Signature
J = 1 - 3 / (4 * m - 1); E_d = delta / J; Bias_d = E_d - delta
| Inputs | Definition | Unit |
|---|---|---|
m | Degrees of freedom of the standard deviation used to standardise, n_1 plus n_2 minus 2 when pooled or the comparator group size minus 1 for the control group standard deviation | count |
delta | True difference in population means divided by the common population standard deviation | standard deviation units |
J | Hedges' approximate small-sample correction factor at m degrees of freedom | dimensionless |
|---|---|---|
E_d | Mean of the uncorrected estimate d over repeated samples of the same size | standard deviation units |
Bias_d | Mean amount by which d exceeds delta in size | 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
Expected value and bias of Cohen's d in three cells
With the degrees of freedom in a cell named DegFreedomSD and the population standardised difference in TrueEffect, the formulas return the factor in CorrectionJBias, the expected value of d in ExpectedD and its bias in BiasD.
=1-3/(4*DegFreedomSD-1); =TrueEffect/CorrectionJBias; =ExpectedD-TrueEffect
Assumptions
Normal outcomes and a known population effect for the expected bias of d
The result holds under Hedges' Theorem 1, normal outcomes with a common variance in both groups. delta is a population value, so the formula shows how far d overstates a given effect on average; applied to an observed d it would not give g, which is d times J (HE-FM-HEDG-001).
Degrees of freedom set by the standard deviation in the expected bias of d
m is the number of degrees of freedom of the standard deviation used, as in HE-AS-HEDG-002. Hedges' own example of five subjects per group with 4 degrees of freedom corresponds to the control group standard deviation; with the pooled standard deviation the same trial has 8 degrees of freedom.
Worked examples
Hedges' example of five subjects per group with four degrees of freedom
Hedges states that with five subjects per group and 4 degrees of freedom d overestimates delta by approximately 25 percent. With an illustrative delta of 0.5 the approximate J of 0.8 gives a mean d of 0.625, an overstatement of 0.125; the exact factor 0.79788 of Hedges' Table I gives about 0.6267, about 25.3% (computed here).
m = 4; delta = 0.5; J = 0.8; E_d = 0.625; Bias_d = 0.125
Twenty degrees of freedom as in the article's trial
Computed here for illustration with 20 degrees of freedom, the article's trial, and an illustrative delta of 0.8. The mean d is about 0.8316, an overstatement of about 0.0316 or 3.9%, which the factor J of about 0.962 removes on average.
m = 20; delta = 0.8; J = 0.962025; E_d = 0.8316; Bias_d = 0.0316
Zero population effect gives no expected bias in Cohen's d
With delta equal to 0 the mean of d is 0 at any number of degrees of freedom, a limiting case, because the bias is proportional to delta.
m = 20; delta = 0; J = 0.962025; E_d = 0; Bias_d = 0
Common errors
Bias of Cohen's d taken as delta times one minus J
The reduction from d to g is d times one minus J, but the expected bias of d is delta times (1/J minus 1), which is larger. At 4 degrees of freedom the first reading gives 20% instead of 25%, and at 20 degrees of freedom about 3.8% instead of about 3.9%, computed here for illustration.
Expecting pooling of many small studies to remove the bias of Cohen's d
Hedges shows that the average of uncorrected estimates from many small studies is not a consistent estimator of delta, because each estimate carries the same bias while the variance of the average falls, so the average estimates the wrong quantity more precisely. Ten studies with 4 degrees of freedom each and a delta of 0.5 have an expected average d of about 0.627 whatever their number, computed here for illustration with the exact factor.
Sources
Hedges on the expectation and bias of the uncorrected standardised mean difference
Hedges LV. Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics. 1981;6(2):107-128. Theorem 1 (page 111), equation 6a, which gives the expectation of the uncorrected estimate as delta over c(m); the Remark on the consistency of G (page 112), which states that with five subjects per group and 4 degrees of freedom the estimator overestimates delta by approximately 25 percent, that as the number of experiments increases the bias of the averaged estimator is unchanged while its variance tends to zero, and section 4 with Figure 1 (pages 112 and 113), which plots the ratio of the expectation of the uncorrected estimate to delta against m.
Canonical Identity
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