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GEEPERs: Principal Stratification using Principal Scores and Stacked Estimating Equations

This paper introduces GEEPERs, a robust and accessible method for estimating principal stratification effects in one-way noncompliance scenarios that utilizes principal scores and stacked estimating equations to avoid strong distributional assumptions while remaining implementable via conventional regression techniques.

Original authors: Adam C. Sales, Kirk P. Vanacore, Erin R. Ottmar

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam C. Sales, Kirk P. Vanacore, Erin R. Ottmar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a teacher trying to figure out if a new, fancy online math platform actually helps students learn faster. You run a fair experiment: you flip a coin to decide who gets the new platform (the Treatment) and who keeps using the old way (the Control).

But here's the catch: just because a student is assigned the new platform doesn't mean they actually use it. Some might ignore it, some might get stuck and quit, and some might use it exactly as intended.

The Problem: The "Hidden Groups"

In statistics, we usually compare the average results of the whole Treatment group against the whole Control group. But that misses the nuance. You might really want to know: "How much did the platform help the students who actually used the hints?"

The problem is, we can't see the "hidden groups."

We can see the students in the Treatment group who used the hints.
But we cannot see the students in the Control group who would have used the hints if they had been given the option. They are invisible to us.

This is like trying to judge how well a new pair of running shoes works for "fast runners" by only looking at the people who bought them. You don't know who in the "no-shoes" group would have been fast runners if they had the shoes.

The Old Way: The "Crystal Ball" Approach

For a long time, statisticians tried to solve this using complex models (called Parametric Mixture Models). Think of these models as trying to build a crystal ball. You have to guess the exact shape of the data (e.g., "the errors must look like a perfect bell curve") and then use heavy math to guess the hidden groups.

The paper argues this is like trying to bake a soufflé without a recipe: if your guess about the shape of the data is even slightly wrong, the whole thing collapses, and your results are misleading. Plus, it requires a PhD in math to even attempt it.

Another method (Principal Score Weighting) tries to guess the hidden groups by looking at student characteristics (like past grades). But this relies on a very strict assumption: that once you know a student's past grades, their future performance is totally random. The paper says this is a dangerous assumption to make.

The New Solution: "GEEPERS"

The authors introduce a new method called GEEPERS (General Estimating Equations for Principal Effects using Regressions).

Think of GEEPERS not as a crystal ball, but as a smart detective that uses a three-step process:

The Prediction Step: First, the detective looks at the students who did get the treatment and used the hints. It asks, "What kind of students are these?" It builds a simple profile (a "score") to predict who is likely to use the hints.
The Guessing Step: The detective then looks at the Control group (who didn't get the hints). Using the profile from Step 1, it says, "Okay, based on their past grades, Student A has an 80% chance of being a 'hint-user' if they had the option, and Student B has a 20% chance." It doesn't say they are hint-users; it just assigns them a probability.
The Calculation Step: Finally, it runs a standard, simple math class (called Ordinary Least Squares regression) that everyone knows. It uses those probabilities to calculate the effect of the hints on the "likely hint-users" versus the "likely non-users."

Why is GEEPERS Better?

The paper claims GEEPERS is the "Goldilocks" of statistics:

It's Accessible: You don't need a crystal ball or a PhD. You just need to know how to run a standard regression (a tool almost every social scientist already has).
It's Robust: Unlike the old "crystal ball" methods, GEEPERS doesn't care if the data is perfectly shaped like a bell curve. It's flexible. If the data is messy, GEEPERS doesn't break.
It's Honest: The paper shows through simulations that when the old methods make a wrong guess about the data shape, they give you wrong answers. GEEPERS stays steady.

The Real-World Test

The authors tested this on real data from ASSISTments, an online math homework platform. They wanted to know if "video scaffolding" (short videos that help students when they get stuck) actually helped students learn faster.

The Result: They found that for the students who would use the videos, the videos helped them learn significantly faster.
The Comparison: When they compared GEEPERS to the old methods, the old methods gave weird results (like suggesting the videos helped students who never watched them, which makes no sense). GEEPERS gave a result that made logical sense: the videos helped the students who actually used them.

The Bottom Line

If you are a researcher trying to figure out the effect of a treatment on a specific subgroup of people who didn't all follow the rules, GEEPERS is a new, simpler, and sturdier tool. It lets you use the standard math tools you already know to solve a problem that used to require complex, fragile, and hard-to-use models. It's about making advanced causal inference accessible to regular researchers without sacrificing accuracy.

Technical Summary: GEEPERs – Principal Stratification using Principal Scores and Stacked Estimating Equations

Problem Statement
Principal stratification is a causal inference framework designed to estimate treatment effects for subgroups defined by post-treatment variables (e.g., compliance behavior) that may themselves be influenced by the treatment. A primary challenge in this field is that principal strata (e.g., "compliers" vs. "never-takers") are unobserved counterfactuals; researchers cannot observe what a subject's compliance status would have been under a different treatment assignment.

Existing estimators for principal effects often rely on strong structural assumptions or complex modeling techniques that limit their accessibility to applied researchers. Specifically:

Parametric Mixture Models: These often require Bayesian Markov Chain Monte Carlo (MCMC) methods, are difficult to specify, and can yield severely biased estimates if distributional assumptions (e.g., normality) are violated.
Principal Score Weighting: While easier to implement, these methods rely on the "principal ignorability" assumption (that principal stratum membership is independent of potential outcomes conditional on covariates). This is a strong, untestable assumption analogous to those required in observational studies, which randomized trials are specifically designed to avoid.

Methodology: GEEPERs
The authors introduce GEEPERs (General Estimating Equations for Principal Effects using Regressions), a new estimator for one-way noncompliance scenarios where principal stratum membership is observed in the treatment group but not the control group. The method relies on M-estimation and stacked estimating equations to combine the estimation of principal scores with the estimation of principal effects.

The GEEPERs procedure follows a three-step logic, implemented as a single regression model:

Estimate Principal Scores: Fit a model (e.g., logistic regression) to predict the probability of belonging to the "complier" stratum ( $S_T=1$ ) using pre-treatment covariates ( $x$ ) and observed compliance data from the treatment group. Let this probability be $e(x)$ .
Impute Missing Strata: Construct a variable $R_i$ $R_{i}$ that serves as a proxy for the unobserved stratum membership in the control group:
- If $Z_i=1$ (treatment), $R_i = S_i$ (observed compliance).
- If $Z_i=0$ (control), $R_i = e(x_i)$ (estimated principal score).
Estimate Effects via OLS: Fit an Ordinary Least Squares (OLS) regression of the outcome $Y$ on the treatment indicator $Z$ , the imputed stratum proxy $R$ , and their interaction, potentially including covariates $g(x)$ to relax independence assumptions:
$Y_i = \beta_0 + \beta_1 R_i + \beta_2 Z_i + \beta_3 (Z_i R_i) + \gamma' g(x)_i + \epsilon_i$
The principal effects are recovered as linear combinations of the coefficients: $\hat{\tau}_0 = \hat{\beta}_2$ and $\hat{\tau}_1 = \hat{\beta}_2 + \hat{\beta}_3$ .

Key Assumptions and Technical Properties

Strong Monotonicity: Assumes one-way noncompliance (e.g., $S_{Ci}=0$ for all $i$ ), meaning no subjects would comply if assigned to control but not if assigned to treatment.
Residualized Covariate Independence: The method relaxes the strict "Covariate Independence" assumption by allowing for a linear adjustment of covariates. It assumes that after adjusting for covariates, the potential outcomes are mean-independent of the covariates within strata.
No Distributional Assumptions: Unlike mixture models, GEEPERs does not assume a specific distribution for the regression errors.
Standard Errors: Because the method involves a two-step process (estimating scores then effects), the authors utilize stacked estimating equations. This allows for the calculation of "sandwich" standard errors that correctly propagate uncertainty from the first step (score estimation) into the second step (effect estimation).

Simulation Results
The authors conducted a simulation study comparing GEEPERs against Parametric Mixture Modeling (PMM) and Principal Score Weighting (PSW) under various conditions:

Robustness to Distributional Violations: GEEPERs performed consistently well regardless of whether regression errors were normal, lognormal, or uniform. In contrast, PMM suffered from severe bias, bimodal sampling distributions, and poor confidence interval coverage when errors were non-normal (specifically uniform).
Robustness to Unmeasured Confounding: When the "principal ignorability" assumption was violated (due to an unmeasured covariate predicting both compliance and outcome), PSW exhibited significant bias. GEEPERs remained less biased than PMM and PSW, though it showed some sensitivity to interactions between covariates and treatment assignment.
Sample Size and Predictability: GEEPERs was approximately unbiased across sample sizes. While it had slightly higher variance than PMM in small samples or when covariates poorly predicted compliance, this efficiency gap diminished as sample size and the predictive power of covariates increased.
Sensitivity Analysis: The authors demonstrated that if interactions between covariates and treatment assignment (effect moderators) are suspected, excluding those specific covariates from the analysis can restore the validity of the GEEPERs estimates, offering a practical sensitivity check.

Real-Data Application
The method was applied to a re-analysis of three randomized controlled trials (RCTs) from the ASSISTments online math platform, evaluating the effect of "video scaffolding" on student mastery speed.

Findings: GEEPERs estimated a positive effect for "compliers" (students who used scaffolding) across all model specifications, with estimates ranging from 0.30 to 0.50 standard deviations. Estimates for "never-takers" were close to zero with large standard errors, consistent with no effect.
Comparison: PMM produced similar point estimates for compliers but suggested a surprisingly large positive effect for never-takers, which the authors flagged as suspicious due to model misspecification checks. PSW produced a large positive effect for never-takers and a smaller effect for compliers, likely due to the violation of principal ignorability (students who needed help were inherently struggling more).

Significance and Claims
The paper claims that GEEPERs addresses three critical gaps in the existing literature:

Accessibility: By reducing principal stratification to standard logistic and linear regression, the method is accessible to applied researchers without advanced training in Bayesian MCMC or likelihood optimization.
Robustness: The method avoids the fragility of parametric mixture models regarding distributional assumptions and the strong, untestable assumptions of principal ignorability required by weighting methods.
Flexibility: It provides a unified framework that utilizes widely available regression tools while offering a rigorous way to handle the uncertainty of two-step estimation via sandwich variance estimators.

The authors conclude that while GEEPERs is not immune to all violations (particularly those involving unmodeled effect moderators), it offers a robust, interpretable, and computationally straightforward alternative for estimating principal effects in educational and other randomized trials with one-way noncompliance.