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.

A never-taker is a unit whose potential treatment receipts are $D(0)=D(1)=0$: it receives no treatment under either value of a binary assignment or encouragement $Z$. This is the $(0,0)$ principal stratum, defined before observing which assignment actually occurs.

For example, a unit may decline participation whether or not it receives an invitation. Among units with observed $Z=0,D=0$, never-takers are mixed with compliers, so a single observation does not reveal the principal stratum. Under an exclusion restriction through treatment receipt, assignment has no outcome effect on a never-taker because receipt stays fixed; that is an additional causal assumption, not part of the definition.^[ref]

A complier is a unit with potential treatment receipts $D(0)=0$ and $D(1)=1$: binary encouragement $Z$ changes its receipt from no treatment to treatment. This $(0,1)$ pair defines a principal stratum, rather than a behavior inferred from one observed assignment.

For example, a unit that participates exactly when invited is a complier. Under instrumental-variable monotonicity, which excludes defiers,Thus, with randomized assignment, the difference in treatment-receipt rates estimates the complier fraction. To interpret an outcome-rate ratio as the local average treatment effect, one additionally needs the relevant independence and exclusion assumptions and a nonzero complier fraction. An observed encouraged participant may instead be an always-taker.^[ref]

A defier is a unit with potential treatment receipts $D(0)=1$ and $D(1)=0$: it takes the treatment without encouragement but does not take it when encouraged. This is the $(1,0)$ principal stratum. The word specifies potential behavior under a particular binary intervention; it does not describe a general personality trait.

A defier contributes $-1$ to $D(1)-D(0)$, whereas a complier contributes $+1$. Consequently,The usual instrumental-variable monotonicity assumption sets the second probability to zero. A positive average effect of encouragement on receipt does not prove that there are no defiers: positive and negative individual changes can coexist.^[ref]

An always-taker is a unit with potential treatment receipts $D(0)=D(1)=1$: it takes the treatment under either value of binary encouragement $Z$. This $(1,1)$ pair is a principal stratum defined by both potential receipts.

For example, a unit may participate whether or not it receives an invitation. Encouraged participants combine always-takers and compliers; unencouraged participants combine always-takers and defiers. Under instrumental-variable monotonicity, the latter group consists only of always-takers. Even then, comparing observed participants across assignment arms mixes different strata and is not automatically a within-stratum causal effect. The exclusion restriction, when imposed, says assignment cannot affect an always-taker's outcome through any channel other than the treatment whose receipt is unchanged.^[ref]