Mantel-Haenszel and Stratified Analysis

Stratified analysis controls for a confounding categorical variable by splitting the data into strata defined by that variable, analysing the exposure-outcome association within each stratum, and then combining the strata into a single summary. The Mantel-Haenszel method is the classical procedure for doing this with a series of 2×2 tables: it provides both a pooled test of association and a weighted summary estimate of the odds ratio or risk ratio adjusted for the stratifying variable.

Definition

Mantel-Haenszel stratified analysis combines a series of 2×2 tables — one per level of a stratifying (confounding) variable — into a single weighted summary measure of association and an associated test, yielding an estimate of the exposure-outcome effect adjusted for the stratifying variable.

Scope

This entry covers why stratification controls confounding, how the Mantel-Haenszel summary estimator weights the per-stratum tables, the accompanying chi-squared test of association across strata, the standard variance estimator used for confidence intervals, and how stratification reveals effect modification when stratum-specific estimates differ. It frames these as methods for analysing data, not as clinical guidance.

Core questions

• How does splitting the data into strata remove the influence of a confounding variable?
• How are the separate per-stratum tables weighted and combined into one summary odds ratio or risk ratio?
• How is the pooled association tested, and how is a confidence interval obtained?
• When should the strata not be pooled because the effect differs across them (effect modification)?

Key concepts

• Stratification to control confounding
• Per-stratum 2×2 tables
• Weighted pooled (summary) estimate
• Mantel-Haenszel test of association
• Variance estimator for the summary estimate
• Homogeneity of effect across strata
• Effect modification (interaction)
• Crude versus adjusted estimate

Mechanisms

The data are divided into strata defined by the levels of a potential confounder, giving one 2×2 table of exposure against outcome per stratum. Within each stratum the association is free of confounding by that variable because the variable is held constant. The Mantel-Haenszel summary odds ratio is a weighted average of the stratum-specific cross-products, with weights that give more influence to larger and more informative strata; an analogous estimator exists for the risk ratio. A single chi-squared test sums the observed-minus-expected exposed cases across strata to test the pooled association while preserving the stratification. Confidence intervals use a variance estimator that is valid both when there are many small strata (sparse data) and when there are few large strata, the dual-consistency property established by Robins, Breslow and Greenland; Greenland and Robins gave related results for sparse follow-up data. If the stratum-specific estimates are similar, pooling is appropriate and the summary estimate is the confounding-adjusted effect; if they differ substantially, the stratifying variable is an effect modifier and a single pooled number can be misleading.

Clinical relevance

Adjusted associations in observational health research are frequently produced or checked by stratified Mantel-Haenszel analysis, the transparent non-model precursor to regression adjustment, so understanding it clarifies how confounding is handled and how a crude estimate becomes an adjusted one. It is a method for analysing and interpreting evidence and not a basis for individual diagnostic or treatment decisions.

Epidemiology

Mantel-Haenszel methods are a staple of epidemiologic analysis for cohort and case-control data and are also the basis of the fixed-effect Mantel-Haenszel method widely used to pool 2×2 tables across studies in meta-analysis. They remain the canonical illustration of confounding control before, or alongside, logistic and Poisson regression.

History

Mantel and Haenszel introduced their stratified test and summary estimator in 1959 in the context of retrospective (case-control) studies of disease, and the method quickly became central to chronic-disease epidemiology, codified in Breslow and Day's 1980 monograph. The variance estimators needed for valid confidence intervals across both sparse and large-strata settings were supplied by Greenland and Robins (1985) and Robins, Breslow and Greenland (1986), completing the inferential framework.

Debates

Pooling versus reporting effect modification

When stratum-specific estimates diverge, a single Mantel-Haenszel summary can hide a real interaction; analysts must judge whether the strata are homogeneous enough to pool or whether stratum-specific effects should be reported instead.

Key figures

• Nathan Mantel
• William Haenszel
• Sander Greenland
• James Robins
• Norman Breslow
• Kenneth Rothman

Seminal works

• mantel-haenszel-1959
• greenland-robins-1985
• robins-breslow-greenland-1986

Frequently asked questions

What does the Mantel-Haenszel method control for?

It controls for a confounding categorical variable by analysing the exposure-outcome association within strata of that variable and then combining the strata, so the summary estimate is adjusted for the stratifying variable.

What if the odds ratio differs across strata?

Markedly different stratum-specific estimates indicate effect modification, meaning the association genuinely varies by the stratifying variable; in that case a single pooled summary can be misleading and the stratum-specific results should be reported.