Statistical & Clinical Analysis
Higgins I² & H² Meta-Analysis Heterogeneity Formula from Cochran’s Q
Quantify percentage between-study inconsistency I² = MAX(0, (Q - (k - 1)) / Q) * 100% and variance-inflation factor H² = MAX(1, Q / (k - 1)) in Cochrane meta-analyses.
Why Cochran’s Q depends on trial count k—whereas Julian Higgins’s I² is scale-invariant!
Raw Cochran’s homogeneity statistic Q = Σ w_i * (θ_i - θ̂_FE)^2 has low power when k = 3 studies and excessive rejection rates when k = 50 studies! Julian P. T. Higgins and Simon G. Thompson (2002, Statistics in Medicine) solved this by comparing Q against its null chi-square expectation df = k - 1 to isolate the percentage of total observed dispersion caused by genuine between-study clinical heterogeneity rather than within-study sampling error: I^2 = MAX(0, (Q - (k - 1)) / Q) * 100%!
Alongside I^2, Higgins defined H^2 = MAX(1, Q / (k - 1))—the ratio of observed Q dispersion to degrees of freedom (whose square root H inflates standard-error confidence intervals under Paule-Mandel / quasi-likelihood overdispersion)! For Q = 18.6 across k = 5 trials (df = 4), I^2 = 78.49% (substantial/considerable heterogeneity >= 75%) and H^2 = 4.650—clamping cleanly to I^2 = 0% when Q = 3.2 < df = 4!
| Cochran’s Q & Study Count k | Higgins Inconsistency I² (%) | Dispersion Ratio H² = MAX(1, Q/df) | Cochrane Handbook Tier |
|---|---|---|---|
| Q = 18.6, k = 5 Studies (df = 4) | 78.49% | 4.650 (H = 2.156) | Considerable Heterogeneity (75%–100%) |
| Q = 3.2, k = 5 Studies (Q < df) | 0.00% (Truncated at Zero) | 1.000 | Homogeneous / Sampling Noise Only |
| Invalid k < 2 or Q <= 0 | Error Guard | "Requires k >= 2 Studies and Q > 0" | Meta-Analysis Bounds Guard |
Higgins & Thompson Percentage Heterogeneity Index I² (%)
Evaluates MAX(0, (Q - (k - 1)) / Q) * 100 from Cochran’s Q statistic (B2) and number of independent studies k >= 2 (B3).
=IF(OR(B2 <= 0, B3 < 2), "Requires k >= 2 Studies and Q > 0", ROUND(MAX(0, (B2 - (B3 - 1)) / B2) * 100, 2))Higgins Overdispersion Variance Ratio H² = MAX(1, Q / (k - 1))
Computes the truncated chi-square dispersion ratio Q / df.
=IF(OR(B2 <= 0, B3 < 2), "Requires k >= 2 Studies and Q > 0", ROUND(MAX(1, B2 / (B3 - 1)), 3))Try the example
Use the original values or change them to test your understanding.