9.2 Effect modification
Definition 9.2.1. A variable \(C\) is an effect modifier on a given scale if the measure of association between \(E\) and \(D\) on that scale differs between the levels of \(C\). The measure is then said to be heterogeneous; if it is the same in every stratum it is homogeneous.
Confounding and effect modification are different things. Confounding is a bias, to be removed. Effect modification is a feature of the world, to be described: if a drug works in men but not in women, the most useful report says so, and a single average hides it. A variable can be a confounder, an effect modifier, both, or neither.
Effect modification depends on the scale of measurement.
Proposition 9.2.2. Suppose the exposure has the same risk ratio \(\theta \neq 1\) in two strata whose baseline risks \(p_{01} \neq p_{02}\) differ. Then the risk differences in the two strata differ. Conversely, a common risk difference with different baseline risks implies different risk ratios.
Proof. The risk differences are \((\theta - 1)p_{01}\) and \((\theta - 1)p_{02}\), unequal since \(\theta \neq 1\) and \(p_{01} \neq p_{02}\). For the converse, a common difference \(\delta \neq 0\) gives risk ratios \(1 + \delta /p_{01}\) and \(1 + \delta /p_{02}\), which differ. □
So “is there effect modification?” has no answer until the scale is named. An exposure that triples the risk in a stratum with baseline risk \(0.01\) and in one with baseline \(0.1\) has a homogeneous risk ratio of \(3\) but risk differences of \(0.02\) and \(0.2\). For public health the second scale often matters more: removing the exposure prevents ten times as many cases per person in the high-risk stratum.
When the effect modifier is itself a second exposure, a standard summary of departure from additivity of risks is the relative excess risk due to interaction, \[ \text {RERI} = \RR _{11} - \RR _{10} - \RR _{01} + 1 , \] where \(\RR _{ij}\) is the risk ratio for exposure levels \((i, j)\) relative to \((0, 0)\). It is \(0\) when the excess risks of the two exposures add, and positive when the joint excess exceeds the sum. With \(\RR _{10} = 2\), \(\RR _{01} = 2.5\) and \(\RR _{11} = 6\), the risk ratios combine almost multiplicatively (\(2 \times 2.5 = 5\)) and \(\text {RERI} = 2.5\): more than additive. Additive interaction is the scale that bears on how many cases joint exposure causes, and so on which groups an intervention should target.
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