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Table 2 Performance of approximations of \( \mathrm{SE}\left({\hat{\beta}}_{m-a}\right) \) \( SE\left(\hat{\beta_{m-a}}\right) \) of meta-analytic mean coefficient over 51 cities (\( \hat{\beta} \) =2.19%)

From: Sample size issues in time series regressions of counts on environmental exposures

Estimator (text expression number)Estimated \( \mathrm{SE}\left({\hat{\beta}}_{m-a}\right) \) (%)% error w.r.t. actual FE% error w.r.t. actual RE
Fixed effect model
 Actual \( S{E}_{F-E}\ \left({\hat{\beta}}_{m-a}\right) \)0.0540.0− 46.7
 Poisson (4)0.050−7.6−50.8
 Q-Poisson (5)0.051−6.2−50.0
Random effects model
 Actual \( S{E}_{R-E}\ \left({\hat{\beta}}_{m-a}\right) \)0.10287.60.0
 τ and \( S{E}_{Poiss}^{\ast }\ \left({\hat{\beta}}_j\right) \) known (6)0.101NA−1.4
 I2 and \( S{E}_{Poiss,F-E}^{\ast}\left({\hat{\beta}}_{m-a}\right) \) known (7)0.085NA−16.4
 extreme heterogeneity assumed (8)0.076NA−25.3
  1. RE Random effects, FE Fixed effects, w.r.t. With respect to
  2. The first column shows the standard error of the meta-analytic mean, estimated by each method. The remaining two columns show the % error in each estimate of SE(β ̂) with respect to gold standards actual FE estimate (column 2) and actual RE estimate (column 3) = 100*[approximation-(true value)]/(true value)]