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Marginal and conditional summary measures: transportability and compatibility across studies

This paper clarifies the distinct interpretations and properties of marginal versus conditional summary measures, demonstrating that their non-coincidence—even for collapsible measures under effect modification—poses significant challenges for transportability and evidence synthesis, often leading to bias if incompatible measures are naively pooled without access to individual patient data.

Original authors: Antonio Remiro-Azócar, David M. Phillippo, Nicky J. Welton, Sofia Dias, A. E. Ades, Anna Heath, Gianluca Baio

Published 2026-05-07

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CC BY 4.0

Original authors: Antonio Remiro-Azócar, David M. Phillippo, Nicky J. Welton, Sofia Dias, A. E. Ades, Anna Heath, Gianluca Baio

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 decide which of two different cars (Treatment A and Treatment B) is better. You can't test them both at the same time, so you look at two separate reports: one comparing Car A to a standard sedan (Control), and another comparing Car B to that same standard sedan. To figure out which is better, you have to combine these reports.

This paper is a warning label for that process. It argues that when we combine these reports, we often make a critical mistake: we are comparing apples to oranges without realizing it.

Here is the breakdown of the paper's main points using simple analogies.

1. The Two Ways to Measure "Success"

The paper explains that there are two main ways to measure how well a treatment works, and they don't always give the same number.

The "Group Average" (Marginal): Imagine you take a photo of the entire crowd of people who took the drug and ask, "On average, how much better did the group do?" This is a Marginal measure. It looks at the big picture.
The "Specific Profile" (Conditional): Now, imagine you zoom in on a specific type of person in that crowd—say, "people who are 30 years old and smoke." You ask, "How much better did this specific group do?" This is a Conditional measure.

The Problem: In the real world, these two numbers are rarely the same. Just because the "average person" improved by 10 points doesn't mean the "30-year-old smoker" improved by 10 points.

2. The "Magic Lens" Analogy

The paper introduces a concept called Collapsibility. Think of this as a special lens you look through to measure the treatment effect.

Directly Collapsible (The Clear Lens): Some lenses (like measuring simple "Risk Difference" or "Mean Difference") are clear. If you look at the specific groups and then average them up, you get the exact same number as if you looked at the whole crowd at once.
- Analogy: If you measure the height of every student in a class and average them, you get the same result as measuring the whole class together.
Non-Collapsible (The Distorting Lens): Other lenses (like "Odds Ratios" or "Hazard Ratios," often used for survival or binary outcomes) distort the image. Even if you look at specific groups, when you average them up, the number changes.
- Analogy: Imagine a funhouse mirror. If you look at a specific person, they look tall. If you look at the whole group, the mirror distorts the average height in a way that doesn't match the sum of the individuals.

The Paper's Big Surprise: Most people think this distortion only happens with the "funhouse mirrors" (non-collapsible measures). The paper argues that even with clear lenses, if the treatment works differently for different types of people (effect modification), the "Group Average" and the "Specific Profile" will still disagree.

3. The "Average Person" Trap

The paper highlights a confusing term often used in statistics: "At the Mean."

Imagine a study where people are measured by their "smoking status" (Yes/No).
The "Average" smoker might be 0.5 (half a person who smokes).
The paper warns that calculating the effect for this "0.5 person" is nonsense. It's like trying to calculate the fuel efficiency of a car that is half a sedan and half a truck. It doesn't exist.
The paper insists we must be careful not to confuse the effect for the "Average Person" (who doesn't exist) with the "Average Effect" across all real people.

4. The "Indirect Comparison" Disaster

This is where the paper gets practical. In healthcare, we often have to compare Treatment A vs. Treatment B, but we only have data for A vs. Control and B vs. Control. We have to do an "Indirect Treatment Comparison."

The paper says: If you mix the wrong types of measurements, your conclusion will be biased.

Scenario: Study 1 (A vs. Control) reports a "Group Average" result. Study 2 (B vs. Control) reports a "Specific Profile" result (adjusted for age, gender, etc.).
The Mistake: If you simply subtract Study 2's number from Study 1's number to compare A and B, you are mixing apples and oranges.
The Consequence: You might conclude Treatment A is better when it's actually worse, or vice versa, simply because the math doesn't line up.

5. The Solution: Full Access or Careful Math

The paper suggests two ways to fix this:

The Gold Standard: Have access to the raw data for every single patient in every study (Individual Patient Data). This allows statisticians to recalculate everything to match perfectly.
The Careful Approach: If you only have summary numbers (like "100 people took the drug, 20 got better"), you must use very specific mathematical methods (like ML-NMR) that know how to translate between "Group Averages" and "Specific Profiles" without breaking the math.

Summary

The paper is a call to stop being lazy with statistics. When comparing medical treatments:

Don't assume that an average effect is the same as a specific effect.
Don't assume that different studies are reporting the same thing just because they are measuring the same disease.
Check your lenses: Make sure you are comparing "Group Averages" to "Group Averages," and "Specific Profiles" to "Specific Profiles." If you mix them, your medical decisions could be wrong.

The authors aren't saying one type of measurement is "better" than the other; they are saying you must know exactly which one you are using and ensure you aren't accidentally mixing them up.

Technical Summary: Marginal and Conditional Summary Measures: Transportability and Compatibility Across Studies

Problem Statement
The paper addresses a critical, often overlooked source of heterogeneity in evidence synthesis: the incompatibility between marginal (unconditional) and conditional summary measures. While it is well-established that non-collapsible measures (e.g., odds ratios, hazard ratios) do not coincide with their marginal counterparts even in the absence of effect modification, this work argues that the issue extends to collapsible measures (e.g., risk ratios) when effect modification is present.

The core problem is that marginal and conditional estimands correspond to different decision questions and possess different interpretations. In evidence synthesis, particularly in indirect treatment comparisons (ITCs) and meta-analyses, the "naïve pooling" of these incompatible measures can introduce bias. This is exacerbated when studies report different summary measures or when covariate adjustment methods are applied to transport treatment effects from a trial population to a target population. A specific gap identified is that covariates not conventionally described as effect modifiers (purely prognostic covariates) can still modify population-level treatment effects for non-collapsible measures, thereby compromising transportability.

Methodology
The authors employ a theoretical framework based on the potential outcomes notation and the ICH E9 (R1) estimand definition. They do not present new empirical data but rather clarify the mathematical properties and interpretations of various estimands under different hypothetical outcome-generating mechanisms.

The methodology involves:

Formal Definitions: Defining four key estimands:
- Marginal Treatment Effect (MTE): A contrast of functions of marginal outcome expectations ( $E[Y^1] - E[Y^0]$ ).
- Conditional Treatment Effect (CTEX): A contrast of conditional expectations given specific covariate values ( $E[Y^1|X=x] - E[Y^0|X=x]$ ).
- Conditional Treatment Effect at the Means (CTEM): The CTEX evaluated at the mean covariate value ( $\bar{X}$ ).
- Population-Average Conditional Treatment Effect (PACTE): The average of the CTEX across the population distribution of covariates ( $E_X[CTEX]$ ).
Generative Modeling: The authors analyze these estimands under three hypothetical parametric outcome-generating models within a Generalized Linear Model (GLM) framework:
- Homogeneous CTEX: Treatment effects are constant across individuals (no interaction).
- Linear Heterogeneous CTEX: Treatment effects vary linearly with a covariate (interaction present).
- Non-linear (Quadratic) Heterogeneous CTEX: Treatment effects vary non-linearly (quadratically) with a covariate.
Collapsibility Analysis: The paper examines how the order of operations (averaging before or after applying the link function $g(\cdot)$ ) affects the equivalence of estimands, distinguishing between directly collapsible measures (mean difference, risk difference), collapsible but not directly collapsible measures (risk ratio), and non-collapsible measures (odds ratio, hazard ratio).
Application to ITCs: The theoretical findings are applied to the context of covariate adjustment methods used in indirect treatment comparisons (e.g., MAIC, STC, ML-NMR), mapping specific methods to the estimands they produce.

Key Contributions and Results
The paper provides a taxonomy of marginal estimands and clarifies the conditions under which different summary measures coincide or diverge:

Non-Equivalence of Estimators: The authors demonstrate that MTE, CTEM, and PACTE are generally distinct.
- In homogeneous models, MTE equals CTEM and PACTE for directly collapsible measures (identity link) but diverges for non-collapsible measures (logit link), where the MTE depends on the full joint distribution of prognostic covariates.
- In linear heterogeneous models, MTE equals CTEM and PACTE for directly collapsible measures. However, for non-directly collapsible measures (e.g., log risk ratio), the MTE cannot be expressed as a simple function of model coefficients and covariate means; it requires the full joint distribution of effect modifiers and prognostic covariates.
- In non-linear heterogeneous models, CTEM and PACTE are fundamentally distinct. Due to Jensen's inequality, the PACTE (average of non-linear functions) does not equal the CTEM (function of the average). The MTE coincides with the PACTE for the identity link but diverges for non-linear links.
Role of Prognostic Covariates: A significant finding is that purely prognostic covariates (those not interacting with treatment) can modify the marginal treatment effect for non-collapsible measures. Consequently, transportability is threatened not only by differences in effect modifiers but also by differences in the distribution of prognostic factors.
Method-Specific Estimands: The paper maps existing covariate adjustment methods to their target estimands:
- MAIC and STC-G estimate the MTE.
- STC-P, NMI, and cross-NMR estimate the CTEM.
- ML-NMR can estimate PACTE or MTE.
Incompatibility Risks: The analysis highlights that combining estimates from different methods (e.g., an MTE from MAIC with a CTEM from STC-P) in an indirect comparison leads to estimand incompatibility and potential bias. This is particularly acute when trials report conditional estimates based on different sets of covariates.

Significance and Claims
The paper claims that the choice of summary measure is a fundamental component of the research question, yet it is often treated as a secondary consideration in frameworks like PICO. The authors argue that:

Clarification of Terminology: The paper sets out precise terminology to distinguish between marginal and conditional estimands, preventing the confusion often found in literature where terms like "marginal effects" are used ambiguously (e.g., conflating CTEM with PACTE).
Warning Against Naïve Pooling: The authors emphasize that naïve pooling of marginal and conditional measures in evidence synthesis produces bias. They assert that care must be taken to ensure that synthesis methods combine compatible summary measures.
Need for Individual Patient Data (IPD): The paper suggests that resolving estimand incompatibility is significantly easier with full access to individual patient data. IPD allows for the use of methods like ML-NMR or IPD meta-regression, which can produce any desired marginal or conditional summary measure without incompatibility issues and can adjust for covariate distributions in any target population.
Re-evaluation of Transportability: The work challenges the conventional view that only effect modifiers impact transportability, showing that prognostic covariates can alter marginal effects for non-collapsible measures.

In conclusion, the paper serves as a methodological guide for evidence synthesis, urging researchers to explicitly define and align estimands across studies to ensure valid transportability and compatibility, particularly in the context of indirect treatment comparisons and health technology assessments.