Table 4.

Logarithmic Transformation Methods.

Forward-Transformation (raw to log10)
	Mean	Standard error
	${\Theta }_i={log}_{10}({\theta }_i)-(\frac{se{({\Theta }_i)}^2}{2})$	$se({\Theta }_i)=\sqrt{{log}_{10}(\frac{se{({\theta }_i)}^2}{{{\theta }_i}^2}+1)}$
Back-Transformation (log10 to raw)
Method	Mean	Standard error
Geometric	$\widehat{\theta }=10^{\widehat{\Theta }}$	$\pm C{I}_{1-\alpha /2}(\widehat{\theta })=10^{\widehat{\Theta }\pm v_{1-\alpha /2}·se(\widehat{\Theta })}$$se(\widehat{\theta })=\frac{(+C{I}_{1-\alpha /2}(\widehat{\theta }))-(-C{I}_{1-\alpha /2}(\widehat{\theta }))}{2v_{1-\alpha /2}}$Where v1−α/2 corresponds to critical value
Naïve approximately	$\widehat{\theta }=10^{\left(\widehat{\Theta }+\frac{se{\left(\widehat{\Theta }\right)}^2}{2}\right)}$	$se\left(\widehat{\theta }\right)=\frac{1}{\sqrt{n_i}}\left({10}^{sd{(\widehat{\Theta })}^2)}-1\right){10}^{2\widehat{\Theta }+sd{(\widehat{\Theta })}^2}$
Tailor Series approximately	$\widehat{\theta }=10^{\left(\widehat{\Theta }+\frac{se{\left(\widehat{\Theta }\right)}^2}{2}\right)}$	$se\left(\widehat{\theta }\right)=\sqrt{\frac{1}{n_i}10^{\left(2\widehat{\Theta }+{sd\left(\widehat{\Theta }\right)}^2\right)}{sd\left(\widehat{\Theta }\right)}^2\left(1+\left(\frac{sd{\left(\widehat{\Theta }\right)}^2}{2}\right)\right)}$

Forward-transformation of study-level estimates θ_i to corresponding log-transformed estimates Θ_i, and back-transformation of meta-analysis outcome $\widehat{\Theta }$ to the corresponding outcome $\widehat{\theta }$ on the raw scale (Higgins et al., 2008). v_1−α/2: confidence interval critical value at significance level α.