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• W1992875249 abstract "To the Editor: In our recent article,1 we showed how it was possible to decompose the joint effects of two exposures, G and E, into 3 components: (1) the effect attributable to G alone, (2) the effect attributable to E alone, and (3) the effect due to their interaction. In this letter, we discuss some alternative decompositions for attributing effects to interactions. With 2 binary exposures, G and E, and a binary outcome Y, let pge = P(Y = 1 | G = g, E = e) denote the probability of the outcome when G = g and E = e. For simplicity, we assume here that the effects of both exposures on the outcome are unconfounded, or that analysis is done within strata of covariates C that suffice to control for confounding. We noted in our article1 that the joint effect of both exposures, p11 − p00, could be decomposed as follows: where the first component, (p10 − p00), is the effect of G alone; the second component, (p01 − p00), is the effect of E alone; and the third component, (p11 − p10 − p01 + p00), is a measure of interaction on the additive scale. We could then also compute the proportion of the joint effects due to G alone, , due to E alone, , and due to their interaction, . On the risk ratio scale, if we let denote the risk ratios for the effects of the exposures, we noted that it was possible to decompose the excess relative risk for both exposures, RR11−1, into (1) the excess relative risk for G alone, (2) for E alone, and (3) the excess relative risk due to interaction, RERI. We have that: We could then likewise compute the proportion of the effect due to G alone, , due to E alone, , and due to their interaction and we have for the 3 proportion measures: We noted that this final measure, , which we will denote by AP*, was the proportion of the joint effect attributable to interaction. This differs from what is often called the attributable proportion (AP) due to interaction, defined by , which instead captures the proportion of disease, in the doubly exposed group, which is due to interaction. Both measures were described in Rothman’s early textbook,2 but has come to be the more common measure in the literature. Both measures may be of interest, but they capture different proportion measures: the proportion of the joint effect attributable to interaction for , versus the proportion of disease in the doubly exposed group that is due to interaction for . The measure will be 100% if neither exposure has an effect in the absence of the other, whereas the measure may be less than 100% in this setting because some of risk of disease may be present even in the absence of both exposures.1,3 If one is interested in the proportion of the disease among the doubly exposed due to interaction, ie, in , then an alternative decomposition for this doubly exposed risk, p11, is possible. For the risk in the doubly exposed group, we have: and for relative risks we have: and from this, dividing the decomposition by RR11 we can compute four proportion measures: The first component, , is the proportion of the risk in the doubly exposed that is due to neither of the exposures; the second component, , is that due to the first exposure alone; the third component, , is that due to the second exposure alone; and the fourth component, , is that due to their interaction. The 4 components will always sum to 100% and these 4 components provide a decomposition for the risk in the doubly exposed. Whereas the decomposition in our paper1 for the joint effects of both exposures had 3 components, as above, the decomposition for the risk in the doubly exposed has 4 components. Finally, we might also be interested in the decomposing not simply the risk of disease in the doubly exposed into the proportions attributable to the various exposures and their interaction, but we might instead perhaps be interested in similar measures for the overall risk of disease. We show in the eAppendix (https://links.lww.com/EDE/A884) that the overall risk of disease in the population, P(Y = 1), can be decomposed as follows: If we divide this by P(Y = 1), we have the following 4 proportion measures: where first component, , is the proportion of the overall risk of disease that is due to neither of the exposures; the second component, , is the proportion of risk due to the first exposure alone; the third component, , is the proportion of risk due to the second exposure alone; and the fourth component , is the proportion of the overall risk that is due to the interaction between the 2 exposures. In the eAppendix (https://links.lww.com/EDE/A884), we show how these various decompositions can be carried out using logistic regression, and how control can be made for confounding variables; we give SAS and Stata code to estimate the proportions and their standard errors and 95% confidence intervals. We describe how this can be done with case–control data as well. We believe these various decompositions provide intuitive measures to assess the importance of interaction. In studies of interaction, one or several of these decompositions could be reported. Each of these decompositions may be of interest, but their interpretations are somewhat different according to what is being decomposed: either the joint effects, , in decomposition (1); or the risk among the doubly exposed, p11, in decomposition (2); or the overall risk of disease, P(Y = 1), in decomposition (3). The first decomposition is arguably of more etiologic relevance as it concerns effects; the second and third decompositions are perhaps of more relevance from a public health perspective as they concern the proportions of disease. In the genetic epidemiology example from our recent article, with a genetic variant on chromosome 15 as G, smoking as E, and lung cancer as the outcome, the proportion of the joint effects attributable to interaction was 48% (95% CI = 33%, 62%); the proportion of disease in the doubly exposed group attributable to interaction is somewhat lower at 40% (28%, 52%) because some cases of lung cancer are not due to either exposure; and finally the overall proportion of disease attributable to interaction is only 22% (14%, 30%), which is again somewhat lower still as, in this example, only about a third of the population have both exposures present. ACKNOWLEDGEMENTS Tyler J. VanderWeele was supported by National Institutes of Health Grant ES017876. Tyler J. VanderWeele Eric J. Tchetgen Tchetgen Departments of Epidemiology and Biostatistics Harvard School of Public Health Boston, MA [email protected]" @default.
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• W1992875249 date "2015-05-01" @default.
• W1992875249 modified "2023-09-25" @default.
• W1992875249 title "Alternative Decompositions for Attributing Effects to Interactions" @default.
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• W1992875249 doi "https://doi.org/10.1097/ede.0000000000000263" @default.
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