Signature
N = n_1 + n_2; SE_RM = sqrt(N / (n_1 * n_2) + g^2 / (2 * (N - 3.94)))
| 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 |
g | Hedges' g from HE-FM-HEDG-001, the bias-corrected standardised mean difference | standard deviation units |
N | Number of participants in both groups together | people |
|---|---|---|
SE_RM | Standard error of Hedges' adjusted g as Review Manager computes it | 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
Review Manager standard error of Hedges' g in two cells
With the named cells of HE-IM-HEDG-001, the first formula returns the total sample size in a cell named TotalSize and the second the Review Manager standard error in StdErrRevMan.
=SizeIntervention+SizeComparator; =SQRT(TotalSize/(SizeIntervention*SizeComparator)+HedgesG^2/(2*(TotalSize-3.94)))
Assumptions
Hedges' adjusted g as the standardised mean difference of Cochrane reviews
The Cochrane Handbook states that the standardised mean difference used in Cochrane reviews is Hedges' adjusted g, with a pooled standard deviation in the denominator that assumes similar standard deviations in the two groups. Review Manager computes g with the factor 1 minus 3 over (4N minus 9), the same factor as HE-FM-HEDG-001.
Inverse-variance weights from the Review Manager standard error
Review Manager weights each study by the reciprocal of the square of this standard error in its inverse-variance methods. Mixing it with SE_g of HE-FM-HEDG-002 in one analysis gives studies of the same size slightly different weights.
Worked examples
Article rehabilitation trial under the Review Manager standard error
Computed here for illustration with the article's g of 0.8048, 10 and 12 patients. SE_RM is about 0.4486, about 4.5% above the 0.4294 of HE-FM-HEDG-002, the small-trial difference the article mentions.
n_1 = 10; n_2 = 12; g = 0.8048; N = 22; SE_RM = 0.4486
Fifty patients per arm bring the two standard errors of Hedges' g together
Computed here for illustration with g of 0.8302 and 50 patients per arm. SE_RM is about 0.2088 against 0.2070 from HE-FM-HEDG-002, a gap under 1%.
n_1 = 50; n_2 = 50; g = 0.8302; N = 100; SE_RM = 0.2088
Zero Hedges' g reduces the Review Manager standard error to the group-size term
With g equal to 0 and 30 patients per arm, SE_RM is the square root of 60 over 900, about 0.2582, a limiting case.
n_1 = 30; n_2 = 30; g = 0; N = 60; SE_RM = 0.2582
Common errors
Two standard error formulas for Hedges' g mixed in one meta-analysis
Taking SE_RM for some studies and SE_g of HE-FM-HEDG-002 for others weights like studies unequally. For a trial the size of the article's, the Review Manager form gives a weight about 8% lower (0.4294 squared over 0.4486 squared, about 0.916), computed here for illustration.
Correction factor applied again to the Review Manager standard error
Multiplying SE_RM by J, as the What Works Clearinghouse form does for its own expression, mixes two conventions and shrinks the Review Manager value. For the article's trial it gives about 0.4316 instead of 0.4486, computed here for illustration.
Sources
Review Manager algorithms on the standard error of Hedges' adjusted g
Deeks JJ, Higgins JPT, on behalf of the Cochrane Statistical Methods Group. Statistical algorithms in Review Manager. Cochrane; May 2022. Section Standardized difference in means (standardized mean difference) (page 3), which gives the standard error of the SMD as the square root of N over n_1 n_2 plus SMD squared over 2(N minus 3.94), and the section on inverse-variance methods, which weights each study by the reciprocal of the square of the standard error from the individual study section.
Cochrane Handbook on Hedges' adjusted g as the Cochrane standardised mean difference
Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 6.5.1.2, The standardized mean difference, which states that the SMD used in Cochrane reviews is the effect size known in social science as Hedges' (adjusted) g, with a pooled standard deviation in the denominator that assumes the standard deviations in the two groups are similar.
Canonical Identity
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