Somers' D (asymmetric ordinal, Dxy + Dyx)
Coming soonAdvanced & specialized
This test is implemented and is currently going through StatMinds’ production verification: every statistic is independently checked against a trusted reference (scipy / R), locked with regression tests, and the screen is exercised across assumption-met/violated and significant/non-significant scenarios before it opens up.
See what’s live nowAsymmetric ordinal association coefficient.
D(Y|X) = (C − D) / (N₀ − Ty) — how well x predicts y's rank. D(X|Y) = (C − D) / (N₀ − Tx) — how well y predicts x's rank. They differ when the marginal tie totals on the two axes differ (Tx ≠ Ty). Reports both directional forms with asymptotic z + 95% CI, plus the symmetric harmonic form. The pipeline surfaces a primary-voice qualifier when |D(Y|X) − D(X|Y)| > 0.30 — substantial directional asymmetry that the symmetric form would obscure.
Worked example
How well does one ordinal variable predict another (directional)?
Somers' D is a directional ordinal association — the asymmetric counterpart to gamma, treating one variable as the outcome.
The predictor accounted for ordinal variation in the outcome, Somers' D = .31, p < .001.
Somers' D indicated a moderate directional ordinal association, D = .31, p < .001.
When to use it
- Ordinal predictor → ordinal outcome (directional question)A clinician cross-tabulates pain-severity tier (mild / moderate / severe / very severe) against treatment-response category (no response / partial / full).
- Ordinal logistic regression companion / AUC for binary outcomesA diagnostic-test developer computes D(Disease | TestScore) for an ordinal screening test.
When NOT to — use instead
- Symmetric concordance question (no predict-from direction)When direction doesn't matter substantively, use the symmetric Kendall τ instead. → Kendall's Tau (\u03c4-b concordance / tie-aware)
- Tie-blind concordance summaryGoodman-Kruskal γ ignores tied pairs entirely and gives a denominator-cleaner concordance measure. → Goodman-Kruskal's \u03b3 (ordinal, ignores ties)
- Two nominal variables (no order)Somers' d requires order on both axes. → Goodman-Kruskal \u03bb \u2014 nominal PRE association
- Continuous predictor and continuous outcomeSomers' d is for ordinal cross-tabs. → Pearson Correlation (linear association)
Hypotheses
Parameter tested: population Somers' Dxy and Dyx
Assumptions (and what to do if they fail)
Check: Report Dxy, Dyx, and |Dxy − Dyx|.
If violated: Identical Dxy and Dyx indicate symmetric marginal-tie structure — in that case τ-b or γ may be equivalent and more interpretable.
Check: Report all four; sign disagreement flags non-monotonicity.
If violated: Sign disagreement between the four measures indicates a non-monotonic pattern.
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