Mantel-Haenszel Method

Mantel-Haenszel Method for Pooling Binary Outcomes Across Strata

The Mantel-Haenszel method combines several 2x2 tables, one per stratum or trial, into a single summary effect while keeping every comparison inside its own stratum. In a meta-analysis of a binary outcome it gives a pooled odds ratio, risk ratio or risk difference under a fixed effect model, and the Cochrane Handbook prefers it to inverse-variance weighting for fixed-effect analyses because it behaves better when events are few. In health technology assessment the pooled effect is applied to a baseline risk to give event probabilities in a decision tree or Markov model, so its weighting and zero-cell handling reach the cost-effectiveness result.

Stratified tables and the problem of confounding

Mantel and Haenszel published the method in 1959 for retrospective studies of disease, now called a case-control design. Comparing like with like within strata, such as age groups, removes the part of a crude association due to confounding by that factor.

In evidence synthesis each trial is a stratum. Collapsing trials into one table ignores the randomisation within each trial, and chapter 15 of the Cochrane Handbook warns that risk differences computed from aggregated totals may be seriously misleading when any trial used unbalanced randomisation. Collapsing can even reverse an association, the Simpson paradox.

Pooled odds ratio, risk ratio and risk difference

Each estimator is a weighted average of study-specific effects with weights suited to the measure. RevMan, the Cochrane review software, documents them for study $i$ with $a_i$ events and $b_i$ non-events among $n_{1i}$ people in the experimental group, $c_i$ events and $d_i$ non-events among $n_{2i}$ comparator patients, and $N_i = n_{1i} + n_{2i}$.

$$OR_{MH} = \frac{\sum_i a_i d_i / N_i}{\sum_i b_i c_i / N_i}, \qquad RR_{MH} = \frac{\sum_i a_i n_{2i} / N_i}{\sum_i c_i n_{1i} / N_i}, \qquad RD_{MH} = \frac{\sum_i w_i RD_i}{\sum_i w_i}$$

where $OR_{MH}$, $RR_{MH}$ and $RD_{MH}$ are the pooled odds ratio, risk ratio and risk difference, $RD_i = a_i / n_{1i} - c_i / n_{2i}$ is the risk difference in study $i$ and $w_i = n_{1i} n_{2i} / N_i$ is its weight. The odds ratio form is equivalent to weighting each study's odds ratio by $b_i c_i / N_i$.

The weights depend on counts and group sizes, not on an estimated variance for each study, which is why the method copes with sparse data. Greenland and Robins showed in 1985 that the Mantel-Haenszel risk ratio and risk difference stay consistent in sparse stratifications where maximum likelihood and weighted least squares estimators are biased, as Breslow had shown for the odds ratio, and that variance estimators valid in both sparse and large strata exist for all of them. RevMan uses the variance proposed by Robins, Breslow and Greenland in 1986 for the log odds ratio.

The Cochran-Mantel-Haenszel test of association

The Cochran-Mantel-Haenszel test examines the null hypothesis that exposure and outcome are independent within every stratum. Conditional on the margins of each table, $a_i$ follows a hypergeometric distribution, and the test sums observed-minus-expected counts across strata.

$$\chi^2_{MH} = \frac{\left[\sum_i \left(a_i - E_i\right)\right]^2}{\sum_i V_i}, \qquad E_i = \frac{n_{1i}(a_i + c_i)}{N_i}, \qquad V_i = \frac{n_{1i} n_{2i} (a_i + c_i)(b_i + d_i)}{N_i^2 (N_i - 1)}$$

where $E_i$ is the expected number of experimental-group events in stratum $i$ under the null hypothesis and $V_i$ is its variance. The statistic is compared with a chi-square distribution on one degree of freedom. The name also credits Cochran, who published a closely related statistic in 1954. The FDA's May 2023 guidance on covariate adjustment in randomised trials accepts Cochran-Mantel-Haenszel methods for binary endpoints when the target is a conditional effect assumed constant across levels of a discrete covariate.

Worked example with two trials of unequal allocation

All numbers in this example are illustrative. Two placebo-controlled trials of a new drug report hospital admission within 12 months. Trial A randomised 1 to 1 in a lower-risk population and trial B randomised 2 to 1 in a higher-risk population.

1. Pool the odds ratio. The terms $a_i d_i / N_i$ are 1,400 / 200 = 7 for trial A and 4,000 / 300 = 13.333 for trial B, and the terms $b_i c_i / N_i$ are 2,400 / 200 = 12 and 6,000 / 300 = 20.

$$OR_{MH} = \frac{7 + 13.333}{12 + 20} = \frac{20.333}{32} = 0.635$$

where 0.635 is the pooled odds ratio. The Robins, Breslow and Greenland standard error of the log odds ratio (-0.454) is 0.198, giving a 95% confidence interval of 0.431 to 0.936.

2. Pool the risk ratio and risk difference. The terms $a_i n_{2i} / N_i$ are 10 and 26.667, and the terms $c_i n_{1i} / N_i$ are 15 and 33.333.

$$RR_{MH} = \frac{10 + 26.667}{15 + 33.333} = \frac{36.667}{48.333} = 0.759$$

where 0.759 is the pooled risk ratio, with a 95% interval of 0.601 to 0.958. Both trials have a risk difference of -0.100, so the pooled risk difference is -0.100 whatever the weights.

3. Test the association. The expected drug-arm events are 25 in trial A and 86.667 in trial B, so observed minus expected is -5 and -6.667, with variances 9.422 and 16.425.

$$\chi^2_{MH} = \frac{(-11.667)^2}{9.422 + 16.425} = \frac{136.11}{25.847} = 5.27$$

where 5.27 on one degree of freedom gives a p-value of about 0.022.

4. Interpret. Within each trial the drug lowers the risk of admission by 10 percentage points. The collapsed table understates this at -0.067 because two thirds of treated patients, but half of placebo patients, came from the higher-risk trial.

From a pooled odds ratio to a model transition probability

The NICE manual (PMG36, section 4.6.16) describes quantifying the baseline risk of health outcomes with the comparator, which can come from observational studies, and then applying relative treatment effects from randomised trials to it. For an odds ratio the conversion passes through the odds.

$$p_1 = \frac{OR \times p_0}{1 - p_0 + OR \times p_0}$$

where $p_0$ is the event probability with the comparator over the period of the evidence, $OR$ is the pooled odds ratio and $p_1$ is the probability with the new treatment. Chapter 15 of the Cochrane Handbook uses the same step at an assumed comparator risk.

In the illustrative example, a comparator probability of 0.25 gives odds of 0.333, treated odds of 0.635 times 0.333, or 0.212, and a treated probability of 0.212 / 1.212 = 0.175, about 75 admissions avoided per 1,000. The pooled risk ratio instead gives 0.759 times 0.25, or 0.190, about 60 avoided per 1,000. A constant odds ratio and a constant risk ratio are different assumptions, so the choice needs justifying. The result becomes a transition probability, adjusted to the cycle length through rate to probability conversion when the cycle differs from the trial follow-up. For probabilistic sensitivity analysis the NICE manual (section 4.7.10) asks for distributions representing the precision of mean parameters, so the log odds ratio would be sampled with mean -0.454 and standard error 0.198.

Zero cells, rare events and continuity corrections

Rare outcomes give trials with no events in an arm. Chapter 10 of the Cochrane Handbook explains that software commonly adds 0.5 to every cell of such a table to avoid dividing by zero in inverse-variance methods, whereas Mantel-Haenszel methods need a correction only when the same cell is zero in every study. Many programs apply it to Mantel-Haenszel analyses anyway, and it usually biases estimates towards no difference.

Bradburn and colleagues found the Mantel-Haenszel odds ratio with a 0.5 correction among the most biased methods. Below 1% event rates the Peto odds ratio did best with balanced arms and moderate effects, and otherwise the uncorrected Mantel-Haenszel odds ratio, logistic regression and an exact method performed similarly. Sweeting, Sutton and Lambert found Mantel-Haenszel estimates with alternative corrections least biased across group-size imbalances and advocated sensitivity analysis across methods. Tsujimoto and colleagues reanalysed 885 Cochrane meta-analyses containing single-zero trials and found substantial differences with and without the correction in about 30%. In 2025 they reported that RevMan Web still could not omit it, so a rare-event estimate taken from a review may need recomputing for a model.

Assumptions and choices behind the pooled estimate

As a fixed-effect method it estimates one effect assumed common to all studies on the chosen scale. Chapter 10 of the Cochrane Handbook notes that fixed-effect methods such as Mantel-Haenszel give more robust estimates of the average effect when studies are few, at the cost of ignoring heterogeneity. A random effects model is the alternative when trials differ.

A common effect on one scale implies variation on the others. Chapter 15 notes that relative effects tend to be substantially more stable across risk groups than absolute effects, which favours pooling a ratio and converting it at the decision population's baseline risk. When the effect is not common, the formula above shows that the Mantel-Haenszel risk difference still estimates a weighted average of the stratum-specific risk differences.

Mantel-Haenszel method compared with nearby approaches

The choice among methods turns on sparsity and heterogeneity. Inverse-variance pooling suits large studies, the Peto odds ratio suits event rates below 1% with balanced arms and moderate effects, and logistic regression handles continuous covariates and sparse data without corrections. A random-effects analysis is the usual choice when heterogeneity is clear.

For continuous outcomes the counterpart is the mean difference, and categorical data with more than two categories need ordinal or multinomial methods. Network meta-analysis replaces pairwise pooling when several treatments must be compared, and time-to-event outcomes use the log-rank test and hazard ratios.

Limitations and common misreadings

The method is simple and well behaved. Most errors come from over-reading its output.

1. Adding a continuity correction by default. The estimators rarely need one, and a fixed 0.5 correction biases rare-event odds ratios towards no effect.
2. Pooling raw counts instead of stratifying. A crude table can understate or reverse the effect when allocation or baseline risk differs between trials.
3. Ignoring heterogeneity. A fixed-effect interval can be too narrow when trials differ.
4. Using an odds ratio as a risk ratio. At common event rates they diverge, and a model needs the conversion through odds.
5. Treating stratification as full adjustment. Only the stratifying factor is controlled.

Sources

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