= Principal strata
{synonym}

A <principal stratum> is defined by the joint <potential outcomes> of an intermediate variable under all compared <treatments>, rather than by its one observed value. For binary assignment $A$ and intermediate response $S$, a basic <principal stratum> is the set of units with a specified pair $(S(0),S(1))$. Each unit belongs to the same <principal stratum> whichever assignment it receives, although only $S(A)$ is observed.<Principal Stratification in Causal Inference by Constantine E. Frangakis and Donald B. Rubin>[Summary and section 2: cross-classification by joint intermediate potential values]{ref}

For example, let $S$ record whether a unit takes a <treatment> when offered encouragement. The pairs $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$ define <never-takers>, <compliers>, <defiers> and <always-takers>, respectively. An <instrumental-variable monotonicity> assumption rules out <defiers>. A comparison conditional on observed receipt $S=1$ generally mixes <compliers> with <always-takers> and can mix them in different proportions across assignment arms. It therefore does not automatically identify a <causal effect> within either <principal stratum>.


 Principal stratum (source code)