Potential treatment receipt

With binary assignment or encouragement $Z$, potential treatment receipt $D(z)$ is the treatment a unit would actually receive under assignment $z$. It differs from the assignment itself when there is noncompliance. Only $D=D(Z)$ is observed for that unit, under the usual consistency in causal inference convention.

The pair $(D(0),D(1))$ defines four principal strata: never-takers $(0,0)$, compliers $(0,1)$, defiers $(1,0)$ and always-takers $(1,1)$. A patient who takes a drug only when encouraged is a complier; one who obtains it under either assignment is an always-taker. Randomization of $Z$ does not itself randomize actual treatment receipt. If $D(1)\ge D(0)$ for every unit, defiers are excluded andunder randomized assignment. Thus the effect of encouragement on uptake identifies the complier fraction, not automatically the effect of receiving treatment on outcomes.^[ref]