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. Author manuscript; available in PMC: 2019 Jun 1.
Published in final edited form as: Res Synth Methods. 2018 Mar 24;9(2):261–272. doi: 10.1002/jrsm.1293

Table 2.

The simulation results produced by Models M, H, M0, and U in various settings of within-study variances and correlations under different missingness scenarios. The results outside parentheses were obtained using the Bayesian method; those inside parentheses were obtained using the restricted maximum likelihood method.

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
Model Bias RMSE CP (%) Bias RMSE CP (%) Bias RMSE CP (%) Bias RMSE CP (%) Bias RMSE CP (%)
τ = 1, σ = 1, ρB = 0.5, ρW ~ U(0.6, 0.9):
No missing data (I)
M 0.00 (0.00) 0.26 (0.26) 94 (94) 0.00 (0.00) 0.26 (0.26) 92 (93) 0.00 (0.00) 0.25 (0.25) 94 (95) 0.01 (0.01) 0.26 (0.26) 93 (93) 0.01 (0.01) 0.25 (0.25) 94 (95)
H 0.00 (0.00) 0.26 (0.26) 95 (94) 0.00 (0.00) 0.26 (0.26) 95 (93) 0.00 (0.00) 0.25 (0.25) 96 (95) 0.01 (0.01) 0.26 (0.26) 95 (93) 0.01 (0.01) 0.25 (0.25) 96 (94)
M0 0.00 (0.00) 0.26 (0.26) 96 (95) 0.00 (0.00) 0.26 (0.26) 96 (94) 0.00 (0.00) 0.25 (0.25) 97 (96) 0.01 (0.01) 0.26 (0.26) 96 (95) 0.01 (0.01) 0.25 (0.25) 97 (96)
U 0.00 (0.00) 0.26 (0.26) 95 (94) 0.00 (0.00) 0.26 (0.26) 94 (94) 0.00 (0.00) 0.25 (0.25) 96 (95) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.01 (0.01) 0.25 (0.25) 96 (95)
Factors 1, 3, and 5 in 10 studies were missing completely at random (II)
M 0.00 (0.00) 0.29 (0.30) 95 (93) 0.00 (0.00) 0.26 (0.26) 93 (94) 0.01 (0.00) 0.28 (0.30) 95 (93) 0.01 (0.01) 0.26 (0.26) 94 (93) 0.01 (0.01) 0.28 (0.29) 94 (93)
H 0.00 (0.00) 0.29 (0.31) 96 (92) 0.00 (0.00) 0.26 (0.26) 94 (93) 0.01 (0.00) 0.29 (0.30) 96 (92) 0.01 (0.01) 0.26 (0.26) 95 (93) 0.00 (0.01) 0.28 (0.30) 96 (92)
M0 0.00 (0.00) 0.29 (0.30) 97 (96) 0.00 (0.00) 0.26 (0.26) 95 (94) 0.01 (0.00) 0.28 (0.29) 98 (95) 0.01 (0.01) 0.26 (0.26) 96 (94) 0.01 (0.01) 0.28 (0.29) 97 (96)
U 0.00 (−0.01) 0.32 (0.32) 95 (94) 0.00 (0.00) 0.26 (0.26) 94 (94) 0.02 (−0.01) 0.31 (0.32) 96 (94) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.01 (0.01) 0.31 (0.32) 95 (93)
Factors 1, 3, and 5 in 10 studies were missing not at random (III)
M 0.48 (0.46) 0.56 (0.54) 54 (51) 0.00 (0.00) 0.26 (0.26) 94 (94) 0.48 (0.46) 0.55 (0.54) 52 (48) 0.01 (0.01) 0.26 (0.26) 94 (94) 0.48 (0.47) 0.56 (0.55) 51 (47)
H 0.51 (0.49) 0.58 (0.57) 54 (43) 0.00 (0.00) 0.26 (0.26) 96 (94) 0.52 (0.48) 0.59 (0.56) 49 (41) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.53 (0.50) 0.60 (0.57) 48 (41)
M0 0.58 (0.56) 0.64 (0.62) 53 (44) 0.00 (0.00) 0.26 (0.26) 95 (94) 0.61 (0.55) 0.67 (0.62) 46 (41) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.62 (0.56) 0.67 (0.63) 46 (41)
U 0.75 (0.75) 0.80 (0.80) 19 (16) 0.00 (0.00) 0.26 (0.26) 94 (94) 0.75 (0.75) 0.80 (0.80) 18 (14) 0.01 (0.01) 0.26 (0.26) 94 (94) 0.75 (0.75) 0.80 (0.80) 17 (13)
τ = 1, σ = 1, ρB = 0.5, ρW ~ U(0, 0.3):
Factors 1, 3, and 5 in 10 studies were missing not at random (III)
M 0.69 (0.68) 0.74 (0.74) 26 (23) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.70 (0.68) 0.75 (0.73) 23 (22) 0.02 (0.02) 0.26 (0.26) 94 (93) 0.70 (0.68) 0.75 (0.74) 25 (22)
H 0.68 (0.67) 0.74 (0.73) 30 (21) 0.01 (0.01) 0.26 (0.26) 96 (94) 0.69 (0.67) 0.74 (0.73) 27 (21) 0.02 (0.02) 0.26 (0.26) 95 (93) 0.69 (0.67) 0.74 (0.73) 30 (21)
M0 0.71 (0.69) 0.76 (0.75) 26 (22) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.72 (0.70) 0.77 (0.75) 24 (21) 0.02 (0.02) 0.26 (0.26) 94 (93) 0.72 (0.70) 0.77 (0.75) 25 (22)
U 0.75 (0.75) 0.80 (0.80) 19 (16) 0.01 (0.01) 0.26 (0.26) 95 (94) 0.75 (0.75) 0.80 (0.80) 16 (12) 0.02 (0.02) 0.26 (0.26) 94 (93) 0.75 (0.75) 0.80 (0.80) 18 (14)
τ = 1, σ = 0.5, ρB = 0.5, ρW ~ U(0.6, 0.9):
Factors 1, 3, and 5 in 10 studies were missing not at random (III)
M 0.47 (0.44) 0.52 (0.50) 36 (31) 0.00 (0.00) 0.21 (0.21) 95 (93) 0.46 (0.43) 0.51 (0.49) 35 (30) 0.01 (0.01) 0.21 (0.21) 95 (93) 0.47 (0.45) 0.52 (0.50) 33 (28)
H 0.46 (0.45) 0.51 (0.50) 43 (30) 0.00 (0.00) 0.21 (0.21) 96 (93) 0.46 (0.44) 0.51 (0.49) 39 (29) 0.01 (0.01) 0.21 (0.21) 96 (93) 0.47 (0.45) 0.52 (0.50) 37 (26)
M0 0.45 (0.45) 0.51 (0.51) 39 (31) 0.00 (0.00) 0.21 (0.21) 95 (93) 0.46 (0.44) 0.51 (0.49) 37 (30) 0.01 (0.01) 0.21 (0.21) 95 (93) 0.47 (0.46) 0.52 (0.51) 35 (27)
U 0.60 (0.59) 0.64 (0.64) 15 (13) 0.00 (0.00) 0.21 (0.21) 95 (93) 0.59 (0.59) 0.63 (0.63) 14 (12) 0.01 (0.01) 0.21 (0.21) 94 (93) 0.60 (0.60) 0.64 (0.64) 14 (12)
τ = 1, σ = 2, ρB = 0.5, ρW ~ U(0.6, 0.9):
Factors 1, 3, and 5 in 10 studies were missing not at random (III)
M 0.57 (0.58) 0.73 (0.74) 74 (70) 0.01 (0.01) 0.41 (0.42) 96 (96) 0.59 (0.59) 0.74 (0.74) 73 (70) 0.02 (0.02) 0.41 (0.41) 95 (96) 0.59 (0.59) 0.74 (0.74) 73 (70)
H 0.63 (0.60) 0.78 (0.76) 74 (66) 0.01 (0.01) 0.42 (0.42) 97 (96) 0.65 (0.61) 0.80 (0.76) 72 (65) 0.02 (0.02) 0.41 (0.41) 97 (97) 0.67 (0.62) 0.81 (0.77) 71 (64)
M0 0.96 (0.90) 1.05 (1.00) 62 (56) 0.01 (0.01) 0.42 (0.42) 97 (96) 1.01 (0.90) 1.10 (1.00) 56 (55) 0.02 (0.02) 0.41 (0.41) 97 (97) 1.02 (0.91) 1.11 (1.01) 56 (54)
U 1.18 (1.18) 1.26 (1.26) 29 (25) 0.01 (0.01) 0.42 (0.42) 96 (94) 1.19 (1.19) 1.27 (1.27) 28 (24) 0.02 (0.02) 0.41 (0.41) 96 (94) 1.19 (1.19) 1.27 (1.27) 28 (24)
Factors 1, 3, and 5 in 25 studies were missing not at random (III′)
M 1.57 (1.60) 1.72 (1.76) 76 (32) 0.01 (−0.01) 0.42 (0.41) 95 (95) 1.58 (1.60) 1.73 (1.77) 75 (31) 0.02 (0.02) 0.41 (0.42) 96 (94) 1.60 (1.59) 1.74 (1.76) 75 (34)
H 2.36 (1.46) 2.75 (1.64) 65 (26) 0.01 (−0.02) 0.42 (0.43) 96 (93) 2.45 (1.47) 2.73 (1.67) 65 (27) 0.02 (0.01) 0.41 (0.43) 96 (93) 2.50 (1.48) 2.79 (1.66) 67 (28)
M0 3.05 (2.92) 3.11 (2.99) 73 (4) 0.01 (−0.01) 0.42 (0.41) 96 (96) 3.14 (2.95) 3.20 (3.02) 62 (3) 0.02 (0.02) 0.41 (0.42) 97 (96) 3.11 (2.93) 3.17 (3.00) 60 (3)
U 3.21 (3.19) 3.28 (3.26) 11 (1) 0.01 (−0.01) 0.42 (0.41) 95 (94) 3.23 (3.21) 3.30 (3.27) 9 (1) 0.02 (0.02) 0.41 (0.42) 96 (94) 3.21 (3.18) 3.28 (3.24) 9 (1)

RMSE: root mean square error; CP: 95% CI/CrI coverage probability.