| |
Instrument strength
|
---|
| |
Weak
|
Mod
|
Strong
|
---|
Level
|
Method
|
rB
|
sd
|
rMSE
|
pval
|
rB
|
sd
|
rMSE
|
pval
|
rB
|
sd
|
rMSE
|
pval
|
---|
30000
|
High
|
Tr
|
0.19
|
0.12
|
0.12
|
0.05
|
0.21
|
0.10
|
0.10
|
0.06
|
0.14
|
0.08
|
0.08
|
0.04
|
|
Conv
|
29.28
|
0.12
|
0.89
|
1.00
|
27.53
|
0.10
|
0.83
|
1.00
|
25.10
|
0.08
|
0.76
|
1.00
|
|
2SRI
|
− 11.44
|
0.86
|
0.92
|
0.06
|
− 3.44
|
0.73
|
0.74
|
0.04
|
1.92
|
0.58
|
0.58
|
0.04
|
|
GMM
|
29.37
|
0.23
|
0.91
|
0.38
|
28.39
|
0.19
|
0.87
|
0.77
|
26.69
|
0.13
|
0.81
|
0.93
|
Med
|
Tr
|
0.23
|
0.13
|
0.13
|
0.06
|
0.12
|
0.10
|
0.10
|
0.05
|
0.19
|
0.09
|
0.09
|
0.05
|
|
Conv
|
15.95
|
0.13
|
0.50
|
1.00
|
14.90
|
0.11
|
0.46
|
1.00
|
13.10
|
0.10
|
0.40
|
1.00
|
|
2SRI
|
− 9.75
|
0.97
|
1.01
|
0.07
|
− 6.35
|
0.80
|
0.82
|
0.05
|
− 3.40
|
0.66
|
0.66
|
0.05
|
|
GMM
|
15.35
|
0.18
|
0.49
|
0.29
|
15.72
|
0.17
|
0.50
|
0.56
|
15.28
|
0.15
|
0.48
|
0.89
|
Low
|
Tr
|
0.10
|
0.14
|
0.14
|
0.06
|
− 0.13
|
0.11
|
0.11
|
0.04
|
0.18
|
0.10
|
0.10
|
0.04
|
|
Conv
|
4.92
|
0.15
|
0.21
|
0.72
|
4.39
|
0.12
|
0.18
|
0.67
|
3.95
|
0.12
|
0.17
|
0.56
|
|
2SRI
|
− 6.32
|
1.04
|
1.06
|
0.06
|
− 5.49
|
0.86
|
0.87
|
0.04
|
− 4.23
|
0.73
|
0.75
|
0.04
|
|
GMM
|
3.80
|
0.14
|
0.18
|
0.30
|
4.15
|
0.14
|
0.19
|
0.39
|
4.61
|
0.12
|
0.18
|
0.84
|
20000
|
High
|
Tr
|
0.27
|
0.14
|
0.14
|
0.04
|
0.05
|
0.12
|
0.12
|
0.05
|
0.30
|
0.10
|
0.10
|
0.06
|
|
Conv
|
29.53
|
0.14
|
0.90
|
1.00
|
27.51
|
0.11
|
0.83
|
1.00
|
25.21
|
0.11
|
0.76
|
1.00
|
|
2SRI
|
− 11.03
|
1.05
|
1.10
|
0.05
|
−5.51
|
0.91
|
0.92
|
0.04
|
1.34
|
0.70
|
0.70
|
0.04
|
|
GMM
|
29.01
|
0.22
|
0.90
|
0.32
|
28.03
|
0.17
|
0.86
|
0.72
|
27.28
|
0.19
|
0.84
|
0.92
|
Med
|
Tr
|
0.09
|
0.16
|
0.16
|
0.06
|
0.22
|
0.13
|
0.13
|
0.06
|
0.19
|
0.11
|
0.11
|
0.07
|
|
Conv
|
15.94
|
0.16
|
0.50
|
0.99
|
15.06
|
0.14
|
0.47
|
1.00
|
13.31
|
0.13
|
0.42
|
0.99
|
|
2SRI
|
− 10.08
|
1.19
|
1.23
|
0.06
|
−6.69
|
1.00
|
1.02
|
0.05
|
− 3.67
|
0.79
|
0.80
|
0.04
|
|
GMM
|
15.01
|
0.18
|
0.48
|
0.25
|
15.64
|
0.19
|
0.50
|
0.53
|
15.40
|
0.17
|
0.49
|
0.85
|
Low
|
Tr
|
− 0.24
|
0.12
|
0.12
|
0.06
|
0.11
|
0.14
|
0.14
|
0.06
|
−
|
−
|
−
|
−
|
|
Conv
|
3.58
|
0.15
|
0.19
|
0.42
|
4.71
|
0.16
|
0.21
|
0.61
|
−
|
−
|
−
|
−
|
|
2SRI
|
0.04
|
0.88
|
0.89
|
0.04
|
−5.92
|
1.09
|
1.10
|
0.04
|
−
|
−
|
−
|
−
|
|
GMM
|
4.41
|
0.16
|
0.22
|
0.79
|
4.15
|
0.16
|
0.20
|
0.39
|
−
|
−
|
−
|
−
|
10000
|
High
|
Tr
|
0.03
|
0.21
|
0.21
|
0.05
|
0.49
|
0.16
|
0.16
|
0.05
|
0.43
|
0.14
|
0.14
|
0.06
|
|
Conv
|
29.69
|
0.20
|
0.91
|
1.00
|
28.22
|
0.16
|
0.86
|
1.00
|
25.70
|
0.15
|
0.79
|
1.00
|
|
2SRI
|
− 14.47
|
1.64
|
1.70
|
0.07
|
−2.71
|
1.27
|
1.27
|
0.04
|
1.19
|
1.04
|
1.04
|
0.05
|
|
GMM
|
28.86
|
0.25
|
0.90
|
0.30
|
28.61
|
0.24
|
0.89
|
0.68
|
27.45
|
0.22
|
0.85
|
0.90
|
Med
|
Tr
|
−
|
−
|
−
|
−
|
0.58
|
0.18
|
0.18
|
0.04
|
0.48
|
0.15
|
0.16
|
0.05
|
|
Conv
|
−
|
−
|
−
|
−
|
15.83
|
0.19
|
0.51
|
0.97
|
13.78
|
0.17
|
0.45
|
0.93
|
|
2SRI
|
−
|
−
|
−
|
−
|
−7.42
|
1.41
|
1.43
|
0.05
|
−3.57
|
1.13
|
1.13
|
0.03
|
|
GMM
|
−
|
−
|
−
|
−
|
15.74
|
0.23
|
0.52
|
0.50
|
15.66
|
0.21
|
0.51
|
0.80
|
Low
|
Tr
|
−
|
−
|
−
|
−
|
0.08
|
0.20
|
0.20
|
0.05
|
−0.36
|
0.17
|
0.17
|
0.07
|
|
Conv
|
−
|
−
|
−
|
−
|
5.27
|
0.22
|
0.27
|
0.55
|
3.88
|
0.20
|
0.23
|
0.37
|
|
2SRI
|
−
|
−
|
−
|
−
|
−6.12
|
1.61
|
1.62
|
0.05
|
−4.37
|
1.31
|
1.31
|
0.04
|
|
GMM
|
−
|
−
|
−
|
−
|
3.77
|
0.24
|
0.27
|
0.36
|
3.85
|
0.21
|
0.24
|
0.74
|
- Legend: Tr = True model, Conv = Conventional model, 2SRI = Two-Stage Residual Inclusion, GMM = Generalized Method of Moment. Low, Med (Medium), High denote the level of confounding whereas Weak, Mod (Moderate), Strong stand for instrument strength. For the criteria, rB = relative bias (%), sd = standard deviation, rMSE = root Mean Squares Error and pval = non-coverage probabilities. The numbers 10000, 20000 and 30000 stand for different sample sizes