Proximal causal inference — the full family¶

When you cannot observe the confounder, Proximal Causal Inference (PCI) uses two proxies of the unobserved confounder to identify the ATE. StatsPAI ships the base estimator plus four 2025-2026 frontier variants that are robust to bridge-function misspecification, policy-level interventions, bidirectional confounding, and short→long-term extrapolation.

This guide walks through the whole PCI family with a single running example and tells you which variant to reach for in which situation. Every function below lives at top level: sp.proximal, sp.fortified_pci, sp.bidirectional_pci, sp.pci_mtp, sp.double_negative_control, sp.proximal_surrogate_index, sp.select_pci_proxies.

The setup¶

You have outcome Y, treatment D, observed covariates X, and an unobserved confounder U. Back-door adjustment on X does not identify the ATE because U → D and U → Y after conditioning on X.

Proximal identification (Miao-Geng-Tchetgen, 2018; Tchetgen et al., 2024) works if you can find two proxies of U:

Z (treatment-inducing confounding proxy) — depends on U, possibly on D; conditional on (U, D, X), Z is independent of Y.
W (outcome-inducing confounding proxy) — depends on U, possibly on Y; conditional on (U, X), W is independent of D.

Plus: a completeness condition linking (D, Z) to W (roughly, W carries enough information about U to predict it nonparametrically given (D, X)).

Choosing proxies is the hard part — see sp.select_pci_proxies below.

Canonical estimator: `sp.proximal`¶

Two-stage-least-squares on the outcome bridge. Good default when bridges are plausibly linear and you have one Z and one W.

r = sp.proximal(
    data=df, y="Y", treat="D",
    proxy_z=["Z"], proxy_w=["W"],
    covariates=["X1", "X2"],
    bridge="linear",     # or "loglinear"
    n_boot=500,
)
r.summary()

Reference: Miao, Geng, Tchetgen Tchetgen (2018), Biometrika 105(4); Cui, Pu, Miao, Kennedy, Tchetgen (2024), JASA.

Fortified PCI — `sp.fortified_pci` (Yang & Schwartz, 2025)¶

Standard PCI fails hard when either bridge is misspecified. Fortified PCI adds a stability constraint: at each proxy choice, we fit both bridges with a shared penalty that drives the solution toward an overlap region where both moments are jointly satisfied. Empirically this is the PCI variant most robust to moderate bridge-function misspecification (Yu, Shi & Tchetgen Tchetgen 2025 Tables 2-4).

r = sp.fortified_pci(
    data=df, y="Y", treat="D",
    proxy_z=["Z"], proxy_w=["W"],
    covariates=["X1", "X2"],
)

When to reach for it: your point estimate from sp.proximal moves noticeably when you add/remove a single covariate, or when bridge="linear" vs bridge="loglinear" give very different answers.

Citation: Yu, Shi & Tchetgen Tchetgen (2025), arXiv:2506.13152.

Bidirectional PCI — `sp.bidirectional_pci` (Shi, Miao & Tchetgen, 2025)¶

Standard PCI solves the outcome bridge (regress Y on D, W, X) and then inverts via Z. Bidirectional PCI fits the outcome bridge and the treatment bridge simultaneously in a single two-way regression, which:

Reduces finite-sample bias when either bridge is weakly identified.
Produces a natural doubly-robust GMM-style objective.
Makes the identification assumptions symmetric — useful when you cannot decide which proxy is Z and which is W.

r = sp.bidirectional_pci(
    data=df, y="Y", treat="D",
    proxy_z=["Z"], proxy_w=["W"],
    covariates=["X1", "X2"],
)

Citation: Min, Zhang & Luo (2025), arXiv:2507.13965.

PCI for modified treatment policies — `sp.pci_mtp` (Park & Ying, 2025)¶

Suppose you do not want the ATE under a static "D=1 vs D=0" contrast; you want the effect of shifting the treatment distribution by some amount δ — e.g., "raise the dose by 10% for everyone." PCI-MTP identifies that modified treatment policy effect under the same two- proxy structure as base PCI.

r = sp.pci_mtp(
    data=df, y="Y", treat="dose",
    proxy_z=["Z"], proxy_w=["W"],
    delta=0.10,                     # ← shift treatment by +10%
    covariates=["X1", "X2"],
)

Why it matters: in dose-response or continuous-treatment settings, the "contrast between fixed levels" estimand is often uninteresting. MTP answers the policy-relevant question "what happens if we nudge treatment a bit?" under unobserved confounding.

Citation: Olivas-Martinez, Gilbert & Rotnitzky (2025), arXiv:2512.12038.

Double negative controls — `sp.double_negative_control`¶

A simpler PCI special case used in epidemiology: one negative control exposure (NCE, a treatment-like variable we know cannot affect Y) plus one negative control outcome (NCO, an outcome-like variable we know cannot be affected by D). These together identify the treatment effect under additive structural assumptions.

r = sp.double_negative_control(
    data=df, y="Y", treat="D",
    nce="prior_dental_visits",       # placebo treatment
    nco="baseline_weight",           # placebo outcome
    covariates=["age", "sex"],
)

Use case: lower-tech than full PCI — good for initial analyses when you have only one of each type of proxy. References: sp.negative_control_outcome and sp.negative_control_exposure for the single-control versions.

Long-term surrogate + PCI — `sp.proximal_surrogate_index` (Imbens-Kallus-Mao-Wang, 2025)¶

You ran a randomised experiment for 3 months but care about the 2-year outcome. The classical solution (Athey-Chetty-Imbens-Kang, NBER WP 26463, 2019 surrogate index) needs the surrogates to fully mediate the long-term effect — a strong assumption. Imbens, Kallus, Mao & Wang show that combining the surrogate index with a PCI layer on the observational data lets you drop the full-mediation requirement: short-term surrogates play the role of W, observational proxies play the role of Z, and the two together identify the long-term ATE.

r = sp.proximal_surrogate_index(
    experimental=df_exp,           # short-term RCT
    observational=df_obs,          # long-follow-up observational cohort
    treatment="feature_flag",
    surrogates=["dau_90d", "retention_90d"],   # → W
    proxies=["pre_dau", "pre_purchase"],        # → Z
    long_term_outcome="revenue_24mo",
    covariates=["country", "cohort"],
)

This is also available as the surrogate_pci bridge in sp.bridge — see Bridging theorems.

Citation: Imbens, Kallus, Mao & Wang (2025). "Long-term Causal Inference Under Persistent Confounding via Data Combination." Journal of the Royal Statistical Society Series B 87(2), 362-388. arXiv:2202.07234.

Picking the proxies: `sp.select_pci_proxies`¶

When you have a list of candidate proxy variables, this helper scores each on two PCI-relevant axes:

Z score — how strongly is the candidate predicted by D after partialling out X? (Want this high for a good Z.)
W score — how strongly does the candidate predict Y after partialling out D, X? (Want this high for a good W.)

It returns a ranked table of candidates so you can choose the top-scorers for each role.

ranks = sp.select_pci_proxies(
    data=df, y="Y", treat="D",
    candidates=["V1", "V2", "V3", "V4", "V5"],
    covariates=["X1", "X2"],
    top_k=2,
)
print(ranks.z_table)   # best Z candidates
print(ranks.w_table)   # best W candidates

When to use which — decision guide¶

Got exactly one Z and one W, bridges look roughly linear
  → sp.proximal (default)

Bridges sensitive to specification choice
  → sp.fortified_pci

Unsure which proxy should play Z vs W
  → sp.bidirectional_pci

Continuous treatment / want "shift policy" effect
  → sp.pci_mtp(delta=...)

Only have a negative control exposure + negative control outcome
  → sp.double_negative_control

Want long-term effect from short-term experiment
  → sp.proximal_surrogate_index

Have a pile of candidate proxies, not sure which to use
  → sp.select_pci_proxies first, then the above

Diagnostics every PCI analysis should report¶

Bridge completeness: can Z predict W? (If not, identification fails.) Run sp.regress("W ~ Z + D + X", data=df) and check the F-stat.
Proxy independence: Z ⊥ Y | D, U, X. You cannot test this directly (because U is unobserved), but you can test the weaker Z ⊥ Y | D, X and interpret a non-zero coefficient as suggestive of U-mediated dependence.
Bridge stability: rerun with bridge="linear" and bridge="loglinear" — substantial disagreement suggests misspecification.
Sensitivity via sp.bridge(kind="cb_ipw", ...) on the (D, X) pair — if the back-door-on-X-only estimate equals the PCI estimate, your unobserved confounder might not be doing much work in this sample.

This guide is current for StatsPAI ≥ 1.5.0. All functions are stable and registered in sp.list_functions().

For Agents¶

Pre-conditions - at least one treatment-side proxy Z (independent of outcome given U, X) - at least one outcome-side proxy W (independent of treatment given U, X) - proxy_z and proxy_w measure the same unmeasured confounder U from different angles - n ≥ 1000 — 2SLS on proxies is noisy

Identifying assumptions - Existence of an outcome bridge function h(w, a, x) that recovers E[Y(a) | U, X] - Z and W are conditionally independent given U and (A, X) - Z ⊥ Y | U, A, X (exclusion on Z) - W ⊥ A | U, X (exclusion on W) - Z is relevant for W given A, X (bridge first stage)

Failure modes → recovery

Symptom	Exception	Remedy	Try next
First-stage (Z → W) too weak	`statspai.AssumptionWarning`	Try richer Z or more proxies; without first-stage strength the bridge is underidentified.	`sp.iv`
Proxies collapse to nearly-constant	`statspai.DataInsufficient`	Proxy variation insufficient — redesign measurement or fall back to sensitivity (sp.sensemakr).	`sp.sensemakr`
Estimate highly sensitive to bridge specification	`statspai.AssumptionWarning`	Report multiple bridge families; compare with sp.negative_control_outcome / _exposure.	`sp.negative_control_outcome`

Alternatives (ranked) - sp.negative_control_outcome - sp.negative_control_exposure - sp.double_negative_control - sp.iv - sp.sensemakr

Typical minimum N: 1000

Proximal causal inference — the full family¶

The setup¶

Canonical estimator: sp.proximal¶

Fortified PCI — sp.fortified_pci (Yang & Schwartz, 2025)¶

Bidirectional PCI — sp.bidirectional_pci (Shi, Miao & Tchetgen, 2025)¶

PCI for modified treatment policies — sp.pci_mtp (Park & Ying, 2025)¶

Double negative controls — sp.double_negative_control¶

Long-term surrogate + PCI — sp.proximal_surrogate_index (Imbens-Kallus-Mao-Wang, 2025)¶

Picking the proxies: sp.select_pci_proxies¶