Biostatistics & Meta-Analysis
Hedges’ g Small-Sample Bias Correction and Glass’s Delta Formula
Correct upward small-sample bias in Cohen’s d using Hedges’ J(df) = 1 - 3 / (4*df - 1) factor and switch to Glass’s Delta when treatment alters group variance.
Spot two classic Cohen’s d traps: small-df upward bias and treatment-inflated variance
Raw Cohen’s d divides (M_treat - M_ctrl) by pooled standard deviation s_p, which overestimates population effect size in small samples and distorts the yardstick if the intervention itself widens score dispersion. Hedges (1981) removes small-sample bias via J(df) = 1 - 3 / (4*df - 1) where df = n1 + n2 - 2; Glass (1976) divides strictly by the untouched control-group SD_ctrl.
Verify by hand on a pilot trial: Treatment n1 = 6 (B2), M1 = 82 (B3), SD1 = 12 (B4) vs Control n2 = 6 (B5), M2 = 70 (B6), SD2 = 6 (B7). Here df = 10, pooled variance = (5*144 + 5*36) / 10 = 90 (s_p = 9.4868), and Cohen’s d = 12 / SQRT(90) = 1.2649. Hedges’ factor is J(10) = 1 - 3 / 39 = 12 / 13 = 0.9231, trimming g to 1.1676 (-7.7% bias correction!), while Glass’s Delta = (82 - 70) / 6 = 2.0000.
| Effect Size Estimator | Formula Definition | Pilot Output (n1=n2=6) |
|---|---|---|
| Pooled Cohen’s d | (M1 - M2) / SQRT(((n1-1)*SD1^2 + (n2-1)*SD2^2) / df) | 1.2649 (Biased high at df = 10) |
| Hedges Correction J(df) | 1 - 3 / (4 * df - 1) = 36 / 39 = 12 / 13 | 0.9231 |
| Unbiased Hedges’ g | J(df) * d_Cohen (with asymptotic SE_g) | 1.1676 (SE_g = 0.6246) |
| Glass’s Control Delta | (M1 - M2) / SD_ctrl | 2.0000 (Untouched baseline SD = 6) |
Unbiased Hedges’ g Formula
Guard df < 1 or non-positive standard deviations before pooling variance and applying J(df) = 1 - 3 / (4*df - 1).
=LET(n_1, B2, m_1, B3, sd_1, B4, n_2, B5, m_2, B6, sd_2, B7, df_p, n_1 + n_2 - 2, IF(OR(n_1 < 2, n_2 < 2, sd_1 <= 0, sd_2 <= 0), "Invalid Group Stats", LET(sp_val, SQRT(((n_1 - 1) * sd_1^2 + (n_2 - 1) * sd_2^2) / df_p), d_raw, (m_1 - m_2) / sp_val, j_cor, 1 - 3 / (4 * df_p - 1), ROUND(j_cor * d_raw, 4))))Glass’s Delta & Large-Sample Convergence Check
Compute Glass’s Delta = (M1 - M2) / SD2 alongside Hedges’ meta-analytic standard error SQRT((n1 + n2)/(n1*n2) + g^2 / (2*(n1 + n2))). Increase n1 and n2 to 100 each: J(198) rises to 0.9962 so Hedges’ g (1.2601) converges toward Cohen’s d (1.2649).
=IF(B7 <= 0, "Invalid Group Stats", ROUND((B3 - B6) / B7, 4))Try the example
Use the original values or change them to test your understanding.