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Let $Y(1)$ and $Y(0)$ be the potential outcomes for the same individual under well-defined interventions setting the binary exposure to 1 and 0. The average treatment effect, also called the average causal effect on the additive scale, isThe average is over the target population. This compares hypothetical outcomes under two interventions, rather than the observed difference $\mathbb{E}[Y\mid X=1]-\mathbb{E}[Y\mid X=0]$, which can be distorted by confounding. For binary $Y$, the displayed contrast is a causal risk difference. Consistency in causal inference and a well-defined intervention, including an appropriate no-interference in causal inference assumption, give these potential outcomes their meaning.

In potential outcome notation, a valid instrumental variable $Z$ must satisfy:

In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, includingEquivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating instrumental-variable independence; it is not a valid marginal instrument.

The no unmeasured confounding assumption, or conditional exchangeability, isAlso assume consistency of potential outcomes, no interference in causal inference, positivity in causal inference, and finite expectations. For $a\in \{0,1\}$, the law of total expectation, exchangeability, and consistency givePositivity ensures that the observed conditional means exist on the covariate support being averaged. Subtracting the two cases identifies the average treatment effect as

Let $Z$ indicate a recent birthday event, $A$ indicate a social gathering, and $Y$ denote subsequent household COVID-19 infection. The standard instrumental variable conditions are:

The usual consistency in causal inference, well-defined exposure, and absence of relevant interference in causal inference are also needed to interpret the result causally.