The Mantel Haenszel Method

The most popular method used to compute a weighted risk ratio or odds ratio is the Mantel-Haenszel method, which can be used for risk ratios or rate ratios.

From the following table

		Cases	Total
Stratum 1	Exposed	a1	Te1
	Unexposed	c1	Tu1
	Total		T1
Stratum 2	Exposed	a2	Te2
	Unexposed	c2	Tu2
	Total		T2

The Mantel Haenszel risk ratio (RRMH) can be computed as follows:

In which: a and c are the number of cases exposed and unexposed in a stratum Te and Tu are the total number exposed and unexposed in a stratum T is the total of a stratum The sums ∑ are calculated for the i strata.

Returning to the example of the cohort study with vaccinated girls and boys:

Crude RR

Gender	Cases	Total	Attack Rate	RR
Boys	819	1000	82%	4.52
Girls	181	1000	18%	ref

Stratified RRs

	Gender	Cases	Total	Attack Rate	RR
Unvaccinated	Boys	814	950	86%	1.00
	Girls	86	100	86%	ref
			1050
Vaccinated	Boys	5	50	10%	0.95
	Girls	95	900	11%	ref
			950

In our example, the crude measure of effect (the risk ratio) was 4.5. The weighted measure of effect calculated with the Mantel Haenszel method is close to 1. It is obtained as follows:

The relative difference between the weighted and the crude measures of effect is more than 15% (4.5/0.99 *100 = 450%) therefore suggesting that, in our hypothetical study, vaccination is confounding (is a confounding factor for) the association between gender and disease. Had a stratified analysis been omitted, the data may lead to the conclusion that being a boy was a risk factor for the disease.

The adjusted RR 0.99 is presented, which concludes that this is the measure of association between gender and disease. This differs from effect modification, where two RRs would be presented.

Mathematically, the adjusted estimate is a weighted average of the stratum-specific measures of the risk ratio. It will, therefore, always lie within the range of the stratum-specific measures of the effect. (i.e. in the example above; 0.99 is between the range 0.95 and 1.00 - the stratum-specific RRs).

For a case-control study, the Mantel Haenszel odds ratio (ORMH) can be computed as follows:

Stratified	Risk Factor	Cases	Controls	Totals
Stratum 1	Exposed	a1	b1
	Unexposed	c1	d1
				T1
Stratum 2	Exposed	a2	b2
	Unexposed	c2	d2
				T2

In which: a and c are the number of cases exposed and unexposed in a stratum, b and d are the number of controls exposed and unexposed in a stratum. T is the total for a stratum The sums ∑ are calculated for the i strata.

It can become customary to 'eyeball' the data: comparing the crude measure to the range of the stratum-specific measures. If the crude measure is not included in the range between stratum-specific measures, confounding may exist.

A watertight method for identifying confounding variables exists. It requires the construction of a causal diagram summarizing the knowledge and assumptions between all exposures, confounders and disease outcome, which is then analysed using graphical algorithms .