Background
Methods
Data Generation
Parameters | Values |
---|---|
IPD-Meta-analyses generated: | M = 1000 |
(Number of studies, number of subjects per study, total average sample sizes)b: | (K, n
i
, N) ∈ {(5,100,500), (15,33,500), (15,200,3000), (5,357,500), (15,98,500), (15,588,3000)} |
Fixed effects (intercepts): |
β
0 = − 0.85 |
Prevalence of the outcome |
π = 30% |
Fixed effects (Slopes): |
β
1 = 0.18 |
Random effects distribution: | Normal |
Random effects variances: | {τ
0
2
, τ
1
2
} ∈ (0.05, 1, 4) |
Correlation between random effects: |
ρ ∈ (0,0.5) |
Models
Two-stage IPD methods
One-stage IPD methods
Estimation Procedures and Approximations
-
Two-stage approach (Models 1 and 2)
-
One-stage approach via GLMMs (Models 3 and 4) estimated with PQL.
-
One-stage approach via GLMMs (Models 3 and 4) estimated with AGHQ.
Assessment criteria
Results
Data generation | |||||
---|---|---|---|---|---|
Performance measuresc
| Random-study and treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | |||
Two-staged
| One-stage | Two-stage | One-stage | ||
(τ
0
2
, τ
1
2
) = (4, 4) e
| AB (β
1) | 0.04 (0.02 0.06) | 0.04 (0.02, 0.07) | 0.04 (0.02, 0.06) | 0.04 (0.01, 0.07) |
RMSE (β
1) |
1.11 (0.49, 1.94)
| 1.19 (0.53, 2.12) |
1.18 (0.59, 1.96)
| 1.23 (0.61, 2.14) | |
Coverage (β
1) | 89.3 |
91.8
| 92 |
92.6
| |
AB (τ
1
2
) | 0.23 (0.14,0.30) |
0.16 (0.08, 0.24)
|
0.15 (0.08,0.24)
| 0.24 (0.20, 0.27) | |
RMSE (τ
1
2
) | 7.26 (4.39,7.51) |
4.93 (2.56, 7.51)
|
4.81 (2.38,7.42)
| 7.47 (6.28, 8.64) | |
Coverage (τ
1
2
) f
| NA | NA | NA | NA | |
Convergence |
100
| 97.7 |
100
| 99.8 | |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (β
1) | 0.02 (0.01, 0.04) | 0.02 (0.01, 0.04) | 0.03 (0.01, 0.04) | 0.03 (0.01, 0.04) |
RMSE (β
1) |
0.73 (0.35, 1.29)
| 0.75 (0.37, 1.33) | 0.80 (0.37, 1.30) |
0.79 (0.39, 1.34)
| |
Coverage (β
1) | 89.1 |
90.6
| 91.1 |
91.6
| |
AB (τ
1
2
) | 0.06 (0.03, 0.08) |
0.04 (0.02, 0.1)
| 0.05 (0.03, 0.08) |
0.03 (0.01, 0.07)
| |
RMSE (τ
1
2
) | 1.73 (0.85, 2.65) |
1.22 (0.53, 3.16)
| 1.59 (0.80, 2.46) |
1.06 (0.46, 2.06)
| |
Coverage (τ
1
2
) | NA | NA | NA | NA | |
Convergence |
100
| 90.4 | 100 | 100 |
Data generation | |||||
---|---|---|---|---|---|
Performance measuresc
| Random-study and treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | |||
Two-staged
| One-stage | Two-stage | One-stage | ||
(τ
0
2
, τ
1
2
) = (4, 4) e
| AB (β
1) | 0.03 (0.02 0.06) | 0.03 (0.02, 0.06) | 0.04 (0.01, 0.06) | 0.04 (0.01, 0.06) |
RMSE (β
1) |
1.02 (0.50, 1.85)
| 1.07 (0.49, 1.84) | 1.15 (0.57, 1.93) |
1.11 (0.58, 1.87)
| |
Coverage (β
1) | 91.9 |
92.3
| 92.4 |
93.6
| |
AB (τ
1
2
) | 0.14 (0.07,0.22) |
0.12 (0.06, 0.20)
|
0.11 (0.05,0.17)
| 0.22 (0.20, 0.25) | |
RMSE (τ
1
2
) | 4.36 (2.22,6.80) |
3.87 (1.81, 6.21)
|
3.40 (1.70,5.47)
| 6.99 (6.20, 7.80) | |
Coverage (τ
1
2
) f
| NA | NA | NA | NA | |
Convergence |
100
| 98.3 |
100
| 89.9 | |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (β
1) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) |
RMSE (β
1) | 0.61 (0.30, 1.04) |
0.59 (0.29, 1.05)
|
0.61 (0.30, 1.02)
| 0.63 (0.30, 1.03) | |
Coverage (β
1) | 91.2 |
91.9
| 93 |
93.3
| |
AB (τ
1
2
) | 0.03 (0.02, 0.06) | 0.03 (0.02, 0.05) | 0.03 (0.02, 0.05) |
0.02 (0.01, 0.03)
| |
RMSE (τ
1
2
) | 1.08 (0.53, 1.73) |
1.03 (0.49, 1.68)
| 1.00 (0.51, 1.68) |
0.57 (0.27, 1.00)
| |
Coverage (τ
1
2
) | NA | NA | NA | NA | |
Convergence |
100
| 96.5 |
100
| 88.8 |
Performance measuresc
| Data generation | ||||
---|---|---|---|---|---|
Random-study and treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | ||||
AGHQd
| PQLd
| AGHQ | PQL | ||
(τ
0
2
, τ
1
2
) = (4, 4)e
| AB (\( \beta \)
1) | 0.05 (0.02, 0.08) |
0.04 (0.02, 0.07)
| 0.04 (0.02, 0.07) | 0.04 (0.02, 0.07) |
RMSE (\( \beta \)
1) | 1.42 (0.64, 2.52) |
1.19 (0.53, 2.12)
| 1.35 (0.65, 2.27) |
1.23 (0.61, 2.14)
| |
Coverage (\( \beta \)
1) |
93.2
| 91.8 | 91.7 |
92.6
| |
AB (τ
1
2
) | 0.18 (0.09,0.29) |
0.16 (0.08, 0.24)
|
0.15 (0.08,0.24)
| 0.24 (0.20, 0.27) | |
RMSE (τ
1
2
) | 5.76 (2.80,9.07) |
4.93 (2.56, 7.51)
|
4.79 (2.37,7.62)
| 7.47 (6.28, 8.64) | |
Coverage (τ
1
2
) |
85.5
| 76.2 |
81.4
| 4.6 | |
Convergence |
99
| 97.7 | 96.7 |
99.8
| |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (\( \beta \)
1) | 0.03 (0.01, 0.05) |
0.02 (0.01, 0.04)
| 0.03 (0.01, 0.04) | 0.03 (0.01, 0.04) |
RMSE (\( \beta \)
1) | 0.79 (0.41, 1.42) |
0.75 (0.37, 1.33)
| 0.84 (0.42, 1.38) |
0.79 (0.39, 1.34)
| |
Coverage (\( \beta \)
1) |
92.3
| 90.6 |
93.4
| 91.6 | |
AB (τ
1
2
) | 0.06 (0.03, 0.09) |
0.04 (0.02, 0.1)
| 0.05 (0.02, 0.08) |
0.03 (0.02, 0.07)
| |
RMSE (τ
1
2
) | 1.76 (0.84, 2.70) |
1.22 (0.53, 3.16)
| 1.54 (0.72, 2.40) |
1.06 (0.46, 2.06)
| |
Coverage (τ
1
2
) | 74.5 |
81.1
| 71.6 |
77.2
| |
Convergence |
96.8
| 90.4 | 85.8 |
100
|
Performance measuresc
| Data generation | ||||
---|---|---|---|---|---|
Random-study and treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | ||||
AGHQd
| PQLd
| AGHQ | PQL | ||
(τ
0
2
, τ
1
2
) = (4, 4)e
| AB (\( \beta \)
1) | 0.04 (0.02, 0.06) |
0.03 (0.02, 0.06)
| 0.04 (0.01, 0.06) | 0.04 (0.01, 0.06) |
RMSE (\( \beta \)
1) | 1.20 (0.55, 1.99) |
1.07 (0.49, 1.84)
| 1.16 (0.58, 1.95) |
1.11 (0.58, 1.87)
| |
Coverage (\( \beta \)
1) | 92.2 |
92.3
| 92.1 |
93.6
| |
AB (τ
1
2
) | 0.13 (0.07,0.21) |
0.12 (0.06, 0.20)
|
0.11 (0.05,0.18)
| 0.22 (0.20, 0.25) | |
RMSE (τ
1
2
) | 4.12 (2.06,6.77) |
3.87 (1.81, 6.21)
|
3.42 (1.71,5.77)
| 6.99 (6.20, 7.80) | |
Coverage (τ
1
2
) |
81.9
| 78.9 |
84.7
| 1.0 | |
Convergence |
100
| 98.3 |
100
| 89.9 | |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (\( \beta \)
1) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) |
RMSE (\( \beta \)
1) | 0.60 (0.30, 1.06) |
0.59 (0.29, 1.05)
| 0.63 (0.30, 1.05) | 0.63 (0.30, 1.03) | |
Coverage (\( \beta \)
1) | 91.7 |
91.9
| 92.4 |
93.3
| |
AB (τ
1
2
) | 0.04 (0.02, 0.06) |
0.03 (0.02, 0.05)
| 0.03 (0.02, 0.05) |
0.02 (0.01, 0.03)
| |
RMSE (τ
1
2
) | 1.09 (0.52, 1.75) |
1.03 (0.49, 1.68)
| 1.01 (0.49, 1.69) |
0.57 (0.27, 1.00)
| |
Coverage (τ
1
2
) |
83.6
| 82.5 |
82.5
| 76.5 | |
Convergence |
99.5
| 96.5 |
99.1
| 88.8 |
Performance measuresd
| Data generation | ||||
---|---|---|---|---|---|
Random-study and -treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | ||||
Stratified-intercept | Random-intercept | Stratified-intercept | Random-intercept | ||
(τ
0
2
, τ
1
2
) = (4, 4)e
| AB (\( \beta \)
1) | 0.04 (0.02, 0.08) | 0.04 (0.02, 0.07) | 0.05 (0.02, 0.07) |
0.04 (0.01, 0.07)
|
RMSE (\( \beta \)
1) | 1.24 (0.49, 2.44) |
1.19 (0.53, 2.12)
| 1.43 (0.70, 2.32) |
1.23 (0.61, 2.14)
| |
Coverage (\( \beta \)
1) |
99.1
| 91.8 |
97.4
| 92.6 | |
AB (τ
1
2
) | 0.16 (0.07, 0.25) | 0.16 (0.08, 0.24) |
0.15 (0.07,0.24)
| 0.24 (0.20, 0.27) | |
RMSE (τ
1
2
) | 5.01 (2.35,7.95) |
4.93 (2.56, 7.51)
|
4..75 (2.23,7.64)
| 7.47 (6.28, 8.64) | |
Coverage (τ
1
2
) | 11.6 |
76.2
|
28.4
| 4.6 | |
Convergence | 13.8 |
97.7
| 32.3 |
99.8
| |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (\( \beta \)
1) | 0.03 (0.01, 0.04) |
0.02 (0.01, 0.04)
| 0.03 (0.01, 0.05) | 0.03 (0.01, 0.04) |
RMSE (\( \beta \)
1) | 0.83 (0.41, 1.38) |
0.75 (0.37, 1.33)
| 0.90 (0.42, 1.47) |
0.79 (0.39, 1.34)
| |
Coverage (\( \beta \)
1) |
96.4
| 90.6 |
94
| 91.6 | |
AB (τ
1
2
) | 0.05 (0.03, 0.09) |
0.04 (0.02, 0.1)
| 0.05 (0.02, 0.08) |
0.03 (0.01, 0.07)
| |
RMSE (τ
1
2
) | 1.72 (0.85, 2.78) |
1.22 (0.53, 3.16)
| 1.55 (0.75, 1.61) |
1.06 (0.46, 2.06)
| |
Coverage (τ
1
2
) | 37.3 |
81.1
| 54.4 |
77.2
| |
Convergence | 42.6 |
90.4
| 62.3 |
100
|
Performance measuresd
| Data generation | ||||
---|---|---|---|---|---|
Random-study and -treatment effect (Eq. 1) | Stratified-study effect (Eq. 2) | ||||
Stratified-intercept | Random-intercept | Stratified-intercept | Random-intercept | ||
(τ
0
2
, τ
1
2
) = (4, 4)e
| AB (\( \beta \)
1) | 0.04 (0.02, 0.06) |
0.03 (0.02, 0.06)
| 0.04 (0.02, 0.06) | 0.04 (0.01, 0.06) |
RMSE (\( \beta \)
1) | 1.11 (0.55, 1.94) |
1.07 (0.49, 1.84)
| 1.15 (0.58, 1.98) |
1.11 (0.58, 1.87)
| |
Coverage (\( \beta \)
1) |
95.2
| 92.3 | 92.3 |
93.6
| |
AB (τ
1
2
) | 0.13 (0.06, 0.20) |
0.12 (0.06, 0.20)
|
0.11 (0.05,0.18)
| 0.22 (0.20, 0.25) | |
RMSE (τ
1
2
) | 4.05 (1.85,6.25) |
3.87 (1.81, 6.21)
|
3.39 (1.57,5.56)
| 6.99 (6.20, 7.80) | |
Coverage (τ
1
2
) | 53.7 |
78.9
|
89.8
| 1.0 | |
Convergence | 63.8 |
98.3
|
99.7
| 89.9 | |
(τ
0
2
, τ
1
2
) = (1, 1) | AB (\( \beta \)
1) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) | 0.02 (0.01, 0.03) |
RMSE (\( \beta \)
1) | 0.63 (0.29, 1.07) |
0.59 (0.29, 1.05)
| 0.63 (0.30, 1.05) | 0.63 (0.30, 1.03) | |
Coverage (\( \beta \)
1) | 91.8 |
91.9
| 93.1 |
93.3
| |
AB (τ
1
2
) | 0.03 (0.02, 0.06) | 0.03 (0.02, 0.05) | 0.03 (0.02, 0.05) |
0.02 (0.01, 0.03)
| |
RMSE (τ
1
2
) | 1.06 (0.52, 1.74) |
1.03 (0.49, 1.68)
| 0.98 (0.48, 1.69) |
0.57 (0.27, 1.00)
| |
Coverage (τ
1
2
) |
86.3
| 82.5 |
87.9
| 76.5 | |
Convergence | 95.3 |
96.5
|
99.2
| 88.8 |