Statistical & Clinical Analysis
Begg & Mazumdar Rank Correlation Test: Kendall’s τ Funnel Asymmetry Z-Score
Test meta-analytic publication bias nonparametrically using Colin Begg and Madhuchhanda Mazumdar’s (1994) Kendall rank correlation τ and exact variance σ_S² = k*(k - 1)*(2*k + 5) / 18.
Nonparametric rank-correlation alternative to Egger’s regression when outlier trials skew OLS slopes!
Colin B. Begg and Madhuchhanda Mazumdar (1994, Biometrics) proposed the first formal statistical test of funnel-plot publication bias: standardize each trial’s centered effect t_i* = (θ_i - θ̂_FE) / SQRT(v_i - 1 / Σw_j), pair it against sampling variance v_i across all k*(k - 1)/2 trial pairs, and count concordant pairs C (where larger variance accompanies more extreme standardized effect) minus discordant pairs D!
Kendall’s rank correlation is τ = (C - D) / [k*(k - 1)/2], and under the null hypothesis of symmetric independence, S = C - D has exact variance σ_S^2 = k * (k - 1) * (2*k + 5) / 18, giving Z = (C - D) / SQRT(k*(k - 1)*(2*k + 5)/18)! Across k = 8 trials (28 pairs: C = 22 concordant, D = 6 discordant), τ = +0.5714 and Z = +1.979 (p < 0.05 two-sided rank asymmetry, dropping to τ = 0 when C = D = 14)!
| Meta-Analysis Pair Counts (k = 8 Studies, 28 Pairs) | Begg Kendall’s τ | Asymptotic Z-Statistic | Publication Bias Inference |
|---|---|---|---|
| C = 22 Concordant, D = 6 Discordant (S = +16) | +0.5714 | Z = +1.979 (|Z| > 1.96) | Significant Rank Asymmetry (p = 0.048) |
| C = 14 Concordant, D = 14 Discordant (S = 0) | 0.0000 | Z = 0.000 | Symmetric Funnel Rank Order |
| Invalid k < 3 or C + D > k*(k - 1)/2 | Error Guard | "Invalid Study Pair Counts" | Combinatorial Bounds Guard |
Begg & Mazumdar Kendall Rank Correlation τ
Computes (C - D) / [k * (k - 1) / 2] from study count k >= 3 (B2), concordant pairs C (B3), and discordant pairs D (B4).
=IF(OR(B2 < 3, B3 < 0, B4 < 0, B3 + B4 > B2 * (B2 - 1) / 2), "Invalid Study Pair Counts", ROUND((B3 - B4) / (B2 * (B2 - 1) / 2), 4))Begg-Mazumdar Rank-Correlation Z-Statistic Z = (C - D) / σ_S
Divides S = C - D by exact tie-free Kendall standard error SQRT(k * (k - 1) * (2*k + 5) / 18).
=IF(OR(B2 < 3, B3 < 0, B4 < 0, B3 + B4 > B2 * (B2 - 1) / 2), "Invalid Study Pair Counts", ROUND((B3 - B4) / SQRT(B2 * (B2 - 1) * (2 * B2 + 5) / 18), 3))Try the example
Use the original values or change them to test your understanding.