1 A Problem LATE Cannot Solve

The Local Average Treatment Effect (LATE) framework handles one specific problem: non-compliance. When a randomised experiment assigns some units to treatment but they refuse to comply, and assigns others to control but they obtain treatment anyway, the LATE identifies the average effect for compliers—those who take treatment when assigned to treatment and refuse it when assigned to control [Angrist et al., 1996].

LATE's power rests on the exclusion restriction: the instrument (assignment) affects the outcome only through the treatment, not directly. But what happens when the instrument affects a post-treatment variable that then affects the outcome? What if some subjects die between randomisation and outcome measurement, and whether they survive depends on their treatment? What if a training programme causes some participants to get promoted before the wage outcome is measured?

These situations create a problem that LATE—and, more generally, standard IV—cannot handle. The exclusion restriction fails not because the instrument has a direct effect on the outcome, but because the set of units for whom we can measure the outcome is itself affected by the treatment. This is the domain of principal stratification.

2 A Motivating Example: HIV Vaccines and Mortality

Consider a randomised trial of an HIV vaccine. Some subjects die of AIDS before the trial ends. Whether a subject dies before outcome measurement depends on their treatment status: the vaccine may keep subjects alive long enough to be observed. At the end of the trial, we can only measure CD4 counts (the immune health outcome) for survivors.

Let Sᵢ(1) denote whether unit i survives if assigned to vaccine, and Sᵢ(0) whether they survive if assigned to placebo. Define four groups based on survival potential outcomes‍


In a vaccine trial, the "killed-by-vaccine" stratum is assumed empty (the vaccine should not increase mortality—this is the monotonicity assumption for survival). Under this assumption, there are three relevant strata: always-die, saved-by-vaccine, and always-survive. The naively estimated effect among observed survivors is:

where Z is the instrument (vaccine assignment) and S is observed survival. The problem: the set of survivors in the treatment arm (Z = 1) includes both "always-survive" and "saved-by-vaccine" units, while the set in the control arm (Z = 0) includes only "always-survive" units. These groups are not comparable, making the naive comparison biased.

3 Principal Stratification: The Core Idea

Frangakis and Rubin [2002] proposed defining causal effects within principal strata, where a principal stratum is defined by the joint potential outcomes of the intermediate (post-treatment) variable. Formally: Let Sᵢ(z) be the potential value of the post-treatment variable when the instrument equals z ∈ {0, 1}. The principal stratum of unit i is the pair (Sᵢ(0), Sᵢ(1)).

Because Sᵢ(0) and Sᵢ(1) are characteristics of the unit determined before treatment assignment (even though we only observe one of them), principal strata are pre-treatment subgroups. This is the crucial property: treatment effects within principal strata are causal effects for a well-defined subpopulation, not conditional effects for a sample selected by treatment. The Survivor Average Causal Effect (SACE) is the average treatment effect among always-survivors:

The SACE answers: among units who would survive regardless of treatment assignment, what is the effect of treatment on the outcome? This is a well-defined causal quantity even though we cannot identify this stratum from the data without additional assumptions.

4 Connection to the LATE Framework

Principal stratification and the LATE framework are deeply connected. In the standard IV-noncompliance setting, define:

• Compliers: Dᵢ(0) = 0, Dᵢ(1) = 1
• Always-takers: Dᵢ(0) = 1, Dᵢ(1) = 1
• Never-takers: Dᵢ(0) = 0, Dᵢ(1) = 0
• Defiers: Dᵢ(0) = 1, Dᵢ(1) = 0

These are exactly the principal strata defined by the joint potential values of the post-randomisation variable Dᵢ. The LATE is the average treatment effect within the complier stratum:‍


So LATE is a principal stratum causal effect. The principal stratification framework generalises LATE to settings where the intermediate variable may be continuous or multi-valued, and where the exclusion restriction may not hold.

5 A Figure: The Principal Strata in a Vaccine Trial

6 Identification: What Assumptions Are Needed?

The SACE is not generally identified without additional assumptions, because we cannot observe which stratum any unit belongs to. Common identifying assumptions include:

• Monotonicity: Sᵢ(1) ≥ Sᵢ(0) for all i—treatment can only help, not harm, on the intermediate variable. This rules out the "killed-by-vaccine" stratum.
• Stochastic dominance: Among survivors, outcomes in the treatment arm stochastically dominate those in the control arm. Combined with monotonicity, this provides partial identification bounds on the SACE.
• Latent ignorability: Conditional on observed pre-treatment covariates Xᵢ, the stratum membership is independent of the potential outcomes. This allows parametric or semiparametric estimation of stratum probabilities and hence the SACE.
• Constant treatment effect within strata: Assumes Yᵢ(1) - Yᵢ(0) is constant within each stratum, which is strong but sometimes defensible in specific applications.

7 Beyond Survival: Applications of Principal Stratification

Principal stratification applies whenever a post-treatment variable creates problems for standard analysis:

• Job training and employment: A training programme may create employment; conditioning on employment to measure wages is selection bias. SACE for always-employed is the right target.
• Education experiments: A scholarship may cause students to take more rigorous courses (intermediate variable); the effect on earnings among those who would have taken rigorous courses regardless is a SACE.
• Medical trials with truncation-by-death: Rubin [2006] developed the formal framework for this application.
• Non-compliance in experiments: As discussed above, LATE is a principal stratum causal effect.

8 Relationship to Mediation Analysis

Principal stratification is sometimes confused with mediation analysis, but they answer fundamentally different questions. Mediation analysis decomposes the total effect into a direct effect (not through the mediator) and an indirect effect (through the mediator). Principal stratification conditions on the stratum defined by the mediator's potential values, focusing attention on a specific subpopulation.

The two approaches use different assumptions. Mediation analysis requires the cross-world independence assumption, which cannot be tested even with ideal data [Pearl, 2001]. Principal stratification avoids this cross-world assumption, instead requiring assumptions about the distribution of strata and outcomes within strata.

9 Common Mistakes

1. Conditioning on post-treatment variables: Comparing outcomes for treated and control survivors without accounting for the differential selection this creates is a classic error. Always check whether the conditioning variable S is itself affected by treatment.
2. Confusing strata with observations: Principal strata are latent groups—we never observe which stratum a unit belongs to. Any analysis that treats stratum membership as observed is misspecified.
3. Applying LATE when the exclusion restriction fails: If the instrument has a direct effect on the outcome (not through D), LATE is invalid. Principal stratification provides a framework for analysis even when the exclusion restriction is relaxed, but it requires different identifying assumptions.

10 Where to Learn More

The foundational paper is Frangakis and Rubin [2002]. Rubin [2006] provides an accessible treatment focused on clinical trials. For the connection to LATE and IV, see Angrist et al. [1996]. VanderWeele [2011] reviews applications in epidemiology. For Bayesian estimation of SACE, the sensitivityPStrat package in R implements several approaches.

11 Conclusion

Principal stratification extends the potential outcomes framework to settings where post-treatment variables create complications for standard analysis. By defining causal effects within subpopulations characterised by their joint potential values of the intermediate variable, it provides a rigorous approach to causal questions that would otherwise be unanswerable. Understanding principal stratification illuminates why LATE is valid and what it means and sheds light on the many adjacent settings where LATE-style reasoning breaks down.

References

1. Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91(434):444-455.
2. Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics, 58(1):21-29.
3. Pearl, J. (2001). Direct and indirect effects. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, pages 411-420.
4. Rubin, D. B. (2006). Causal inference through potential outcomes and principal stratification: Application to studies with "censoring" due to death. Statistical Science, 21(3):299-309.
5. VanderWeele, T. J. (2011). Principal stratification—uses and limitations. International Journal of Biostatistics, 7(1):1-14.
6. Imbens, G. W. and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences. Cambridge University Press.