📊 statistics

Identification of complier and noncomplier average causal effects in the presence of latent missing-at-random (LMAR) outcomes: a unifying view and choices of assumptions

This paper provides a unifying framework for identifying complier and noncomplier average causal effects in the presence of latent missing-at-random outcomes by demonstrating that, beyond treatment assignment ignorability and latent missingness, two additional assumptions regarding the missingness mechanism and principal identification are generally required, while also proposing modifications to existing assumptions and illustrating the approach with real-world data.

Original authors: Trang Quynh Nguyen, Michelle C. Carlson, Elizabeth A. Stuart

Published 2026-06-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Trang Quynh Nguyen, Michelle C. Carlson, Elizabeth A. Stuart

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 trying to figure out if a new volunteering program actually makes people feel more fulfilled. You have a group of people who signed up (the Treatment Group) and a group who didn't (the Control Group).

But there are two big problems with your experiment:

The "Slacker" Problem (Noncompliance): Not everyone in the Treatment Group actually did the volunteering. Some showed up and worked hard (the "Compliers"), while others signed up but did very little (the "Noncompliers"). In the Control Group, nobody was asked to volunteer, so you have no idea who would have been a hard worker if they had been asked.
The "Ghost" Problem (Missing Data): Some people dropped out of the study or forgot to fill out their final happiness survey. Their results are missing.

This paper is about how to solve the math puzzle of figuring out the true effect of the program when you have both "Slackers" and "Ghosts."

The Two Types of People (Principal Strata)

The authors divide people into two invisible groups based on how they would behave if asked:

Compliers: People who will do the work if asked.
Noncompliers: People who won't do the work even if asked.

We can see who is who in the Treatment group (because they actually did it or didn't). But in the Control group, everyone looks the same because no one was asked. This makes it hard to calculate the specific effect on the "Compliers" (CACE) and the "Noncompliers" (NACE).

The "Ghost" Problem: Why Missing Data is Tricky

Usually, statisticians assume that missing data is random (like a coin flip). But here, the authors assume something more complex called LMAR (Latent Missing at Random).

Think of it like this:

In the Treatment Group, missing data might be random.
In the Control Group, missing data might depend on whether you would have been a Complier or a Noncomplier.
- Example: Maybe the "Compliers" (who are naturally more responsible) are more likely to fill out the survey, while the "Noncompliers" (who are less engaged) are more likely to lose the survey.

Because we can't see who is a Complier in the Control group, we can't tell if the missing surveys are skewing our results.

The Big Breakthrough: The "Two-Recipe" Puzzle

The authors realized that trying to solve this problem is like trying to bake a cake when you don't know two key ingredients. They showed that the math boils down to two connected equations (or recipes) that are stuck together:

The "Response Recipe": This tries to figure out who is missing the survey in the Control group. (Are the missing people mostly Compliers or Noncompliers?)
The "Outcome Recipe": This tries to figure out what the missing people's scores would have been.

The problem is that you can't solve the "Outcome Recipe" until you solve the "Response Recipe," but they are tangled together.

The Solution: A Flexible Template

The paper proposes a unifying template (a flexible toolkit) to untangle these recipes. The authors say you need to make two separate choices to solve the puzzle:

Choice 1: The "Principal Identification" Assumption
This is your guess about how the program works on the "Slackers" (Noncompliers).

Option A (Exclusion Restriction): "The program has zero effect on people who don't do the work." (If you don't volunteer, the program does nothing for you).
Option B (Principal Ignorability): "If we look at people with similar backgrounds, the 'Slackers' and 'Hard Workers' would have had the same happiness scores in the Control group."

Choice 2: The "Missingness" Assumption
This is your guess about why the surveys are missing.

Option A (Stable Response): "The rate at which people drop out is the same in the Treatment group as it is in the Control group." (e.g., If 20% of Hard Workers drop out in the Treatment group, we assume 20% drop out in the Control group too).
Option B (Response Ignorability): "In the Control group, the 'Slackers' and 'Hard Workers' are equally likely to fill out the survey."

The Magic Trick:
The authors discovered that if you choose Option B for both choices (Principal Ignorability + Response Ignorability), the "Ghost" problem disappears entirely. The missing data becomes simple and random, and you don't need to make any extra guesses about the missing people!

A Warning About Old Rules

The paper also found a flaw in some old math rules (called "Stable Response"). Sometimes, these old rules can lead to impossible math answers, like saying "20% of people are missing" when the math actually implies "-5% are missing" (which is impossible).

To fix this, the authors suggest a "Near-Stable" version. It's like a safety guardrail: if the math tries to give an impossible answer, the rule gently bumps it back to the nearest possible number (like 0% or 100%) instead of breaking the whole calculation.

The Real-World Test

The authors tested their new template on data from the Baltimore Experience Corps (a real program where seniors volunteer in schools). They ran the numbers using different combinations of their "Choice 1" and "Choice 2."

What they found:

The results changed depending on which assumptions they picked (which is expected).
However, when they used the "Principal Ignorability" assumption, the results were very stable and didn't need extra guesses about the missing data.
They confirmed that their new "Near-Stable" rules prevented the impossible math errors found in older methods.

Summary

This paper gives researchers a flexible map for navigating the messy world of missing data and non-compliance. Instead of being stuck with one rigid rule, researchers can now mix and match different assumptions to see how robust their results are. Most importantly, they found a special combination of assumptions that makes the missing data problem vanish, and they fixed a safety hazard in the old math tools.

Technical Summary: Identification of Complier and Noncomplier Average Causal Effects in the Presence of Latent Missing-at-Random (LMAR) Outcomes

1. Problem Statement

The study of treatment effects is frequently complicated by two simultaneous challenges: noncompliance (where participants do not adhere to their assigned treatment) and missing outcome data. This paper focuses on the one-sided noncompliance setting, where noncompliance occurs only in the treatment arm (participants may refuse the active treatment, but control participants cannot "refuse" the control).

In this setting, researchers are often interested in Principal Causal Effects (PCEs): the Complier Average Causal Effect (CACE) and the Noncomplier Average Causal Effect (NACE). These effects are defined based on latent principal strata (compliers vs. noncompliers), which are unobservable in the control condition.

The specific problem addressed is the identification of these effects when outcomes are Missing at Random (MAR) in the treatment arm but Missing Not at Random (MNAR) in the control arm, a condition termed Latent Missing at Random (LMAR) or latent ignorability. Under LMAR, the probability of missingness may depend on the unobserved compliance type. While methods exist for handling LMAR within the Instrumental Variable (IV) framework (which relies on the Exclusion Restriction), no general solution has been proposed for non-IV approaches (such as Principal Ignorability) or for combining different identification strategies with various missingness mechanisms.

2. Methodology and Framework

2.1 Base Assumptions

The authors establish a baseline framework relying on two core assumptions:

Treatment Assignment Ignorability (A0): Conditional on baseline covariates $X$ , treatment assignment $Z$ is independent of potential outcomes and compliance type.
Latent Missing at Random (B0): Conditional on $X$ , $Z$ , and compliance type $C$ , the missingness indicator $R$ is independent of the outcome $Y$ . Crucially, this allows missingness to depend on $C$ in the control arm.

2.2 Characterization of Non-Identifiability

The paper demonstrates that under A0 and B0 alone, the PCEs are unidentifiable. The non-identifiability stems from the inability to recover the conditional outcome means under control for each stratum, denoted $\mu_{0c}(X) = E[Y | X, Z=0, C=c]$ .

The authors derive that this problem reduces to a system of two connected mixture equations:

The Response Mixture Equation: Relates the overall response probability in the control arm ( $\lambda_0(X)$ ) to the stratum-specific response probabilities ( $\varpi_{0c}(X)$ ).
$\lambda_0(X) = \varpi_{01}(X)\pi_1(X) + \varpi_{00}(X)\pi_0(X)$
The Outcome Mixture Equation: Relates the observed outcome mean in the control arm ( $\kappa^R_0(X)$ ) to the stratum-specific outcome means ( $\mu_{0c}(X)$ ) weighted by the stratum-specific response probabilities.
$\kappa^R_0(X) = \mu_{01}(X)\pi^R_{01}(X) + \mu_{00}(X)\pi^R_{00}(X)$

The unknowns are the stratum-specific response probabilities ( $\varpi_{0c}$ ) and the stratum-specific outcome means ( $\mu_{0c}$ ). The equations are connected because the response probabilities determine the weights in the outcome equation.

2.3 The Unifying Template

The paper proposes a two-step identification template:

Step 1 (Missingness Mechanism): Introduce a specific assumption about the missingness mechanism to solve the Response Mixture Equation and identify the response probabilities ( $\varpi_{0c}$ ).
Step 2 (Principal Identification): Introduce a principal identification assumption to solve the Outcome Mixture Equation for the outcome means ( $\mu_{0c}$ ).

The authors emphasize that these are separate processes; a specific missingness assumption does not need to align with a specific principal identification assumption (e.g., one can pair the Exclusion Restriction with a non-ER missingness assumption).

3. Key Contributions

3.1 New Missingness Assumptions and Modifications

The paper categorizes specific missingness assumptions into two types:

Type 1: Comparing Missingness Across Treatment Conditions.
- Includes existing assumptions like Stable Noncomplier Response (SNR/rER) and Stable Complier Response (SCR).
- Critical Finding: The authors identify a limitation in existing SNR/SCR assumptions: they can imply implausible response probabilities (negative or $>1$ ) depending on the observed data distribution.
- Proposal: They propose modified versions, Near-SNR and Near-SCR, which constrain probabilities to valid ranges (e.g., $[\epsilon, 1]$ ) when the exact assumption yields invalid values.
Type 2: Comparing Missingness Across Principal Strata.
- Response Principal Ignorability (rPI): Assumes response probability is independent of compliance type in the control arm ( $R \perp C | X, Z=0$ ).
- Proportional Response Odds (rPO): Assumes the odds ratio of response between compliers and noncompliers is the same in both arms.
- These assumptions do not suffer from the implausible probability issue inherent in Type 1.

3.2 Identification Formulas

The paper derives explicit identification formulas for CACE, NACE, and ATE under various pairings of:

Principal Identification Assumptions: Exclusion Restriction (ER), Principal Ignorability (PI), and sensitivity analysis assumptions (PIsens-GOR, PIsens-MR, PIsens-SMD).
Specific Missingness Assumptions: Near-SNR, Near-SCR, rPI, and rPO.

Key Theoretical Result: If Principal Ignorability (PI) is used as the principal identification assumption, the PCEs are identified without needing a specific missingness mechanism beyond LMAR. PI provides a solution to the outcome mixture equation that bypasses the need to solve the response mixture equation. Furthermore, the combination of PI and LMAR implies that the outcome is MAR.

3.3 Empirical Illustration

The authors apply their framework to data from the Baltimore Experience Corps Trial (a study on elderly volunteering).

Data: 623 participants (284 intervention, 339 control) with 22.6% overall missingness in the outcome "perceived generativity achievement."
Analysis: They estimated effects under all combinations of the three principal assumptions (ER, PI, PIsens-SMDe) and four missingness assumptions.
Findings:
- Effect estimates varied significantly based on the choice of the principal identification assumption (e.g., ER vs. PI).
- Within a fixed principal identification assumption, estimates were relatively robust to the choice of the specific missingness assumption (except for increased uncertainty under Near-SNR).
- When PI was assumed, the results were identical regardless of the missingness mechanism, confirming the theoretical result that PI obviates the need for specific missingness assumptions.

4. Significance and Claims

The paper claims to provide a unifying view and a flexible template for effect identification in the presence of LMAR outcomes. Its primary significance lies in:

Decoupling Assumptions: Clarifying that identifying PCEs generally requires two distinct assumptions (one for missingness, one for principal strata) that can be chosen independently based on substantive justification.
Expanding Options: Moving beyond the IV/ER framework to include Principal Ignorability and sensitivity analysis approaches, which are increasingly used in practice.
Correcting Existing Methods: Highlighting and fixing the issue of implausible probabilities in widely used stable response assumptions (SNR/SCR) by proposing "near" versions.
Special Case Identification: Demonstrating that under PI, the complex problem of LMAR simplifies to a standard MAR problem, removing the need for additional missingness assumptions.

The authors conclude that while the template offers immediate identification formulas, future work is needed to develop robust estimation and inference methods (e.g., variance estimation) for these new formulas, particularly under the "near" stable response assumptions. They also note that the current work is limited to LMAR and does not address missingness dependent on the outcome itself (MNAR beyond LMAR).