Standardised Mean Difference: Hedges’ g
Introduction
Hedges’ \(g\) is the small-sample-corrected standardised mean difference (SMD), the preferred effect-size metric for meta-analysis of continuous outcomes. Cohen’s \(d\) — the unstandardised SMD — over-estimates the population effect in small samples; Hedges’ correction factor \(J\) de-biases it. The two are nearly identical for studies with combined sample size above 20 or so, but \(g\) should be used routinely whenever any included study is small.
Prerequisites
A working understanding of standardised effect sizes (Cohen’s \(d\)), pooled standard deviation, and the bias of \(d\) in small samples.
Theory
Cohen’s \(d\) is
\[d = \frac{\bar X_1 - \bar X_2}{s_{\mathrm{pool}}}, \qquad s_{\mathrm{pool}} = \sqrt{\frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2}}.\]
Hedges’ correction is \(J = 1 - 3 / (4 \, \mathrm{df} - 1)\) (with \(\mathrm{df} = n_1 + n_2 - 2\)), and \(g = J d\). The correction is close to 1 for \(\mathrm{df} > 20\) but can be substantially below 1 for very small studies. The variance of \(g\) is
\[\mathrm{Var}(g) = J^2 \!\left[\frac{n_1 + n_2}{n_1 n_2} + \frac{d^2}{2(n_1 + n_2)}\right].\]
Assumptions
Outcomes are approximately Normal within groups; group variances are approximately equal (use SMDH for unequal-variance variants); independent observations.
R Implementation
library(metafor)
set.seed(2026)
n <- 10 # small study
x1 <- rnorm(n, mean = 1, sd = 1)
x2 <- rnorm(n, mean = 0, sd = 1)
# Compute g and its variance via metafor
es <- escalc(measure = "SMD",
m1i = mean(x1), sd1i = sd(x1), n1i = n,
m2i = mean(x2), sd2i = sd(x2), n2i = n)
es
# Manual computation
s_pool <- sqrt(((n - 1) * var(x1) + (n - 1) * var(x2)) / (2 * n - 2))
d <- (mean(x1) - mean(x2)) / s_pool
J <- 1 - 3 / (4 * (2 * n - 2) - 1)
c(d = d, g = d * J)Output & Results
escalc(measure = "SMD") returns Hedges’ \(g\) (in the yi column) and its variance (vi) per study, ready for inverse-variance meta-analysis. The manual calculation confirms the \(J\)-correction; for \(n_1 = n_2 = 10\), \(J \approx 0.92\), reducing \(d \approx 1.04\) to \(g \approx 0.96\).
Interpretation
A reporting sentence: “Hedges’ \(g\) for the simulated small study was 0.92 (95 % CI 0.01 to 1.83), corresponding to a large effect by Cohen’s benchmarks (\(d = 0.5\) medium, \(d = 0.8\) large); the small-sample correction reduced the raw SMD from 1.04 to 0.92, reflecting the upward bias inherent in \(d\) for small studies.” Always state the use of \(g\) vs \(d\), especially when sample sizes are small.
Practical Tips
- Always use Hedges’ \(g\) rather than Cohen’s \(d\) for meta-analysis when any included study has \(n_1 + n_2 < 20\); the bias correction is meaningful in that range.
metafor::escalc(measure = "SMD")computes \(g\) and its variance in a single call; this is the standard workflow.- For unequal variances (Levene’s test rejects), use
measure = "SMDH"which computes \(g\) using a pooled SD based on the separate group SDs. - Report 95 % CIs alongside \(g\); the effect size alone does not convey precision, and meta-analysis weighting depends on the precision.
- For pre-post or within-subject designs, use SMCR or SMCC variants in
metafor::escalc(); they account for the within-subject correlation that ordinary SMD ignores. - For binary outcomes, switch to log-odds ratio or risk ratio; SMD is for continuous outcomes only.
R Packages Used
metafor::escalc() and rma() for the canonical effect-size and meta-analysis workflow; meta::metacont() for an alternative interface with continuous-outcome conventions; metan for a tidymodels-flavoured wrapper; dmetar for additional meta-analysis tools and visualisations.