 Principal stratum

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.^[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.