Table 1 Simulation comparison of the IPW estimator ${\widehat{\beta }}_{I1}$ , the AIPW estimator ${\widehat{\beta }}_{A2}$ and the complete-case (CC) estimator ${\widehat{\beta }}_C$ under various sample sizes and selection probabilities

Model 1: π _i =.4

50

IPW

-.0415

.3561

.1839

.851

-.2175

1.6737

.3354

.864

-.1610

1.2201

.2962

.847

AIPW

-.0110

.2213

.1664

.890

-.0062

.3099

.2873

.943

-.0186

.3076

.2551

.929

CC

-.0246

.3398

.2738

.938

-.1515

1.6082

.4730

.968

-.1038

1.1709

.4187

.959

100

IPW

-.0650

.1815

.1404

.870

-.0548

.3120

.2458

.891

-.0249

.2653

.2161

.898

AIPW

-.0094

.1376

.1187

.914

-.0024

.2284

.1988

.926

.0027

.1994

.1780

.925

CC

-.0240

.1728

.1685

.948

-.0086

.3086

.2981

.960

.0031

.2556

.2581

.946

300

IPW

-.0368

.0936

.0874

.931

-.0209

.1535

.1460

.946

-.0022

.1419

.1286

.929

AIPW

-.0027

.0732

.0712

.946

-.0028

.1233

.1165

.940

.0005

.1130

.1046

.938

CC

-.0092

.0919

.0935

.960

-.0012

.1627

.1634

.952

.0040

.1438

.1432

.952

500

IPW

-.0183

.0698

.0671

.938

-.0172

.1159

.1128

.943

-.0083

.1069

.0993

.933

AIPW

.0022

.0566

.0550

.936

-.0022

.0956

.0902

.943

-.0068

.0867

.0811

.930

CC

.0006

.0704

.0716

.949

-.0059

.1268

.1255

.949

-.0046

.1135

.1103

.942

800

IPW

-.0126

.0538

.0527

.942

-.0134

.0862

.0889

.950

-.0029

.0759

.0779

.947

AIPW

.0011

.0433

.0435

.952

-.0047

.0720

.0713

.956

-.0020

.0638

.0640

.951

CC

.0002

.0562

.0565

.948

-.0051

.0974

.0990

.958

-.0013

.0844

.0869

.958

Model 2: π _i =.6

50

IPW

-.0316

.2079

.1714

.926

-.0934

.8426

.3112

.944

-.0563

.3320

.2690

.937

AIPW

-.0072

.1723

.1591

.941

-.0105

.2893

.2772

.950

-.0172

.2653

.2440

.948

CC

-.0126

.1973

.1949

.959

-.0594

.8369

.3512

.967

-.0278

.3213

.3044

.959

100

IPW

-.0366

.1399

.1259

.926

-.0420

.2363

.2192

.944

-.0100

.2103

.1911

.925

AIPW

-.0121

.1206

.1133

.941

-.0107

.2069

.1921

.944

.0078

.1764

.1700

.940

CC

-.0142

.1370

.1345

.947

-.0216

.2379

.2370

.961

.0030

.2103

.2072

.949

300

IPW

-.0138

.0742

.0728

.944

-.0194

.1267

.1250

.957

-.0049

.1064

.1096

.964

AIPW

-.0030

.0650

.0651

.948

-.0044

.1136

.1093

.949

.0005

.0960

.0974

.956

CC

-.0017

.0763

.0759

.946

-.0118

.1345

.1328

.951

-.0035

.1147

.1169

.957

500

IPW

-.0069

.0571

.0555

.945

-.0096

.0946

.0965

.947

-.0094

.0866

.0844

.953

AIPW

.0029

.0495

.0496

.942

-.0032

.0856

.0841

.947

-.0076

.0757

.0749

.955

CC

.0013

.0577

.0581

.947

-.0034

.1024

.1019

.949

-.0086

.0906

.0899

.954

800

IPW

-.0072

.0437

.0438

.954

-.0069

.0754

.0763

.956

-.0025

.0692

.0664

.947

AIPW

-.0011

.0401

.0393

.951

-.0019

.0693

.0665

.943

-.0015

.0626

.0590

.931

CC

-.0012

.0452

.0460

.958

-.0026

.0805

.0806

.952

-.0024

.0723

.0709

.952

Full data: π _i =1

50

-.0079

.1510

.1466

.952

-.0182

.2691

.2618

.948

-.0104

.2264

.2263

.957

100

-.0079

.1068

.1024

.943

-.0075

.1841

.1798

.948

-.0039

.1560

.1577

.949

300

-.0019

.0596

.0583

.950

-.0081

.1032

.1023

.936

.0001

.0934

.0898

.932

500

.0006

.0452

.0450

.951

-.0041

.0783

.0788

.950

.0014

.0656

.0693

.960

800

-.0004

.0343

.0356

.951

.0025

.0612

.0622

.938

-.0006

.0532

.0547

.955