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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.

An instrumental variable $G$ supplies variation in an exposure $B$ that can be used for causal inference about its effect on $Y$. Its three core requirements are:

Relevance, independence, and exclusion are the core instrumental-variable assumptions. They do not by themselves identify an arbitrary population-average effect. For a binary instrument and treatment, instrumental-variable monotonicity and the usual consistency in causal inference assumptions can identify a local average treatment effect; stronger effect-homogeneity or structural-model assumptions are needed for other effect interpretations.

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.

The birth-period comparison requires consistency, no interference between individuals, and a real discontinuity in sugar exposure at the end of rationing. Near the cutoff, potential diabetes outcomes must otherwise vary continuously with birth date: no simultaneous policy, nutritional, diagnostic, seasonal, or demographic discontinuity may affect them. Birth dates must not be manipulable around the cutoff, and survival and UK Biobank participation must not create differential selection bias. These assumptions make cutoff assignment locally exchangeable and give a regression discontinuity design an exclusion restriction in which birth period affects diabetes through sugar exposure.

The raw difference is generally not a causal effect. Existing UTCs may differ systematically from non-UTCs in location, admissions, pupil composition, prior attainment, resources, and the local apprenticeship market. These are possible confounders of school type and outcome. The comparison also weights the 43 UTCs and roughly 3500 controls very differently from a clearly defined target population. Without conditional exchangeability, positivity, and consistency, the difference mixes the effect of UTC status with selection bias.