Bivariate random-effects meta-analysis and the estimation of between-study correlation

Riley, Richard D; Abrams, Keith R; Sutton, Alexander J; Lambert, Paul C; Thompson, John R

doi:10.1186/1471-2288-7-3

BMC Medical Research Methodology

Table 4 Simulation results of the normal BRMA and URMA models for scenarios (i), (ii), (viii), and (ix)

From: Bivariate random-effects meta-analysis and the estimation of between-study correlation

Meta-analysis model	n	No. of the 1000 simulations that converged	Bias of mean $\hat{β}$ ₁	Mean s.e. of $\hat{β}$ ₁	MSE of $\hat{β}$ ₁	Coverage of the 95% CIs for $\hat{β}$ ₁	Bias of mean $\hat{β}$ ₂	Mean s.e. of $\hat{β}$ ₂	MSE of $\hat{β}$ ₂	Coverage of the 95% CIs for $\hat{β}$ ₂	Bias of mean ${\hat{τ}}_{1}^{2}$ (no. of ${\hat{τ}}_{1}^{2}$ = 0)	Bias of mean ${\hat{τ}}_{2}^{2}$ (no. of ${\hat{τ}}_{2}^{2}$ = 0)	Bias of mean $\hat{ρ}$ _B	% of $\hat{ρ}$ _B= -1	% of $\hat{ρ}$ _B= 1
Scenario (i): Complete data – zero correlation; within-study variance similar to between-study variance
URMA	50	1000	-0.005	0.102	0.010	94.8%	0.001	0.107	0.0108	95.4%	-0.003 (0)	0.005 (1)	-	-	-
BRMA	50	1000	-0.005	0.101	0.010	94.7%	0.001	0.106	0.0108	95.5%	-0.003 (0)	0.005 (0)	-0.001	0.2%	0.4%
URMA	5	1000	-0.002	0.267	0.081	96.0%	-0.006	0.267	0.0887	94.0%	-0.006 (89)	0.015 (81)	-	-	-
BRMA	5	998	-0.002	0.274	0.081	96.7%	-0.006	0.269	0.0894	95.3%	0.008 (10)	0.024 (0)	-0.027	29.6%	29.0%
Scenario (ii): Complete data – no within-study correlation, high between-study correlation; within-study variance similar to between-study variance
URMA	50	1000	-0.004	0.102	0.010	95.6%	0.001	0.106	0.0114	94.4%	0 (0)	-0.004 (1)	-	-	-
BRMA	50	1000	-0.004	0.100	0.010	95.3%	0	0.104	0.0107	95.4%	0.001 (0)	-0.001 (0)	-0.005	0%	25.2%
URMA	5	999	-0.002	0.271	0.077	97.3%	-0.005	0.263	0.0826	94.2%	0.004 (80)	0.005 (104)	-	-	-
BRMA	5	1000	-0.002	0.279	0.077	98.1%	-0.008	0.268	0.0819	95.7%	0.024 (15)	0.024 (0)	-0.161	10.3%	60.5%
Scenario (viii): Missing data – no within-study correlation, high between-study correlation; within-study variance smaller than between-study variance
URMA	50	1000	-0.004	0.071	0.005	94.9%	0	0.099	0.0101	95.0%	-0.006 (0)	-0.005 (0)	-	-	-
BRMA	50	1000	-0.004	0.071	0.005	94.9%	0	0.082	0.0068	95.2%	-0.006 (0)	-0.007 (0)	-0.001	0%	0%
URMA	10	1000	-0.002	0.154	0.028	94.1%	-0.003	0.209	0.0576	93.7%	-0.006 (0)	-0.006 (0)	-	-	-
BRMA	10	1000	-0.002	0.154	0.028	94.1%	-0.001	0.174	0.0427	93.3%	-0.006 (0)	0.006 (0)	-0.040	0%	3.9%
Scenario (ix): Missing data – no within-study correlation, high between-study correlation; within-study variance similar to between-study variance
URMA	50	1000	-0.004	0.102	0.010	95.6%	-0.001	0.145	0.0228	94.2%	0 (0)	-0.003 (6)	-	-	-
BRMA	50	1000	-0.004	0.101	0.010	95.8%	-0.003	0.137	0.0203	94.7%	0.001 (0)	0.003 (0)	-0.012	0.1%	35.6%
URMA	10	1000	-0.001	0.218	0.045	93.9%	-0.005	0.263	0.0825	94.2%	0.006 (45)	0.005 (84)	-	-	-
BRMA	10	997	-0.001	0.222	0.045	96.5%	-0.007	0.255	0.0797	95.6%	0.020 (0)	0.025 (0)	-0.164	10.1%	60.2%

MSE = mean-square-error, n = number of studies in each meta-analysis, CIs = confidence intervals, s.e. = standard error;

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