Table 3 Examples of dependence between two sampling variances (v₁ and v₂) and their covariance for four common effect size statistics

Proportion

Shared measurement

\({v}_{1}=\frac{{y}_{1}\left({n}_{1}-{y}_{1}\right)}{{n}_{1}^{3}}\)

\({v}_{2}=\frac{{y}_{2}\left({n}_{2}-{y}_{2}\right)}{{n}_{2}^{3}}\)

\(\rho \sqrt{\frac{{y}_{1}\left({n}_{1}-{y}_{1}\right)}{{n}_{1}^{3}}\frac{{y}_{2}\left({n}_{2}-{y}_{2}\right)}{{n}_{2}^{3}}}\)

Zr

Shared measurement

\({v}_{1}=\frac{1}{2}{\text{ln}}\left(\frac{1+{r}_{1}}{1-{r}_{1}}\right)\)

\({v}_{2}=\frac{1}{2}{\text{ln}}\left(\frac{1+{r}_{2}}{1-{r}_{2}}\right)\)

\(\rho \sqrt{\frac{1}{4}{\text{ln}}\left(\frac{1+{r}_{1}}{1-{r}_{1}}\right){\text{ln}}\left(\frac{1+{r}_{2}}{1-{r}_{2}}\right)}\)

lnRR

Shared measurement

\(v_{1} = \frac{{s_{1C}^{2} }}{{n_{1C} \overline{x}_{1C}^{2} }} + \frac{{s_{1T}^{2} }}{{n_{1T} \overline{x}_{1T}^{2} }}\)

\(v_{2} = \frac{{s_{2C}^{2} }}{{n_{2C} \cdot \overline{x}_{2C}^{2} }} + \frac{{s_{2T}^{2} }}{{n_{2T} \cdot \overline{x}_{2T}^{2} }}\)

\(\rho \sqrt {\left( {\frac{{s_{1C}^{2} }}{{n_{1C} \overline{x}_{1C}^{2} }} + \frac{{s_{1T}^{2} }}{{n_{1T} \overline{x}_{1T}^{2} }}} \right)\left( {\frac{{s_{2C}^{2} }}{{n_{2C} \overline{x}_{2C}^{2} }} + \frac{{s_{2T}^{2} }}{{n_{2T} \overline{x}_{2T}^{2} }}} \right)}\)

Shared control

\(v_{1} = \frac{{s_{1C}^{2} }}{{n_{1C} \overline{x}_{1C}^{2} }} + \frac{{s_{1T}^{2} }}{{n_{1T} \overline{x}_{1T}^{2} }}\)

\(v_{2} = \frac{{s_{1C}^{2} }}{{n_{1C} \overline{x}_{1C}^{2} }} + \frac{{s_{2T}^{2} }}{{n_{2T} \overline{x}_{2T}^{2} }}\)

\(\frac{{s_{1C}^{2} }}{{n_{1C} \overline{x}_{1C}^{2} }}\)

SMD

Shared measurement

\({v}_{1}=\frac{1}{{n}_{1C}}+\frac{1}{{n}_{1T}}+\frac{{d}_{1}^{2}}{2\left({n}_{1C}+{n}_{1T}\right)}\)

\({v}_{2}=\frac{1}{{n}_{2C}}+\frac{1}{{n}_{2T}}+\frac{{d}_{1}^{2}}{2\left({n}_{2C}+{n}_{2T}\right)}\)

\(\rho \sqrt{\left(\frac{1}{{n}_{1C}}+\frac{1}{{n}_{1T}}+\frac{{d}_{1}^{2}}{2\left({n}_{1C}+{n}_{1T}\right)}\right)\left(\frac{1}{{n}_{2C}}+\frac{1}{{n}_{2T}}+\frac{{d}_{2}^{2}}{2\left({n}_{2C}+{n}_{2T}\right)}\right)}\)

Shared control

\({v}_{1}=\frac{1}{{n}_{1C}}+\frac{1}{{n}_{1T}}+\frac{{d}_{1}^{2}}{2\left({n}_{1C}+{n}_{1T}+{n}_{2T}\right)}\)

\({v}_{2}=\frac{1}{{n}_{1C}}+\frac{1}{{n}_{2T}}+\frac{{d}_{2}^{2}}{2\left({n}_{1C}+{n}_{1T}+{n}_{2T}\right)}\)

\(\frac{1}{{n}_{1C}}+\frac{{d}_{1}{d}_{2}}{2\left({n}_{1C}+{n}_{1T}+{n}_{2T}\right)}\)