# B Công thức tính mức độ ảnh hưởng

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) $$p = \dfrac{k}{n}$$ $$\text{SE}_{p} = \sqrt{\dfrac{p(1-p)}{n}}$$
Proportion (3.2.2) $$p_{\text{logit}} = \log_{e}\left(\dfrac{p}{1-p}\right)$$ $$\text{SE}_{p_{\text{logit}}} = \sqrt{\dfrac{1}{np}+\dfrac{1}{n(1-p)}}$$
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
Product-Moment Correlation (3.2.3.1) $$z = 0.5\log_{e}\left(\dfrac{1+r}{1-r}\right)$$ $$\text{SE}_{z} = \dfrac{1}{\sqrt{n-3}}$$
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) $$\text{SMD}= \dfrac{\bar{x}_{t_2} - \bar{x}_{t_1}}{s_{\text{t}_1}}$$ $$\text{SE}_{\text{SMD}} = \sqrt{\dfrac{2(1-r_{t_1t_2})}{n}+\dfrac{\text{SMD}^2_{\text{within}}}{2n}}$$
Binary Outcome Effect Size
Risk Ratio (3.3.2.1) $${p_{e}}_{\text{treat}} =\dfrac{a}{n_{\text{treat}}}$$ $$\text{SE}_{\log \text{RR}} = \sqrt{\dfrac{1}{a}+\dfrac{1}{c} - \dfrac{1}{a+b} - \dfrac{1}{c+d}}$$
Risk Ratio (3.3.2.1) $${p_{e}}_{\text{control}} = \dfrac{c}{n_{\text{control}}}$$
Risk Ratio (3.3.2.1) $$\text{RR} = \dfrac{{p_{e}}_{\text{treat}}}{{p_{e}}_{\text{control}}}$$
Risk Ratio (3.3.2.1) $$\log \text{RR} = \log_{e}(\text{RR})$$
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
Odds Ratio (3.3.2.2) $$\text{Odds}_{\text{control}} = \dfrac{c}{d}$$
Odds Ratio (3.3.2.2) $$\text{OR} = \dfrac{a/b}{c/d}$$
Odds Ratio (3.3.2.2) $$\log \text{OR} = \log_{e}(\text{OR})$$
Incidence Rate Ratio (3.3.3) $$\text{IRR} = \dfrac{ E_{\text{treat}}/T_{\text{treat}} }{E_{\text{control}}/T_{\text{control}}}$$ $$\text{SE}_{\log \text{IRR}} = \sqrt{\dfrac{1}{E_{\text{treat}}}+\dfrac{1}{E_{\text{control}}}}$$
Incidence Rate Ratio (3.3.3) $$\log \text{IRR} = \log_{e}(\text{IRR})$$
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) $${r_{xy}}_{c} = \dfrac{r_{xy}}{\sqrt{r_{xx}}}$$ $$\text{SE}_c = \dfrac{\text{SE}}{\sqrt{r_{xx}}}$$
Unreliability (3.4.2) $${r_{xy}}_{c} = \dfrac{r_{xy}}{\sqrt{r_{xx}}\sqrt{r_{yy}}}$$ $$\text{SE}_c = \dfrac{\text{SE}}{\sqrt{r_{xx}}\sqrt{r_{yy}}}$$
Unreliability (3.4.2) $$\text{SMD}_c = \dfrac{\text{SMD}}{\sqrt{r_{xx}}}$$
Range Restriction (3.4.3) $$\text{U} = \dfrac{s_{\text{unrestricted}}}{s_{\text{restricted}}}$$ $$\text{SE}_{{r_{xy}}_c} = \dfrac{{r_{xy}}_c}{r_{xy}}\text{SE}_{r_{xy}}$$
Range Restriction (3.4.3) $${r_{xy}}_c = \dfrac{\text{U}\times r_{xy}}{\sqrt{(\text{U}^2-1)r_{xy}^2+1}}$$
Range Restriction (3.4.3) $$\text{SMD}_c = \dfrac{\text{U}\times \text{SMD}}{\sqrt{(\text{U}^2-1)\text{SMD}^2+1}}$$ $$\text{SE}_{{\text{SMD}}_c} = \dfrac{{\text{SMD}}_c}{\text{SMD}}\text{SE}_{\text{SMD}}$$
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 $$s_{\text{pooled}} = \sqrt{\dfrac{(n_1-1)s^2_1+(n_2-1)s^2_2}{(n_1-1)+(n_2-1)}}$$.