. Author manuscript; available in PMC: 2025 Jun 30.

Published in final edited form as: Stat Med. 2022 May 16;41(18):3466–3478. doi: 10.1002/sim.9428

Table 2.

Bivariate trim and fill simulation study results with various values of between-study heterogeneity (quantified by τ² and I²), number of observed studies n and number of suppressed studies k₀. Half of the k₀ studies with the smallest y₁ + y₂ values are randomly selected to be suppressed (i.e., non-deterministically). The correlation ρ of the two outcomes y₁ and y₂ is specified as their estimated correlation. The estimator used in the T&F to calculate the estimated ${\hat{k}}_{0}$ is R₀. Listed are the bias of the estimation of the first outcome’s effect size ( ${\hat{μ}}_{1} - μ_{1}$ ), and the number of suppressed studies ( ${\hat{k}}_{0}$ ) for unadjusted multivariate meta-analysis (unadjusted), univariate T&F (tf.uni), proposed bivariate T&F along the true projection direction (tf.biv.true) and the proposed bivariate T&F with direction-searching algorithm (tf.biv). The empirical standard errors (SEs) across the 1000 replicates are also listed in the parentheses.

τ² (I²)	n	k ₀	Unadjusted	tf.uni		tf.biv.true		tf.biv
τ² (I²)	n	k ₀	${\hat{μ}}_{1} - μ_{1}$ (SE)	${\hat{μ}}_{1} - μ_{1}$ (SE)	${\hat{k}}_{0}$ (SE)	${\hat{μ}}_{1} - μ_{1}$ (SE)	${\hat{k}}_{0}$ (SE)	${\hat{μ}}_{1} - μ_{1}$ (SE)	${\hat{k}}_{0}$ (SE)
0.2 (42%)	25	0	0 (0.13)	0 (0.20)	1.64 (3.18)	−0.03 (0.15)	0.64 (1.60)	−0.03 (0.16)	2.33 (2.56)
	25	10	0.09 (0.12)	0.08 (0.17)	1.62 (3.16)	0.08 (0.14)	1.11 (2.79)	0.08 (0.21)	4.94 (5.50)
	25	20	0.12 (0.11)	0.10 (0.15)	1.53 (3.15)	0.10 (0.11)	0.47 (1.52)	0.11 (0.15)	3.33 (4.44)
	50	0	0 (0.09)	0 (0.12)	1.28 (2.22)	−0.01 (0.09)	0.49 (1.28)	0 (0.11)	3.84 (4.05)
	50	10	0.07 (0.08)	0.05 (0.11)	1.46 (2.60)	0.05 (0.10)	1.13 (2.89)	0.05 (0.12)	4.80 (6.13)
	50	20	0.10 (0.08)	0.08 (0.10)	1.33 (2.81)	0.08 (0.09)	0.75 (1.55)	0.08 (0.13)	4.04 (5.42)
0.4 (70%)	25	0	0.02 (0.16)	0.03 (0.27)	1.93 (3.57)	−0.01 (0.16)	0.51 (1.37)	0 (0.20)	2.71 (2.94)
	25	10	0.13 (0.15)	0.11 (0.25)	1.87 (3.93)	0.10 (0.15)	0.62 (2.10)	0.12 (0.27)	5.17 (6.38)
	25	20	0.16 (0.13)	0.15 (0.20)	1.63 (3.44)	0.14 (0.16)	0.47 (2.11)	0.15 (0.28)	5.40 (7.30)
	50	0	−0.01 (0.11)	0 (0.18)	1.60 (3.48)	−0.02 (0.11)	0.81 (1.89)	−0.01 (0.14)	3.31 (3.04)
	50	10	0.08 (0.01)	0.07 (0.15)	1.57 (3.28)	0.06 (0.12)	0.88 (2.89)	0.06 (0.16)	4.50 (6.54)
	50	20	0.13 (0.10)	0.11 (0.14)	1.55 (2.82)	0.12 (0.11)	0.67 (1.77)	0.12 (0.15)	5.19 (7.50)
1.0 (86%)	25	0	−0.02 (0.23)	−0.01 (0.49)	2.38 (4.32)	−0.01 (0.24)	0.43 (1.31)	−0.02 (0.33)	2.50 (2.78)
	25	10	0.19 (0.20)	0.18 (0.37)	1.93 (4.17)	0.15 (0.25)	0.82 (2.94)	0.20 (0.41)	5.53 (7.08)
	25	20	0.25 (0.18)	0.23 (0.36)	2.04 (4.48)	0.22 (0.19)	0.54 (2.38)	0.27 (0.44)	6.40 (8.76)
	50	0	0 (0.15)	0.01 (0.28)	1.80 (4.38)	−0.01 (0.16)	0.43 (1.32)	0.02 (0.20)	3.22 (3.66)
	50	10	0.13 (0.14)	0.12 (0.23)	1.67 (3.99)	0.11 (0.16)	0.86 (2.46)	0.12 (0.31)	6.05 (8.76)
	50	20	0.19 (0.14)	0.17 (0.22)	1.47 (3.39)	0.17 (0.15)	0.62 (1.80)	0.20 (0.23)	5.27 (8.97)