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 5 simulation results for meta-analysis of proportions

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
Complete data – we set 25 true positives and 25 true negatives in each study, with the pooled logit-sensitivity and pooled logit-specificity set at 1.386 (i.e. a sensitivity and specificity of 0.8). The between-study correlation was set as -0.8, and the between-study variation was set as 1 for both sensitivity and specificity. Simulations were performed as in Chu and Cole [12].
Normal URMA	10	995	-0.152	0.311	0.120	93.1%	-0.174	0.310	0.119	93.2%	-0.313 (17)	-0.309 (5)	-	-	-
Normal BRMA	10	995	-0.124	0.318	0.118	93.8%	-0.144	0.317	0.115	94.5%	-0.245 (7)	-0.245 (2)	0.067	53.7%	0%
Generalised URMA	10	1000	-0.013	0.330	0.124	93.8%	-0.037	0.328	0.116	94.0%	-0.186 (20)	-0.189 (21)	-	-	-
Generalised URMA	10	603*	-0.012	0.339	0.119	94.9%	-0.030	0.342	0.118	95.8%	-0.136 (0)	-0.114 (0)	-	-	-
Generalised BRMA	10	603	-0.009	0.339	0.119	95.4%	-0.029	0.341	0.116	96.2%	-0.134 (0)	-0.113 (0)	-0.084	0%	0%
Normal URMA	50	1000	-0.175	0.141	0.049	77.2%	-0.182	0.141	0.050	75.5%	-0.337 (0)	-0.335 (0)	-	-	-
Normal BRMA	50	1000	-0.151	0.142	0.042	81.4%	-0.157	0.143	0.043	81.3%	-0.285 (0)	-0.282 (0)	0.087	17.0%	0%
Generalised URMA	50	1000	-0.018	0.157	0.024	95.1%	-0.026	0.157	0.023	96.1%	-0.091 (0)	-0.084 (0)	-	-	-
Generalised URMA	50	973*	-0.019	0.157	0.024	95.1%	-0.022	0.157	0.023	96.1%	-0.087 (0)	-0.080 (0)	-	-	-
Generalised BRMA	50	973	-0.016	0.157	0.024	96.0%	-0.020	0.158	0.023	96.2%	-0.078 (0)	-0.071 (0)	0.019	0%	0%

MSE = mean-square-error, n = number of studies in each meta-analysis, CIs = confidence intervals, s.e. = standard error;
Restricted maximum likelihood estimation was used for the normal models, whereas maximum likelihood estimation was used for the generalised models.
* These were a subset of the 1000 URMA simulations that converged and were the same ones that converged in the equivalent BRMA; this helps fairly compare the URMA and BRMA results.

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ISSN: 1471-2288

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