# The fundamental issue in determining entity level causality

Let $Y_i(t)$ denote the outcome, where $i$ denotes the entity being examined (for instance, the individual) and $t$ denotes the treatment. Individual Treatment Effect (ITE) is then defined as the difference in the outcome between the two scenarios: with treatment and without treatment. Mathematically, it is represented as $f(x)$.

However, there is a fundamental problem with such a computation. Typically, we cannot have both the treatment and control (no treatment) condition done on the same subject. For instance, if we are interested in understanding the impact of sunlight on skin acne, we can allocate an individual to just one condition: either treatment (exposure to sunlight) or control (no exposure to sunlight). As such, an entity level (individual) treatment effect cannot be computed. Instead, the approach commonly used adopts an average perspective across multiple individuals to calculate the average treatment effect (ATE).

The third (in)equation above, $ATE ≠ \mathbb{E}[Y|T=1]-\mathbb{E}[Y|T=0]$, stems from the rationale that association does not imply causation.

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