Categorical › Contingency / proportions

Cochran-Mantel-Haenszel (stratified 2×2)

The Cochran–Mantel–Haenszel test pools the within-stratum odds ratios of a 2 × 2 × k table into a single common OR while controlling for the stratifying variable.

What is Cochran-Mantel-Haenszel (stratified 2×2)?

When the same 2 × 2 exposure × outcome contingency is observed in several strata (centres, age bands, sex), CMH gives one combined test and one common OR. The pooled OR is the Mantel-Haenszel estimator OR_MH = Σ aᵢdᵢ/nᵢ / Σ bᵢcᵢ/nᵢ, which is consistent under sparse-data asymptotics.

CMH is more powerful than a single chi-square on the marginal (collapsed) table when the strata have different baseline outcome rates — the marginal table risks Simpson's paradox, while CMH respects the stratification.

A companion homogeneity test (Breslow-Day or Tarone) checks whether the per-stratum ORs are equal; if they're not, pooling into a single common OR isn't meaningful and you should report per-stratum effects.

When should I use Cochran-Mantel-Haenszel (stratified 2×2)?

  • Multi-centre trials with binary outcome.
  • Pooling across age / sex / region strata in observational data.
  • Meta-analysis of paired 2 × 2 tables (though dedicated meta-analytic tools usually do better).

What data does it need?

Binary exposure column + binary outcome column + categorical stratum column.

What does it report?

CMH χ² statistic, p-value, Mantel-Haenszel common OR + CI, Breslow-Day homogeneity test.

What does it assume?

  • Independent observations.
  • Strata are exchangeable (no effect modification by stratum) — check via Breslow-Day.

How do I interpret the result?

Breslow-Day p < 0.05 ⇒ the OR differs across strata ⇒ don't report a single pooled OR; show per-stratum effects + an interaction model instead.

See also

References

  • Mantel & Haenszel (1959). Statistical aspects of the analysis of data from retrospective studies of disease. JNCI 22(4).