Which practical interventions does the do-operator refer to in causal inference? Illustration on the example of obesity and cancer
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For exposures X like obesity, no precise and unambiguous definition exists for the hypothetical intervention do(X = x_0). This has raised concerns about the relevance of causal effects estimated from observational studies for such exposures. Under the framework of structural causal models, we study how the effect of do(X = x_0) relates to the effect of interventions on causes of X. We show that for interventions focusing on causes of X that affect the outcome through X only, the effect of do(X = x_0) equals the effect of the considered intervention. On the other hand, for interventions on causes W of X that affect the outcome not only through X, we show that the effect of do(X = x_0) only partly captures the effect of the intervention. In particular, under simple causal models (e.g., linear models with no interaction), the effect of do(X = x_0) can be seen as an indirect effect of the intervention on W.
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