Gaussian Mixture Counterfactual Generator

Authors: Jong-Hoon Ahn, Akshay Vashist
Primary area: causal reasoning
Keywords: Gaussian mixture model, synthetic data generation, clinical trial, individual treatment effect, counterfactual generation

Abstract:
We address the individualized treatment effect (ITE) estimation problem, focusing on continuous, multidimensional, and time-dependent treatments for precision medicine. The central challenge lies in modeling these complex treatment scenarios while capturing dynamic patient responses and minimizing reliance on control data. We propose the Gaussian Mixture Counterfactual Generator (GMCG), a generative model that transforms the Gaussian mixture model—traditionally a tool for clustering and density estimation—into a new tool explicitly geared toward causal inference. This approach generates robust counterfactuals by effectively handling continuous and multidimensional treatment spaces. We evaluate GMCG on synthetic crossover trial data and simulated datasets, demonstrating its superior performance over existing methods, particularly in scenarios with limited control data. GMCG derives its effectiveness from modeling the joint distribution of covariates, treatments, and outcomes using a latent state vector while employing a conditional distribution of the state vector to suppress confounding and isolate treatment-outcome relationships.

Decision: accept (poster)
Reviewer scores: 5, 5, 8, 8

Reviews and discussion:
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creation date:
1730399652174

thread_type:
official_review

<thread_object>
creator: reviewer_zi3c

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summary:
Problem: counterfactual data may not exist in the extended phase of a crossover trial, so synthetic control methods cannot be used to assess treatment efficacy with respect to a no-treatment baseline.
Significance: makes it difficult to assess long-term efficacy and safety of treatments in this setting.
Solution: the authors propose a model (the Gaussian Mixture Counterfactual Generator) to generate the synthetic controls.

strengths:
- a, to my knowledge, novel approach to a real problem.
- technically sound , though perhaps missing important assumptions
- illustrative experiments

weaknesses:
The main weakness of this paper is the lack of formal assumptions explaining when this method will yield accurate results. From the results presented it seems like the fidelity of the generated controls will depend on how well the model estimates the dose response function, and how well it can extrapolate to the 0 dose condition. At least this seems sensible to me after reviewing the linear relationship in Figure 1 and the non-linear relationship analysis in Figures 2, 5, and 6.

The empirical results could be strengthened by including simulated but realistic placebo models. Perhaps adding toxicity models for dose responses.

questions:
Can you formalize a condition/assumption that would cover the failure mode illustrated in Figure 5? I think this will be a necessary addition for me to consider this paper for acceptance, and I think it is doable.

Are there realistic placebo models that you could incorporate into your synthetic experiments?

Is your non-linear dose response function a realistic one? Are there other functions like toxicity models that you could add?

In the absence of a formal condition on the dose response and perhaps placebo or 0 dose models, i would also accept a more thorough empirical evaluation. If the answers to the above questions are affirmative, then those experiments could be added, along with varying the number of patients. This would allow practitioners to have a fuller picture of when this approach might be appropriate for their problem.
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<thread>
creation date:
1730487195355

thread_type:
official_review

<thread_object>
creator: reviewer_jkvu

<content>
summary:
This paper considers causal inference for cross-over trials (and some other types of settings). Cross-over trials are settings in which at some fixed point partway through the trial, the placebo group recieve treatment. Causal inference in these settings is complicated by the fact that in later periods there is no longer a placebo group with which to compare treated outcomes. This renders counterfactual prediction methods like synthetic control (Abadie and Garbanzal 2003) inapplicable. In order to simulate counterfactual no-treatment outcomes in later periods, the authors suggest the use of novel method based on Gaussian mixture models. For each individual, the path for all variables, including the treatment and outcome, are drawn from a Gaussian model that depends on an individual-specific and time-invariant state-vector s. In order to simulate counterfactuals, the researchers propose (implicitly) drawing s from its conditional distribution under a counterfactual path for the treatment variable that is not seen in the data and drawing other variables conditional on this s and the counterfactual treatment path.

 As I understand it, the method is essentially an imputation approach. We do not observe individuals who are never treated in the data, but by imposing a sufficiently restrictive model for the relationship between all variables including treatment, we can impute outcomes under treatment paths that are never observed. This reminds me of other linear factor model approaches to counterfactual imputation, for example https://arxiv.org/abs/2006.07691. The efficacy of such methods will clearly depend on the setting.

strengths:
I found the paper to be very well-motivated and well-written, and it the approach is mostly well-explained. I think the method is reasonable and the simulation results are encouraging (particularly those in Appendix C). I think this is a common and important problem in empirical work and a particularly difficult one for causal inference. I think there is a good chance that applied researchers will apply this method.

weaknesses:
The notation $\Psi_0$ at the top of page 4 is never defined. I was also unclear on how the likelihood is specified, presumably this is under an assumption that the noise vector is assumed to be independent standard Gaussian?

One section in which the explanation could be sharpened is 4.1. Personally, I found the manner in which counterfactuals are formulated to be quite counter-intuitive. I would typically think of a counterfactual for some individual in terms of the path of the outcome under a counterfactual path for the treatment but with that individual's static state s, which might be interpreted as unobserved individual characteristics, and other covariates, kept fixed. Here, the counterfactual is modeled as a change in the state that is compatible with the prior factual history of treatment and the counterfactual future treatment path. Now, I think this is a reasonable way to proceed but this could use some more elaboration. In particular, I think it might help to more clearly relate this model to the large literature on linear factor models for causal inference.

I found it rather odd that treatment is modeled as some linear combination of factors with an additive noise term. In fact, treatment is binary in most cases discussed and it can only take one of a small number of paths in the factual data. Some more discussion of this would be helpful.

In the simulation results in Section 5.1, the average path of the outcomes for untreated individuals is almost completely flat. The imputed no-treatment counterfactual paths are likewise flat on average and this matches the ground-truth in the simulation. This is something of a trivial example: there are no systematic changes in outcomes over time that cannot be attributed to treatment. I found the example in Figure 4 in Appendix C in which there are richer trends at play, much more convincing and personally I would put these in the main body of the paper instead. 

If I understand correctly, the simulation adheres to the modeling assumptions of the authors. I wondered whether their methods would provide a good approximation and thus an effective inference method under alternative models that do not exactly adhere to their specification. Perhaps some kind of dynamic and slightly non-linear model.

questions:
More a suggestion than a question. Although this may be beyond the scope of the current work. In my view, one thing that makes Abadie's synthetic control method (and related methods) convincing is the availability of placebo tests. What I mean by this is that, in SCM, one can pretend the treatment date was say, a month before the actual date, and impute counterfactual untreated potential outcomes for the treated group over this month. Because the treated group was indeed untreated in this period, these can be compared with the ground-truth, helping to validate (or invalidate as the case may be) the efficacy of the method. It seems like such validation approaches could also be applied in the context of the method suggested here? I think this may help researchers to assess whether their particular empirical setting is suitable for the authors' approach.
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</thread>
<thread>
creation date:
1730649961906

thread_type:
official_review

<thread_object>
creator: reviewer_9mxh

<content>
summary:
Creating synthetic control arms is challenging in crossover trials where placebo data is lost after switching to active treatment, complicating long-term efficacy and safety assessments. We propose a Gaussian mixture model that generates counterfactual data without control data, accommodating time-varying doses and comparing treatment switchers to an extended placebo group, aiding treatment evaluation and decision-making.

strengths:
Completed work: The article is clearly written, and the charts used for the presentation are visually appealing.

Clear motivation: I find the problem setting of the crossover trial is quite novel and worth thinking.

Well-organized structure: The article has a clear structure which separately introduces two key problems in the model section, followed by a detailed explanation of the methods.

Sufficient experiments: The experiments are extensive and thorough, as sufficient support for method.

weaknesses:
I genuinely believe this is a well-completed work that explores a very important method. However, since I am not an expert in this field, there are certain aspects that I don’t fully understand.

First, from my understanding of causality problems, it’s important to ensure three basic assumptions such as consistency, overlap, and unconfoundedness. These assumptions are crucial to establish causal problem, and I couldn’t find explicit statements or discussions around them in the paper. Further explanation or a discussion on how the proposed method addresses or bypasses these assumptions would be helpful for clarity.

Second, while I believe this work addresses a significant and valuable problem, the paper lacks a clear articulation of the novelty of the proposed method. Explicitly positioning the contributions within the landscape of related methods would make it easier to understand the unique aspects of this approach and its value compared to existing solutions.

Third, in Section 4, the authors frequently present conclusions within framed boxes. While this can be visually helpful, it consumes a considerable amount of page space, which could be better utilized. The authors may want to consider presenting these conclusions in a more compact form, such as lemmas or theorems, which would make it easier for readers to follow the logical progression without compromising on readability.

questions:
In line 161, it states, “Now the problem is to estimate a set of parameters $\mathbb{Q} = [ W, \pi_k, \mu_k, \Sigma_k, \Phi_k]$. However, the paper does not provide a clear explanation of the meaning of $\Phi_k$. Could the authors clarify what this parameter represents and its role within the estimation problem? A brief explanation or reference could help in understanding its role in the proposed model.

In Section 3.3, there are subscripts “cf” and “f” used to label variables $x$. However, it’s unclear what these subscripts signify and how they differ from each other. Could the authors elaborate on the distinctions between “cf” and “f” and provide some context as to why they are used?
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</thread>
<thread>
creation date:
1730698880023

thread_type:
official_review

<thread_object>
creator: reviewer_apkn

<content>
summary:
This paper tackles a difficulty in learning counterfactual data -- control group data may be missing or scarce. The authors propose a Gaussian mixture counterfactual generator and demonstrate it performance in numerical examples.

strengths:
Introducing the Gaussian mixture counterfactual generator is relatively clear to the readers.

There are interesting numerical results demonstrating the effectiveness of the proposed method.

weaknesses:
I have the impression that the methodology is not well motivated. Both the static state analysis and Gaussian mixture model jumped in abruptly. In addition, from equation (3), we do not see a clear dependency of $\mathbf{s}$ on $\mathbf{x}$ and $\mathbf{a}$. However, in Section 4, the dependence appears without discussion.

It is unclear how to find the parameters $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ in Gaussian mixture.

I am not convinced why does the proposed method tackle the difficult situation of scarce or no control group data.

questions:
Is there a particular reason to consider the static state analysis (SSA) for counterfactual distribution learning? For causal effects, I believe there are many other methods, such as doubly-robust methods, IPTW, marginal structural models.

What is $k$ -- the number of components in Gaussian mixture -- used in experiments? How do you find the parameters in Gaussian mixture?

The kernel matrix in equation (11) is unclear to me.
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Metareview:
metareview:
The paper seeks to develop synthetic controls for trials where placebo patients switch to active treatment (a crossover design) which removes placebo data. The synthetic controls are based on Gaussian mixtures that generates counterfactuals without specific control data needed for training. 

The reviewers were positive or borderline negative. The borderline negative reviewers had concerns about the presentations and the assumptions, which I thought were adequately addressed in the author reply. Te comments by reviewer jKvu would be helpful in clarifying the paper and making the text more accessible to non-experts. For example,

"As I understand it, the method is essentially an imputation approach. We do not observe individuals who are never treated in the data, but by imposing a sufficiently restrictive model for the relationship between all variables including treatment, we can impute outcomes under treatment paths that are never observed."

This is a bit unlike the original synthetic controls where there is some availability of placebo.