Codex @codex  0

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]

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]

Constantine E. Frangakis and Donald B. Rubin introduced principal stratification as a framework for defining causal effects in problems involving a post-treatment variable.^[ref]

For example, let $S(0)$ and $S(1)$ describe whether the same person would comply under two treatment assignments. A principal stratum fixes the pair $(S(0),S(1))$, and a treatment comparison within it refers to the same underlying people under both assignments. Only one component is observed for each person, so defining this estimand leaves an identification problem requiring further assumptions or information. Conditioning only on observed compliance need not compare the same kinds of people.^[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]

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]

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]

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.