Statistical Thinking for Non-Statisticians in Drug Regulation

STATISTICAL THINKING FOR NON-STATISTICIANS IN DRUG REGULATION

Statistical methods in the pharmaceutical industry are accepted as a key element in the design and analysis of clinical studies. Increasingly, the medical and scientific community are aligning with the regulatory authorities and recognizing that correct statistical methodology is essential as the basis for valid conclusions. In order for those correct and robust methods to be successfully employed there needs to be effective communication across disciplines at all stages of the planning, conducting, analyzing and reporting of clinical studies associated with the development and evaluation of new drugs and devices.

Statistical Thinking for Non-Statisticians in Drug Regulation provides a comprehensive in-depth guide to statistical methodology for pharmaceutical industry professionals, including physicians, investigators, medical science liaisons, clinical research scientists, medical writers, regulatory personnel, statistical programmers, senior data managers and those working in pharmacovigilance. The author’s years of experience and up-to-date familiarity with pharmaceutical regulations and statistical practice within the wider clinical community make this an essential guide for the those working in and with the industry.

The third edition of Statistical Thinking for Non-Statisticians in Drug Regulation includes:

• A detailed new chapter on Estimands in line with the 2019 Addendum to ICH E9
• Major new sections on topics including Combining Hierarchical Testing and Alpha Adjustment, Biosimilars, Restricted Mean Survival Time, Composite Endpoints and Cumulative Incidence Functions, Adjusting for Cross-Over in Oncology, Inverse Propensity Score Weighting, and Network Meta-Analysis
• Updated coverage of many existing topics to reflect new and revised guidance from regulatory authorities and author experience

Statistical Thinking for Non-Statisticians in Drug Regulation is a valuable guide for pharmaceutical and medical device industry professionals, as well as statisticians joining the pharmaceutical industry and students and teachers of drug development.

Richard Kay, PhD is a Visiting Professor at the School of Pharmacy and Pharmaceutical Medicine, Cardiff University, UK, and a longtime statistical consultant for the pharmaceutical industry. He provides consultancy and training services for pharmaceutical companies and research institutions.

Preface to the second edition, xv

Preface to the first edition, xvii

Abbreviations, xxi

1             Basic ideas in clinical trial design, 1

1.1         Historical perspective, 1

1.2         Control groups, 2

1.3         Placebos and blinding, 3

1.4         Randomisation, 3

1.4.1      Unrestricted randomisation, 4

1.4.2      Block randomisation, 4

1.4.3      Unequal randomisation, 5

1.4.4      Stratified randomisation, 6

1.4.5      Central randomisation, 7

1.4.6      Dynamic allocation and minimisation, 8

1.4.7      Cluster randomisation, 9

1.5         Bias and precision, 9

1.6         Between- and within-patient designs, 11

1.7         Crossover trials, 12

1.8         Signal, noise and evidence, 13

1.8.1      Signal, 13

1.8.2      Noise, 13

1.8.3      Signal-to-noise ratio, 14

1.9         Confirmatory and exploratory trials, 15

1.10       Superiority, equivalence and non-inferiority trials, 16

1.11       Data and endpoint types, 17

1.12       Choice of endpoint, 18

1.12.1    Primary variables, 18

1.12.2    Secondary variables, 19

1.12.3    Surrogate variables, 20

1.12.4    Global assessment variables, 21

1.12.5    Composite variables, 21

1.12.6    Categorisation, 21

2             Sampling and inferential statistics, 23

2.1         Sample and population, 23

2.2         Sample statistics and population parameters, 24

2.2.1      Sample and population distribution, 24

2.2.2      Median and mean, 25

2.2.3      Standard deviation, 25

2.2.4      Notation, 26

2.2.5      Box plots, 27

2.3         The normal distribution, 28

2.4         Sampling and the standard error of the mean, 31

2.5         Standard errors more generally, 34

2.5.1      The standard error for the difference between two means, 34

2.5.2      Standard errors for proportions, 37

2.5.3      The general setting, 37

3             Confidence intervals and p-values, 38

3.1         Confidence intervals for a single mean, 38

3.1.1      The 95 per cent Confidence interval, 38

3.1.2      Changing the confidence coefficient, 40

3.1.3      Changing the multiplying constant, 40

3.1.4      The role of the standard error, 41

3.2         Confidence interval for other parameters, 42

3.2.1      Difference between two means, 42

3.2.2      Confidence interval for proportions, 43

3.2.3      General case, 44

3.2.4      Bootstrap Confidence interval, 45

3.3         Hypothesis testing, 45

3.3.1      Interpreting the p-value, 46

3.3.2      Calculating the p-value, 47

3.3.3      A common process, 50

3.3.4      The language of statistical significance, 53

3.3.5      One-sided and two-sided tests, 54

4             Tests for simple treatment comparisons, 56

4.1         The unpaired t-test, 56

4.2         The paired t-test, 57

4.3         Interpreting the t-tests, 60

4.4         The chi-square test for binary data, 61

4.4.1      Pearson chi-square, 61

4.4.2      The link to a ratio of the signal to the standard error, 64

4.5         Measures of treatment benefit, 64

4.5.1      Odds ratio, 65

4.5.2      Relative risk, 65

4.5.3      Relative risk reduction, 66

4.5.4      Number needed to treat, 66

4.5.5      Confidence intervals, 67

4.5.6      Interpretation, 68

4.6         Fisher’s exact test, 69

4.7         Tests for categorical and ordinal data, 71

4.7.1      Categorical data, 71

4.7.2      Ordered categorical (ordinal) data, 73

4.7.3      Measures of treatment benefit, 74

4.8         Extensions for multiple treatment groups, 75

4.8.1      Between-patient designs and continuous data, 75

4.8.2      Within-patient designs and continuous data, 76

4.8.3      Binary, categorical and ordinal data, 76

4.8.4      Dose-ranging studies, 77

4.8.5      Further discussion, 77

5             Adjusting the analysis, 78

5.1         Objectives for adjusted analysis, 78

5.2         Comparing treatments for continuous data, 78

5.3         Least squares means, 82

5.4         Evaluating the homogeneity of the treatment effect, 83

5.4.1      Treatment-by-factor interactions, 83

5.4.2      Quantitative and qualitative interactions, 85

5.5         Methods for binary, categorical and ordinal data, 86

5.6         Multi-centre trials, 87

5.6.1      Adjusting for centre, 87

5.6.2      Significant treatment-by-centre interactions, 87

5.6.3      Combining centres, 88

6             Regression and analysis of covariance, 89

6.1         Adjusting for baseline factors, 89

6.2         Simple linear regression, 89

6.3         Multiple regression, 91

6.4         Logistic regression, 94

6.5         Analysis of covariance for continuous data, 94

6.5.1      Main effect of treatment, 94

6.5.2      Treatment-by-covariate interactions, 96

6.5.3      A single model, 98

6.5.4      Connection with adjusted analyses, 98

6.5.5      Advantages of ANCOVA, 99

6.5.6      Least squares means, 100

6.6         Binary, categorical and ordinal data, 101

6.7         Regulatory aspects of the use of covariates, 103

6.8         Baseline testing, 105

7             Intention-to-treat and analysis sets, 107

7.1         The principle of intention-to-treat, 107

7.2         The practice of intention-to-treat, 110

7.2.1      Full analysis set, 110

7.2.2      Per-protocol set, 112

7.2.3      Sensitivity, 112

7.3         Missing data, 113

7.3.1      Introduction, 113

7.3.2      Complete cases analysis, 114

7.3.3      Last observation carried forward, 114

7.3.4      Success/failure classification, 114

7.3.5      Worst-case/best-case classification, 115

7.3.6      Sensitivity, 115

7.3.7      Avoidance of missing data, 116

7.3.8      Multiple imputation, 117

7.4         Intention-to-treat and time-to-event data, 118

7.5         General questions and considerations, 120

8             Power and sample size, 123

8.1         Type I and type II errors, 123

8.2         Power, 124

8.3         Calculating sample size, 127

8.4         Impact of changing the parameters, 130

8.4.1      Standard deviation, 130

8.4.2      Event rate in the control group, 130

8.4.3      Clinically relevant difference, 131

8.5         Regulatory aspects, 132

8.5.1      Power >80 per cent, 132

8.5.2      Powering on the per-protocol set, 132

8.5.3      Sample size adjustment, 133

8.6         Reporting the sample size calculation, 134

9             Statistical significance and clinical importance, 136

9.1         Link between p-values and Confidence intervals, 136

9.2         Confidence intervals for clinical importance, 137

9.3         Misinterpretation of the p-value, 139

9.3.1      Conclusions of similarity, 139

9.3.2      The problem with 0.05, 140

9.4         Single pivotal trial and 0.05, 140

10           Multiple testing, 142

10.1       Inflation of the type I error, 142

10.1.1    False positives, 142

10.1.2    A simulated trial, 142

10.2       How does multiplicity arise?, 143

10.3       Regulatory view, 144

10.4       Multiple primary endpoints, 145

10.4.1    Avoiding adjustment, 145

10.4.2    Significance needed on all endpoints, 145

10.4.3    Composite endpoints, 146

10.4.4    Variables ranked according to clinical importance: Hierarchical testing, 146

10.5       Methods for adjustment, 149

10.5.1    Bonferroni correction, 149

10.5.2    Hochberg correction, 150

10.5.3    Interim analyses, 151

10.6       Multiple comparisons, 152

10.7       Repeated evaluation over time, 153

10.8       Subgroup testing, 154

10.9       Other areas for multiplicity, 156

10.9.1    Using different statistical tests, 156

10.9.2    Different analysis sets, 156

10.9.3    Pre-planning, 157

11           Non-parametric and related methods, 158

11.1       Assumptions underlying the t-tests and their extensions, 158

11.2       Homogeneity of variance, 158

11.3       The assumption of normality, 159

11.4       Non-normality and transformations, 161

11.5       Non-parametric tests, 164

11.5.1    The Mann–Whitney U-test, 164

11.5.2    The Wilcoxon signed rank test, 166

11.5.3    General comments, 167

11.6       Advantages and disadvantages of non-parametric methods, 168

11.7       Outliers, 169

12           Equivalence and non-inferiority, 170

12.1       Demonstrating similarity, 170

12.2       Confidence intervals for equivalence, 172

12.3       Confidence intervals for non-inferiority, 173

12.4       A p-value approach, 174

12.5       Assay sensitivity, 176

12.6       Analysis sets, 178

12.7       The choice of Δ, 179

12.7.1    Bioequivalence, 179

12.7.2    Therapeutic equivalence, 180

12.7.3    Non-inferiority, 180

12.7.4    The 10 per cent rule for cure rates, 182

12.7.5    The synthesis method, 183

12.8       Biocreep and constancy, 184

12.9       Sample size calculations, 184

12.10     Switching between non-inferiority and superiority, 186

13           The analysis of survival data, 189

13.1       Time-to-event data and censoring, 189

13.2       Kaplan-Meier curves, 190

13.2.1    Plotting Kaplan-Meier curves, 190

13.2.2    Event rates and relative risk, 192

13.2.3    Median event times, 192

13.3       Treatment comparisons, 193

13.4       The hazard ratio, 196

13.4.1    The hazard rate, 196

13.4.2    Constant hazard ratio, 197

13.4.3    Non-constant hazard ratio, 197

13.4.4    Link to survival curves, 198

13.4.5    Calculating Kaplan-Meier curves, 199

13.5       Adjusted analyses, 199

13.5.1    Stratified methods, 200

13.5.2    Proportional hazards regression, 200

13.5.3    Accelerated failure time model, 201

13.6       Independent censoring, 202

13.7       Sample size calculations, 203

14           Interim analysis and data monitoring committees, 205

14.1       Stopping rules for interim analysis, 205

14.2       Stopping for efficacy and futility, 206

14.2.1    Efficacy, 206

14.2.2    Futility and conditional power, 207

14.2.3    Some practical issues, 208

14.2.4    Analyses following completion of recruitment, 209

14.3       Monitoring safety, 210

14.4       Data monitoring committees, 211

14.4.1    Introduction and responsibilities, 211

14.4.2    Structure and process, 212

14.4.3    Meetings and recommendations, 214

15           Bayesian statistics, 215

15.1       Introduction, 215

15.2       Prior and posterior distributions, 215

15.2.1    Prior beliefs, 215

15.2.2    Prior to posterior, 217

15.2.3    Bayes theorem, 217

15.3       Bayesian inference, 219

15.3.1    Frequentist methods, 219

15.3.2    Posterior probabilities, 219

15.3.3    Credible intervals, 220

15.4       Case study, 221

15.5       History and regulatory acceptance, 222

15.6       Discussion, 224

16           Adaptive designs, 225

16.1       What are adaptive designs?, 225

16.1.1    Advantages and drawbacks, 225

16.1.2    Restricted adaptations, 226

16.1.3    Flexible adaptations, 227

16.2       Minimising bias, 228

16.2.1    Control of type I error, 228

16.2.2    Estimation, 229

16.2.3    Behavioural issues, 230

16.2.4    Exploratory trials, 232

16.3       Unblinded sample size re-estimation, 232

16.3.1    Product of p-values, 232

16.3.2    Weighting the two parts of the trial, 233

16.3.3    Rationale, 234

16.4       Seamless phase II/III studies, 234

16.4.1    Standard framework, 234

16.4.2    Aspects of the p-value calculation, 235

16.4.3    Logistical challenges, 236

16.5       Other types of adaptation, 236

16.5.1    Changing the primary endpoint, 236

16.5.2    Focusing on a sub-population, 237

16.5.3    Dropping the placebo arm in a non-inferiority trial, 237

16.6       Further regulatory considerations, 238

16.6.1    Impact on power, 238

16.6.2    Non-standard experimental settings, 239

17           Observational studies, 241

17.1       Introduction, 241

17.1.1    Non-randomised comparisons, 241

17.1.2    Study types, 241

17.1.3    Sources of bias, 243

17.1.4    An empirical investigation, 244

17.1.5    Selection bias in concurrently controlled studies: An empirical evaluation, 245

17.1.6    Selection bias in historically controlled studies: An empirical evaluation, 246

17.1.7    Some conclusions, 246

17.2       Guidance on design, conduct and analysis, 247

17.2.1    Regulatory guidance, 247

17.2.2    Strengthening the Reporting of Observational Studies in Epidemiology, 248

17.3       Evaluating and adjusting for selection bias, 249

17.3.1    Baseline balance, 249

17.3.2    Adjusting for imbalances using stratification and analysis of covariance, 250

17.3.3    Propensity scores, 250

17.3.4    Different methods for adjustment: An empirical evaluation, 253

17.3.5    Some conclusions, 256

17.4       Case–control studies, 257

17.4.1    Background, 257

17.4.2    Odds ratio and Relative risk, 259

18           Meta-analysis, 261

18.1       Definition, 261

18.2       Objectives, 263

18.3       Statistical methodology, 264

18.3.1    Methods for combination, 264

18.3.2    Confidence intervals, 265

18.3.3    Fixed and random effects, 265

18.3.4    Graphical methods, 266

18.3.5    Detecting heterogeneity, 266

18.3.6    Robustness, 269

18.3.7    Rare events, 269

18.3.8    Individual patient data, 269

18.4       Case study, 270

18.5       Ensuring scientific validity, 271

18.5.1    Planning, 271

18.5.2    Assessing the risk of bias, 273

18.5.3    Publication bias and funnel plots, 273

18.5.4    Preferred Reporting Items for Systematic Reviews and Meta-Analyses, 275

18.6       Further regulatory aspects, 275

19           Methods for the safety analysis and safety monitoring, 277

19.1       Introduction, 277

19.1.1    Methods for safety data, 277

19.1.2    The rule of three, 278

19.2       Routine evaluation in clinical studies, 279

19.2.1    Types of data, 280

19.2.2    Adverse events, 281

19.2.3    Laboratory data, 284

19.2.4    ECG data, 287

19.2.5    Vital signs, 288

19.2.6    Safety summary across trials, 288

19.2.7    Specific safety studies, 289

19.3       Data monitoring committees, 289

19.4       Assessing benefit–risk, 290

19.4.1    Current approaches, 290

19.4.2    Multi-criteria decision analysis, 291

19.4.3    Quality-Adjusted Time without Symptoms or Toxicity, 297

19.5       Pharmacovigilance, 299

19.5.1    Post-approval safety monitoring, 299

19.5.2    Proportional reporting ratios, 300

19.5.3    Bayesian shrinkage, 302

20           Diagnosis, 304

20.1       Introduction, 304

20.2       Measures of diagnostic performance, 304

20.2.1    Sensitivity and specificity, 304

20.2.2    Positive and negative predictive value, 305

20.2.3    False positive and false negative rates, 306

20.2.4    Prevalence, 306

20.2.5    Likelihood ratio, 307

20.2.6    Predictive accuracy, 307

20.2.7    Choosing the correct cut-point, 307

20.3       Receiver operating characteristic curves, 308

20.3.1    Receiver operating characteristic, 308

20.3.2    Comparing ROC curves, 309

20.4       Diagnostic performance using regression models, 310

20.5       Aspects of trial design for diagnostic agents, 312

20.6       Assessing agreement, 313

20.6.1    The kappa statistic, 313

20.6.2    Other applications for kappa, 314

21           The role of statistics and statisticians, 316

21.1       The importance of statistical thinking at the design stage, 316

21.2       Regulatory guidelines, 317

21.3       The statistics process, 321

21.3.1    The statistical methods section of the protocol, 321

21.3.2    The statistical analysis plan, 322

21.3.3    The data validation plan, 322

21.3.4    The blind review, 322

21.3.5    Statistical analysis, 323

21.3.6    Reporting the analysis, 323

21.3.7    Pre-planning, 324

21.3.8    Sensitivity and robustness, 326

21.4       The regulatory submission, 327

21.5       Publications and presentations, 328

References, 331

Index, 339

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