Table 3 Impact of Auxiliary Relationship Under Conditions 1, 2, and 3.

A: No Auxiliary

X1 (X2 = 0)

(True effect = 1.0)

Full

0.998

0.211

-0.002

0.043

0.403

95.2 (93.9,96.5)

CC

0.999

0.323

-0.001

0.106

1.000

95.3 (94.0,96.6)

MI

1.106

0.321

0.106

0.101

0.951

95.4 (94.1,96.7)

X1 (X2 = 1)

(True effect = 2.5)

Full

2.542

0.378

0.042

0.161

0.031

95.2 (94.0,96.6)

CC

2.890

11.419

0.390

5.271

1.000

95.7 (94.4,97.0)

MI

2.330

0.615

-0.170

0.317

0.060

92.8 (91.2,94.4)

Interaction (True effect = 1.5)

Full

1.544

0.434

0.044

0.195

0.037

96.0 (94.8,97.2)

CC

1.891

11.500

0.391

5.321

1.000

96.4 (95.2,97.6)

MI

1.223

0.690

-0.277

0.382

0.072

94.6 (93.2,96.0)

B: Moderate

Auxiliary

X1 (X2 = 0)

(True effect = 1.0)

Full

0.986

0.211

-0.014

0.043

0.410

96.1 (94.9,97.3)

CC

0.984

0.323

-0.016

0.105

1.000

95.7 (94.4,97.0)

MI

1.157

0.312

0.157

0.110

1.039

94.9 (93.5,96.3)

X1 (X2 = 1) (True effect = 2.5)

Full

2.53

0.376

0.03

0.149

0.066

95.9 (94.7,97.1)

CC

2.698

4.867

0.198

2.28

1.000

96.1 (94.9,97.3)

MI

2.438

0.600

-0.062

0.251

0.110

96.3 (95.1,97.5)

Interaction (True effect = 1.5)

Full

1.544

0.432

0.044

0.195

0.080

95.7 (94.4,97.0)

CC

1.714

4.949

0.214

2.437

1.000

96.8 (95.7,97.9)

MI

1.281

0.671

-0.219

0.317

0.130

96.7 (95.6,97.8)

C: Strong Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

0.997

0.211

-0.003

0.046

0.403

95.4 (94.1,96.7)

CC

0.999

0.324

-0.001

0.113

1.000

94.4 (93.0,95.8)

MI

1.027

0.223

0.027

0.051

0.447

95.3 (94.0,96.6)

X1 (X2 = 1)

True effect = 2.5)

Full

2.546

0.378

0.046

0.158

0.027

96 (94.8,97.2)

CC

2.968

12.65

0.468

5.941

1.000

96.9 (95.8,98.0)

MI

2.555

0.407

0.055

0.176

0.030

96.5 (95.0,97.4)

Interaction (True effect = 1.5)

Full

1.549

0.434

0.049

0.200

0.034

96.1 (94.9,97.3)

CC

1.969

12.730

0.469

5.971

1.000

96.6 (95.5,97.7)

MI

1.528

0.462

0.028

0.205

0.034

96.2 (95.0,97.4)

b. Condition 2

Scenario

Variable

Method

Mean β

Mean SE

Mean Bias

MSE

RelMSE

Coverage

D: No Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

1.008

0.108

0.008

0.011

0.496

95.6 (94.3,96.9)

CC

1.013

0.146

0.013

0.023

1.000

93.8 (92.3,95.3)

MI

1.090

0.145

0.090

0.029

1.254

92.0 (90.3,93.7)

X1 (X2 = 1) (True effect = 2.5)

Full

2.551

0.280

0.051

0.083

0.869

95.6 (94.3,96.9)

CC

2.550

0.299

0.050

0.096

1.000

95.4 (94.1,96.7)

MI

2.357

0.293

-0.143

0.085

0.888

93.1 (91.5,94.7)

Interaction (True effect = 1.5)

Full

1.543

0.300

0.043

0.093

0.804

94.9 (93.5,96.3)

CC

1.537

0.333

0.037

0.116

1.000

95.5 (94.2,96.8)

MI

1.267

0.325

-0.233

0.126

1.084

90.2 (88.4,92.0)

E: Moderate

Auxiliary

X1 (X2 = 0)

(True effect = 1.0)

Full

1.011

0.108

0.010

0.012

0.533

95.4 (94.1,96.7)

CC

1.009

0.146

0.009

0.023

1.000

94.5 (93.1,95.9)

MI

1.163

0.142

0.163

0.046

2.039

81.4 (79.0,83.8)

X1 (X2 = 1)

(True effect = 2.5)

Full

2.558

0.280

0.058

0.090

0.911

94.2 (92.8,95.6)

CC

2.555

0.298

0.055

0.099

1.000

94.5 (93.5,96.3)

MI

2.571

0.296

0.071

0.085

0.855

96.7 (95.6,97.8)

Interaction

(True effect = 1.5)

Full

1.547

0.300

0.047

0.103

0.845

94.9 (93.5,96.3)

CC

1.545

0.333

0.045

0.122

1.000

94.7 (93.3,96.1)

MI

1.408

0.328

-0.092

0.102

0.832

95.1 (93.8,96.4)

F: Strong

Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

1.002

0.108

0.002

0.012

0.518

94.5 (93.1,95.9)

CC

1.004

0.146

0.004

0.024

1.000

94.2 (92.8,95.6)

MI

1.053

0.114

0.053

0.016

0.693

93.0 (91.4,94.6)

X1 (X2 = 1) (True effect = 2.5)

Full

2.530

0.278

0.030

0.079

0.860

95.2 (93.9,96.5)

CC

2.530

0.297

0.030

0.092

1.000

95.0 (93.6,96.4)

MI

2.573

0.280

0.073

0.084

0.908

95.7 (93.8,96.4)

Interaction (True effect = 1.5)

Full

1.527

0.298

0.027

0.092

0.788

95.0 (93.6,96.4)

CC

1.526

0.331

0.026

0.117

1.000

94.9 (93.5,96.3)

MI

1.520

0.303

0.020

0.092

0.791

95.1 (93.8,96.4)

c. Condition 3

Scenario

Variable

Method

Mean β

Mean SE

Mean Bias

MSE

RelMSE

Coverage

G: No Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

1.006

0.108

0.006

0.011

0.066

96.2 (95.0,97.4)

CC

0.608

0.119

-0.392

0.168

1.000

10.2 (8.3,12.1)

MI

0.680

0.119

-0.320

0.116

0.691

23.9 (21.3,26.5)

X1 (X2 = 1)

(True effect = 2.5)

Full

2.549

0.279

0.049

0.085

0.693

94.6 (93.2,96.0)

CC

2.322

0.288

-0.178

0.122

1.000

85.3 (83.1,87.5)

MI

1.974

0.274

-0.526

0.324

2.654

49.9 (46.8,53.0)

Interaction (True effect = 1.5)

Full

1.543

0.299

0.043

0.097

0.640

95.1 (93.8,96.4)

CC

1.714

0.312

0.214

0.152

1.000

92.3 (90.6,94.0)

MI

1.294

0.297

-0.206

0.093

0.609

93.5 (92.0,95.0)

H: Moderate

Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

1.003

0.108

0.004

0.012

0.073

94.5 (93.1,95.9)

CC

0.610

0.119

-0.390

0.167

1.000

11.9 (9.9,13.9)

MI

0.793

0.116

-0.207

0.057

0.340

54.4 (51.3,57.5)

X1 (X2 = 1) (True effect = 2.5)

Full

2.535

0.277

0.035

0.085

0.657

93.5 (92.0,95.0)

CC

2.309

0.287

-0.191

0.129

1.000

85.1 (82.9,87.3)

MI

2.222

0.279

-0.278

0.141

1.093

80.9 (78.5,83.3)

Interaction (True effect = 1.5)

Full

1.531

0.298

0.031

0.096

0.658

94.3 (92.9,95.7)

CC

1.699

0.310

0.198

0.146

1.000

91.4 (89.7,93.1)

MI

1.428

0.301

-0.072

0.074

0.508

95.4 (94.1,96.7)

I: Strong Auxiliary

X1 (X2 = 0) (True effect = 1.0)

Full

0.999

0.108

-0.001

0.012

0.069

95.4 (94.1,96.7)

CC

0.603

0.118

-0.397

0.172

1.000

11.1 (9.2,13.0)

MI

0.986

0.109

-0.014

0.012

0.072

94.8 (93.4,96.2)

X1 (X2 = 1) (True effect = 2.5)

Full

2.533

0.277

0.033

0.081

0.635

95.0 (93.6,96.4)

CC

2.303

0.287

-0.197

0.128

1.000

84.2 (81.9,86.5)

MI

2.533

0.278

0.033

0.081

0.628

95.2 (93.9,96.5)

Interaction (True effect = 1.5)

Full

1.533

0.298

0.033

0.091

0.638

95.3 (94.0,96.6)

CC

1.700

0.310

0.200

0.142

1.000

92.8 (91.2,94.4)

MI

1.547

0.299

0.047

0.091

0.640

95.8 (94.6,97.0)