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Principal Stratification

Frangakis & Rubin (2002). Principal strata classify units by the joint potential values of a post-treatment variable — most often compliance type or survival status — and the resulting principal causal effects (PCEs) are conditional on stratum membership.

Scenario Post-treatment variable S
Encouragement design / noncompliance Actual take-up
Survival / truncation-by-death Alive-at-follow-up indicator
Employment after training Employed indicator
Dropout / missingness mechanism Response indicator

StatsPAI ships two identification strategies via sp.principal_strat(..., method=...):

  1. 'monotonicity' — Angrist-Imbens-Rubin (1996) + Zhang-Rubin (2003) sharp bounds.
  2. 'principal_score' — Ding & Lu (2017) principal-score weighting (point identification under principal ignorability).

Plus a convenience wrapper sp.survivor_average_causal_effect(...) for the classical truncation-by-death problem.

Monotonicity method — complier LATE + SACE bounds

Under AIR monotonicity (S(1) ≥ S(0) almost surely, i.e. no defiers), the three surviving strata have observable mixture decompositions:

Stratum Proportion Name
(S(0)=1, S(1)=1) P(S=1 \| D=0) Always-taker / always-survivor
(S(0)=0, S(1)=1) P(S=1 \| D=1) - P(S=1 \| D=0) Complier / harmed
(S(0)=0, S(1)=0) P(S=0 \| D=1) Never-taker / never-survivor

Complier PCE (= LATE) is point-identified by the Wald-type ratio, and the always-survivor SACE has sharp Zhang-Rubin bounds.

res = sp.principal_strat(
    df, y='wage', treat='treat', strata='employed',
    method='monotonicity',
    n_boot=500, seed=0,
)

res.strata_proportions         # dict: always / complier / never shares
res.effects                    # DataFrame — LATE row with SE / CI / p-value
res.bounds                     # DataFrame — SACE lower / upper bounds
print(res.summary())           # pretty-printed

Zhang-Rubin bounds on E[Y(1) - Y(0) | always-survivor] come from worst/best-case slicing of the (D=1, S=1) cell, which is a mixture of always-survivors (share q = π_always / P(S=1|D=1)) and compliers. When π_always ≈ 0 the bounds degenerate to NaN (flagged explicitly).

Principal-score method — point identification via covariates

Under principal ignorability (Y(d) ⊥ stratum | X within D=d) + monotonicity, stratum membership is identified from the observable cell probabilities p11(X) = P(S=1|D=1,X) and p10(X) = P(S=1|D=0,X):

e_always(X)   = p10(X)
e_complier(X) = p11(X) - p10(X)
e_never(X)    = 1 - p11(X)

The stratum-specific ATEs are then estimated by covariate-weighted slicing of the observed cells (Ding & Lu 2017).

res = sp.principal_strat(
    df, y='wage', treat='treat', strata='employed',
    covariates=['age', 'edu', 'tenure'],
    method='principal_score',
    n_boot=500, seed=0,
)
res.effects                     # DataFrame: complier / always / never PCE

Returns per-stratum point estimate + bootstrap SE/CI/p-value for all three strata.

Monotonicity diagnostics: when the fitted p11(x) < p10(x) for more than 5 % of units, a RuntimeWarning fires and model_info records:

res.model_info['mono_violation_frac']     # fraction with p11<p10
res.model_info['mono_min_raw_complier']   # min(p11(x) - p10(x))

Principal ignorability is a strong assumption — pair this with sensitivity analysis before publishing.

Survivor Average Causal Effect wrapper

Classical truncation-by-death problem (Zhang-Rubin 2003):

sace = sp.survivor_average_causal_effect(
    df, y='qol_score', treat='treat', survival='alive',
    alpha=0.05, n_boot=500, seed=0,
)
sace.model_info['sace_lower'], sace.model_info['sace_upper']
sace.ci    # Imbens-Manski-style union confidence interval

Returns a CausalResult with the midpoint of the sharp bounds as the point estimate and the bound endpoints in model_info. pvalue is NaN because the estimand is partially identified and a point-null Wald test is not well-defined.

Two-layer IV / encouragement design

The current release does not support passing both an instrument and a take-up variable — that two-layer setup requires a separate estimator that combines IV + principal strata. Use sp.dml(model='iivm') for the classical AIR LATE in an encouragement design. An explicit instrument= argument to sp.principal_strat raises NotImplementedError.

References

  • Frangakis, C.E. & Rubin, D.B. (2002). Principal stratification in causal inference. Biometrics.
  • Zhang, J.L. & Rubin, D.B. (2003). Estimation of causal effects via principal stratification when some outcomes are truncated by "death". JEBS.
  • Angrist, J.D., Imbens, G.W. & Rubin, D.B. (1996). Identification of causal effects using instrumental variables. JASA.
  • Ding, P. & Lu, J. (2017). Principal stratification analysis using principal scores. JRSS-B.
  • Jo, B. & Stuart, E.A. (2009). On the use of propensity scores in principal causal effect estimation. Stat. Med.