Are principal strata observed before treatment?

Their definitions use potential post-treatment values, so membership is generally latent even though it is conceptually unaffected by assignment.

Does randomization identify every principal effect?

No. It identifies some assignment comparisons but not all contrasts within hidden strata.

Is the complier effect necessarily the average effect for everyone?

No. It is specific to the complier stratum under the stated design and assumptions.

AI Glossary / Statistics and probability / Advanced

Principal stratification

Definition

Principal stratification is a causal-inference framework that defines groups using the joint potential values of a post-treatment variable under each treatment assignment. For binary assignment Z and an intermediate variable S, a principal stratum is defined by the pair (S(0), S(1)). A principal effect compares potential outcomes within such a stratum. Because the stratum is defined by both potential values, its definition does not change with actual assignment. However, those joint values are generally not fully observed for any individual.

What it means

Conditioning on an observed post-treatment variable can compare different kinds of people across treatment groups, even when assignment was randomized. Principal stratification instead defines a subgroup by what the intermediate variable would be under each possible assignment. This supplies a precise causal estimand, but it does not automatically identify that estimand from data. Only one potential intermediate value is observed per person. Additional assumptions, bounds, models or information may be needed, and the scientific usefulness of the resulting subgroup effect depends on the question.

How it works

Specify the assignment, post-treatment variable and final outcome, with potential outcomes under each assignment. Define the strata and the causal contrast of interest before choosing an estimation method. For binary treatment receipt, the four pairs are never-takers (0,0), compliers (0,1), defiers (1,0) and always-takers (1,1). Randomization can support comparisons by assignment but does not reveal every person's stratum. Assumptions such as monotonicity or an exclusion restriction can provide identification in particular instrumental-variable settings. Explain their meaning and assess sensitivity rather than treating the framework itself as an identification theorem for every application.

Three practical examples

Noncompliance analysis: define an effect among people whose treatment receipt would follow their randomized assignment.
Outcome truncation: consider an effect among units for whom an outcome would be defined under both assignments, with careful attention to identification.
Digital experiments: distinguish an effect of offering a feature from an effect within a potential-usage stratum, rather than comparing observed users with nonusers as if they were randomized groups.

Think of it this way

Think of classifying people by how they would respond to either of two possible invitations. You observe only the invitation each person actually received, so the complete response type is hidden. The classification can make the question precise, but precision of the question does not mean the hidden type has become directly measurable.

Worked example

Consider an original hypothetical randomized encouragement study. Let 20% of units be always-takers, 60% compliers and 20% never-takers, with no defiers. Treatment receipt is then 80% when encouraged and 20% when not encouraged, a difference of 0.60. Suppose the assignment effect on the outcome is 12 units. Under random assignment, consistency and no interference, a relevant encouragement, monotonicity and an exclusion restriction that assignment affects the outcome only through treatment receipt, the usual Wald ratio gives a complier treatment effect of 12/0.60 = 20 units. These are stipulated assumptions and numbers, not an empirical finding. Observed treated units in the encouraged arm mix always-takers and compliers: their proportions are 0.20/0.80 = 25% and 0.60/0.80 = 75%. They are not a directly observed pure complier group. Without the identification assumptions, the ratio need not have that causal interpretation.

Illustrative case study

In a fictional case, Cedar Learning randomizes access to a reminder intended to increase use of a training tool. Some people would use the tool regardless of the reminder, and others would never use it. Analysts first report the effect of reminder assignment. They then consider a complier effect, explicitly discussing whether the reminder could influence the outcome through motivation independently of tool use. Because that possibility challenges exclusion, the report separates the randomized assignment evidence from the more assumption-dependent subgroup interpretation.

Common misconceptions and mistakes

Principal strata are not simply groups formed by observed post-treatment values. A causal estimand being well defined does not make it identifiable. Monotonicity rules out a particular response type and cannot usually be verified individual by individual from observed assignments. Exclusion restrictions are substantive assumptions, not automatic consequences of randomization. A complier effect may differ from the population-average effect and may concern a subgroup that changes with the encouragement mechanism. Principal-stratum effects also do not generally answer every mediation question or describe what would happen if the intermediate variable were directly manipulated.

Business use

Businesses evaluating adoption, engagement or compliance should begin with the effect of the assigned intervention and clarify any additional question. Principal stratification can prevent an imprecise comparison of observed users and nonusers from being mistaken for a causal result. Its value depends on whether the stratum-specific estimand is relevant to decisions and whether assumptions are credible. Report uncertainty, possible nonidentification and sensitivity to assumptions. Do not label people as definite compliers from incomplete observations or deploy individual targeting rules as if latent principal membership had been directly measured.

Check your understanding

A person receives treatment when encouraged. Does that prove the person is a complier?

Answer: No. The person could be an always-taker who would also receive treatment without encouragement.

Related terms

Instrumental variables
Monotonicity
Potential outcomes
noncompliance
instrumental variable
local average treatment effect
exclusion restriction
post-treatment bias

Learn next

Learn the distinction between defining a causal estimand and identifying it, then examine instrumental-variable assumptions and sensitivity analysis for effects within latent subgroups.

Sources

Frangakis and Rubin, Principal Stratification in Causal Inference: https://pmc.ncbi.nlm.nih.gov/articles/PMC4137767/ VanderWeele, Principal Stratification: Uses and Limitations: https://pmc.ncbi.nlm.nih.gov/articles/PMC3154088/

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