Goodman-Kruskal's γ (ordinal, ignores ties)
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 nowOrdinal concordance measure for two ordered variables.
γ = (C − D) / (C + D) — the proportion of concordant minus discordant pairs AMONG UNTIED pairs. Range [-1, +1]. γ ignores ties entirely (they're excluded from the denominator), which makes |γ| larger than |τ-b| whenever ties are common — a known interpretive caveat the engine surfaces in the primary-voice qualifier.
Worked example
How strongly are two ordinal variables associated?
Goodman-Kruskal gamma measures ordinal association from concordant versus discordant pairs (ignoring ties).
The variables were strongly associated, γ = .58, p < .001.
Goodman-Kruskal gamma indicated a strong positive ordinal association, γ = .58, p < .001.
When to use it
- Concordance strength among ordered categoriesA health researcher cross-tabulates self-rated health (poor / fair / good / excellent) against physical activity level (sedentary / moderate / active) in 400 adults.
- Strength-of-association after χ² rejected independenceCross-tab of treatment-arm tier (1 / 2 / 3 / 4) against treatment-response category (no response / partial / full) in a clinical trial.
When NOT to — use instead
- One nominal and one ordinal variableγ requires order on both axes. → Goodman-Kruskal \u03bb \u2014 nominal PRE association
- Asymmetric prediction (one variable predicts the other)γ is symmetric. → Somers' D (asymmetric ordinal, Dxy + Dyx)
- Tie-heavy data where τ-b is the more conservative summaryγ inflates relative to τ-b when ties dominate; if the audience expects a tie-aware coefficient, use τ-b. → Kendall's Tau (\u03c4-b concordance / tie-aware)
- Continuous variables with linear associationγ is for ordinal × ordinal cross-tabs. → Pearson Correlation (linear association)
Hypotheses
Parameter tested: population Goodman-Kruskal γ
Assumptions (and what to do if they fail)
Check: Report (C + D) as a percentage of N₀.
If violated: When almost all pairs are tied, γ has very few pairs contributing — large standard error, wide CI.
Check: Check sign agreement between γ, τ-b, and Dxy.
If violated: Disagreement in sign between γ, τ-b, and Dxy signals non-monotonic behaviour.
Ready to run a Goodman-Kruskal's γ (ordinal, ignores ties) on your own data?
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