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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 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.