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Table 2 Selected list of effect size measures and their sampling variances, belonging to three types: (1) single-group effect, (2) comparative effect and (3) association effect

From: Quantitative evidence synthesis: a practical guide on meta-analysis, meta-regression, and publication bias tests for environmental sciences

Type

Effect size

Point estimate

Sampling variance estimate

Reference

Single group

Mean

\(\overline{x}_{i}\)

\({s}_{i}^{2}/{n}_{i}\)

[134]

Single group

Proportion

\({p}_{i}=\frac{{y}_{i}}{{n}_{i}}\)

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

[134]

Single group

Log standard deviation (lnSD)

\({\text{ln}}{s}_{i}\)

\(\frac{1}{2\left({n}_{i}-1\right)}\)

[27]

Single group

Log coefficient of variation (lnCV)

\(\ln \left( {\frac{{S_{i} }}{{\overline{x}_{i} }}} \right)\)

\(\frac{{s_{i}^{2} }}{{n_{i} \overline{x}_{i}^{2} }} + \frac{1}{{2\left( {n_{i} - 1} \right)}}\)

[27]

Comparative

Mean difference (MD)

\(\overline{x}_{iT} - \overline{x}_{iC}\)

\(\frac{{s}_{iC}^{2}}{{n}_{iC}}+\frac{{s}_{iT}^{2}}{{n}_{iT}}\)

[134]

Comparative

Standardised mean difference (SMD)

\(d_{i} = \frac{{\overline{x}_{iT} - \overline{x}_{iC} }}{{\sqrt {\frac{{\left( {n_{iC} - 1} \right)s_{iC}^{2} + \left( {n_{iT} - 1} \right)s_{iT}^{2} }}{{n_{iC} + n_{iT} - 2}}} }}\)

\(\frac{1}{{n}_{iC}}+\frac{1}{{n}_{iT}}+\frac{{d}_{i}^{2}}{2\left({n}_{iC}+{n}_{iT}\right)}\)

[25]

Comparative

Risk (proportion) difference (RD)

\(\frac{{y}_{iT}}{{n}_{iT}}-\frac{{y}_{iC}}{{n}_{iC}}\)

\(\frac{{y}_{iT}\left({n}_{iT}-{y}_{iT}\right)}{{n}_{iT}^{3}}+\frac{{y}_{iC}\left({n}_{iC}-{y}_{iC}\right)}{{n}_{iC}^{3}}\)

[134]

Comparative

Log odds ratio (lnOR)

\({\text{ln}}\left(\frac{{y}_{iT}}{{n}_{iT}-{y}_{iT}}\right)-{\text{ln}}\left(\frac{{y}_{iC}}{{n}_{iC}-{y}_{iC}}\right)\)

\(\frac{1}{{y}_{iT}}+\frac{1}{{n}_{iT}-{y}_{iT}}+\frac{1}{{y}_{iC}}+\frac{1}{{n}_{iC}-{y}_{iC}}\)

[134]

Comparative

Log response ratio (lnRR)

\({\text{ln}}\left( {\frac{{\overline{x}_{iT} }}{{\overline{x}_{iC} }}} \right)\)

\(\frac{{s_{iC}^{2} }}{{n_{iC} \overline{x}_{iC}^{2} }} + \frac{{s_{iT}^{2} }}{{n_{iT} \overline{x}_{iT}^{2} }}\)

[135]

Comparative

Log variability ratio (lnVR)

\({\text{ln}}\left(\frac{{s}_{iT}}{{s}_{iC}}\right)\)

\(\frac{1}{2\left({n}_{iC}-1\right)}+\frac{1}{2\left({n}_{iT}-1\right)}\)

[27]

Comparative

Log coefficient of variation ratio (lnCVR)

\({\text{ln}}\left( {\frac{{s_{iT} }}{{\overline{x}_{iT} }}} \right) - {\text{ln}}\left( {\frac{{s_{iC} }}{{\overline{x}_{iC} }}} \right)\)

\(\frac{{s_{iC}^{2} }}{{n_{iC} \overline{x}_{iC}^{2} }} + \frac{1}{{2\left( {n_{iC} - 1} \right)}} + \frac{{s_{iT}^{2} }}{{n_{iT} \overline{x}_{iT}^{2} }} + \frac{1}{{2\left( {n_{iT} - 1} \right)}}\)

[27]

Association

Fisher’s z-transformation of correlation, r (Zr)

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

\(\frac{1}{{n}_{i}-3}\)

[134]

  1. For the column 3rd and 4th, notations represent: \(\overline{x}\) (mean), s (standard deviation), n (sampling size), y (the number of events), the subscript T (treatment group), the subscript C (control group) and the subscript i (the ith effect size or study)
  2. Note that better estimators may be found in the relevant references; for example, SMD can be best estimated by multiplying by \(\left(1-\frac{3}{4\left({n}_{iC}+{n}_{iT}-2\right)-1}\right)\), and see also [43]