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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 β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 1 Mean s.e. of β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 1 MSE of β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 1 Coverage of the 95% CIs for β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 1 Bias of mean β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 2 Mean s.e. of β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 2 MSE of β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 2 Coverage of the 95% CIs for β ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqcaaaa@2E64@ 2 Bias of mean τ ^ 1 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFepaDgaqcamaaDaaaleaacqaIXaqmaeaacqaIYaGmaaaaaa@3097@ (no. of τ ^ 1 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFepaDgaqcamaaDaaaleaacqaIXaqmaeaacqaIYaGmaaaaaa@3097@ = 0) Bias of mean τ ^ 2 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFepaDgaqcamaaDaaaleaacqaIYaGmaeaacqaIYaGmaaaaaa@3099@ (no. of τ ^ 2 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFepaDgaqcamaaDaaaleaacqaIYaGmaeaacqaIYaGmaaaaaa@3099@ = 0) Bias of mean ρ ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFbpGCgaqcaaaa@2E83@ B % of ρ ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFbpGCgaqcaaaa@2E83@ B = -1 % of ρ ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFbpGCgaqcaaaa@2E83@ 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%
  1. MSE = mean-square-error, n = number of studies in each meta-analysis, CIs = confidence intervals, s.e. = standard error;