Statistical & Psychometrics
Goodman and Kruskal’s Gamma (γ) Ordinal Association Formula
Calculate tie-free ordinal monotonic association γ = (N_c - N_d) / (N_c + N_d) across Likert scales and clinical staging tables alongside tie-penalized Kim’s d_y.x.
Measuring monotonic ordinal association across coarse 3×3 or 5×5 Likert tables where ties abound!
When cross-tabulating coarse ordinal variables—such as tumor stage (I–IV) against histologic grade (1–3) or 5-point Likert satisfaction surveys—massive numbers of tied pairs attenuate Kendall’s τ_a even under perfect weak monotonicity! Leo A. Goodman and William H. Kruskal (JASA, 1954) defined Gamma strictly over untied pairs: γ = (N_c - N_d) / (N_c + N_d) on [-1, +1], representing the proportional reduction in error when predicting the rank order of an untied pair!
Evaluate an oncology staging cohort with N_c = 1,840 concordant pairs, N_d = 460 discordant pairs, and T_y = 520 pairs tied on outcome Y: Goodman-Kruskal γ = (1840 - 460) / (1840 + 460) = +0.6000 (60% excess concordance among untied pairs), whereas asymmetric Somers’ d_y|x = (N_c - N_d) / (N_c + N_d + T_y) = +0.4894 accounts for outcome ties!
| Pair Counts [Concordant Nc, Discordant Nd, Tied Ty] | Goodman-Kruskal γ | Asymmetric d_y|x (With Y-Ties) | Ordinal Interpretation |
|---|---|---|---|
| Nc = 1,840, Nd = 460, Ty = 520 | +0.6000 | +0.4894 | Strong Positive Monotonic Trend |
| Nc = 920, Nd = 920, Ty = 520 | 0.0000 | 0.0000 | Ordinal Independence |
| Nc + Nd = 0 (All Pairs Tied) | "No Untied Pairs" | Error Guard | Prevents #DIV/0! |
Goodman and Kruskal’s Symmetric Ordinal Gamma γ
Evaluates (N_c - N_d) / (N_c + N_d) from concordant pair count B2 and discordant pair count B3.
=IF(OR(B2 < 0, B3 < 0, B2 + B3 = 0), "No Untied Pairs", ROUND((B2 - B3) / (B2 + B3), 4))Tie-Penalized Directional Concordance Ratio (N_c - N_d) / (N_c + N_d + T_y)
Includes pairs tied on dependent variable Y (B4) in the denominator to quantify tie dilution relative to γ.
=IF(OR(B2 < 0, B3 < 0, B4 < 0, B2 + B3 + B4 = 0), "No Untied Pairs", ROUND((B2 - B3) / (B2 + B3 + B4), 4))Try the example
Use the original values or change them to test your understanding.