Medical Statistics

STEPHEN J. WALTERS is Professor of Medical Statistics and Clinical Trials in the School of Health and Related Research (ScHARR) at the University of Sheffield, UK. Stephen is a prolific researcher and writer, including the popular textbooks How to Display Data and How to Design, Analyse and Report Cluster Randomised Trials in Medicine and Health Related Research. He is a National Institute for Health Research (NIHR) Senior Investigator, and has developed several courses on teaching medical statistics to medical and health science students, clinicians and allied health professionals.

MICHAEL J. CAMPBELL is Emeritus Professor of Medical Statistics in the School of Health and Related Research (ScHARR) at the University of Sheffield, UK. Mike is a leading researcher in medical statistics and clinical trials with a national and international reputation. A prolific writer, Mike has written many best-selling textbooks on medical statistics and clinical trials including: Statistics at Square One, Statistics at Square Two, Sample Size Tables for Clinical Studies, and How to Design, Analyse and Report Cluster Randomised Trials in Medicine and Health Related Research.

DAVID MACHIN is Emeritus Professor of Medical Statistics in the School of Health and Related Research (ScHARR) at the University of Sheffield, UK. He was Foundation Director of the National Medical Research Council, Clinical Trials and Epidemiology Research Unit, Singapore, and Head of the MRC Cancer Trials Office, Cambridge, UK. He has published more than 250 peer reviewed articles, and several books on a wide variety of topics in statistics and medicine. His earlier experience included posts at the Universities of Wales, Leeds, Stirling, Southampton and Sheffield, a period with the European Organisation for Research and Treatment of Cancer in Brussels, Belgium, and at the World Health Organization in Geneva, Switzerland.

Preface xi

1 Uses and Abuses of Medical Statistics 1

1.1 Introduction 2

1.2 Why Use Statistics? 2

1.3 Statistics is About Common Sense and Good Design 3

1.4 How a Statistician Can Help 5

2 Displaying and Summarising Data 9

2.1 Types of Data 10

2.2 Summarising Categorical Data 13

2.3 Displaying Categorical Data 15

2.4 Summarising Continuous Data 17

2.5 Displaying Continuous Data 24

2.6 Within-Subject Variability 28

2.7 Presentation 30

2.8 Points When Reading the Literature 31

2.9 Technical Details 32

2.10 Exercises 33

3 Summary Measures for Binary Data 37

3.1 Summarising Binary and Categorical Data 38

3.2 Points When Reading the Literature 46

3.3 Exercises 46

4 Probability and Distributions 49

4.1 Types of Probability 50

4.2 The Binomial Distribution 54

4.3 The Poisson Distribution 55

4.4 Probability for Continuous Outcomes 57

4.5 The Normal Distribution 58

4.6 Reference Ranges 63

4.7 Other Distributions 64

4.8 Points When Reading the Literature 66

4.9 Technical Section 66

4.10 Exercises 67

5 Populations, Samples, Standard Errors and Confidence Intervals 71

5.1 Populations 72

5.2 Samples 73

5.3 The Standard Error 74

5.4 The Central Limit Theorem 75

5.5 Standard Errors for Proportions and Rates 77

5.6 Standard Error of Differences 79

5.7 Confidence Intervals for an Estimate 80

5.8 Confidence Intervals for Differences 83

5.9 Points When Reading the Literature 84

5.10 Technical Details 85

5.11 Exercises 86

6 Hypothesis Testing, P-values and Statistical Inference 91

6.1 Introduction 92

6.2 The Null Hypothesis 92

6.3 The Main Steps in Hypothesis Testing 94

6.4 Using Your P-value to Make a Decision About Whether to Reject, or Not Reject, Your Null Hypothesis 96

6.5 Statistical Power 99

6.6 One-sided and Two-sided Tests 101

6.7 Confidence Intervals (CIs) 101

6.8 Large Sample Tests for Two Independent Means or Proportions 104

6.9 Issues with P-values 107

6.10 Points When Reading the Literature 108

6.11 Exercises 108

7 Comparing Two or More Groups with Continuous Data 111

7.1 Introduction 112

7.2 Comparison of Two Groups of Paired Observations - Continuous Outcomes 113

7.3 Comparison of Two Independent Groups - Continuous Outcomes 119

7.4 Comparing More than Two Groups 127

7.5 Non-Normal Distributions 130

7.6 Degrees of Freedom 131

7.7 Points When Reading the Literature 132

7.8 Technical Details 132

7.9 Exercises 140

8 Comparing Groups of Binary and Categorical Data 145

8.1 Introduction 146

8.2 Comparison of Two Independent Groups - Binary Outcomes 146

8.3 Comparing Risks 151

8.4 Comparison of Two Groups of Paired Observations - Categorical Outcomes 152

8.5 Degrees of Freedom 153

8.6 Points When Reading the Literature 154

8.7 Technical Details 154

8.8 Exercises 160

9 Correlation and Linear Regression 163

9.1 Introduction 164

9.2 Correlation 165

9.3 Linear Regression 171

9.4 Comparison of Assumptions Between Correlation and Regression 178

9.5 Multiple Regression 179

9.6 Correlation is not Causation 181

9.7 Points When Reading the Literature 182

9.8 Technical Details 182

9.9 Exercises 190

10 Logistic Regression 193

10.1 Introduction 194

10.2 Binary Outcome Variable 194

10.3 The Multiple Logistic Regression Equation 196

10.4 Conditional Logistic Regression 200

10.5 Reporting the Results of a Logistic Regression 201

10.6 Additional Points When Reading the Literature When Logistic Regression Has Been Used 202

10.7 Technical Details 202

10.8 The Wald Test 204

10.9 Evaluating the Model and its Fit: The Hosmer-Lemeshow Test 204

10.10 Assessing Predictive Efficiency (1): 2 x 2 Classification Table 205

10.11 Assessing Predictive Efficiency (2): The ROC Curve 206

10.12 Investigating Linearity 206

10.13 Exercises 207

11 Survival Analysis 211

11.1 Time to Event Data 212

11.2 Kaplan-Meier Survival Curve 214

11.3 The Logrank Test 217

11.4 The Hazard Ratio 221

11.5 Modelling Time to Event Data 223

11.6 Points When Reading Literature 226

11.7 Exercises 229

12 Reliability and Method Comparison Studies 233

12.1 Introduction 234

12.2 Repeatability 234

12.3 Agreement 237

12.4 Validity 239

12.5 Method Comparison Studies 240

12.6 Points When Reading the Literature 243

12.7 Technical Details 243

12.8 Exercises 245

13 Evaluation of Diagnostic Tests 249

13.1 Introduction 250

13.2 Diagnostic Tests 250

13.3 Prevalence, Overall Accuracy, Sensitivity, and Specificity 251

13.4 Positive and Negative Predictive Values 252

13.5 The Effect of Prevalence 253

13.6 Confidence Intervals 254

13.7 Functions of a Screening and Diagnostic Test 255

13.8 Likelihood Ratio, Pre-test Odds and Post-test Odds 256

13.9 Receiver Operating Characteristic (ROC) Curve 257

13.10 Points When Reading the Literature About a Diagnostic Test 261

13.11 Exercises 262

14 Observational Studies 265

14.1 Introduction 266

14.2 Risk and Rates 266

14.3 Taking a Random Sample 272

14.4 Questionnaire and Form Design 273

14.5 Cross-sectional Surveys 274

14.6 Non-randomised Studies 275

14.7 Cohort Studies 278

14.8 Case-Control Studies 282

14.9 Association and Causality 287

14.10 Modern Causality Methods and Big Data 287

14.11 Points When Reading the Literature 288

14.12 Technical Details 288

14.13 Exercises 290

15 The Randomised Controlled Trial 293

15.1 Introduction 294

15.2 The Protocol 294

15.3 Why Randomise? 295

15.4 Methods of Randomisation 296

15.5 Design Features 298

15.6 Design Options 303

15.7 Meta-analysis 309

15.8 Checklists for Design, Analysis and Reporting 309

15.9 CONSORT 311

15.10 Points When Reading the Literature About a Trial 311

15.11 Exercises 311

16 Sample Size Issues 313

16.1 Introduction 314

16.2 Study Size 315

16.3 Continuous Data 318

16.4 Binary Data 319

16.5 Prevalence 321

16.6 Subject Withdrawals 322

16.7 Other Aspects of Sample Size Calculations 323

16.8 Points When Reading the Literature 325

16.9 Technical Details 325

16.10 Exercises 327

17 Other Statistical Methods 331

17.1 Analysing Serial or Longitudinal Data 332

17.2 Poisson Regression 341

17.3 Missing Data 343

17.4 Bootstrap Methods 350

17.5 Points When Reading the Literature 353

17.6 Exercises 353

18 Meta-analysis 355

18.1 Introduction 356

18.2 What is a Meta-analysis? 356

18.3 Meta-analysis Methods 358

18.4 Example: Mobile Phone Based Intervention for Smoking Cessation 359

18.5 Discussion 363

18.6 Technical Details 363

18.7 Exercises 365

19 Common Mistakes and Pitfalls 369

19.1 Introduction 370

19.2 Misleading Graphs and Tables 370

19.3 Plotting Change Against Initial Value 376

19.4 Within Group Versus Between Group Analyses 380

19.5 Analysing Paired Data Ignoring the Matching 381

19.6 Unit of Analysis 382

19.7 Testing for Baseline Imbalances in an RCT 382

19.8 Repeated Measures 383

19.9 Clinical and Statistical Significance 387

19.10 Post Hoc Power Calculations 387

19.11 Predicting or Extrapolating Beyond the Observed Range of Data 388

19.12 Exploratory Data Analysis 390

19.13 Misuse of P-values 391

19.14 Points When Reading the Literature 391

Appendix: Statistical Tables 393

Solutions to Multiple-Choice Exercises 403

References 413

Index 423