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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]

Principal stratification groups units by the joint values of their potential values of an intermediate variable under every treatment assignment. With a binary instrument and binary treatment, the principal strata are always-takers, never-takers, compliers, and defiers.