📊 statistics

Causal Inference Using Augmented Epidemic Models

This paper distinguishes between data-generating and causal interpretations of epidemic models to establish a framework for estimating intervention effects through causal inference that accounts for confounders and time-varying measures.

Original authors: Heejong Bong, Valérie Ventura, Larry Wasserman

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Heejong Bong, Valérie Ventura, Larry Wasserman

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 specific action, like wearing a mask or staying home, actually stops a virus from spreading. You have a history of how the virus behaved (the data) and a list of actions people took over time. To understand the "cause and effect," scientists build mathematical models.

This paper is about a common trap in building these models and offers a better way to solve it.

The Two Ways to Look at the Model

The authors say that when we add "interventions" (like vaccines or lockdowns) to an epidemic model, we usually treat the model in one of two ways, but we often mix them up:

The "Storyteller" (Data Generating Model): This model tries to tell the story of what actually happened. It says, "Given what happened yesterday and the actions we took, here is how the virus spread today."
The "What-If" Machine (Causal Model): This model tries to answer a hypothetical question. It asks, "If we had changed our actions yesterday, what would have happened today?"

The Problem: The paper argues that if you use the "Storyteller" model to predict the "What-If" scenario, you get the wrong answer. It's like trying to predict the weather next week by looking at a map of traffic jams from today. The tools are designed for different jobs.

The "Ghost" Problem (Phantom Bias)

The paper introduces a sneaky problem called the "g-null paradox" caused by "Phantom Variables."

The Analogy: Imagine you are studying why a car crashes. You look at the driver's speed (Intervention) and the crash (Outcome). But there is a "Ghost" (a Phantom Variable) you can't see, like a sudden gust of wind.
- The wind causes the car to swerve (affecting the outcome).
- The wind also makes the driver hit the brakes harder (affecting the intervention).
- The wind does not directly cause the crash in a way that links the two in a simple line.

If you use standard math (Maximum Likelihood) to analyze this, the math gets confused. It sees that "braking" and "crashing" happen together and concludes that braking causes crashes, even if it doesn't. The "Ghost" (the wind) creates a fake connection. In the paper's terms, the model thinks there is a causal effect when there is none, or it exaggerates the effect that does exist.

The Four Methods: A Menu of Solutions

The authors propose four ways to handle this. They recommend one specific method as the best balance of simplicity and accuracy.

Method 1: The "Standard" Way (Don't Do This)

How it works: You build the "Storyteller" model, fit it to the data using standard math, and then pretend it answers "What-If" questions.
The Result: This is the trap. It leads to the "Ghost" problem, giving you fake or exaggerated results. The paper says: Avoid this.

Method 2: The "Heavy Lifter"

How it works: You build the "Storyteller" model, but then you use a complex mathematical recipe (called the g-formula) to manually extract the "What-If" answer.
The Result: It works and avoids the Ghost problem, but it is very hard to do. The "What-If" answer is buried in a black box and is hard to explain to people.

Method 3: The "Recommended" Way (The Sweet Spot)

How it works: Instead of trying to tell the whole story of the past, you treat your model strictly as a "What-If" machine from the start. You acknowledge that the past data has "noise" (confounders) and use a special statistical tool (called estimating equations) to filter that noise out.
The Result: This is the simplest and most natural approach. It gives you a clear, interpretable answer about the effect of the intervention without getting tripped up by the "Ghosts." It's like using a specialized filter to see the truth, rather than trying to reconstruct the whole universe.

Method 4: The "Architect"

How it works: You build a "What-If" model first, and then you build a new, very complex "Storyteller" model that is perfectly consistent with it.
The Result: This avoids the Ghost problem and doesn't require the complex math of Method 2. However, it is extremely difficult to build and requires making many assumptions about how the world works.

The Real-World Test

The authors tested these ideas using two things:

Fake Data: They created computer simulations of a virus outbreak with "Ghosts" hidden in the background.
- Result: The standard method (Method 1) failed, seeing effects where there were none. Method 3 worked perfectly, ignoring the ghosts.
Real Data: They looked at real COVID-19 death data from 30 US states and the effect of "mobility" (how much people moved around).
- Result: The standard method suggested that staying home had a huge effect on reducing deaths. Method 3 suggested the effect was smaller (or sometimes non-existent). The authors argue the standard method was likely "hallucinating" a stronger effect because of the hidden "Ghosts" (like unmeasured weather patterns or healthcare capacity) that influenced both movement and death rates.

The Bottom Line

If you want to know if an intervention (like a vaccine or lockdown) actually works based on time-series data:

Don't just fit a standard model and hope for the best; it will likely lie to you about the cause-and-effect relationship.
Instead, use Method 3. Treat your model as a tool to answer "What-If" questions directly, and use specific statistical techniques to clean out the hidden variables that confuse the picture.

The paper concludes that while all models are imperfect, Method 3 is the most practical way to get a truthful answer about the causal effects of public health interventions without falling into the trap of "phantom" biases.

Technical Summary: Causal Inference Using Augmented Epidemic Models

Problem Statement

The paper addresses the challenge of inferring the causal effects of time-varying interventions ( $A_t$ ) on epidemic outcomes ( $Y_t$ ) using a single observed time series. While epidemic models are frequently modified to include intervention variables to predict scenarios or estimate effects, the literature often conflates two distinct interpretations of these "augmented" models:

Data Generating Models (DGMs): Models describing the conditional density of observed random variables.
Marginal Structural Models (MSMs): Causal models describing the distribution of counterfactual random variables $Y_t(a_1, \dots, a_t)$ .

The central problem is that treating an augmented epidemic model as a DGM to estimate parameters via Maximum Likelihood Estimation (MLE) or Bayesian methods, and then using those same parameters to make causal inferences (as if it were an MSM), leads to inconsistent estimators. This inconsistency manifests as the g-null paradox: even when there is no true causal effect ( $\beta_A = 0$ ), the estimated causal effect is non-zero. This bias arises due to phantom variables—unobserved variables that affect the outcome but not the intervention—which induce conditional dependence between the treatment and outcome in non-linear parametric models, causing likelihood-based estimates to conflate correlation with causation.

Methodology

The authors propose and analyze four distinct approaches to handling augmented epidemic models, distinguishing between those that treat the model as a DGM versus an MSM.

Method 1: Maximum Likelihood/Bayesian Estimation (The Flawed Approach)

Procedure: The augmented model is treated as a DGM. Parameters are estimated via MLE or Bayesian inference. The fitted model is then used to compute causal effects (e.g., via the g-formula) or scenario predictions.
Issue: This approach suffers from the g-null paradox. In the presence of phantom variables, the likelihood function cannot separate causal effects from conditional dependence. Consequently, estimates of causal parameters are inconsistent, yielding non-zero effects even when the true effect is zero.

Method 2: Causal Effect Extraction via Estimating Equations

Procedure:
1. Treat the augmented model as a DGM and include confounders ( $X_t$ ).
2. Extract the causal effect $\psi(a_t)$ from the DGM using the g-formula (Eq. 9).
3. Estimate the parameters of the resulting causal model using estimating equations (Eq. 14) rather than likelihood.
Mechanism: This method uses the propensity score (derived from the DGM) to weight the estimating equations, effectively adjusting for confounding.
Limitation: It requires specifying the full joint distribution of observables, which is computationally demanding and often intractable. The resulting causal model may lack interpretability.

Method 3: Augmented Models as Marginal Structural Models (Recommended)

Procedure:
1. Interpret the augmented epidemic model directly as an MSM for the counterfactual $Y_t(a_t)$ .
2. Specify a separate model for the propensity score (the probability of the intervention given past history).
3. Estimate parameters using estimating equations (Eq. 14).
Mechanism: By treating the epidemic model as a causal model for counterfactuals, the authors avoid the variation dependence between causal and dependence parameters that plagues likelihood-based methods. The causal effect $\psi(a_t; \theta)$ is often approximated via Monte Carlo simulation (Eq. 15) if a closed form is unavailable.
Advantage: This is the simplest approach, aligns with standard causal inference practice, and handles confounders easily via the propensity score. It avoids the g-null paradox.

Method 4: Causal-Preserving Data Generating Models

Procedure:
1. Use the augmented model as an MSM.
2. Construct a full DGM for the observables $(X_t, A_t, Y_t)$ that is consistent with the specified MSM using copulas (specifically "frugal parameterization" as described by Evans and Didelez, 2024).
3. Estimate all parameters via Maximum Likelihood.
Mechanism: By explicitly separating the parameterization of causation (via the MSM) and dependence (via the copula), the model becomes variation independent, avoiding the g-null paradox without dividing by the propensity score.
Limitation: This is technically the most challenging approach, requiring complex modeling of joint distributions and copulas, particularly for single time series.

Key Contributions and Results

Theoretical Contributions

Identification of Phantom Bias: The paper rigorously demonstrates that phantom variables (unobserved factors affecting outcomes but not interventions) cause likelihood-based estimates in non-linear epidemic models to be inconsistent, even in the absence of unmeasured confounding.
Distinction of Interpretations: It clarifies that DGMs and MSMs are generally not interchangeable in time-series settings and that conflating them leads to the g-null paradox.
Methodological Framework: The authors provide a structured comparison of four methods, highlighting that Methods 2, 3, and 4 are valid for causal inference, while Method 1 is not.

Empirical Results

The authors validate their theoretical claims through simulated data and an observational study:

Semi-Mechanistic Model Simulation:
- Method 1 (MLE): Showed significant phantom bias. When the true causal effect $\beta_A = 0$ , the MLE estimated a non-zero effect.
- Method 3 (Estimating Equations): Produced unbiased estimates, correctly recovering $\beta_A = 0$ when the true effect was null.
SEIR Model Simulation:
- Replicated the findings: MLE estimates were biased due to phantom variables, while estimating equation estimates (Method 3) were consistent.
Observational Study (US COVID-19 Data):
- Analyzed the effect of mobility (proportion of full-time work) on COVID-19 deaths across 30 US states.
- Comparison: Method 3 estimates were significantly lower than Method 1 (MLE) estimates in 24 out of 30 states ( $p < 0.001$ ).
- Interpretation: The authors suggest that the higher MLE estimates likely overestimated the causal effect of mobility on deaths due to phantom bias, whereas the estimating equation approach provided a more conservative and likely accurate estimate.

Significance and Claims

The paper claims that standard practices in epidemic modeling—specifically fitting augmented models via likelihood and interpreting them causally—are fundamentally flawed due to the g-null paradox. The authors argue that:

MLE and Bayesian estimates should not be used for causal inference in these contexts because they yield inconsistent estimators.
Method 3 is the most practical and natural solution. It requires fewer modeling assumptions than Method 4 and avoids the intractability of Method 2, while successfully eliminating phantom bias.
Model Misspecification vs. Phantom Bias: While all methods suffer from potential model misspecification, only Methods 2, 3, and 4 eliminate the specific structural bias introduced by phantom variables in non-linear models.

The authors conclude that while Method 4 offers a principled, fully specified alternative that avoids dividing by propensity scores, Method 3 remains the recommended approach for its simplicity and interpretability. They emphasize the critical need to distinguish between data generating models and causal models in epidemic research.