Signature
V_d = (n_1 + n_2) / (n_1 * n_2) + d^2 / (2 * (n_1 + n_2)); V_g = J^2 * V_d; SE_g = sqrt(V_g)
| Inputs | Definition | Unit |
|---|---|---|
n_1 | Number of participants in group 1, usually the intervention group | people |
n_2 | Number of participants in group 2, usually the comparator group | people |
d | Uncorrected standardised mean difference from HE-FM-CD-001 | standard deviation units |
J | Small-sample correction factor from HE-FM-HEDG-001 | dimensionless |
V_d | Approximate sampling variance of the uncorrected standardised mean difference | squared standard deviation units |
|---|---|---|
V_g | Approximate sampling variance of Hedges' g, the inverse of its inverse-variance weight | squared standard deviation units |
SE_g | Square root of the variance of Hedges' g | standard deviation units |
Function
Hedges' g small-sample correction, sampling variance and bias of the standardised mean difference
Maps Cohen's d, the uncorrected standardised mean difference of HE-FM-CD-001, and the two group sizes to Hedges' g, the same difference multiplied by a correction factor J that removes the upward small-sample bias of d, and to the variance and standard error of g that an inverse-variance meta-analysis needs. It also gives the standard error that Review Manager attaches to the same statistic, and the expected overstatement of a population standardised difference by d at a given number of degrees of freedom. The pooled standard deviation and d itself are HE-FM-CD-001 and are not restated. A confidence interval for g is the large-sample interval HE-FM-AN-001 applied with the standard error of g, the logistic conversion to a log odds ratio is HE-FM-CD-002 with g in place of d, and pooling g across studies is HE-FM-ADMA-001 for a common effect or HE-FM-ADMA-005 for random effects. Notation follows the Hedges' g article.
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Implementations
Excel
Variance and standard error of Hedges' g in three cells
With the named cells of HE-IM-HEDG-001, the first formula returns the variance of d in a cell named VarianceD, the second the variance of g in VarianceG and the third the standard error in StdErrG.
=(SizeIntervention+SizeComparator)/(SizeIntervention*SizeComparator)+CohensD^2/(2*(SizeIntervention+SizeComparator)); =CorrectionJ^2*VarianceD; =SQRT(VarianceG)
Assumptions
Large-sample approximation to the variance of Hedges' g
The expression is a large-sample approximation with the observed d standing in for the unknown population effect, which the What Works Clearinghouse handbook notes makes every such formula an approximation. Hedges' own normal approximation uses delta squared over 2m in the second term, and his Theorem 2 gives the exact variance of the corrected estimate. For the article's trial, with delta set to 0.8048, the exact variance is about 0.2067 against 0.1844 here and 0.1995 for Hedges' approximation (computed here and confirmed by simulation), so in very small trials this standard error is somewhat too small.
Independent groups with a pooled standard deviation for the variance of Hedges' g
The two groups are independent and d uses the pooled standard deviation, as in HE-FM-CD-001. Crossover trials, change scores and cluster-randomised trials need other variance formulas, and the What Works Clearinghouse handbook gives separate formulas for cluster-level assignment.
Worked examples
Article rehabilitation trial variance and standard error of Hedges' g
From the article's trial, V_d is about 0.1992, V_g about 0.1844 and SE_g about 0.4294, shown as 0.199, 0.184 and 0.429 in the article. HE-FM-AN-001 with z of 1.96 then gives an interval of about minus 0.037 to 1.646 around g of 0.8048, the article's minus 0.04 to 1.65, which includes no effect.
n_1 = 10; n_2 = 12; d = 0.8366; J = 0.962025; V_d = 0.1992; V_g = 0.1844; SE_g = 0.4294
Fifty patients per arm narrow the standard error of Hedges' g
Computed here for illustration with the same d and 50 patients in each arm, V_d is about 0.0435, V_g about 0.0428 and SE_g about 0.2070, under half the standard error of the 22-patient trial.
n_1 = 50; n_2 = 50; d = 0.8366; J = 0.992327; V_d = 0.0435; V_g = 0.0428; SE_g = 0.2070
Zero effect leaves only the group-size term in the variance of Hedges' g
With d equal to 0 and 30 patients per arm, V_d is 60 over 900, about 0.0667, V_g about 0.0649 and SE_g about 0.2548, a limiting case in which the effect term vanishes.
n_1 = 30; n_2 = 30; d = 0; J = 0.987013; V_d = 0.0667; V_g = 0.0649; SE_g = 0.2548
Common errors
Variance of Cohen's d attached to Hedges' g
Using V_d as the variance of g ignores the factor J squared. For the article's trial the standard error becomes about 0.4464 instead of 0.4294 and the study's inverse-variance weight about 7.5% lower than it should be, computed here for illustration.
One over the total sample size in place of the group-size term for Hedges' g
Writing the first term as 1 over (n_1 + n_2) instead of (n_1 + n_2) over n_1 n_2 shrinks it about fourfold with equal arms. For the article's trial the standard error falls to about 0.2383 and the lower 95% limit rises to about 0.338, so a trial compatible with no effect appears clearly beneficial, computed here for illustration.
Sources
What Works Clearinghouse handbook on the standard error of Hedges' g
What Works Clearinghouse. What Works Clearinghouse Procedures and Standards Handbook, Version 5.0 (WWC 2022008). US Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance; August 2022. Appendix E (page 164), equation E.6, which gives the standard error of Hedges' g from unadjusted means as omega times the square root of (n_i + n_c)/(n_i n_c) plus g squared over 2(n_i + n_c), citing Borenstein and Hedges 2019, and footnote 35, which states that Borenstein and Hedges use the uncorrected d in that right-hand term, that both versions are approximations because the population variance formula uses the population effect size, and that the left-hand term is usually much the larger.
Hedges on the variance of the corrected estimator and its normal approximation
Hedges LV. Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics. 1981;6(2):107-128. Section 4 (page 114), which states that the variance of the corrected estimator is c(m) squared times the variance of the uncorrected one and so smaller; Theorem 2 (page 116), which gives its exact variance; and section 4.1, Confidence intervals for the effect size (page 117), which treats the corrected estimator as approximately normal with variance 1 over n tilde plus delta squared over 2m, where n tilde is n^E n^C over (n^E + n^C).
Canonical Identity
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