 541

Index 585