B Effect Size Formulas


Effect Size (\(\hat\theta\)) Standard Error (SE) Function
Arithmetic Mean (3.2.1) \(\bar{x} = \dfrac{\sum^{n}_{i=1}x_i}{n}\) \(\text{SE}_{\bar{x}} = \dfrac{s}{\sqrt{n}}\) mean
Proportion (3.2.2) \(p = \dfrac{k}{n}\) \(\text{SE}_{p} = \sqrt{\dfrac{p(1-p)}{n}}\)
\(p_{\text{logit}} = \log_{e}\left(\dfrac{p}{1-p}\right)\) \(\text{SE}_{p_{\text{logit}}} = \sqrt{\dfrac{1}{np}+\dfrac{1}{n(1-p)}}\)
Correlation
Product-Moment Correlation (3.2.3.1) \(r_{xy} = \dfrac{\sigma^{2}_{xy}}{\sigma_x \sigma_y}\) \(SE_{r_{xy}} = \frac{1-r_{xy}^2}{\sqrt{n-2}}\) cor
\(z = 0.5\log_{e}\left(\dfrac{1+r}{1-r}\right)\) \(\text{SE}_{z} = \dfrac{1}{\sqrt{n-3}}\)
Point-Biserial Correlation1 (3.2.3.2) \({r_{\text{pb}}}= \dfrac{(\bar{y_1}-\bar{y_2})\sqrt{\dfrac{n_1}{N}\left(1-\dfrac{n_1}{N}\right)}}{s_y}\) cor
(Standardized) Mean Difference
Between-Group Mean Difference (3.3.1.1) \(\text{MD} = \bar{x}_1 - \bar{x}_2\) \(\text{SE}_{\text{MD}} = {s_{\text{pooled}}}^*\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2}}\)
Between-Group Standardized Mean Difference (3.3.1.2) \(\text{SMD} = \dfrac{\bar{x}_1 - \bar{x}_2}{{s_{\text{pooled}}}^*}\) \(\text{SE}_{\text{SMD}} = \sqrt{\dfrac{n_1+n_2}{n_1n_2} + \dfrac{\text{SMD}^2_{\text{between}}}{2(n_1+n_2)}}\) esc_mean_sd
Within-Group Mean Difference (3.3.1.3) \(\text{MD} = \bar{x}_{t_2} - \bar{x}_{t_1}\) \(SE_{\text{MD}}=\sqrt{\dfrac{s^2_{\text{t}_{\text{2}}}+s^2_{\text{t}_{\text{1}}}-(2r_{\text{t}_{\text{1}}\text{t}_{\text{2}}}s_{\text{t}_{\text{1}}}s_{\text{t}_{\text{2}}})}{n}}\)
Within-Group Standardized Mean Difference (3.3.1.3) \(\text{SMD}= \dfrac{\bar{x}_{t_2} - \bar{x}_{t_1}}{\left({\dfrac{\sqrt{(s^2_{t_1}+s^2_{t2})/2}}{\sqrt{2(1-r_{t_1t_2})}}}\right)}\) \(\text{SE}_{\text{SMD}} = \sqrt{\dfrac{2(1-r_{t_1t_2})}{n}+\dfrac{\text{SMD}^2_{\text{within}}}{2n}}\)
Binary Outcome Effect Size
Risk Ratio (3.3.2.1) \({p_{e}}_{\text{treat}} =\dfrac{a}{n_{\text{treat}}}\) \(\text{SE}_{\log \text{RR}} = \sqrt{\dfrac{1}{a}+\dfrac{1}{c} - \dfrac{1}{a+b} - \dfrac{1}{c+d}}\)
\({p_{e}}_{\text{control}} = \dfrac{c}{n_{\text{control}}}\)
\(\text{RR} = \dfrac{{p_{e}}_{\text{treat}}}{{p_{e}}_{\text{control}}}\)
\(\log \text{RR} = \log_{e}(\text{RR})\)
Odds Ratio (3.3.2.2) \(\text{Odds}_{\text{treat}} = \dfrac{a}{b}\) \(\text{SE}_{\log \text{OR}} = \sqrt{\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}}\) esc_2x2
\(\text{Odds}_{\text{control}} = \dfrac{c}{d}\)
\(\text{OR} = \dfrac{a/b}{c/d}\)
\(\log \text{OR} = \log_{e}(\text{OR})\)
Incidence Rate Ratio (3.3.3) \(\text{IRR} = \dfrac{ E_{\text{treat}}/T_{\text{treat}} }{E_{\text{control}}/T_{\text{control}}}\) \(\text{SE}_{\log \text{IRR}} = \sqrt{\dfrac{1}{E_{\text{treat}}}+\dfrac{1}{E_{\text{control}}}}\)
\(\log \text{IRR} = \log_{e}(\text{IRR})\)
Effect Size Correction
Small Sample Bias (3.4.1) \(g = \text{SMD} \times \left(1-\dfrac{3}{4n-9}\right)\) hedges_g
Unreliability (3.4.2) \({r_{xy}}_{c} = \dfrac{r_{xy}}{\sqrt{r_{xx}}}\) \(\text{SE}_c = \dfrac{\text{SE}}{\sqrt{r_{xx}}}\)
\({r_{xy}}_{c} = \dfrac{r_{xy}}{\sqrt{r_{xx}}\sqrt{r_{yy}}}\) \(\text{SE}_c = \dfrac{\text{SE}}{\sqrt{r_{xx}}\sqrt{r_{yy}}}\)
\(\text{SMD}_c = \dfrac{\text{SMD}}{\sqrt{r_{xx}}}\)
Range Restriction (3.4.3) \(\text{U} = \dfrac{s_{\text{unrestricted}}}{s_{\text{restricted}}}\) \(\text{SE}_{{r_{xy}}_c} = \dfrac{{r_{xy}}_c}{r_{xy}}\text{SE}_{r_{xy}}\)
\({r_{xy}}_c = \dfrac{\text{U}\times r_{xy}}{\sqrt{(\text{U}^2-1)r_{xy}^2+1}}\)
\(\text{SMD}_c = \dfrac{\text{U}\times \text{SMD}}{\sqrt{(\text{U}^2-1)\text{SMD}^2+1}}\) \(\text{SE}_{{\text{SMD}}_c} = \dfrac{{\text{SMD}}_c}{\text{SMD}}\text{SE}_{\text{SMD}}\)
1 Point-biserial correlations may be converted to SMDs for meta-analysis (see Chapter 3.2.3.2).
* The pooled standard deviation is defined as \(s_{\text{pooled}} = \sqrt{\dfrac{(n_1-1)s^2_1+(n_2-1)s^2_2}{(n_1-1)+(n_2-1)}}\).