Table 3. Formulas for the meta-analysis...

Table 3.

Formulas for the meta-analysis assuming fixed-effects and random-effects models

Name	Formula	Description
Fixed-effects model
Effect size using correlation	$E S_{r} = r$	ES is the effect size and r is the product-moment correlation coefficient
Effect size using t-statistic	$E S_{r} = \sqrt{\frac{t^{2}}{(t^{2} + d f)}}$	ES obtained as a function of the t-statistic (t) and df that is the degrees of freedom given by n–1 where n is the sample size
Effect size using z-statistic	$E S_{r} = \sqrt{\frac{Z^{2}}{N}}$	ES obtained as a function of the z-statistic (Z) and the total sample size (N)
Z-transform effect size	$Z_{r} = 0.5 * l o g_{e} (\frac{1 + E S_{r}}{1 - E S_{r}})$	Fisher’s transformed effect size
Standard error	$S E_{Z_{r}} = \frac{1}{\sqrt{n_{i} - 3}}$	Standard error of the transformed effect sizes
Inverse variance	$w_{i} = \frac{1}{S E_{Z_{r}}^{2}}$	Weight associated with the sample size of each study
Weighted mean effect size	$\bar{E S} = \frac{\sum^{} (w_{i} E S_{i})}{\sum^{} w_{i}}$	Mean effect size calculated for all effect sizes in the meta-analysis
Standard error of the mean	$S E_{\bar{E S}} = \frac{1}{\sqrt{\sum^{} w_{i}}}$	Standard error of the mean effect size
Random-effects model
Chi-square statistic	$Q = \sum^{} w_{i} {(E S_{i} - \bar{E S})}^{2}$	Q-statistic for the test for heterogeneity, which is distributed as a chi-square with k–1 degrees of freedom. K is the number of effect sizes in the study
Tau squared	$τ^{2} = \frac{Q - d f}{\sum^{} w_{i} - \frac{\sum^{} w_{i}^{2}}{\sum^{} w_{i}}}$	Between-study variance, where Q is the Q-statistic and df is the degrees of freedom
Weight	$w_{i}^{} = \frac{1}{v_{i}^{}}$	Weight assigned to each study
Total variance	$v_{i}^{*} = v_{i} + τ^{2}$	Total variance obtained as the sum of within-study variance for study i and the between-studies variance (τ²)
Weighted mean effect size	${\bar{E S}}^{} = \frac{\sum^{} (w_{i}^{} E S_{i})}{\sum^{} w_{i}^{*}}$	Mean effect size in a random-effects model
Variance of mean effect size	$v^{} = \frac{1}{\sum^{} w_{i}^{}}$	Variance of mean effect size determined as the reciprocal of the sum of the weights
Standard error of mean effect size	$S E_{{\bar{E S}}^{}} = \sqrt{v^{}}$	Standard error of mean effect size
Lower limit	${\bar{E S}}_{i}^{} = {\bar{E S}}^{} - 1.96 (S E_{{\bar{E S}}^{*}})$	Lower limit of the confidence interval for the mean effect size
Upper limit	${\bar{E S}}_{i}^{} = {\bar{E S}}^{} + 1.96 (S E_{{\bar{E S}}^{*}})$	Upper limit of the confidence interval for the mean effect size
z-statistic	$Z^{} = \frac{\| {\bar{E S}}^{} \|}{S E_{{\bar{E S}}^{*}}}$	Z-value for the test of the significance of the mean effect size
Fail-safe number	$X = \frac{k [k * {(Z^{*})}^{2} - 2, 706]}{2, 706}$	Number of studies that would be required to overturn significant results into insignificant. k is the number of studies in the meta-analysis and Z* is the z-statistic

Name	Formula	Description
Fixed-effects model
Effect size using correlation	$E S_{r} = r$	ES is the effect size and r is the product-moment correlation coefficient
Effect size using t-statistic	$E S_{r} = \sqrt{\frac{t^{2}}{(t^{2} + d f)}}$	ES obtained as a function of the t-statistic (t) and df that is the degrees of freedom given by n–1 where n is the sample size
Effect size using z-statistic	$E S_{r} = \sqrt{\frac{Z^{2}}{N}}$	ES obtained as a function of the z-statistic (Z) and the total sample size (N)
Z-transform effect size	$Z_{r} = 0.5 * l o g_{e} (\frac{1 + E S_{r}}{1 - E S_{r}})$	Fisher’s transformed effect size
Standard error	$S E_{Z_{r}} = \frac{1}{\sqrt{n_{i} - 3}}$	Standard error of the transformed effect sizes
Inverse variance	$w_{i} = \frac{1}{S E_{Z_{r}}^{2}}$	Weight associated with the sample size of each study
Weighted mean effect size	$\bar{E S} = \frac{\sum^{} (w_{i} E S_{i})}{\sum^{} w_{i}}$	Mean effect size calculated for all effect sizes in the meta-analysis
Standard error of the mean	$S E_{\bar{E S}} = \frac{1}{\sqrt{\sum^{} w_{i}}}$	Standard error of the mean effect size
Random-effects model
Chi-square statistic	$Q = \sum^{} w_{i} {(E S_{i} - \bar{E S})}^{2}$	Q-statistic for the test for heterogeneity, which is distributed as a chi-square with k–1 degrees of freedom. K is the number of effect sizes in the study
Tau squared	$τ^{2} = \frac{Q - d f}{\sum^{} w_{i} - \frac{\sum^{} w_{i}^{2}}{\sum^{} w_{i}}}$	Between-study variance, where Q is the Q-statistic and df is the degrees of freedom
Weight	$w_{i}^{} = \frac{1}{v_{i}^{}}$	Weight assigned to each study
Total variance	$v_{i}^{*} = v_{i} + τ^{2}$	Total variance obtained as the sum of within-study variance for study i and the between-studies variance (τ²)
Weighted mean effect size	${\bar{E S}}^{} = \frac{\sum^{} (w_{i}^{} E S_{i})}{\sum^{} w_{i}^{*}}$	Mean effect size in a random-effects model
Variance of mean effect size	$v^{} = \frac{1}{\sum^{} w_{i}^{}}$	Variance of mean effect size determined as the reciprocal of the sum of the weights
Standard error of mean effect size	$S E_{{\bar{E S}}^{}} = \sqrt{v^{}}$	Standard error of mean effect size
Lower limit	${\bar{E S}}_{i}^{} = {\bar{E S}}^{} - 1.96 (S E_{{\bar{E S}}^{*}})$	Lower limit of the confidence interval for the mean effect size
Upper limit	${\bar{E S}}_{i}^{} = {\bar{E S}}^{} + 1.96 (S E_{{\bar{E S}}^{*}})$	Upper limit of the confidence interval for the mean effect size
z-statistic	$Z^{} = \frac{\| {\bar{E S}}^{} \|}{S E_{{\bar{E S}}^{*}}}$	Z-value for the test of the significance of the mean effect size
Fail-safe number	$X = \frac{k [k * {(Z^{*})}^{2} - 2, 706]}{2, 706}$	Number of studies that would be required to overturn significant results into insignificant. k is the number of studies in the meta-analysis and Z* is the z-statistic

Sources: Lipsey and Wilson (2001), Opare et al. (2021) and Ahamed et al. (2023)

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