Table 2.

Performance of approximations of $\mathrm{SE}\left({\widehat{\beta }}_{m-a}\right)$ $\mathit{SE}\left(\widehat{{\beta }_{m-a}}\right)$ of meta-analytic mean coefficient over 51 cities ($\widehat{\beta }$ =2.19%)

Estimator (text expression number)	Estimated $\mathrm{SE}\left({\widehat{\beta }}_{m-a}\right)$ (%)	% error w.r.t. actual FE	% error w.r.t. actual RE
Fixed effect model
Actual $S{E}_{F-E}\left({\widehat{\beta }}_{m-a}\right)$	0.054	0.0	− 46.7
Poisson (4)	0.050	−7.6	−50.8
Q-Poisson (5)	0.051	−6.2	−50.0
Random effects model
Actual $S{E}_{R-E}\left({\widehat{\beta }}_{m-a}\right)$	0.102	87.6	0.0
τ and $S{E}_{\mathit{\text{Poiss}}}^{\ast }\left({\widehat{\beta }}_j\right)$ known (6)	0.101	NA	−1.4
I2 and $S{E}_{\mathit{\text{Poiss}},F-E}^{\ast }\left({\widehat{\beta }}_{m-a}\right)$ known (7)	0.085	NA	−16.4
extreme heterogeneity assumed (8)	0.076	NA	−25.3

RE Random effects, FE Fixed effects, w.r.t. With respect to

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)]