= 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 and
$$
\mathbb E[D\mid Z=1]-\mathbb E[D\mid Z=0]=\mathbb P\bigl(D(0)=0,D(1)=1\bigr)
$$
under randomized assignment. Thus the effect of encouragement on uptake identifies the complier fraction, not automatically the effect of receiving treatment on outcomes.<Principal Stratification in Causal Inference by Constantine E. Frangakis and Donald B. Rubin>[Definitions of potential intermediate variables and principal strata]{ref}
Back to article page
