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Table 6 Sample scenarios assuming beta priors p(π 1) and p(π 2) where \(\tau _{1}^{2}=0.08\). Hypothesized values of π 1 and π 2 set equal to m 1 and m 2, respectively, under the traditional design. Two-sided α=0.05 and 1−β=0.80 assumed

From: Sample size determination for a binary response in a superiority clinical trial using a hybrid classical and Bayesian procedure

 

(\(\tau _{1}^{2}=0.08\))

Traditional design

CEP design

 

(m 1,m 2)

\(\tau _{2}^{2}\)

\(\hat {N}\)

Performance

CEP

N

Performance

E(π 2π 1|π 2>π 1)

P(π 2>π 1)

Marginal benefit

(0.1,0.9)

0.001

10

0.127

0.389

70

0.703

0.459

0.913

0.0096

 

0.01

10

0.097

0.356

88

0.707

0.435

0.861

0.0078

 

0.02

10

0.083

0.335

102

0.709

0.417

0.815

0.0068

 

0.03

10

0.075

0.319

118

0.711

0.402

0.774

0.0059

 

0.04

10

0.069

0.306

132

0.712

0.389

0.734

0.0053

 

0.05

10

0.064

0.294

148

0.713

0.378

0.693

0.0047

 

0.06

10

0.060

0.283

166

0.713

0.366

0.650

0.0042

 

0.07

10

0.056

0.273

186

0.714

0.355

0.602

0.0037

(0.1,0.8)

0.001

14

0.147

0.399

114

0.710

0.407

0.825

0.0056

 

0.01

14

0.134

0.387

124

0.710

0.396

0.794

0.0052

 

0.02

14

0.128

0.377

134

0.711

0.386

0.760

0.0049

 

0.03

14

0.124

0.368

146

0.712

0.378

0.728

0.0045

 

0.04

14

0.121

0.360

158

0.713

0.370

0.695

0.0041

 

0.05

14

0.118

0.353

170

0.713

0.362

0.661

0.0038

 

0.06

14

0.116

0.346

184

0.714

0.355

0.625

0.0035

 

0.07

14

0.113

0.339

198

0.715

0.348

0.586

0.0033

(0.1,0.7)

0.001

20

0.174

0.415

172

0.713

0.355

0.731

0.0035

 

0.01

20

0.172

0.414

176

0.713

0.353

0.712

0.0035

 

0.02

20

0.174

0.413

182

0.713

0.350

0.691

0.0033

 

0.03

20

0.177

0.412

188

0.714

0.348

0.669

0.0032

 

0.04

20

0.180

0.411

192

0.714

0.346

0.646

0.0031

 

0.05

20

0.183

0.410

200

0.715

0.344

0.622

0.0030

 

0.06

20

0.185

0.409

206

0.714

0.342

0.596

0.0028

 

0.07

20

0.187

0.408

214

0.715

0.340

0.567

0.0027

(0.2,0.8)

0.001

20

0.264

0.488

118

0.710

0.403

0.821

0.0046

 

0.01

20

0.247

0.475

128

0.711

0.393

0.789

0.0043

 

0.02

20

0.233

0.462

138

0.712

0.384

0.755

0.0041

 

0.03

20

0.223

0.451

150

0.712

0.375

0.723

0.0038

 

0.04

20

0.215

0.442

162

0.713

0.367

0.690

0.0035

 

0.05

20

0.208

0.433

174

0.713

0.360

0.656

0.0033

 

0.06

20

0.201

0.424

188

0.714

0.353

0.620

0.0031

 

0.07

20

0.195

0.416

202

0.715

0.346

0.580

0.0029

(0.1,0.6)

0.001

28

0.193

0.426

252

0.715

0.303

0.632

0.0023

 

0.01

28

0.199

0.431

250

0.715

0.307

0.623

0.0023

 

0.02

28

0.209

0.437

246

0.715

0.311

0.613

0.0023

 

0.03

28

0.220

0.443

244

0.715

0.316

0.601

0.0023

 

0.04

28

0.231

0.449

240

0.715

0.320

0.589

0.0023

 

0.05

28

0.241

0.456

238

0.715

0.324

0.576

0.0023

 

0.06

28

0.252

0.463

234

0.715

0.328

0.562

0.0022

 

0.07

28

0.262

0.469

230

0.715

0.332

0.546

0.0022

(0.2,0.7)

0.001

30

0.302

0.510

178

0.713

0.352

0.725

0.0028

 

0.01

30

0.298

0.507

182

0.713

0.350

0.706

0.0027

 

0.02

30

0.295

0.504

186

0.714

0.348

0.685

0.0027

 

0.03

30

0.293

0.502

192

0.714

0.346

0.663

0.0026

 

0.04

30

0.292

0.499

198

0.714

0.344

0.640

0.0025

 

0.05

30

0.291

0.497

204

0.714

0.342

0.616

0.0024

 

0.06

30

0.291

0.495

210

0.715

0.340

0.590

0.0024

 

0.07

30

0.291

0.493

218

0.715

0.338

0.561

0.0023

(0.1,0.5)

0.001

40

0.213

0.437

364

0.716

0.253

0.531

0.0016

 

0.01

40

0.226

0.448

350

0.716

0.262

0.530

0.0016

 

0.02

40

0.244

0.461

334

0.716

0.272

0.529

0.0016

 

0.03

40

0.264

0.474

318

0.716

0.282

0.528

0.0016

 

0.04

40

0.283

0.488

300

0.715

0.293

0.527

0.0017

 

0.05

40

0.303

0.502

284

0.716

0.303

0.526

0.0017

 

0.06

40

0.323

0.516

266

0.716

0.313

0.525

0.0017

 

0.07

40

0.342

0.530

248

0.716

0.324

0.523

0.0018

(0.2,0.6)

0.001

46

0.343

0.536

258

0.715

0.301

0.625

0.0018

 

0.01

46

0.347

0.539

256

0.715

0.305

0.617

0.0018

 

0.02

46

0.352

0.543

252

0.715

0.309

0.606

0.0018

 

0.03

46

0.359

0.547

250

0.715

0.314

0.595

0.0017

 

0.04

46

0.366

0.552

246

0.715

0.318

0.583

0.0017

 

0.05

46

0.373

0.556

242

0.715

0.322

0.570

0.0017

 

0.06

46

0.382

0.562

238

0.715

0.326

0.556

0.0017

 

0.07

46

0.390

0.567

234

0.716

0.330

0.540

0.0017

(0.3,0.7)

0.001

48

0.435

0.605

182

0.713

0.349

0.718

0.0021

 

0.01

48

0.431

0.602

186

0.713

0.347

0.700

0.0020

 

0.02

48

0.427

0.599

192

0.714

0.345

0.679

0.0020

 

0.03

48

0.423

0.596

196

0.714

0.343

0.657

0.0020

 

0.04

48

0.419

0.592

202

0.714

0.342

0.634

0.0019

 

0.05

48

0.416

0.589

208

0.715

0.340

0.610

0.0019

 

0.06

48

0.414

0.586

214

0.715

0.338

0.584

0.0018

 

0.07

48

0.411

0.583

222

0.715

0.336

0.555

0.0017

(0.1,0.4)

0.001

64

0.258

0.468

532

0.716

0.202

0.428

0.0010

 

0.01

64

0.281

0.486

494

0.716

0.217

0.435

0.0010

 

0.02

64

0.308

0.506

452

0.716

0.234

0.444

0.0011

 

0.03

64

0.336

0.526

412

0.716

0.250

0.454

0.0011

 

0.04

64

0.363

0.546

374

0.716

0.267

0.464

0.0011

 

0.05

64

0.391

0.566

338

0.716

0.283

0.475

0.0012

 

0.06

64

0.418

0.586

302

0.716

0.299

0.487

0.0013

 

0.07

64

0.446

0.607

268

0.716

0.316

0.501

0.0013

(0.2,0.5)

0.001

78

0.404

0.577

374

0.716

0.251

0.524

0.0011

 

0.01

78

0.414

0.585

358

0.716

0.260

0.523

0.0011

 

0.02

78

0.427

0.594

342

0.716

0.270

0.522

0.0011

 

0.03

78

0.441

0.604

326

0.716

0.281

0.521

0.0011

 

0.04

78

0.455

0.615

308

0.716

0.291

0.521

0.0011

 

0.05

78

0.470

0.625

290

0.716

0.301

0.520

0.0012

 

0.06

78

0.486

0.637

272

0.716

0.312

0.519

0.0012

 

0.07

78

0.503

0.649

254

0.716

0.322

0.518

0.0012

(0.3,0.6)

0.001

84

0.497

0.647

266

0.715

0.299

0.618

0.0012

 

0.01

84

0.501

0.649

264

0.715

0.303

0.609

0.0012

 

0.02

84

0.505

0.652

260

0.715

0.307

0.599

0.0012

 

0.03

84

0.509

0.655

256

0.715

0.311

0.588

0.0012

 

0.04

84

0.514

0.658

252

0.715

0.316

0.576

0.0012

 

0.05

84

0.519

0.661

248

0.715

0.320

0.564

0.0012

 

0.06

84

0.524

0.665

244

0.715

0.324

0.550

0.0012

 

0.07

84

0.530

0.668

240

0.716

0.328

0.534

0.0012

(0.1,0.3)

0.001

124

0.339

0.526

808

0.717

0.153

0.323

0.0006

 

0.01

124

0.376

0.554

708

0.717

0.176

0.341

0.0006

 

0.02

124

0.414

0.582

612

0.717

0.200

0.361

0.0006

 

0.03

124

0.450

0.608

530

0.717

0.223

0.382

0.0007

 

0.04

124

0.483

0.632

458

0.717

0.245

0.404

0.0007

 

0.05

124

0.515

0.656

394

0.717

0.266

0.428

0.0007

 

0.06

124

0.546

0.679

338

0.717

0.287

0.452

0.0008

 

0.07

124

0.578

0.702

286

0.716

0.308

0.480

0.0009

(0.2,0.4)

0.001

164

0.502

0.647

546

0.717

0.201

0.420

0.0006

 

0.01

164

0.519

0.659

506

0.716

0.216

0.428

0.0006

 

0.02

164

0.537

0.672

464

0.717

0.233

0.437

0.0006

 

0.03

164

0.557

0.686

422

0.717

0.249

0.447

0.0006

 

0.04

164

0.576

0.700

382

0.716

0.265

0.457

0.0006

 

0.05

164

0.596

0.715

344

0.716

0.282

0.469

0.0007

 

0.06

164

0.617

0.730

308

0.716

0.298

0.482

0.0007

 

0.07

164

0.638

0.745

272

0.716

0.314

0.495

0.0007

(0.3,0.5)

0.001

186

0.597

0.716

384

0.716

0.249

0.516

0.0006

 

0.01

186

0.605

0.721

368

0.716

0.258

0.515

0.0006

 

0.02

186

0.614

0.728

352

0.716

0.269

0.515

0.0006

 

0.03

186

0.623

0.735

334

0.716

0.279

0.514

0.0006

 

0.04

186

0.633

0.742

314

0.716

0.289

0.514

0.0006

 

0.05

186

0.644

0.749

296

0.716

0.299

0.513

0.0007

 

0.06

186

0.655

0.757

278

0.716

0.310

0.513

0.0007

 

0.07

186

0.667

0.766

258

0.716

0.320

0.512

0.0007

(0.4,0.6)

0.001

194

0.662

0.763

274

0.715

0.296

0.610

0.0007

 

0.01

194

0.664

0.764

270

0.715

0.301

0.602

0.0007

 

0.02

194

0.667

0.766

266

0.715

0.305

0.592

0.0007

 

0.03

194

0.669

0.768

262

0.715

0.309

0.581

0.0007

 

0.04

194

0.672

0.770

258

0.715

0.314

0.569

0.0007

 

0.05

194

0.675

0.772

254

0.715

0.318

0.557

0.0007

 

0.06

194

0.679

0.774

250

0.716

0.322

0.543

0.0007

 

0.07

194

0.682

0.777

244

0.716

0.326

0.528

0.0007

(0.1,0.2)

0.001

398

0.506

0.648

1346

0.717

0.104

0.218

0.0002

 

0.01

398

0.563

0.691

978

0.714

0.139

0.247

0.0003

 

0.02

398

0.606

0.720

814

0.717

0.172

0.286

0.0003

 

0.03

398

0.642

0.746

660

0.717

0.201

0.319

0.0003

 

0.04

398

0.673

0.768

542

0.717

0.227

0.352

0.0003

 

0.05

398

0.700

0.788

450

0.717

0.253

0.386

0.0003

 

0.06

398

0.726

0.807

372

0.717

0.277

0.422

0.0004

 

0.07

398

0.751

0.825

304

0.716

0.302

0.460

0.0004

(0.2,0.3)

0.001

588

0.667

0.765

828

0.717

0.152

0.316

0.0002

 

0.01

588

0.691

0.783

694

0.714

0.175

0.330

0.0002

 

0.02

588

0.709

0.794

624

0.717

0.199

0.355

0.0002

 

0.03

588

0.728

0.808

540

0.717

0.222

0.376

0.0002

 

0.04

588

0.747

0.821

466

0.717

0.243

0.398

0.0002

 

0.05

588

0.764

0.834

402

0.717

0.265

0.422

0.0003

 

0.06

588

0.781

0.846

344

0.717

0.286

0.447

0.0003

 

0.07

588

0.799

0.858

292

0.716

0.307

0.474

0.0003

(0.3,0.4)

0.001

712

0.748

0.822

560

0.717

0.200

0.413

0.0002

 

0.01

712

0.761

0.832

502

0.715

0.215

0.416

0.0002

 

0.02

712

0.767

0.836

474

0.717

0.231

0.430

0.0002

 

0.03

712

0.778

0.843

432

0.716

0.248

0.440

0.0002

 

0.04

712

0.788

0.851

390

0.716

0.264

0.451

0.0002

 

0.05

712

0.799

0.859

352

0.716

0.280

0.462

0.0002

 

0.06

712

0.810

0.866

314

0.716

0.296

0.475

0.0002

 

0.07

712

0.821

0.874

278

0.716

0.312

0.490

0.0002

(0.4,0.5)

0.001

776

0.797

0.857

394

0.716

0.247

0.508

0.0002

 

0.01

776

0.803

0.863

368

0.714

0.257

0.503

0.0002

 

0.02

776

0.806

0.863

360

0.716

0.267

0.507

0.0002

 

0.03

776

0.811

0.867

342

0.716

0.277

0.507

0.0002

 

0.04

776

0.816

0.871

322

0.716

0.287

0.507

0.0002

 

0.05

776

0.822

0.875

302

0.716

0.298

0.506

0.0002

 

0.06

776

0.827

0.879

284

0.716

0.308

0.506

0.0002

 

0.07

776

0.834

0.883

264

0.716

0.318

0.506

0.0002