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Table 5 Sample scenarios assuming beta priors p(π 1) and p(π 2) where \(\tau _{1}^{2}=0.001\). 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

 

(\(\tau _{1}^{2}=0.001\))

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)

0.01

10

0.344

0.706

14

0.640

0.728

1

0.0740

 

0.02

10

0.280

0.639

16

0.615

0.682

1

0.0559

 

0.03

10

0.239

0.586

20

0.655

0.640

0.999

0.0416

 

0.04

10

0.208

0.539

24

0.659

0.601

0.997

0.0322

 

0.05

10

0.184

0.497

30

0.671

0.564

0.991

0.0244

 

0.06

10

0.163

0.459

38

0.687

0.528

0.979

0.0187

 

0.07

10

0.144

0.424

50

0.693

0.494

0.955

0.0137

 

0.08

10

0.127

0.389

70

0.703

0.459

0.913

0.0096

(0.1,0.8)

0.01

14

0.439

0.748

16

0.586

0.656

1

0.0735

 

0.02

14

0.389

0.697

20

0.623

0.624

1

0.0390

 

0.03

14

0.353

0.652

24

0.651

0.592

0.999

0.0298

 

0.04

14

0.324

0.611

28

0.667

0.561

0.997

0.0245

 

0.05

14

0.299

0.574

34

0.674

0.533

0.990

0.0187

 

0.06

14

0.278

0.540

44

0.687

0.505

0.975

0.0136

 

0.07

14

0.259

0.509

56

0.695

0.480

0.950

0.0104

 

0.08

14

0.241

0.480

72

0.703

0.455

0.910

0.0080

(0.1,0.7)

0.01

20

0.507

0.773

22

0.603

0.572

1

0.0481

 

0.02

20

0.466

0.732

26

0.623

0.552

1

0.0262

 

0.03

20

0.438

0.695

30

0.643

0.532

0.999

0.0205

 

0.04

20

0.416

0.662

36

0.667

0.513

0.995

0.0157

 

0.05

20

0.397

0.633

42

0.680

0.494

0.986

0.0129

 

0.06

20

0.382

0.607

52

0.691

0.478

0.969

0.0097

 

0.07

20

0.369

0.585

62

0.696

0.464

0.943

0.0078

 

0.08

20

0.358

0.566

76

0.705

0.450

0.906

0.0062

(0.2,0.8)

0.01

20

0.430

0.734

24

0.601

0.561

1

0.0428

 

0.02

20

0.382

0.676

30

0.641

0.528

0.999

0.0259

 

0.03

20

0.349

0.628

38

0.666

0.499

0.995

0.0176

 

0.04

20

0.324

0.589

48

0.681

0.473

0.982

0.0127

 

0.05

20

0.305

0.557

62

0.693

0.452

0.960

0.0092

 

0.06

20

0.289

0.530

76

0.699

0.434

0.927

0.0073

 

0.07

20

0.276

0.508

96

0.706

0.418

0.881

0.0057

 

0.08

20

0.264

0.488

118

0.710

0.403

0.821

0.0046

(0.1,0.6)

0.01

28

0.511

0.770

32

0.604

0.483

1

0.0231

 

0.02

28

0.490

0.737

36

0.631

0.474

1

0.0175

 

0.03

28

0.477

0.708

40

0.651

0.465

0.998

0.0145

 

0.04

28

0.467

0.685

46

0.667

0.457

0.991

0.0111

 

0.05

28

0.461

0.667

54

0.685

0.452

0.978

0.0086

 

0.06

28

0.458

0.653

62

0.691

0.448

0.959

0.0069

 

0.07

28

0.457

0.643

70

0.698

0.446

0.934

0.0057

 

0.08

28

0.458

0.636

78

0.704

0.446

0.903

0.0049

(0.2,0.7)

0.01

30

0.479

0.750

36

0.608

0.477

1

0.0214

 

0.02

30

0.447

0.702

44

0.649

0.458

0.999

0.0144

 

0.03

30

0.426

0.664

54

0.671

0.441

0.991

0.0102

 

0.04

30

0.411

0.636

66

0.686

0.427

0.974

0.0076

 

0.05

30

0.401

0.615

78

0.695

0.417

0.947

0.0061

 

0.06

30

0.395

0.599

92

0.702

0.410

0.912

0.0050

 

0.07

30

0.391

0.587

106

0.706

0.404

0.868

0.0042

 

0.08

30

0.388

0.578

122

0.711

0.400

0.816

0.0035

(0.1,0.5)

0.01

40

0.490

0.751

48

0.614

0.392

1

0.0155

 

0.02

40

0.493

0.725

54

0.643

0.392

0.999

0.0107

 

0.03

40

0.498

0.708

60

0.666

0.395

0.993

0.0084

 

0.04

40

0.504

0.697

66

0.678

0.400

0.981

0.0067

 

0.05

40

0.514

0.692

72

0.689

0.407

0.965

0.0055

 

0.06

40

0.525

0.692

76

0.695

0.417

0.945

0.0047

 

0.07

40

0.538

0.694

80

0.701

0.429

0.923

0.0041

 

0.08

40

0.552

0.699

80

0.705

0.442

0.899

0.0038

(0.2,0.6)

0.01

46

0.487

0.742

56

0.624

0.388

1

0.0137

 

0.02

46

0.475

0.704

68

0.659

0.380

0.996

0.0083

 

0.03

46

0.471

0.679

82

0.678

0.376

0.981

0.0057

 

0.04

46

0.472

0.665

94

0.691

0.376

0.956

0.0046

 

0.05

46

0.477

0.658

106

0.698

0.379

0.925

0.0037

 

0.06

46

0.484

0.655

114

0.704

0.384

0.890

0.0032

 

0.07

46

0.493

0.655

120

0.707

0.390

0.851

0.0029

 

0.08

46

0.502

0.658

126

0.711

0.396

0.810

0.0026

(0.3,0.7)

0.01

48

0.465

0.724

62

0.630

0.379

1

0.0118

 

0.02

48

0.440

0.673

82

0.670

0.363

0.990

0.0068

 

0.03

48

0.429

0.643

104

0.689

0.354

0.963

0.0046

 

0.04

48

0.426

0.626

124

0.698

0.349

0.924

0.0036

 

0.05

48

0.425

0.616

144

0.705

0.347

0.879

0.0029

 

0.06

48

0.427

0.610

158

0.708

0.347

0.829

0.0026

 

0.07

48

0.431

0.607

172

0.711

0.348

0.776

0.0023

 

0.08

48

0.435

0.605

182

0.713

0.349

0.718

0.0021

(0.1,0.4)

0.01

64

0.497

0.735

82

0.635

0.300

0.999

0.0077

 

0.02

64

0.519

0.720

92

0.667

0.312

0.992

0.0053

 

0.03

64

0.540

0.717

98

0.680

0.328

0.977

0.0041

 

0.04

64

0.563

0.722

102

0.691

0.346

0.960

0.0034

 

0.05

64

0.585

0.730

100

0.696

0.366

0.942

0.0031

 

0.06

64

0.608

0.741

96

0.700

0.388

0.925

0.0029

 

0.07

64

0.632

0.754

90

0.703

0.412

0.909

0.0027

 

0.08

64

0.656

0.768

84

0.706

0.437

0.895

0.0025

(0.2,0.5)

0.01

78

0.496

0.726

104

0.644

0.297

0.998

0.0057

 

0.02

78

0.505

0.701

126

0.677

0.302

0.982

0.0036

 

0.03

78

0.521

0.695

140

0.691

0.313

0.954

0.0027

 

0.04

78

0.540

0.698

148

0.699

0.327

0.922

0.0023

 

0.05

78

0.559

0.705

148

0.704

0.342

0.890

0.0021

 

0.06

78

0.579

0.715

144

0.707

0.358

0.860

0.0019

 

0.07

78

0.599

0.726

138

0.709

0.375

0.831

0.0018

 

0.08

78

0.620

0.738

130

0.711

0.393

0.804

0.0018

(0.3,0.6)

0.01

84

0.474

0.708

120

0.652

0.291

0.997

0.0049

 

0.02

84

0.477

0.677

156

0.684

0.290

0.970

0.0029

 

0.03

84

0.489

0.668

180

0.697

0.297

0.928

0.0022

 

0.04

84

0.504

0.669

194

0.704

0.305

0.882

0.0018

 

0.05

84

0.520

0.674

198

0.707

0.315

0.838

0.0016

 

0.06

84

0.536

0.681

198

0.710

0.325

0.795

0.0015

 

0.07

84

0.551

0.689

194

0.712

0.336

0.753

0.0015

 

0.08

84

0.566

0.698

188

0.713

0.346

0.712

0.0014

(0.1,0.3)

0.01

124

0.525

0.718

182

0.671

0.214

0.987

0.0025

 

0.02

124

0.574

0.730

186

0.688

0.242

0.960

0.0018

 

0.03

124

0.614

0.749

174

0.695

0.271

0.937

0.0016

 

0.04

124

0.650

0.768

156

0.699

0.301

0.920

0.0016

 

0.05

124

0.681

0.787

138

0.702

0.332

0.908

0.0015

 

0.06

124

0.710

0.805

120

0.704

0.363

0.899

0.0015

 

0.07

124

0.738

0.822

102

0.704

0.397

0.894

0.0015

 

0.08

124

0.765

0.840

86

0.705

0.433

0.891

0.0016

(0.2,0.4)

0.01

164

0.524

0.711

254

0.678

0.210

0.979

0.0017

 

0.02

164

0.565

0.718

272

0.695

0.233

0.934

0.0012

 

0.03

164

0.602

0.735

260

0.702

0.258

0.895

0.0010

 

0.04

164

0.635

0.753

236

0.706

0.283

0.864

0.0010

 

0.05

164

0.665

0.771

208

0.708

0.309

0.841

0.0010

 

0.06

164

0.692

0.789

182

0.710

0.335

0.822

0.0010

 

0.07

164

0.718

0.805

156

0.710

0.362

0.809

0.0010

 

0.08

164

0.742

0.822

134

0.711

0.389

0.798

0.0010

(0.3,0.5)

0.01

186

0.509

0.697

314

0.683

0.205

0.971

0.0014

 

0.02

186

0.545

0.701

348

0.699

0.224

0.909

0.0010

 

0.03

186

0.580

0.717

336

0.705

0.244

0.857

0.0008

 

0.04

186

0.610

0.734

310

0.708

0.265

0.815

0.0008

 

0.05

186

0.638

0.751

280

0.711

0.285

0.780

0.0008

 

0.06

186

0.663

0.767

250

0.711

0.304

0.751

0.0008

 

0.07

186

0.686

0.782

222

0.713

0.324

0.726

0.0008

 

0.08

186

0.707

0.796

194

0.714

0.343

0.704

0.0008

(0.4,0.6)

0.01

194

0.495

0.684

354

0.686

0.200

0.962

0.0012

 

0.02

194

0.526

0.685

406

0.702

0.214

0.885

0.0008

 

0.03

194

0.556

0.698

402

0.707

0.229

0.820

0.0007

 

0.04

194

0.582

0.712

382

0.710

0.244

0.767

0.0007

 

0.05

194

0.605

0.726

354

0.712

0.258

0.722

0.0007

 

0.06

194

0.626

0.739

326

0.713

0.272

0.682

0.0007

 

0.07

194

0.645

0.751

300

0.714

0.284

0.646

0.0007

 

0.08

194

0.662

0.763

274

0.715

0.296

0.610

0.0007

(0.1,0.2)

0.01

398

0.626

0.752

562

0.701

0.145

0.897

0.0005

 

0.02

398

0.703

0.799

402

0.705

0.190

0.870

0.0005

 

0.03

398

0.751

0.830

302

0.706

0.230

0.861

0.0005

 

0.04

398

0.786

0.853

234

0.706

0.268

0.861

0.0005

 

0.05

398

0.814

0.872

184

0.707

0.305

0.864

0.0005

 

0.06

398

0.837

0.888

146

0.707

0.343

0.870

0.0005

 

0.07

398

0.859

0.903

116

0.707

0.384

0.878

0.0005

 

0.08

398

0.879

0.917

88

0.706

0.429

0.887

0.0006

(0.2,0.3)

0.01

588

0.623

0.747

874

0.704

0.140

0.864

0.0003

 

0.02

588

0.697

0.792

630

0.708

0.180

0.813

0.0003

 

0.03

588

0.743

0.823

470

0.710

0.216

0.791

0.0003

 

0.04

588

0.778

0.846

362

0.711

0.250

0.782

0.0003

 

0.05

588

0.806

0.865

284

0.711

0.283

0.779

0.0003

 

0.06

588

0.829

0.881

224

0.711

0.316

0.780

0.0003

 

0.07

588

0.849

0.895

176

0.711

0.350

0.785

0.0003

 

0.08

588

0.868

0.909

138

0.711

0.386

0.793

0.0004

(0.3,0.4)

0.01

712

0.614

0.739

1126

0.706

0.136

0.841

0.0002

 

0.02

712

0.684

0.783

830

0.710

0.171

0.774

0.0002

 

0.03

712

0.730

0.813

628

0.711

0.202

0.740

0.0002

 

0.04

712

0.764

0.836

488

0.712

0.231

0.720

0.0002

 

0.05

712

0.791

0.854

388

0.713

0.259

0.708

0.0002

 

0.06

712

0.814

0.870

310

0.713

0.286

0.701

0.0003

 

0.07

712

0.834

0.884

250

0.713

0.313

0.697

0.0003

 

0.08

712

0.852

0.896

200

0.714

0.341

0.697

0.0003

(0.4,0.5)

0.01

776

0.605

0.732

1298

0.707

0.132

0.823

0.0002

 

0.02

776

0.672

0.773

984

0.711

0.163

0.742

0.0002

 

0.03

776

0.715

0.802

762

0.712

0.189

0.698

0.0002

 

0.04

776

0.747

0.824

608

0.713

0.213

0.668

0.0002

 

0.05

776

0.773

0.841

494

0.714

0.234

0.646

0.0002

 

0.06

776

0.794

0.856

406

0.714

0.255

0.629

0.0002

 

0.07

776

0.813

0.869

338

0.715

0.275

0.614

0.0002

 

0.08

776

0.829

0.880

282

0.715

0.294

0.602

0.0002