Stuart's τ-c (rectangular ordinal tables)
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 nowRectangular-table ordinal association.
τ-c = 2m(C − D) / (n²(m − 1)), with m = min(r, c). The key advantage over τ-b: |τ-c| can reach ±1 for ANY r × c ordinal table; τ-b's maximum is bounded below 1 whenever r ≠ c, so reading τ-b on a rectangular table understates the strength of the actual association. Reports τ-c, τ-c-max-attainable (the table-shape ceiling), and the ratio of |τ-c| to that ceiling.
Worked example
How associated are two ordinal variables in a rectangular table?
Kendall's tau-c measures ordinal association and, unlike tau-b, can reach ±1 in non-square tables.
The variables were moderately associated, τc = .34, p = .002.
Kendall's tau-c indicated a moderate positive ordinal association, τc = .34, p = .002.
When to use it
- Two ordinal variables with unequal numbers of categoriesA consumer-research firm cross-tabulates customer satisfaction (1–5 Likert, 5 categories) against intent-to-recommend bracket (Detractor / Passive / Promoter, 3 categories).
- Ordered severity categories on rectangular gridsA clinician cross-tabulates BMI category (4 levels: underweight, normal, overweight, obese) against blood-pressure category (3 levels: normal, pre-hypertensive, hypertensive).
When NOT to — use instead
- Square ordinal table (r == c)On square tables τ-b and τ-c are nearly identical and τ-b is the more familiar reference. → Kendall's Tau (\u03c4-b concordance / tie-aware)
- Two continuous variables with no natural categorisationDon't bin continuous data just to use τ-c — you lose information; use Spearman or Pearson on the raw values. → Spearman's Rank-Order Correlation (monotonic / nonparametric)
- Asymmetric prediction (one variable is the 'predictor')Use Somers' d if the predict-from / predict-to direction matters substantively. → Somers' D (asymmetric ordinal, Dxy + Dyx)
- Two binary categorical variables (2×2)On a 2×2 table τ-c collapses to phi — use chi-square / Fisher's exact for the proper test. → Fisher's Exact Test
Hypotheses
Parameter tested: population Stuart's τ-c
Assumptions (and what to do if they fail)
Check: Report number of unique levels on each axis.
If violated: When either variable has many unique levels (essentially continuous), τ-c loses its interpretation as a bounded-table coefficient.
Check: All measures should share the same sign.
If violated: Sign disagreement suggests a non-monotone pattern.
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