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Table 4 Sample scenarios assuming beta priors p(π 1) and p(π 2) where \(\tau _{1}^{2}=\tau _{2}^{2}\). 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

  

Traditional design

CEP design

 

(m 1,m 2)

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

\(\hat {N}\)

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

0.797

0.518

12

0.742

0.783

1

0.1120

 

0.01

10

0.622

0.242

16

0.627

0.674

1

0.0642

 

0.02

10

0.508

0.164

26

0.673

0.585

0.993

0.0318

 

0.03

10

0.433

0.123

40

0.687

0.519

0.965

0.0188

 

0.04

10

0.381

0.099

60

0.698

0.469

0.915

0.0120

 

0.05

10

0.342

0.082

86

0.705

0.429

0.846

0.0082

 

0.06

10

0.311

0.070

120

0.710

0.397

0.761

0.0058

 

0.07

10

0.285

0.060

160

0.713

0.369

0.662

0.0044

 

0.08

10

0.262

0.052

214

0.716

0.342

0.545

0.0033

(0.1,0.8) or (0.2,0.9)

0.001

14

0.804

0.559

14

0.559

0.688

1

0

 

0.01

14

0.656

0.314

22

0.623

0.602

1

0.0386

 

0.02

14

0.553

0.237

34

0.674

0.529

0.989

0.0218

 

0.03

14

0.486

0.195

52

0.692

0.476

0.953

0.0131

 

0.04

14

0.439

0.168

76

0.702

0.438

0.896

0.0086

 

0.05

14

0.405

0.149

102

0.706

0.408

0.823

0.0063

 

0.06

14

0.377

0.134

134

0.711

0.383

0.740

0.0048

 

0.07

14

0.353

0.121

172

0.713

0.361

0.646

0.0037

 

0.08

14

0.331

0.110

218

0.716

0.340

0.540

0.0030

(0.1,0.7) or (0.3,0.9)

0.001

20

0.814

0.616

20

0.616

0.590

1

0

 

0.01

20

0.670

0.364

30

0.636

0.518

0.999

0.0272

 

0.02

20

0.578

0.295

48

0.677

0.463

0.979

0.0136

 

0.03

20

0.523

0.260

72

0.697

0.426

0.931

0.0084

 

0.04

20

0.488

0.238

98

0.705

0.401

0.866

0.0060

 

0.05

20

0.462

0.223

126

0.708

0.382

0.792

0.0046

 

0.06

20

0.442

0.211

154

0.712

0.367

0.712

0.0037

 

0.07

20

0.424

0.200

186

0.714

0.353

0.627

0.0031

 

0.08

20

0.407

0.189

222

0.716

0.338

0.534

0.0026

(0.2,0.8)

0.001

20

0.800

0.537

22

0.644

0.593

1

0.0537

 

0.01

20

0.676

0.372

30

0.643

0.530

0.999

0.0271

 

0.02

20

0.588

0.308

48

0.680

0.474

0.982

0.0133

 

0.03

20

0.532

0.272

70

0.695

0.435

0.936

0.0085

 

0.04

20

0.495

0.247

94

0.703

0.408

0.872

0.0062

 

0.05

20

0.467

0.229

122

0.709

0.387

0.798

0.0047

 

0.06

20

0.445

0.215

152

0.711

0.370

0.717

0.0038

 

0.07

20

0.425

0.202

184

0.714

0.354

0.630

0.0031

 

0.08

20

0.407

0.189

222

0.716

0.339

0.534

0.0026

(0.1,0.6) or (0.4,0.9)

0.001

28

0.804

0.572

28

0.572

0.491

1

0

 

0.01

28

0.655

0.368

48

0.660

0.430

0.996

0.0146

 

0.02

28

0.578

0.319

76

0.689

0.393

0.958

0.0077

 

0.03

28

0.540

0.299

106

0.701

0.374

0.895

0.0052

 

0.04

28

0.518

0.290

132

0.706

0.362

0.824

0.0040

 

0.05

28

0.504

0.285

158

0.710

0.355

0.752

0.0033

 

0.06

28

0.494

0.281

180

0.713

0.349

0.680

0.0028

 

0.07

28

0.486

0.278

202

0.714

0.343

0.606

0.0025

 

0.08

28

0.476

0.272

226

0.715

0.336

0.528

0.0022

(0.2,0.7) or (0.3,0.8)

0.001

30

0.803

0.558

30

0.558

0.494

1

0

 

0.01

30

0.682

0.412

46

0.653

0.446

0.998

0.0151

 

0.02

30

0.609

0.365

72

0.686

0.410

0.966

0.0076

 

0.03

30

0.570

0.342

100

0.700

0.388

0.907

0.0051

 

0.04

30

0.546

0.328

126

0.706

0.374

0.837

0.0039

 

0.05

30

0.529

0.317

150

0.710

0.363

0.763

0.0033

 

0.06

30

0.515

0.309

174

0.713

0.354

0.688

0.0028

 

0.07

30

0.503

0.300

198

0.714

0.346

0.611

0.0025

 

0.08

30

0.491

0.291

226

0.716

0.337

0.529

0.0022

(0.1,0.5) or (0.5,0.9)

0.001

40

0.781

0.488

42

0.566

0.392

1

0.0393

 

0.01

40

0.627

0.358

80

0.675

0.343

0.985

0.0079

 

0.02

40

0.573

0.335

124

0.697

0.326

0.918

0.0043

 

0.03

40

0.553

0.334

160

0.706

0.322

0.841

0.0031

 

0.04

40

0.546

0.339

182

0.709

0.324

0.769

0.0026

 

0.05

40

0.545

0.346

200

0.712

0.327

0.703

0.0023

 

0.06

40

0.545

0.353

210

0.713

0.331

0.642

0.0021

 

0.07

40

0.546

0.359

220

0.715

0.334

0.584

0.0020

 

0.08

40

0.545

0.362

230

0.716

0.334

0.522

0.0019

(0.2,0.6) or (0.4,0.8)

0.001

46

0.791

0.518

48

0.587

0.395

1

0.0345

 

0.01

46

0.668

0.422

80

0.668

0.360

0.991

0.0072

 

0.02

46

0.617

0.401

118

0.694

0.343

0.933

0.0041

 

0.03

46

0.597

0.397

150

0.704

0.339

0.860

0.0030

 

0.04

46

0.588

0.397

172

0.709

0.338

0.787

0.0025

 

0.05

46

0.584

0.399

190

0.712

0.338

0.719

0.0022

 

0.06

46

0.580

0.400

202

0.713

0.337

0.654

0.0020

 

0.07

46

0.577

0.400

216

0.714

0.337

0.589

0.0018

 

0.08

46

0.572

0.398

230

0.715

0.335

0.523

0.0017

(0.3,0.7)

0.001

48

0.793

0.524

50

0.558

0.396

1

0.0167

 

0.01

48

0.678

0.438

80

0.665

0.365

0.992

0.0071

 

0.02

48

0.629

0.419

118

0.695

0.349

0.938

0.0039

 

0.03

48

0.610

0.415

146

0.704

0.344

0.866

0.0029

 

0.04

48

0.601

0.415

168

0.709

0.342

0.794

0.0024

 

0.05

48

0.595

0.415

186

0.711

0.341

0.724

0.0021

 

0.06

48

0.591

0.415

200

0.714

0.340

0.658

0.0020

 

0.07

48

0.586

0.413

214

0.715

0.338

0.591

0.0018

 

0.08

48

0.580

0.409

230

0.715

0.335

0.523

0.0017

(0.1,0.4) or (0.6,0.9)

0.001

64

0.763

0.461

72

0.588

0.292

1

0.0159

 

0.01

64

0.612

0.377

156

0.689

0.262

0.951

0.0034

 

0.02

64

0.588

0.381

216

0.705

0.265

0.850

0.0021

 

0.03

64

0.588

0.397

242

0.709

0.276

0.768

0.0018

 

0.04

64

0.594

0.414

250

0.712

0.289

0.703

0.0016

 

0.05

64

0.603

0.432

250

0.714

0.302

0.649

0.0015

 

0.06

64

0.613

0.449

246

0.715

0.314

0.603

0.0015

 

0.07

64

0.622

0.464

238

0.715

0.325

0.560

0.0014

 

0.08

64

0.628

0.474

234

0.715

0.332

0.516

0.0014

(0.2,0.5) or (0.5,0.8)

0.001

78

0.776

0.491

84

0.584

0.296

1

0.0154

 

0.01

78

0.654

0.441

158

0.686

0.277

0.964

0.0031

 

0.02

78

0.635

0.450

210

0.703

0.282

0.872

0.0019

 

0.03

78

0.635

0.465

232

0.709

0.292

0.791

0.0016

 

0.04

78

0.640

0.480

238

0.711

0.302

0.724

0.0014

 

0.05

78

0.647

0.493

240

0.713

0.312

0.666

0.0014

 

0.06

78

0.653

0.504

238

0.714

0.321

0.615

0.0013

 

0.07

78

0.658

0.514

234

0.715

0.328

0.566

0.0013

 

0.08

78

0.661

0.520

234

0.716

0.332

0.517

0.0013

(0.3,0.6) or (0.4,0.7)

0.001

84

0.775

0.483

92

0.584

0.297

1

0.0126

 

0.01

84

0.666

0.457

162

0.683

0.283

0.968

0.0029

 

0.02

84

0.649

0.471

210

0.702

0.289

0.881

0.0018

 

0.03

84

0.652

0.489

228

0.708

0.300

0.802

0.0015

 

0.04

84

0.658

0.504

234

0.712

0.309

0.735

0.0014

 

0.05

84

0.663

0.516

234

0.713

0.318

0.676

0.0013

 

0.06

84

0.668

0.525

232

0.714

0.324

0.621

0.0013

 

0.07

84

0.671

0.532

232

0.715

0.329

0.569

0.0012

 

0.08

84

0.672

0.536

234

0.716

0.333

0.517

0.0012

(0.1,0.3) or (0.7,0.9)

0.001

124

0.736

0.453

152

0.614

0.194

1

0.0057

 

0.01

124

0.625

0.434

346

0.703

0.193

0.866

0.0012

 

0.02

124

0.636

0.470

374

0.710

0.216

0.750

0.0010

 

0.03

124

0.654

0.501

358

0.712

0.239

0.680

0.0009

 

0.04

124

0.671

0.528

334

0.714

0.261

0.631

0.0009

 

0.05

124

0.687

0.553

306

0.714

0.281

0.594

0.0009

 

0.06

124

0.701

0.575

280

0.715

0.300

0.565

0.0009

 

0.07

124

0.714

0.594

258

0.715

0.317

0.538

0.0009

 

0.08

124

0.725

0.610

240

0.716

0.330

0.510

0.0009

(0.2,0.4) or (0.6,0.8)

0.001

164

0.754

0.487

190

0.605

0.197

1

0.0045

 

0.01

164

0.667

0.496

372

0.702

0.205

0.887

0.0010

 

0.02

164

0.683

0.537

380

0.709

0.229

0.774

0.0008

 

0.03

164

0.701

0.568

356

0.712

0.251

0.702

0.0007

 

0.04

164

0.716

0.593

326

0.714

0.271

0.650

0.0007

 

0.05

164

0.730

0.614

300

0.714

0.289

0.609

0.0007

 

0.06

164

0.742

0.632

276

0.715

0.305

0.574

0.0007

 

0.07

164

0.752

0.647

254

0.715

0.319

0.543

0.0008

 

0.08

164

0.760

0.659

240

0.716

0.330

0.511

0.0008

(0.3,0.5) or (0.5,0.7)

0.001

186

0.755

0.487

214

0.605

0.198

1

0.0042

 

0.01

186

0.679

0.513

394

0.700

0.210

0.896

0.0009

 

0.02

186

0.699

0.560

390

0.709

0.236

0.787

0.0007

 

0.03

186

0.719

0.594

356

0.712

0.259

0.715

0.0007

 

0.04

186

0.735

0.620

324

0.713

0.278

0.662

0.0007

 

0.05

186

0.748

0.640

294

0.714

0.294

0.619

0.0007

 

0.06

186

0.759

0.656

272

0.715

0.308

0.581

0.0007

 

0.07

186

0.768

0.669

252

0.715

0.321

0.546

0.0007

 

0.08

186

0.774

0.679

238

0.716

0.331

0.511

0.0007

(0.4,0.6)

0.001

194

0.757

0.491

222

0.603

0.198

1

0.0040

 

0.01

194

0.683

0.518

402

0.700

0.211

0.898

0.0009

 

0.02

194

0.704

0.567

396

0.709

0.238

0.791

0.0007

 

0.03

194

0.724

0.602

358

0.712

0.261

0.720

0.0007

 

0.04

194

0.741

0.628

322

0.713

0.280

0.666

0.0007

 

0.05

194

0.754

0.648

294

0.714

0.296

0.622

0.0007

 

0.06

194

0.765

0.664

270

0.715

0.310

0.583

0.0007

 

0.07

194

0.773

0.677

252

0.715

0.321

0.547

0.0007

 

0.08

194

0.779

0.686

238

0.716

0.331

0.511

0.0007

(0.1,0.2) or (0.8,0.9)

0.001

398

0.683

0.470

676

0.675

0.098

0.981

0.0007

 

0.01

398

0.712

0.584

794

0.712

0.141

0.708

0.0003

 

0.02

398

0.750

0.642

618

0.714

0.179

0.626

0.0003

 

0.03

398

0.775

0.679

504

0.715

0.211

0.586

0.0003

 

0.04

398

0.794

0.706

424

0.715

0.239

0.562

0.0003

 

0.05

398

0.809

0.728

364

0.715

0.265

0.544

0.0004

 

0.06

398

0.822

0.747

316

0.716

0.288

0.530

0.0004

 

0.07

398

0.833

0.763

276

0.716

0.310

0.518

0.0004

 

0.08

398

0.843

0.777

244

0.716

0.328

0.505

0.0004

(0.2,0.3) or (0.7,0.8)

0.001

588

0.701

0.495

924

0.672

0.100

0.985

0.0005

 

0.01

588

0.746

0.633

922

0.711

0.149

0.727

0.0002

 

0.02

588

0.787

0.695

666

0.714

0.188

0.644

0.0002

 

0.03

588

0.812

0.731

524

0.715

0.218

0.601

0.0003

 

0.04

588

0.829

0.756

432

0.715

0.245

0.573

0.0003

 

0.05

588

0.842

0.775

366

0.716

0.269

0.552

0.0003

 

0.06

588

0.853

0.791

316

0.716

0.291

0.535

0.0003

 

0.07

588

0.863

0.805

276

0.716

0.311

0.520

0.0003

 

0.08

588

0.871

0.816

244

0.716

0.328

0.506

0.0003

(0.3,0.4) or (0.6,0.7)

0.001

712

0.704

0.500

1100

0.671

0.101

0.986

0.0004

 

0.01

712

0.756

0.647

1034

0.711

0.153

0.736

0.0002

 

0.02

712

0.800

0.713

714

0.714

0.193

0.654

0.0005

 

0.03

712

0.825

0.751

544

0.714

0.224

0.611

0.0002

 

0.04

712

0.843

0.776

440

0.715

0.250

0.581

0.0002

 

0.05

712

0.856

0.795

368

0.715

0.273

0.559

0.0002

 

0.06

712

0.867

0.810

314

0.716

0.294

0.540

0.0002

 

0.07

712

0.875

0.823

274

0.716

0.312

0.523

0.0002

 

0.08

712

0.882

0.833

244

0.716

0.329

0.506

0.0003

(0.4,0.5) or (0.5,0.6)

0.001

776

0.706

0.502

1190

0.670

0.101

0.986

0.0004

 

0.01

776

0.760

0.652

1096

0.711

0.154

0.740

0.0002

 

0.02

776

0.804

0.720

744

0.713

0.195

0.659

0.0002

 

0.03

776

0.831

0.758

558

0.714

0.226

0.615

0.0002

 

0.04

776

0.849

0.784

446

0.715

0.253

0.586

0.0002

 

0.05

776

0.862

0.803

368

0.715

0.275

0.562

0.0002

 

0.06

776

0.872

0.818

314

0.716

0.295

0.542

0.0002

 

0.07

776

0.881

0.830

274

0.716

0.313

0.524

0.0002

 

0.08

776

0.887

0.840

244

0.716

0.329

0.506

0.0002