01.12.2006  Research article  Ausgabe 1/2006 Open Access
Planning a cluster randomized trial with unequal cluster sizes: practical issues involving continuous outcomes
 Zeitschrift:
 BMC Medical Research Methodology > Ausgabe 1/2006
Electronic supplementary material
Competing interests
Authors' contributions
Background
Methods and results
Theoretical background
The mixed effects model
Sample size calculations
Impact of cluster size inequality
Simulation study
Cluster size
Analysis
Results
Simulation parameters
^{1}

Type of imbalance

Bias

Mean Square Error

Empirical type I error
^{2}

Empirical power
^{2}



Intraclass correlation coefficient (
ρ)

Number of clusters in each arm (
g)

Total number of subjects in each arm (
N)


0.005

5

485

None

0.0020

0.0062

0.0328

0.7756

Moderate

0.0015

0.0062

0.0300

0.7800


Poisson

0.0003

0.0061

0.0368

0.7814


Pareto

0.0002

0.0100

0.0664

0.6432


0.005

10

326

None

0.0005

0.0070

0.0326

0.7868

Moderate

0.0009

0.0073

0.0402

0.7838


Poisson

0.0010

0.0070

0.0356

0.7884


Pareto

0.0043

0.0100

0.0566

0.6968


0.005

20

282

None

0.0010

0.0072

0.0320

0.7942

Moderate

0.0014

0.0075

0.0398

0.7878


Poisson

0.0010

0.0076

0.0408

0.7802


Pareto

0.0000

0.0090

0.0486

0.7258


0.005

40

265

None

0.0006

0.0078

0.0444

0.7848

Moderate

0.0011

0.0082

0.0458

0.7936


Poisson

0.0017

0.0082

0.0484

0.7772


Pareto

0.0000

0.0086

0.0466

0.7572


0.020

10

629

None

0.0017

0.0070

0.0448

0.8012

Moderate

0.0017

0.0073

0.0544

0.7974


Poisson

0.0009

0.0074

0.0510

0.7992


Pareto

0.0022

0.0118

0.0904

0.6236


0.020

20

353

None

0.0004

0.0073

0.0452

0.8000

Moderate

0.0006

0.0074

0.0408

0.7980


Poisson

0.0007

0.0075

0.0458

0.7968


Pareto

0.0009

0.0115

0.0660

0.6546


0.020

40

290

None

0.0017

0.0080

0.0518

0.7932

Moderate

0.0001

0.0077

0.0466

0.7944


Poisson

0.0003

0.0077

0.0466

0.7912


Pareto

0.0003

0.0101

0.0556

0.7008


0.050

20

743

None

0.0007

0.0075

0.0436

0.7916

Moderate

0.0018

0.0077

0.0540

0.8026


Poisson

0.0003

0.0078

0.0536

0.7950


Pareto

0.0022

0.0115

0.0562

0.6256


0.050

40

361

None

0.0012

0.0080

0.0528

0.7944

Moderate

0.0031

0.0080

0.0510

0.7926


Poisson

0.0001

0.0080

0.0502

0.7904


Pareto

0.0023

0.0121

0.0604

0.6242


0.100

40

652

None

0.0021

0.0076

0.0504

0.7966

Moderate

0.0013

0.0078

0.0458

0.8118


Poisson

0.0022

0.0078

0.0506

0.7946


Pareto

0.0031

0.0121

0.0546

0.6006

Simulation parameters
^{1}

Type of imbalance

Bias

Mean Square Error

Empirical type I error
^{2}

Empirical power
^{2}



Intraclass correlation coefficient (
ρ)

Number of clusters in each arm (
g)

Total number of subjects in each arm (
N)

Type of imbalance


0.005

5

89

None

0.0025

0.0238

0.0190

0.7648

Moderate

0.0010

0.0243

0.0204

0.7622


Poisson

0.0014

0.0243

0.0214

0.7596


Pareto

0.0064

0.0393

0.0256

0.6250


0.005

10

73

None

0.0011

0.0288

0.0328

0.7660

Moderate

0.0015

0.0290

0.0322

0.7718


Poisson

0.0007

0.0298

0.0318

0.7662


Pareto

0.0005

0.0344

0.0352

0.7090


0.005

20

67

None

0.0011

0.0303

0.0384

0.7764

Moderate

0.0038

0.0296

0.0318

0.7700


Poisson

0.0012

0.0301

0.0382

0.7664


Pareto

0.0004

0.0323

0.0334

0.7322


0.005

40

65

None

0.0005

0.0310

0.0446

0.7986

Moderate

0.0021

0.0322

0.0478

0.7896


Poisson

0.0028

0.0305

0.0396

0.7860


Pareto

0.0007

0.0320

0.0382

0.7518


0.020

5

119

None

0.0025

0.0238

0.0190

0.7648

Moderate

0.0011

0.0248

0.0310

0.7856


Poisson

0.0021

0.0250

0.0306

0.7786


Pareto

0.0009

0.0413

0.0674

0.6262


0.020

10

81

None

0.0031

0.0273

0.0320

0.7798

Moderate

0.0006

0.0282

0.0364

0.7772


Poisson

0.0003

0.0288

0.0378

0.7778


Pareto

0.0078

0.0394

0.0550

0.6910


0.020

20

70

None

0.0026

0.0312

0.0476

0.7838

Moderate

0.0026

0.0312

0.0476

0.7838


Poisson

0.0032

0.0306

0.0436

0.7894


Pareto

0.0003

0.0362

0.0460

0.7056


0.020

40

66

None

0.0026

0.0314

0.0498

0.7878

Moderate

0.0026

0.0312

0.0476

0.7838


Poisson

0.0049

0.0326

0.0476

0.7828


Pareto

0.0007

0.0337

0.0422

0.7328


0.050

5

423

None

0.0015

0.0246

0.0482

0.7980

Moderate

0.0001

0.0240

0.0460

0.8004


Poisson

0.0005

0.0237

0.0478

0.7988


Pareto

0.0026

0.0337

0.0768

0.6808


0.050

10

103

None

0.0006

0.0280

0.0426

0.7964

Moderate

0.0012

0.0286

0.0446

0.7952


Poisson

0.0017

0.0293

0.0440

0.7754


Pareto

0.0022

0.0466

0.0770

0.6342


0.050

20

76

None

0.0027

0.0298

0.0436

0.8020

Moderate

0.0018

0.0308

0.0452

0.7784


Poisson

0.0016

0.0323

0.0526

0.7672


Pareto

0.0056

0.0396

0.0528

0.6620


0.050

40

67

None

0.0019

0.0313

0.0468

0.7880

Moderate

0.0012

0.0294

0.0516

0.7740


Poisson

0.0000

0.0335

0.0504

0.7712


Pareto

0.0022

0.0376

0.0516

0.7026


0.100

10

213

None

0.0006

0.0263

0.0426

0.8056

Moderate

0.0015

0.0289

0.0530

0.7940


Poisson

0.0007

0.0287

0.0538

0.8004


Pareto

0.0027

0.0438

0.0730

0.6394


0.100

20

89

None

0.0029

0.0303

0.0470

0.7888

Moderate

0.0020

0.0324

0.0530

0.7760


Poisson

0.0004

0.0316

0.0488

0.7744


Pareto

0.0064

0.0492

0.0674

0.6276


0.100

40

70

None

0.0038

0.0331

0.0510

0.7890

Moderate

0.0031

0.0337

0.0506

0.7738


Poisson

0.0020

0.0332

0.0456

0.7658


Pareto

0.0019

0.0433

0.0536

0.6641

Sample size adjustment for unbalanced trials
Adjusted variance inflation factors
Simulation study
Results
Robustness of sample size adjustment for unbalanced trials with misspecification of the ICC
Method
No correction

Equal weights

Cluster size weights
^{1}

Minimum variance weights



Intraclass correlation coefficient (
ρ)

Number of clusters in each arm (
g)

Sample size

Empirical probabilities

Sample size

Empirical probabilities

Sample size

Empirical probabilities

Sample size

Empirical probabilities


Type I error

Power

Type I error

Power

Type I error

Power

Type I error

Power


0.005

5

485

0.0664

0.6432

1569

0.1028

0.8606







1037

0.0948

0.7992

10

326

0.0566

0.6968

1057

0.0784

0.9386

515

0.0704

0.8106

464

0.0704

0.7806


20

282

0.0486

0.7258

917

0.0624

0.9770

336

0.0450

0.7934

331

0.0458

0.7850


40

265

0.0466

0.7572

861

0.0512

0.9918

287

0.0424

0.7842

286

0.0474

0.7706


0.02

10

629

0.0904

0.6236

2043

0.0638

0.8196







1731

0.0624

0.7968

20

353

0.0660

0.6546

1147

0.0614

0.8924

1852

0.0576

0.9432

677

0.0752

0.7976


40

290

0.0556

0.7008

942

0.0582

0.9486

435

0.0558

0.8092

401

0.0514

0.7960


0.05

20

743

0.0562

0.6256

2414

0.0564

0.8140







2165

0.0480

0.7976

40

361

0.0604

0.6242

1173

0.0500

0.8598







770

0.0550

0.8048


0.10

40

652

0.0546

0.6006

2116

0.0542

0.8090







1881

0.0500

0.8036

No correction

Equal weights

Cluster size weights
^{1}

Minimum variance weights



Intraclass correlation coefficient (
ρ)

Number of clusters in each arm (
g)

Sample size

Empirical probabilities

Sample size

Empirical probabilities

Sample size

Empirical probabilities

Sample size

Empirical probabilities


Type I error

Power

Type I error

Power

Type I error

Power

Type I error

Power


0.005

5

89

0.0256

0.6250

288

0.0536

0.9330

111

0.0270

0.7156

108

0.0324

0.6906

10

73

0.0352

0.7090

236

0.0528

0.9768

79

0.0306

0.7370

79

0.0306

0.7370


20

67

0.0334

0.7322

218

0.0470

0.9912

70

0.0390

0.7524

70

0.0390

0.7524


40

65

0.0382

0.7518

210

0.0394

0.9970

66

0.0400

0.7558

66

0.0400

0.7558


0.02

5

119

0.0674

0.6262

387

0.1072

0.8642







256

0.0954

0.7946

10

81

0.0550

0.6910

261

0.0900

0.9346

127

0.0654

0.7990

115

0.0672

0.7856


20

70

0.0460

0.7056

226

0.0684

0.9752

83

0.0492

0.7680

82

0.0482

0.7680


40

66

0.0422

0.7328

212

0.0534

0.9908

71

0.0390

0.7540

71

0.0390

0.7540


0.05

5

423

0.0768

0.6808

1375

0.0578

0.8130







1311

0.0556

0.7962

10

103

0.0770

0.6342

335

0.0824

0.8600







230

0.0920

0.7952


20

76

0.0528

0.6620

245

0.0652

0.9284

136

0.0706

0.8252

115

0.0628

0.7872


40

67

0.0516

0.7026

217

0.0590

0.9712

83

0.0578

0.7694

81

0.0488

0.7772


0.15

10

213

0.0730

0.6394

691

0.0548

0.8042







631

0.0572

0.8002

20

89

0.0674

0.6276

290

0.0644

0.8646







193

0.0638

0.7888


40

70

0.0536

0.6641

225

0.0564

0.9316

122

0.0506

0.8208

104

0.0578

0.7838

Results
Practical implications
General considerations
Adaptation of the VIF for a Pareto like imbalance
Discussion
Conclusion
Appendix A: corrected variance inflation factor (VIF) for an a prioripostulated Pareto imbalance
Number of clusters by intervention arm

Number of patients belonging to the clusters

Mean cluster size



Small clusters

0.8
g

0.2
$\stackrel{\xaf}{m}$
g

(0.2
$\stackrel{\xaf}{m}$
g)/(0.8
g) = 0.25
$\stackrel{\xaf}{m}$

Big clusters

0.2
g

0.8
$\stackrel{\xaf}{m}$
g

(0.8
$\stackrel{\xaf}{m}$
g)/(0.2
g) = 4
$\stackrel{\xaf}{m}$

Appendix B: minimum variance weightscorrected variance inflation factor (VIF) for an a prioripostulated Paretolike imbalance
Number of clusters by intervention arm

Number of patients belonging to the clusters

Mean cluster size



Small clusters

(1
γ)
g

(1 
τ)
$\stackrel{\xaf}{m}$
g

$\frac{1\tau}{1\gamma}\stackrel{\xaf}{m}$

Big clusters

γg

τ
$\stackrel{\xaf}{m}$
g

$\frac{\tau}{\gamma}\stackrel{\xaf}{m}$
