Mantel Haenszel Test Model Overview

Mantel Haenszel Test is used when the same 2 × 2 association is observed repeatedly across several strata and the analyst wants a single adjusted test and a pooled effect estimate. Instead of collapsing everything into one table and ignoring the stratifying variable, the method respects the stratum structure and asks whether there is a common association after controlling for the stratification factor.

What question does the Mantel Haenszel Test answer?

In this worked example, the research question is whether the association between sex and G3 pass remains after adjusting for studytime. Each study-time level forms a separate 2 × 2 table, and the Mantel Haenszel procedure combines them into a common adjusted result.

What does the common odds ratio mean?

The Mantel Haenszel common odds ratio summarizes the adjusted odds association across strata. A value of 1 indicates no association after adjustment, a value below 1 indicates lower adjusted odds in the numerator group, and a value above 1 indicates higher adjusted odds. In the current workbook, the male-versus-female adjusted common odds ratio is 0.7999.

Worked example at a glance

The four study-time strata contain separate male/female by pass/fail tables. The verified Mantel Haenszel statistic is 0.9333 with p = 0.3340. The common odds ratio is 0.7999 with a 95% confidence interval of 0.5102 to 1.2542. Because the interval includes 1 and the p-value is well above .05, the adjusted association is not statistically significant.

Quick Answer: Mantel Haenszel Test Result

The Mantel Haenszel Test combines four study-time-specific 2 × 2 tables to estimate one adjusted association between sex and pass status. The common-effect estimate is below 1, but the evidence is not statistically significant.

Table of Contents

1. Research question, hypotheses and data design
2. When to use the Mantel Haenszel Test
3. Mantel Haenszel Test formula and calculation
4. Variables and data dictionary
5. Stratum tables and complete results
6. Mantel Haenszel versus chi-square and crude analysis
7. Mantel Haenszel Test calculator workflow
8. Python and R chart interpretations
9. Python, R, SPSS and Excel workflows
10. Expandable software code
11. Assumptions, limitations and advanced interpretation
12. APA reporting for the Mantel Haenszel Test
13. Common mistakes and how to correct them
14. Reports and worked Excel download
15. Related Salar Cafe guides
16. Frequently asked questions
17. Conclusion

Research Question, Hypotheses and Data Design

Research question

Is there an association between sex and final-pass status after adjusting for differences in study-time strata?

Null and alternative hypotheses

Why this is a stratified analysis

Each study-time level creates its own 2 × 2 table. A simple chi-square test of independence on the collapsed table would ignore possible confounding by study time. The Mantel Haenszel Test preserves the stratum structure and produces an adjusted estimate.

Current data design

The workbook contains 649 students. The observed strata are studytime 1, 2, 3 and 4. Within each stratum, the table is ordered as male pass a, male fail b, female pass c and female fail d. That orientation matters because it determines the direction of the odds ratio.

When to Use the Mantel Haenszel Test

Simple selection logic

Useful companion topics are chi-square test, chi-square test of homogeneity, Cramér’s V and logistic regression.

Mantel Haenszel Test Formula and Calculation

The Mantel Haenszel procedure uses each stratum's observed cell count and expected count under the null, then combines them into one chi-square statistic. The common odds ratio is computed from weighted cross-products.

Step 1: List the 2 × 2 tables

Step 2: Compute stratum expectations and variances

Step 3: Combine the deviations

The sum of the deviations is −4.2193 and the sum of the variances is 19.0760. Substituting into the formula gives the verified Mantel Haenszel chi-square:

Step 4: Obtain the p-value

The Mantel Haenszel statistic is compared with a chi-square distribution with 1 degree of freedom. The verified p-value is 0.3340182, which is not statistically significant at α = .05.

Step 5: Compute the common odds ratio

Using the weighted cross-product formula, the common odds ratio is:

The 95% confidence interval is 0.5101770 to 1.2541894. Because the interval includes 1, the adjusted common effect is not statistically distinguishable from no association.

Variables and Data Dictionary

Stratum-specific pass rates

The raw strata show some variation. In strata 1, 2 and 4 the male odds are lower, whereas stratum 3 shows a male odds ratio above 1. This is exactly why a common-effect procedure should be accompanied by a heterogeneity check such as the Breslow Day Test.

Worked Stratified 2 × 2 Tables and Complete Results

Crude versus adjusted story

The unstratified male-versus-female odds ratio from the collapsed table is 0.6486, whereas the adjusted Mantel Haenszel common odds ratio is 0.7999. The move toward 1 after adjustment suggests that study time explains part of the crude association. That is exactly the kind of confounding pattern for which a stratified adjusted analysis is useful.

Mantel Haenszel Test Versus Crude Chi-Square and Logistic Regression

The Mantel Haenszel Test sits between a simple chi-square analysis and a full regression model. It is more informative than a crude association because it controls one categorical confounder, but it is simpler and more transparent than a multivariable model.

Mantel Haenszel Test Calculator: Step-by-Step Workflow

A correct Mantel Haenszel Test calculator should accept multiple 2 × 2 tables, compute the expected counts and variances within each stratum, return the MH statistic, and report the common odds ratio with a confidence interval.

Calculator inputs for the worked example

Calculator steps

• Enter each 2 × 2 table separately rather than collapsing them.
• Keep the same cell orientation in every stratum.
• Review the stratum-specific odds ratios before trusting the common odds ratio.
• Read the common OR and its confidence interval together with the p-value.
• If heterogeneity appears strong, interpret the pooled effect cautiously and consider a separate Breslow Day Test.

Six Python and R Mantel Haenszel Test Chart Interpretations

The Python and R workflows illustrate the same underlying Mantel Haenszel story: how each stratum behaves, how large the stratum tables are, whether the stratum-specific effects look homogeneous, and what the final common-effect result means.

Chart 1: Stratum odds ratios and common effectPython + R

Pattern

Most strata have odds ratios below 1, but one stratum is above 1 and one has a zero cell that forces the raw stratum OR to 0.

Key values

Stratum ORs are 0.7210, 0.9271, 1.8261 and 0.0000. The common OR is 0.7999.

Interpretation

The common-effect summary lies below 1, but the variation across strata reminds us to think about heterogeneity before overinterpreting the pooled result.

Why it matters

This chart is the fastest visual summary of whether a pooled odds ratio is directionally sensible.

Next step: compare the effect pattern with the stratum sample sizes and counts.

Chart 2: Stratum counts and pass totalsPython + R

Pattern

Stratum 2 is the largest and therefore has the strongest influence on the pooled estimate.

Key values

Stratum totals are 212, 305, 97 and 35, with the fourth stratum contributing comparatively little information.

Interpretation

The common effect is not a simple average of the stratum ORs; larger, more informative strata receive more weight.

Why it matters

Understanding the stratum sizes helps explain why the common OR ends up near 0.80 rather than being dominated by the extreme small-stratum OR.

Next step: inspect failure counts and sparse cells because they influence odds-ratio stability.

Chart 3: Heterogeneity profile and fail countsPython + R

Pattern

The fourth stratum contains a zero female-fail cell, and the third stratum has very few male failures.

Key values

Male fails are 32, 15, 1 and 2; female fails are 18, 26, 6 and 0.

Interpretation

Sparse failure cells can create unstable or extreme raw stratum odds ratios, which is why a pooled adjusted estimate can be more informative than reading each OR in isolation.

Why it matters

This is also why the common-effect assumption should be checked and why continuity decisions matter in small strata.

Next step: connect the raw tables with absolute log-effect size and stratum-size context.

Chart 4: Absolute log effects and stratum sizesPython + R

Pattern

The most extreme apparent effect does not occur in the largest stratum.

Key values

Stratum 3 has an OR above 1.8 but only 97 observations, whereas stratum 2 has 305 observations and an OR close to 0.93.

Interpretation

Magnitude alone is not enough; the pooled result balances effect size and information content across strata.

Why it matters

This chart pair explains why the adjusted common OR is moderate and why the significance test is weak.

Next step: compare the signed log-OR pattern with the confidence interval for the common effect.

Chart 5: Log odds ratio by stratum and common-effect intervalPython + R

Pattern

The common-effect interval crosses 1, confirming the non-significant adjusted result.

Key values

Common OR = 0.7999 with 95% CI 0.5102 to 1.2542.

Interpretation

The data are compatible with a meaningful reduction in male pass odds, but also compatible with little or no adjusted difference.

Why it matters

The interval communicates uncertainty better than the p-value alone.

Next step: finish with the summary panel that integrates the statistic, p-value and pooled effect.

Chart 6: Result summaryPython + R

Pattern

The summary chart condenses the adjusted common-effect analysis into one compact result panel.

Key values

MH χ² = 0.9333, p = 0.3340, common OR = 0.7999 and 95% CI 0.5102 to 1.2542.

Interpretation

The adjusted evidence is not statistically significant, and the effect estimate remains uncertain.

Why it matters

This is the best visual for reports, teaching slides and quick communication of the final conclusion.

Next step: connect the chart results to the software workflows and reporting templates below.

Mantel Haenszel Test in Python, R, SPSS and Excel

Expandable Code for the Mantel Haenszel Test

tables = [
    [[91, 32], [71, 18]],
    [[92, 15], [172, 26]],
    [[21, 1], [69, 6]],
    [[12, 2], [21, 0]]
]
# compute stratum-specific values
# combine them into the MH statistic
# compute the common odds ratio and confidence interval

# create a 2 x 2 x 4 array
# run mantelhaen.test(x)
# extract the common odds ratio and confidence interval
# summarize the strata and create plots

E(a) = ((a+b)*(a+c))/n
Var(a) = ((a+b)*(c+d)*(a+c)*(b+d)) / (n^2*(n-1))
MH_stat = (SUM(a_minus_E)^2) / SUM(var_a)
OR_MH = SUM(a*d/n) / SUM(b*c/n)

Assumptions, Limitations and Advanced Interpretation

The biggest interpretive safeguard is to separate common-effect estimation from homogeneity checking. The Mantel Haenszel Test can estimate a common adjusted association, but it does not by itself prove that a single common odds ratio describes every stratum equally well. That is why a separate Breslow Day Test is usually recommended.

APA Reporting for the Mantel Haenszel Test

Common Mantel Haenszel Test Mistakes and How to Correct Them

Mantel Haenszel Test Reports and Worked Excel Download

Related Salar Cafe Guides

Mantel Haenszel Test Conclusion

Mantel Haenszel Test is one of the most useful stratified tools in categorical data analysis because it adjusts a binary exposure-outcome association for one categorical confounder without hiding the data structure. In this worked example, four study-time-specific 2 × 2 tables were combined into one adjusted analysis. The resulting common odds ratio was 0.7999, and the Mantel Haenszel statistic was 0.9333 with p = 0.3340, so the adjusted association was not statistically significant.

The larger lesson is methodological as well as numerical: analysts should move beyond a crude 2 × 2 table when a meaningful stratifier exists, but they should also check whether a common pooled effect is appropriate before making strong substantive claims. If you want to extend the same topic, continue with Breslow Day Test, Chi Square Calculator, Categorical Data Analysis in Python and Logistic Regression.