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Table 4 Simulation study results: bias, variance, mean square error and related root mean square error in the \(\hat{\beta}\) regression coefficient for CCA and the IPPW method (18 response models), for three MNAR response mechanism scenarios

From: Inverse probability weighting to handle attrition in cohort studies: some guidance and a call for caution

Scenarioa

γx

γy

 

CCA

 

IPPW method

Response models

(X), Z1, Z2, Z3, Z4

(X), Z1, Z2, Z3, Z4, Z5, Z6, Z7

(X), Z1, Z3

(X), Z1, Z2, Z3, Z5, Z7

(X), Z1

(X), Z5

(X), Z1, Z5

(X), Z1, Z5, Z7

(X), Z5, Z7

MNAR 1

0.0

0.2

Bias

0.00

Xb

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Variance (10-3)

1.676

X

1.701

1.707

1.694

1.698

1.688

1.685

1.689

1.690

1.686

1.697

1.703

1.689

1.695

1.682

1.677

1.684

1.685

1.679

MSE (10-3)

1.688

X

1.711

1.715

1.703

1.706

1.700

1.698

1.702

1.702

1.697

1.709

1.714

1.702

1.706

1.695

1.690

1.697

1.698

1.691

RRMSE (%)

16.4

X

16.5

16.6

16.5

16.5

16.5

16.5

16.5

16.5

16.5

16.5

16.6

16.5

16.5

16.5

16.4

16.5

16.5

16.5

MNAR 2

0.2

0.2

Bias

−0.01

Xb

−0.01

−0.01

−0.01

−0.01

−0.01

−0.01

−0.01

−0.01

−0.01

− 0.01

− 0.01

− 0.01

− 0.01

− 0.01

− 0.01

− 0.01

− 0.01

− 0.01

Variance (10-3)

1.668

X

1.746

1.751

1.737

1.740

1.728

1.727

1.730

1.730

1.728

1.695

1.703

1.686

1.694

1.678

1.671

1.681

1.682

1.672

MSE (10-3)

1.881

X

1.907

1.897

1.896

1.885

1.942

1.940

1.943

1.925

1.920

1.904

1.907

1.900

1.900

1.892

1.884

1.895

1.896

1.885

RRMSE (%)

17.3

X

17.5

17.4

17.4

17.4

17.6

17.6

17.6

17.5

17.5

17.5

17.5

17.4

17.4

17.4

17.4

17.4

17.4

17.4

MNAR 3

0.5

0.2

Bias

−0.03

Xb

−0.03

−0.02

−0.03

−0.02

−0.03

−0.03

−0.03

−0.03

− 0.03

−0.03

− 0.03

− 0.03

− 0.03

−0.03

− 0.03

−0.03

− 0.03

−0.03

Variance (10-3)

1.756

X

2.044

2.050

2.028

2.032

2.013

2.013

2.014

2.013

2.013

1.793

1.806

1.780

1.794

1.772

1.760

1.777

1.778

1.761

MSE (10-3)

2.572

X

2.708

2.650

2.685

2.630

2.885

2.882

2.885

2.811

2.806

2.611

2.613

2.605

2.607

2.596

2.578

2.604

2.604

2.579

RRMSE (%)

20.3

X

20.8

20.6

20.7

20.5

21.5

21.5

21.5

21.2

21.2

20.4

20.4

20.4

20.4

20.4

20.3

20.4

20.4

20.3

  1. Abbreviations: CCA Complete case analysis, IPPW Inverse probability of participation weighting, MSE Mean square error, RRMSE Relative root mean square error
  2. aScenarios MNAR 1: response depending on outcome and covariates; Scenario MNAR 2, 3: response depending on outcome, exposure, and covariates
  3. bWhether for the bias or the variance, the first line represents the result obtained when the exposure variable X is included in the response model, whereas the second line represents the result obtained without the exposure variable X in the response model