Statistical & Psychometrics
DerSimonian-Laird Random-Effects Variance (τ²) & Higgins I² Meta-Analysis
Calculate between-study random-effects heterogeneity variance τ^2 = MAX(0, (Q - (k - 1)) / (Σw - Σw^2 / Σw)) and Higgins I^2 (%) from Cochran’s Q and fixed-effect weights.
Accounting for genuine between-trial clinical diversity with DerSimonian & Laird’s 1986 τ² estimator!
Fixed-effect meta-analysis assumes every randomized trial estimates the exact same true treatment effect—yielding artificially narrow confidence intervals whenever dosing, patient populations, or follow-up durations vary! Rebecca DerSimonian and Nan Laird (Controlled Clinical Trials, 1986—one of the 100 most-cited papers in all of science!) derived the non-iterative method-of-moments estimator for between-study variance τ^2 from Cochran’s heterogeneity Q, study count k, sum of fixed-effect inverse-variance weights Σw, and sum of squared weights Σ(w^2): τ^2 = MAX(0, (Q - (k - 1)) / (Σw - Σ(w^2) / Σw)), updating each study’s random-effects weight to w*_i = 1 / (v_i + τ^2)!
Evaluate a k = 6 trial meta-analysis with Cochran’s Q = 14.80 (df = 5), Σw = 42.0, and Σ(w^2) = 340.0 (scaling denominator C = 42 - 340/42 = 33.9048): DerSimonian-Laird τ^2 = 0.2890 and Higgins-Thompson inconsistency I^2 = MAX(0, (Q - (k - 1)) / Q) * 100 = 66.22% (substantial heterogeneity; truncating cleanly to τ^2 = 0 when Q = 3.80 < k - 1)!
| Cochran’s Q (k = 6 Studies, df = 5, C = 33.9048) | DerSimonian-Laird τ² | Higgins I² (%) | Meta-Analytic Model |
|---|---|---|---|
| Q = 14.80 (> df = 5) | 0.2890 | 66.22% (Substantial) | Random-Effects w* = 1 / (v_i + 0.2890) |
| Q = 3.80 (<= df = 5) | 0.0000 (Truncated at 0) | 0.00% (Homogeneous) | Reduces Exactly to Fixed-Effect |
| Invalid k < 2 or Non-Positive C | "Invalid Meta-Analysis Weights" | Error Guard | Requires >= 2 Studies |
DerSimonian-Laird Between-Study Variance Estimator τ²
Evaluates MAX(0, (Q - (k - 1)) / (Σw - Σ(w^2) / Σw)) from Cochran’s Q (B2), number of studies k (B3), Σw (B4), and Σ(w^2) (B5).
=IF(OR(B3 < 2, B4 <= 0, B5 <= 0, (B4 - B5 / B4) <= 0), "Invalid Meta-Analysis Weights", ROUND(MAX(0, (B2 - (B3 - 1)) / (B4 - (B5 / B4))), 4))Higgins-Thompson Inconsistency Percentage I² (%)
Quantifies the percentage of total variation across studies due to true heterogeneity rather than sampling chance.
=IF(OR(B3 < 2, B2 <= 0), 0, ROUND(MAX(0, (B2 - (B3 - 1)) / B2) * 100, 2))Try the example
Use the original values or change them to test your understanding.