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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
a_{1}
Te_{1}
Unexposed
c_{1}
Tu_{1}
T_{1}
Stratum 2
a_{2}
Te_{2}
c_{2}
Tu_{2}
T_{2}
The Mantel Haenszel risk ratio (RR_{MH}) can be computed as follows:
In which: a and c are the number of cases exposed and unexposed in a stratumTe and Tu are the total number exposed and unexposed in a stratumT is the total of a stratumThe sums ∑ are calculated for the i strata.
Returning to the example of the cohort study with vaccinated girls and boys:
Gender
Attack Rate
RR
Boys
819
1000
82%
4.52
Girls
181
18%
ref
Unvaccinated
814
950
86%
1.00
86
100
1050
Vaccinated
5
50
10%
0.95
95
900
11%
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:
RR_{MH} =
∑ (a_{i}Tu_{i}/T_{i})
=
[(814*100)/1050)] + [(5*900)/950]
82.2
0.99
∑ (c_{i}Te_{i}/T_{i})
[(86*950)/1050)] + [(95*50) / 950)]
82.8
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 is different 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 (OR_{MH}) can be computed as follows:
Stratified
Risk Factor
Controls
Totals
b_{1}
d_{1}
b_{2}
d_{2}
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 stratumThe 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 [1].
1. Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology 1999 Jan;10(1):37-48.
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