Table 1 Formulas used in the meta-analysis adapted from Borenstein et al. (2009)

Standardized Mean

Difference¹

\(\mathbf{S}\mathbf{M}\mathbf{D}=\frac{\left({{\varvec{M}}}_{\mathbf{T}\mathbf{p}\mathbf{o}\mathbf{s}\mathbf{t}}-\boldsymbol{ }{{\varvec{M}}}_{\mathbf{C}\mathbf{p}\mathbf{o}\mathbf{s}\mathbf{t}}\right)}{{{\varvec{S}}{\varvec{D}}}_{\mathbf{p}\mathbf{o}\mathbf{o}\mathbf{l}\mathbf{e}\mathbf{d}}}\times \left(1-\frac{3}{4\times {\varvec{d}}{\varvec{f}}-1}\right)\)

Pooled Standard Deviation¹

\({{\varvec{S}}{\varvec{D}}}_{\mathbf{p}\mathbf{o}\mathbf{o}\mathbf{l}\mathbf{e}\mathbf{d}}=\sqrt{\frac{\left({{\varvec{n}}}_{\mathbf{T}}-1\right){{\varvec{S}}{\varvec{D}}}_{\mathbf{T}\mathbf{p}\mathbf{o}\mathbf{s}\mathbf{t}}^{2}+\left({{\varvec{n}}}_{\mathbf{C}}-1\right){{\varvec{S}}{\varvec{D}}}_{\mathbf{C}\mathbf{p}\mathbf{o}\mathbf{s}\mathbf{t}}^{2}}{{{\varvec{n}}}_{\mathbf{T}}+{{\varvec{n}}}_{\mathbf{C}}-2}}\)

Variance of SMD

\({{\varvec{V}}}_{\mathbf{S}\mathbf{M}\mathbf{D}}={\left(1-\frac{3}{4\times {\varvec{d}}{\varvec{f}}-1}\right)}^{2}\boldsymbol{ }\times \boldsymbol{ }\left(\frac{{{\varvec{n}}}_{\mathbf{T}}+\boldsymbol{ }{{\varvec{n}}}_{{\varvec{C}}}}{{{\varvec{n}}}_{\mathbf{T}}{{\varvec{n}}}_{\mathbf{C}}}+\frac{\mathbf{S}\mathbf{M}{\mathbf{D}}^{2}}{2\boldsymbol{ }\times \left({{\varvec{n}}}_{\mathbf{T}}+\boldsymbol{ }{{\varvec{n}}}_{{\varvec{C}}}\right)}\right)\)

Standard Error of SMD

\({{\varvec{S}}{\varvec{E}}}_{\mathbf{S}\mathbf{M}\mathbf{D}}=\boldsymbol{ }\sqrt{{{\varvec{V}}}_{\mathbf{S}\mathbf{M}\mathbf{D}}}\)