Stuart-Maxwell / Bowker (paired k-category)
Coming soonCategorical
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 nowTests of symmetry and marginal homogeneity for paired categorical outcomes with ≥ 3 categories.
Bowker's test evaluates whether the full k × k contingency is symmetric (n_ij = n_ji for all i ≠ j); Stuart-Maxwell evaluates the weaker condition of marginal homogeneity (Time-1 row marginals = Time-2 column marginals). Reports both Bowker χ²(k(k−1)/2) and Stuart-Maxwell χ²(k−1). For k = 2 both reduce to McNemar's uncorrected χ². The canonical paired-categorical test when the outcome has 3+ levels — common in pre/post severity ratings, inter-rater agreement, and any longitudinal categorical assessment.
Worked example
Did marginal proportions shift across paired categorical ratings (3+ categories)?
The Stuart-Maxwell test checks marginal homogeneity for a paired square table with three or more categories — the multi-category McNemar.
The marginal distributions shifted between the two occasions, χ²(2) = 8.9, p = .012.
A Stuart-Maxwell test showed a significant change in the marginal distributions, χ²(2) = 8.9, p = .012.
When to use it
- Paired ordered or nominal outcome (k ≥ 3)125 patients are rated as mild / moderate / severe before and after 12 weeks of physical therapy.
- Inter-rater categorical agreement (k ≥ 3)80 X-ray scans rated as normal / mild / severe by two radiologists.
When NOT to — use instead
- Independent groups (different subjects per arm)Stuart-Maxwell requires PAIRING. → Chi-Square Test of Independence
- Binary paired outcome (k = 2)Stuart-Maxwell collapses to McNemar at k = 2. → McNemar's Test (paired 2\u00d72)
- 3+ related binary measurements (not k-category)Cochran's Q is the right test for ≥3 BINARY repeated measures. → Cochran's Q Test (repeated binary)
- Need ordinality / covariate adjustmentStuart-Maxwell ignores ordinality and doesn't adjust covariates. → Ordinal Logistic \u2014 ordered categories + Brant proportional-odds test
Hypotheses
Parameter tested: set of k marginal proportions {π_j·, π_·j}
Assumptions (and what to do if they fail)
Check: Flag when any off-diagonal pair has a total < 5.
If violated: Small off-diagonal totals make the χ² approximation for Bowker and the Wald form for Stuart-Maxwell unreliable.
Check: Sum all off-diagonal cells; report.
If violated: Perfect concordance (all subjects on the diagonal) — the tests are undefined.
Ready to run a Stuart-Maxwell / Bowker (paired k-category) on your own data?
Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.
Run this test →