On the collapsibility of measures of effect in the counterfactual causal framework

A measure of association (such as the risk difference or the risk ratio) is said to be collapsible if the marginal measure of association is equal to a weighted average of the stratum-specific measures of association [1]. The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the literature [2]. In this paper, we argue that the concept of collapsibility can be made clearer if we frame the discussion in terms of causal effect measures based on counterfactual variables.

In all the examples, we are interested in the effect of a binary exposure A (e.g. a drug), on a binary outcome Y (e.g. a side effect). We use superscript to denote counterfactual variables [3]. For example, \(Y^{a=1}\) is an indicator for whether an individual would have got the outcome if, possibly contrary to fact, she had been exposed to the drug. We will make a distinction between measures of association, which compare the distribution of the outcome in the exposed with the distribution of outcome in the unexposed; and causal measures of effect, which compare the counterfactual distribution of the outcome under exposure (that is, the distribution of the outcome if everyone is exposed) with the counterfactual distribution of the outcome under the absence of exposure (that is, the distribution of the outcome if everyone is unexposed). For example, the associational risk difference is \(\text {Pr}(Y=1 \vert A=1) - \text {Pr}(Y=1\vert A=0)\) whereas the causal risk difference (RD) is \(\text {Pr}(Y^{a=1}=1)-\text {Pr}(Y^{a=0}=1)\). These effect measures may be defined within levels of covariates V.

We will adopt Pearl’s definition of collapsibility for measures of association [4]:

Newman [5] showed conditions under which the associational risk difference, risk ratio, and odds ratio are collapsible (or averageable) according to this definition. He also provided corresponding weights. Briefly, we note that: the associational risk difference is collapsible with weights \(\text {Pr}(V=v \vert A=1)\) if \(V\) is not associated with the outcome in the unexposed, or if \(V\) is not associated with the exposure; the associational risk ratio is collapsible with weights \(\text {Pr}(V=v \vert A=1)\times \text {Pr}(Y=1 \vert A=0, V=v)\) under similar conditions; and the associational odds ratio is collapsible with weights \(\text {Pr}(V=v \vert A=1)\times \text {Pr}(Y=0 \vert A=1,V=v)\times \frac{\text {Pr}(Y=1 \vert A=0, V=v)}{1-\text {Pr}(Y=1 \vert A=0,V=v)}\) under certain very limiting conditions, for example if \(\text {Pr}(Y=1\vert A=0, V=v)\) is equal for all values of v. A full discussion of the graphical and probabilistic conditions that lead to collapsibility under this definition is provided by Greenland and Pearl [6].

As we will show in the next section, under Definition 2 it is possible to provide results that guarantee collapsibility of certain effect measures for any data set. Collapsibility can then be understood as a mathematical property of the effect measure, rather than a consequence of certain graphical or probabilistic structures in the particular data set. Consequently, results from Greenland and Pearl do not apply under Definition 2, and measures of effect may be collapsible over \(V\) even if \(V\) is a confounder. Definitions 1 and 2 are not generally equivalent: a set of weights that satisfies Definition 1 may not satisfy Definition 2, and conversely a set of weights that satisfies Definition 2 may not satisfy Definition 1. The definitions are however equivalent if there is neither confounding unconditionally, nor confounding conditional on V  (i.e. if \(Y^{a} {\perp\mkern-10mu\perp} A\) and \(Y^{a} {\perp\mkern-10mu\perp} A \vert V\) for all values of a, respectively).

Finally, we consider a third related concept, discussed by Miettinen [7], who stated (correctly, but without proof) that the “standardized risk ratio” (SRR), which is constructed by standardizing the risk in the exposed and the risk in the unexposed separately with weights Pr(\(V=v\)) and reporting the ratio of these measures (Formula 4 in Miettinen), is equal to a weighted average of the stratum-specific risk ratios under the weights \(\text {Pr}(V=v)\times \text {Pr}(Y=1 \vert A=0, V=v).\) (Formula 6 in Miettinen). Since Miettinen’s SRR is equal to the causal risk ratio if there is no unmeasured confounding, Miettinen’s weights satisfy Definition 2 in the special case of no confounding conditional on V.

If the investigator intends to report an average of the stratum-specific effects as an estimate of the marginal effect, it is necessary to know not only that the effect is collapsible in principle, but also to construct appropriate weights, identify them from the data and apply them in the analysis. The weights for the causal risk ratio \(RR(-)\), \(\hbox {Pr}(V=v \vert Y^{a=0}=1)\), have a counterfactual variable in the conditioning event, and may not be identified from the data. However, we proceed to show that the weights are identified in the absence of unmeasured confounding, i.e if \(Y^{a=0} {\perp\mkern-10mu\perp} A \vert V\)

\(\text {Pr}(Y^{a=0}=1)\) is constant over v and can therefore be factored out of the weights. In the absence of unmeasured confounders, the weights Pr\((V=v \vert Y^{a=0}=1)\) are therefore equivalent to Miettinen’s weights \(\hbox {Pr}(V=v)\times \text {Pr}(Y=1 \vert A=0, V=v)\) as discussed earlier. These weights can be used both to control for confounding due to V, as suggested in a simple example in Table 2, or to standardize the results to a new target population (by taking the risk ratio from the study population, and information on \(\hbox {Pr}(V=v)\) and \(\hbox {P}(Y=1 \vert A=0, V=v)\) from the target population).

An alternative identification of the weights can be used if standardizing experimental results to a population in which everyone is unexposed. In such situations, \(Y^{a=0}= Y\) in all individuals by consistency, and the weights in the target population are identified as \(\text {Pr}(V=v \vert Y=1)\).

We have reviewed well-established results on the collapsibility of measures of association, and shown corresponding results for measures of causal effect. With these causal effect measures, one can disentangle the components of non-collapsibility that are due to the mathematical properties of the effect measure from the components that are due to structural bias and the probabilistic structure of the dataset. We have provided new, simple weights for the causal risk ratio, which guarantee collapsibility over arbitrary baseline covariates, and we showed that such weights do not exist for the causal odds ratio.

Our weights for the causal risk ratio \(RR(-)\) are equivalent to the weights previously discussed by Miettinen when there is no unmeasured confounding; in other words, in all situations where standardizing over V provides a valid estimate of the causal effect. Our formulation allows a simpler presentation of the weights and of the proofs. Furthermore, our formulation highlights pitfalls of using weighted averages: When conditioning on V, the correct weights cannot be estimated from the data if unmeasured confounding is present. In such scenarios, using erroneous weights can possibly amplify the bias that is caused by unmeasured confounding within the strata.

Finally, we note that in many cases we do not need to use results on collapsibility al all, because we can standardize the distributions of \(Y^{a=1}\) and \(Y^{a=0}\) separately. One way to do this is by reporting the overall marginal risk ratio \(RR(-)\) asSince this simple procedure does not depend on collapsibility, analogous procedures are valid for any effect measure, including the odds ratio.