Gaussian Mixture Counterfactual Generator
Authors: Jong-Hoon Ahn
Akshay Vashist
Venue: ICLR 2025 Poster (2025)
Keywords: Gaussian mixture model
synthetic data generation
clinical trial
individual treatment effect
counterfactual generation
Primary area: causal reasoning

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.

Published as a conference paper at ICLR 2025
GAUSSIAN MIXTURE COUNTERFACTUAL GENERATOR
Jong-Hoon Ahn and Akshay Vashist
Otsuka Pharmaceutical Development & Commercializattinn Inc. 508 Carnegie Center Dr, Princeton, NJ 08540 (Jong (Jong-Hoon,
Hoon. Ahn, Akshay. Vashist)8orssua com
ABSTRACT
We address the individualized treatment effect (TTE) estimation problem, focus- ing on continuous, multidimensional, and time-dependent treatments for precision medicine. The central challenge lies in modeling these complex treatment scenar- ios while capturing dynamic patient responses and minimizing reliance on con- trol data. We propose the Gaussian Mixture Counterfactual Generator (GMCG), a generative model that transforms the Gaussian mixture mooell-radiitonns tool for clustering and density estimation into ward causal inference. This approach generates robust counterfactuals by effee- tively handling continuous and multidimensional treatment spaces. We evaluate GMCG on synthetic crossover trial data and simulated datasets, demonstrating its superior performance over existing methods, partieularly 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-outes outcome relationships.
a tool explicitly geared to-
INTRODUCTION
The advent of precision medicine is fundamentally reshaping clinical research, spotlighting the de- velopment of individualized treatment strategies tailored to each patient's distinct characteristics (Collins& & Varmus, 20155] Jameson & Longo 2015). This evolution has heightened the demand for sophisticated methods to estimate treatment effects, a process complicated by the intricate nature of real-works patient data. For instance, clinicians frequently encounter the challenge of optimiz- ing multidimensional treatment regimens where multiple therapeutic interventions are combined to achieve synergistic effects (Chen al., 2019). Equally critical the task of determining precise dosage combinations, which often involve continuous adjustments to maximize efficacy while min- imizing adverse outcomes (Bica et al.) 2020b). Moreover, patient responses are rarely static; they evolve over time due to physiological changes, disease progression, external factors (Xu et al., 2016). These complexities underscore the need for tools capable addressing modern healthcare applications" dynamic, multifaceted demands, such as personalized therapy design, clinical decision support systems, and patient outcome forecasting.
Despite notable advances, many existing approaches struggle tackle these challenges in unified manner. Some methods excel at modeling interactions among multiple treatments (Wang & Blei, 2019), while others focus on refining dosage optimization (Schwab et 2020) or capturing tempo- ral dynamics in patient trajectories (Schulam & Saria 2017). For instance, methods like propensity score matching are limited to fixed treatment categories, restricting their applicability in continuous dosage scenarios (Nagalapatti et al. 2024). However, these solutions often operate in isolation, lack- ing the flexibility to integrate the full spectrum of treatment variables- variables-combiianttee combinations, dosages, and -ime-dependent effects-witth within a single framework (Bica et al., (02T). This fragmentation poses a significant limitation, particularly in clinical scenarios where decisions must simultaneously ac- count for these factors interplay. Additionally, traditional techniques frequently impose simplifying assumptions, such as fixed treatment categories or static patient states, failing to reflect real-world care's 's continuous and adaptive nature. Consequently, critical gap remains for approach that can cohesively address these interwoven dimensions while remaining adaptable to diverse data environ- ments.

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To address this, introduce the Gaussian Mixture Counterfactual Generator (GMCG), novel probabilistic model rooted in the foundational principles of counterfactual generation. GMCG em- ploys a Gaussian mixture model (GMM) to capture the joint distribution of covariates, treatments, and outcomes, generating counterfactuals through probability-driven approach that adheres rigor- ously the core tenets of causal inference. By grounding its methodology in these established prin- ciples, GMCG ensures valid and interpretable estimation of individualized treatment effects (ITE) across complex scenarios, including those involving multidimensional treatments and continuous dosage spaces.
This paper makes three significant contributions to the field. First, it presents GMCG as theoretical advancement that unifies previously disparate elements of treatment effect estimation, repurposing GMM-originall; used for clustering or density estimation-into a robust framework for causal inference. Second, it offers practical methodology for designing and evaluating personalized treat- ment strategies across varied clinical settings. Third, we validate GMCG's performance through synthetic and simulated datasets, benchmarking against state-of-the-an art
extensive experiments techniques. The paper is structured as follows: Section 2 reviews related work and identifies gaps in current approaches; Section 3 details the GMCG framework, modeling treatments observed data and addressing the key challenge- challenge-ccunterffaeee counterfactual generation and ITE computation- wwth specific methods and formulas provided in Section 4; Section 5 presents experimental results and analysis; and Section concludes with implications for causal inference, precision medicine, and future re- search. The appendices present the GMCG EM algorithm and pseudocode, among other details, to facilitate a deeper understanding of its implementation. Through this work, we aim to provide researchers and practitioners with a versatile tool meet the evolving demands of personalized healthcare.
RELATED WORK
In common types of meta-learner, two distinct concepts are prevalent: the S-leareer and the T-leamer (Künzel et al., 2019 Okasa 2022). The term 'S-learner' is derived from the notion of single base learner, while 'Tllaanerr' is an abbreviation for two base learners (Shalit et al. 2017]. In a straight- forward scenario involving a treatment group and a control group within a patient data set, the THlearner estimates two separate base learners: one for the treatment group data and another for the control group data. Subsequently, it computes the difference between these two base learners. Esti- mating ITEs based on Gaussian mixture models in the paper by (Ahn & Vashisi 2024) exemplifies a T-learner in counterfactual generation. The GMM-based counterfactual generator demonstrated superior performance compared to the synthetic control method on simulated low-density lipopro- tein (LDL) data (Qian et al. 2021) and showed robustness even when dealing with heavily biased datasets.
However, the algorithm fundamentally operates as a treatment-conttolledd method, requiring a learned data distribution model on control units to generate counterfactual outcomes. This reliance poses limitations in scenarios where control data are unavailable, such as in crossover trials with no extended control group (Zhou al. 2024). Additionally, its application data sets with a small, finite number of groups As the number of groups increases, preparing dis- tinct base learners, each trained on different group data becomes necessary. A significant limitation of such a T-learnr is its inability to accommodate more complex real-world data, where potential treatments can span multivariate, time-varying, and continuous domain.
T-learner is confined to
Despite this limitation, the algorithm provides an important conceptual foundation for our work. Specifically, functions as a T-learner, assuming separate distribution models for treatment and control data. Adapting the model into S-learer framework allows us to develop unified model capable of counterfactual inference even for unobserved treatment types. This adaptation allows us generate counterfactual data even without relying control data for training. Therefore, our proposed method builds upon the foundational idea, extending its applicability to complex treatment scenarios.
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3 MODEL
In this section, we outline the latent variable model known as static state analysis (SSA), originally used in the paper by (Ahn & Vashist 2024), and present how new ideas incorporated into this model. Building on the methodologies, we introduce two key problems. The first problem involves brief overview of how train the SSA model. The second problem concerns what counterfactual predictions to generate using the trained model.
3.1 STATIC STATE ANALYSIS (SSA)
The static state analysis decomposes a longitudinal dataset {o!"} into virevngyying observer ma- trix W(() and time-independent state vectors 8 with additive noises r7(1) ~O, vp()):
(+)))
W s (t)
The line-independent state is referred to static state. In this paper, longitudinal data (x()) gen- erally concatenate all different types of observed data, including baseline data (e.g., demographic and genetic information), dependent covariates (()), results y and treatments a!!), x!1)) and
(( can be written as (+()) (); (0) y0) and ( [W(), w w for 2 1. Equation T is decomposed into
a (1) w! 8 T
(2)
( w s 7h (9) and ( ( 77
(8) with (0)
W! me (1) Equation (2) is
our new equation that is directly included as observational data, whereas the original SSA model did not include it. Although may appear be mere addition of simple formula, it important to note that this single addition fundamentally alters the nature and scope of the algorithm. Equations (T) and (2) also represent a factor model where the noise model follows diagonal noise covariance matrix. This paper is not the first to model treatments or causes directly using factor model. Wang & Blei| (2019) and Bica et al. (2020a) successfully removed hidden confounding factors by directly modeling multiple causes latent variables.
3.2 PROBLEM I: FITTING OF A MODEL TO DATA SET
(1) / E Ta,n = I,..,, .N C D, our first problem to perform SSA by decomposing the data set ID) into a time-varying matrix w and set of static state vectors $ {s, n 1.... _ C S. To learn the static state space S from the dataset, we employ a Gaussian mixture model (GMM) probabilistic distribution to model the static state as pector-valued random variable, which is given by
Given an observed data set ID {an
p(s) Σ T&N(s: Ex)
(3)
1=1
where K is the number of mixture components, There is no more dependence on treatment notation because treatment data are directly included in the observational data. The problem is estimate a set of parameters Q = {fT)). {1g). {Ex}, }. W, y) that maximizes the expected complete-date log-likelihood by using the expectation and maximization algorithm:
N
= arg max
((a s, (Q))
(4)
TT
where (s..) = [ds|p(ss E,, Q) x s and (s, . = f Ids p(s x, Q) x SS Then, we can also get
the noise vector (n) !0 w (s.). Through GMM, latent states reflect various
clusters (e.g., patient types, conditions), which is essential for predicting various potential outcomes in counterfactual scenarios.
3.3 PROBLEM II: COUNTERFACTUAL GENERATION WITH No BACKWARD CAUSATION
Let + be a maximizer of the log-likelihood function by Equation 4 for factual data set Dy = 1 (1r(1)
1,... .N. The main problem in our paper is to generate synthetic counterfactual
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longitudinal dataset DB (T+6)
far (r+ neet 1,... NN from using Q when we assume that we had (7:7+6) where define the colon notation as instead of anf
assigned alternative treatments an
(+:++6)
concatenating viry-ng data or parameters:
a
(0)
(
(7)
x = x
w
(1)
W ((0r)
(rir-5)
(5)
w
a(r++-11)
Counterfactual pretreatment data must align closely as possible with the provided factual pretreat- ment data (Abadie Gardeazabal 2003 Doudchenko al., 202T). Thus, equivalently, the above
1,..,,,,, N 1, problem can be written as generating an entire dataset Der {0 =
that satisfies
(tr) n.
The subscript "f" stands for factual, and "cf" is derived from the first letters of counter-factual. Data obtained through observation is called factual data. On the other hand, data generated through counterfactual reasoning without being observed in the real world is referred to as counterfactual data in this paper. Counterfactual reasoning involves considering hypothetical scenarios, such as imagining what would happen if patient who took medication at time 7 had not taken it. Before time т, the patient actually did take the medication, so the factual data and counterfactual data are the same, which can be expressed as x nnf counterfactual data. This can be interpreted in the context of clinical trials as virtual patient data or synthetic placebo arm. In the next section, introduce the basic assumptions of counterfactual reasoning and propose a new method for counterfactual prediction.
(:T) The goal of this paper is to generate
4 METHODS
To solve the second problem, this section details specific assumptions and formulas for generat- ing counterfactual predictions.
4.1 ASSUMPTIONS
A probabilistic observation refers to a process of observing outcomes that are not deterministic but rather governed by probability distributions. In other words, instead of single fixed outcome, there are multiple possible outcomes, each associated with a certain probability of occurrence. In this paper, probabilistic observation is formalized using the SSA model as follows:
Probabilistic Observation
Consider patient with pretreatment data a; ((77) who receives course of treatment a (1:++)) Post-treatment data are obtained by monitoring the patient's state vector Sy using observers (T:) and additive noises r7 (T.);
(T)) (T:))) 8r + nn (7:)
(6)
where 8; is realization of
St p(s| (:7) a (1.7+8)).
This formulation implies a probabilistic observation process in which future outcomes are proba- bilistically determined. However, the random generation of a state vector from Equation ( must not affect the past information prior to or the treatment information. Thus, we have:
W( (T:++)
n (T)) (T: Ma
(TiT+8)
The process of probabilistic observation is not limited exclusively to the acquisition of factual data. When an alternative treatment, distinct from the one administered, is posited, this same process can be applied to generate counterfactual data. This is restated counterfactual generation follows:

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Counterfactual Generation
Consider patient with pretreatment data Øf (:7) who, instead of receiving the actual treat- ment a (7:7+)) is assumed to have received an alternative treatment a data would be obtained by monitoring the patient's state vector su using the same observers ((r)) and noises 17 (T:);
(T:T+6), Post-treatment
a cf
(T:) W
Bet + +1 (Ti)
(9)
where Ser realization of
Ser ~ p(s|op
ad (117+)))
(10)
This formulation also implies that the random generation of a counterfactual state vector from Equa- tion 10 must not affect the past information prior to or the treatment information. Thus, we have:
(:T)
( (TT++))
17 (++)
+
(T:T+6)
(11)
ac
where 77 is the noise in the factual data observation from Equations (6) and (8). If the patient were to return exactly to the time т and receive a different treatment, only the patient's 's state vector would change due to the altered treatment, while the additive noise, acting independently of the changed treatment, should remain the same.
The counterfactual generation described above assumes retrospective perspective. It refers to imag- ining alternative outcomes to past events, essentially considering what could have been if different actions or cireumstances had occurred. The current time is T + 5. However, Equations ? and 10) can still be applied from a prospective perspective. That is, it can be written as "If the pulient were to receive an alternative treatment, the post-treatment data would obtained by Equations (9) and (70)." In this case, the current time is If the retrospective perspective is valid for post- analysis (c.g. after completing a clinical trial), the prospective perspective can be valid for ongoing decision making (e.g. virtual twins).
4.2 GENERATING COUNTERFACTUAL STATE
Once we find a maximizer Q estimated by solving problem using EM algorithms (see Appendix
sample factual state 8; from p(s| (sr) a (1:7+5) obtain its expectation s (st) for a new patient with data of
(er) who took the treatment a; (1: As the first step in counterfactual data generation for the patient, we can generate a counterfactual state by realizing vector 8, from Equation 3
K
Sef
TWN Mkx Σn
(12)
1=1
that is conditioned on, by subtracting Equation
from Equation (8)
0
((r)
(T.++5)
(1:++5)
((
(ser 88).
(13)
a
Equivalently, we can write a counterfactual state random vector in the form of
(
O (1:7+6) ker a
([ W
([ )) &
Set 81 (
(T:74))
a
a
81 + 8.
NE
(14)
where dagger and ker() denote the pscudo-inverse and a basis of the kernel of (w); ((r+++))
a respectively, and is a random vector of the following distribution by denoting a state shift by treatment change as 5, and the kernel matrix as N:
K PoN((E; mk, m& Ck)
3 ~
(15)
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with
C₁ = ENT N)-
(16)
ma CLN' Σ (1z 81 - 6a)
(17)
Pk = TWN(Nm: + 81 5a ΣΦ c
(18)
EK= Tg
81 + ba Mgr. EE CCrl
Notice that 8 in Equations (4) and 18 the term derived by introducing Equation and makes crucial contribution to this paper.
4.3 GENERATING COUNTERFACTUAL OUTCOMES
From Equations
and we can generate a counterfactual data vector based on the Gaus-
sian mixture: Det 2r W o + NŞ) where is realization of Equation (15). More formally, we can state from a retrospective perspective:
Gaussian Mixture Counterfactual Generator
(r:7+6) the counter- If a patient with factual data æt had received an alternative treatment aef a factual data that would have obtained is represented random variable given by
Xet
P&N x; 21 ( + Nm,), WNC.NTWT)
(19)
where Pks mk, and 8, are functions of a
(T:++6) as given by Equations 14g
and 18
This formulation reflects probabilistic approach for generating counterfactual outcomes based a single Gaussian mixture model. Note that the GMCG presented in Equation (99 is not a statically fixed GMM model but complex model where Ps. 5,. N. Cs. and m, change by т, 6, and a Equation (19) is expressed in terms of complex mean vectors and covariance matrices, but can be implemented more simply by separating it into sampling process and linear transformation.
Because W(+)) 8, = ( N = 0, the random variable x is always the same as क by Equation 19 However, the difference in outcomes made by particular vector & in the proba- bilistic model in Equation [5] is a time-varying individualized treatment effect (ITE) caused by the difference in treatments
(T:++)) aj
(r:++)) that we can obtain
ITE ya
(m:)
yi (T:) ((r (5. N
(20)
The expected ITE (eITE) we can obtain on average is
E ITE
E ((y
(T))
(:T)
a
(1:7+8) +8]
yi
(
(6, +N++ N
Σ p.m.).
(21)
By using Equations 15
and 18 can calculate the conditional averaged treatment effect
(CATE) from Equation
20
CATE
= Byy (7 +::
x (IT)
a
(1:T+8))
(++6) (sT) (rir+ (1ir+6)
(o) ar
K
= w
(pama - 9kTzs)
(22)
where
Thy
C2N" Ex (mx - 81)
(23)
TNN(NN + SilpAE, Ex)(
(24)
= NWNNNNN + s Exe) C

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5 EXPERIMENTS
5.1 ILLUSTRATIVE EXAMPLE OF A CROSSOVER TRIAL
To demonstrate the effectiveness of GMCG in handling continuous treatment spaces and time- dependent effects the individual patient level, present crossover trial as an illustrative ample. A crossover trial is a clinical study design in which participants receive different treatments in sequential manner, offering robust framework for evaluating the prolonged eflicacy and safety of new therapeutic interventions (Elbourne et al., 2002 Nolan et al. 2016 Dwan et al., 2019). In this design, patients are initially randomized into two groups one receiving the active treatment and the other a placebo- for defined period. Following this, the placebo group transitions to the active treatment during an open-label extension phase. This approach ensures that all participants eventually receive the treatment, addressing ethical considerations while enabling the observation of sustained treatment responses. However, a key challenge arises in the extension phase due to the lack of concurrent control data, making it difficult to distinguish treatment effectt from natural disease progression. By employing the GMCG, we address this issue by generating synthetic coun- terfactual data, capitalizing its ability to model continuous variations in treatment and capture dynamic, dimeddepnddent patient outcomes effectively.
be applied to crossover trial data to generate synthetic control data for illustrative purposes. The data set shown here is not actual but derived from simulation model, the details of which can be found in Appendix C In the left panel of Figure [ data from two arms are depicted using shades of red and blue. The red-shaded arm represents groups that received the assigned doses over an entire 24-week period, while the blue-shaded arm represents groups that received the doses only during the last 12 weeks. Each arm further subdivided into groups based on different doses of mg and 2 mg, resulting in total of four distinct dose groups. Notably, group remains on placebo for the entire duration, necessitating the creation of synthetic control arm to compare to the treatment arms to estimate the efficacy of the treatment. The GMCG model was trained on 400 patients' outcome data across the four groups, each consisting of 100 patients, without additional covariates.
This section demonstrates how the proposed GMCG algorithm by Equation (9)
The four thick lines of different colors on the left are randomly selected and displayed, and they are further dealt with in the four subplots on the right. The first subplot displays data from the 1mg group during the second 12-week period, with the algorithm-generann by the red dashed line. Comparing these synthetic data with the observed data and the ground- truth control data, respectively, we observe a perfect match during the first 12 weeks and a close alignment in the second half. The difference between the observed 12-week mg data and the red dashed synthetic data after the 12th week can be interpreted as the patient's expected individual treatment effect (cITE) estimated by Equation (2I). The critical point is that the synthetic control arm created through the proposed algorithm corresponds individually to the active or crossover arm,
synthetic data marked
12-80ats 1mg
ground 1ing moan
2mg grousic
12-weeks 2mg
wooks 1mg
Syntheric meat
24-neeks 2mg
0
zoweeks
govind cyrthvity
gonnd gothet
12
time bowni)]
Figure 1: Illustrativ crossover trial dataset (linear efficacy) and synthetic control data generation
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forming synthetic individually controlled arm. Generating and utilizing the synthetic individually controlled arm makes it possible to estimate ITEs or elTEs.
This simple example clearly illustrates problem that does not have readily available logical solution despite its apparent simplicity in the reference. For example, look at the blue-colored data. Seemingly, synthetie control methods could create synthetic control data for the data. However, traditional synthetic control methods rely on the availability of control units, which are absent here, approximate it by taking the weighted sum of them. Thus, this algorithm provides innovative way to infer these control data by circumventing the need for explicit control units, highlighting its capability to solve problems that traditional algorithms cannot address.
5.2
SIMULATED DATA
5.2.1 LDL CHOLESTEROL DATA
For quantitative analysis, LDL cholesterol simulated data was utilized. Data for the treatment and control groups were created using the PK/PD model (Faltaos al., [0066; Yokote al. 2008 Kim jet al. 2011) of statin drugs prescribed to hypercholesterolaemii patients. The experiment was con- ducted using data with the same characteristics those generated by SyncTTwin (Qian et al. 2021)
and SSA-GMM (Ahn & Vashist 2024).
As explained in Section on the motivation of this paper, the proposed algorithm was developed by converting SSA-GMM into the S-learner framework. While this greatly enhances its versatility by allowing application to various types of treatments, its performance should remain equivalent to the original E2MM-baed counterfactual generator. However, we achieved better results by finding optimized parameters in the GMCG algorithm. As shown in Table | below, the MAE of SSA- GMM reached a level of 0.07, but GMCG was able to reduce the error to much lower level. When compared to the SyncTTwin, SCM (Synthetic Control Method), and CRN (Bica et 2020b models, the error gap becomes even larger.
Table 1: LDL cholesterol data experiment results
N =
No + N1 = 200 + 200 = 400
N = No + N1 = 1000 + 200 = 1200
Method
Po = 0.1
Po = 0.25
Pn = 0.5
Pn = 0.1
0.25
Pn = 0.5
GMCG
0.042
0.038
0.043
0.040
0.039
0.035
SSA-GMM
0.072
0.072
0.076
0.070
0.073
0.069
SyncTwin
0.308
0.150
0.116
0.178
0.106
0.094
SCM
0.341
0.151
0.150
0.231
0.172
0.158
CRN
0.530
0.631
0.343
0.456
0.404
0.371
5.2.2 LDL CHOLESTEROL DATA WITH No CONTROLS
The reason for using LDL. cholesterol data is not just to show superiority SSA-GMM and the other algorithms. As sketched in the introduction and illustratively shown in Section 5.1 is intended be used for more quantitative analysis of crossover trials. However, the composition of the data differs from that in Table [ In Table [ No 200 control data and N1 = 200 treatment data were used. In addition, No 1000 control data and N2 = 200 treatment data were also used as training data. But now, we aim to evaluate a much more challenging task: using control data all. This is logically unsound because comparing the results of the treatment data with the control data is the golden standard for determining the treatment effect. Without comparison, the effect of the treatment cannot be discussed. Instead, we assumed that we would test the experimental data at various doses. The total duration of the experiment is T = 30. Until time 25, no patient receives the drug. After time = 25, each patient takes a random dose of statin between Omg and 10mg. This marks crossover point at time f. 25.
After training the model using random doses data, we tested on the 10mg test data used in Table I and obtained the results shown in Table 2 When comparing the results for N 1200, the performance was similar to SCM. Increasing N slightly improved the results. However, the data for N 400 did not train properly. The results in Table [ were obtained using control data, while
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the results in Table 2 were obtained by training the model without control data, highlighting the notable performance of GMCG. As explained in the Related Work section, models like SCM and SyncTwin do not function without control data because they create synthetic control by finding linear or nonlinear combinations of control units. Although recurrent network models like CRN can be tested, considering the poor results in Table T. it is not possible to obtain proper results without training with control data. The results in Tables ! land Jare are numerical results in terms of MAE, and examples for visual reference are provided in the Appendix D.
Table 2: LDL cholesterol data experiment results (no control group)
N = 1200
N = 2000
Method
Po = 0.1
Po 0.25
Po = 0.5
Po = 0.1 Po 0.25
Po = 0.5
GMCG
0.251
0. 0.199
0.168
0.184
0.150
0.152
GMM-type algorithms use the EM algorithm to find the local maximum of the given likelihood function. The EM algorithm for GMCG training also finds the local maximum of the likelihood function (see Appendix for the EM algorithm for GMCG). This means that results are influenced by the initial values. In the GMCG algorithm for crossover trials, the choice of initial values affected performance. Applying the EM algorithm from random values may make it difficult to achieve re- sults like those in Table 2 The data structure of crossover trials is fundamentally curved and requires multiple mixture components. The synthetic control data by zero dose is particularly influenced by mixture components derived from low-dose data. Therefore, the GMCG-EM algorithm should be executed using values obtained by first applying a GMM to data created by low doses. The results in Table uued two mixture components for dose data between Omg and 3mg. one for 3mg to 7mg. and one for 7mg to 10mg. Thus, a total of K = 4 mixture components were used.
Although the MAE is larger compared the GMCG results in Table 1, this method has poten- tially powerful implications. First, to the best of the authors' knowledge, this is the first attempt solve causal inference problem without using control data. In the field of clinical trials, there are single-arm trials (Wang et al., 2024), but they make control groups using historical or external controls. However, the GMCG method structurally learns the dose response from the experimental data alone and extrapolates the prediction for zero dose. Second, clinical trials without any control data present a new form of clinical trials that are more economical, easier to recruit patients, and more ethically suitable for patients in many cases where historical or external controls cannot be sufficiently obtained, thus opening up broader possibilities.
CONCLUSION
This study introduces the Gaussian mixture counterfactual generator (GMCG) as a novel approach to estimating individualized treatment effects (ITE) in precision medicine. By adapting the Gaus- sian mixture model (GMM) into a robust framework for causal inference, GMCG addresses chal- lenges that traditional methods struggle with, such continuous treatment doses, multidimensional treatments, and time-varying effects. Its probabilistic nature allows it to effectively capture the complexity of real-world treatment scenarios, suggesting its potential to contribute meaningfully to developing personalized treatment strategies.
The experimental results lend support to GMCG's practical utility. Using synthetic crossover trial data, it reliably generated counterfactuals even without concurrent controls, and it outperformed state-of-the-art methods like SCM, SyncTwin, and CRN on simulated LDL cholesterol and tumor growth datasets. Notably, GMCG maintained strong performance despite limited control data, hint- ing its applicability real clinical settings for personalized treatment planning and decision sup- port. Moving forward, validating GMCG with real-world clinical data and extending the model to handle diverse treatment while relaxing assumptions like Gaussian noise-coulle enhance its flexibility and broaden its impact.
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Ioana Bica, Ahmed M Alaa, Craig Lambert, and Mihaela Van Der Schaar. From real-world patient data individualized treatment effects using machine learning: current and future methods to address underlying challenges. Clinical Pharmacology & Therapeutics, 109(1):87-100, 2021.
Peipei Chen, Wei Dong, Xudong Lu, Uzay Kaymak, Kunlun He, and Zhengxing Huang. Deep rep- resentation learning for individualized treatment effect estimation using electronic health records. Journal of biomedical informatics, 100: 100:103303 2019.
Francis Collins Harold Varmus. A new initiative on precision medicine. New England journal of medicine, 372(9):793-795, 2015.
Nick Doudchenko. Khashayar Khosravi, Jean Pouget-Abadie Sebastien Lahaie, Miles Lubin, Vahab Mirrokni, Jann Spiess, et al. Synthetic design: An optimization approach to experimental design with synthetie controls. Advances in Neural Information Processing Systems, 34:8691-8701, 2021.
Kerry Dwan, Tianjing Li. Douglas G Altman, and Diana Elbourne. Consort 2010 statement: exten- sion to randomised crossover trials. bmj. 366, 2019.
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Jimyon Kim, Byung-Jin Ahn, Hong-Seok Chae, Seunghoon Han, Kichan Doh, Jeongeun Choi, Yong Jun, Yong W Lee, and Dong-Seok Yim. A population pharmacokinetic pharmacodynamic model for simvastatin that predicts low-density choplotein-cholesttrol reduc- tion in patients with primary hyperlipidaemia. Basic & clinical pharmacology & toxicology, 109 (3):156-163, 2011.
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Published as a conference paper at ICLR 2025
A NOTATIONS
Numbers and Arrays
a
scalar (integer or real)
a
vector
A
A matrix
In
Identity matrix with n rows and n columns
Identity matrix with dimensionality implied by context
diag(a)
A square, diagonal matrix with diagonal entries given by a scalar random variable
a
vector-valued random variable
A
A matrix-valued random variable
[a;
a]
Vectors concatenation
a
A
[A;
AJ
Matrices concatenation
A
Sets and Graphs
A
A set
The set of real numbers The set containing 0 and The set of all integers between 0 and
{0,
(0, 1. n2)
Probability and Information Theory
P(a)
A probability distribution over discrete variable
p(a)
probability distribution over a continuous variable, or over variable whose type has not been specified
a
Random variable has distribution
or or Bf(x)
Expectation of f(z) with respect to P(x)
Var(f(z))
Variance of fz) under P(x)
Cor(f(z),g(z)
Covariance of f(x) and g(x)) under P(x)
N(o; u, E)
Gaussian distribution over æ with mean f and covariance
Σ
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B EM ALGORITHMS FOR TRAINING GMCG
For the case of having noise s7(1)) ~ (O, (0) with diagonal matrix 10) Eq. probability distribution over (1)) -space for a given s E RMd of the form
implies a
{ (a(0
(+ 8)
12 12 101 exp
W (1)))) w(00)
(25)
With Gaussian mixture prior over defined by Eq. the form
obtain the marginal distribution of x(()
p(x
K (1) = _ T& 1 V
1
exp {
- (xx u6) v(0 (x - wrllpp))
1
(26)
where the model covariance is v(() (t()) W I (
We can also use the colon notation æ = ( and W w defined by Eq. conditional distribution and the marginal distribution in the form of
Then, we have the
(x Ws) W W
p(a|s)
exp
(27)
Σ Th 12 V A exp { (- Wo. W
p(x)
(28)
where V 4 WI, T By Bayes' rule, this leads the posterior distribution of the form
p()/z)
= p((k)))))
k
[kkk)) |2 exp - (s
m4x) 2
M M. 'a (x= WHmr)
x M (ss (s-14)) m4)) M Ma WHw))
(29)
where Ms = (I + T
Now have the joint distribution of observational data æ, and the latent variables s, in the form of
1/ exp { {-1
- (In - Wsn) T tx - Won)
p(In, sn)
mTk 2 exp { {-1
- -1 ( mal
(30)
By caleulating the expectation values
Ink p(k(arn))
(31)
(#n)) = / p(s ki) Ids)
(32)
(8m) I p(s)or.)) s lds)
(33)
((s, - (s,)) (sx)) = Σ (kk)oon)) M
(34)
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can obtain the expectation of the complete data log-likelihood in the form of
N
K
1
1
(Cc) =
Tan
(In TT
In |Ixl
[r(( (sxx + p I (sn)
z Ix 1x
2
2
k=
In 44
- Fn
Tu
on + x
44
W (8n)) - -r W (1))
2
2
m 1)
(35)
k=
where the expectation values are given by E-steps:
(EE W pk: Va)
Ink EN TO x WA V(
(36)
(8 = Me + M&WW
(xn W
(37)
(sn) E Tha
(38)
s) Σn Yok (M + ((sm)x - (sn)) - (sa))) - (s.))
Ynu
(39)
with V = 4 WI,WT and M (Ix + WW)
Equation 35 is maximized by the following M-step formulas
M-steps:
En Ink
Tk
(40)
N
P4
Σn Inn
Σ Ynu Thn (8m)
(41)
Σ Vin Inn ((8n5))))))
- (Sn) mmu?))
(42)
In Vnk
W
Σ En (8m)
()))")))
(43)
n
12
Diag
In
2W (s,))= ((,, W)
(44)
where Diag(.) returns a diagonal matrix composed of the diagonal elements given square matrix.
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Algorithm Gaussian Mixture Counterfactual Generator (GMCG)
Require: (1) Factual training data set { (2) Factual test data æ; and alternative treatment der
Ensure: (I) GMCG parameters Q {[Ta) [ua) (Ex), W, w (2) Counterfactual data Fe
1: Training SSA model
2: Input: {In.t}
3: Initialize Q
4: repeat
5: E-step
6: for = to N do
7: for k to K do
8:
Compute "ink and (sn)) from Eqs. (36 and (7
9:
end for Compute (8n) and (8,,4) from Eqs. end for M-step
10:
38 and
39
12:
13:
for = to K do
14:
Compute TRk Mgg and Eg from Eqs.
40
41 and (42)
15:
end for
16:
Compute W and v from Eqs. (3) and 44
17:
until convergence
18:
return Q
Inferring new patient's state vector 20: Input: Q, It
for k K do
22:
Compute "/k and (s)x from Eqs.
36 and 37
23: end for
24: Compute (8) from Eq. 38
25: return (s) as ST
26: Generating the patient's counterfactual data
27: Input: Q, 3r, 8r, Oef
28: Compute 5g and N from Eq. (14
for k 1 K do
30:
Compute Ct. my, and pk from Eqs. (66)
and
18
end for
32: Generate from Eq. 199
33: return Per
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C ADDITIONAL MATERIALS FOR CROSSOVER TRIAL EXAMPLE
groundd control
efficacy
crossove
Li-eeeis rcdo
3.0
CUCCOm
0.0
-02
12 time (aneic]
12 time twnek]
timw (omek)
Figure 2: A simple illustrative example of creating crossover trials data.
Data Generation. The linear efficacy data presented in Figure | were created using the method shown in Figure First, illustrated in the left panel, outcome data for 400 patients were created by adding noise mainly within the range of -0.2 to 0.2. Figure used these data as the ground-truth control. The middle panel shows the assumed efficacy for four groups based the duration and dosage of the drug used. It was assumed that all patients in each group exhibited the same efficacy. The assumed curve shape was generated as a function of time, specifically using log(t + 1) for 0 S t < 24. The two blue-shaded curves are horizontal translations of the two red-shaded curves. The final synthetic crossover trial data, shown in the right panel, were created by combining the data from panels and II for each patient. This way represents the simplest form of data and does not accurately reflect accurate clinical trial data. Clinical data include discontinuous values, missing values, and various complex covariate structures. While covariate data can provide additional helpful information for experimental results, they were excluded from this example.
Nonlinear Efficacy. In Figure D the synthetic control data arm could be created with only two different dose groups per arm due to the assumption that the efficacy proportional to the dose administered. However, in a more general case, where the relationship between dose and efficacy nonlinear, the generation of synthetic control data requires more dose groups. Figure J presents an example that addresses this question, showing four dose groups for each arm: shades of red representing four dose groups with 0.5, 1.0, 1.5, and 2.0 mg over the full 24-week period, and shades of blue representing four other groups with the same doses during only the last 12 weeks.
The nonlinear relationship assumed here is modeled by (treatment effect by a dose) = (dose + 0.2 x dose") x (treatment effect by Img dose). Each group consists of 100 individuals, totaling 800 training data. The four small plots on the right show synthetic control data for four hypothetical patients taking 1.0 mg and 2.0 mg in each arm, with the generated synthetic mean control data closely approximating the ground truth control.
12-week 0.5mg
12oeeda 1.0gg
12onsela 200g
12-weoks
ground bodi cential syt Usetic mean
grvand
12-weels 1.5mg
Spithetic moan
12 weeks 2.0mg
24-weet:
24-wooks 1.0mg
26-weees 1 5mg
24 weeks 2.0ma
autcome
24/oeats Long glound DLEY
24-maaks 2.0g
ground buth contre
syrthver
12
time booeic]
Figure 3: Simulated crossover trial dataset (nonlinear efficacy) and synthetic control data generation.
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This problem's nonlinear effect cannot be adequately captured with only two dose groups per The Appendix provides additional results related to this issue. This situation is similar to needing at least three points, rather than just two, determine quadratic function's 's position, direction, and curvature. Therefore, more diverse dose levels are necessary in the experimental data to uncover complex hidden effects in counterfactual predictions. Therefore, more diverse dose levels likely necessary in the experimental data to uncover more complex or hidden effects in counterfac- tual predictions.
This section demonstrates how the proposed GMCG algorithm can be applied to crossover trial data and the types of results can produce. However, it important to note that this paper does not claim guaranteed success for all types of crossover trials. As an algorithm within the GMM family, the GMCG algorithm's clear and concise mathematical model highlights its potential tackle new types of synthetic data generation problems. Discussing this potential alone represents meaningful achievement.
Non-trivial Control The ground-truth control data in Figure | or 2 follows simple linear trend, which may seem trivial. However, the proposed method does not fit the shape or distribution of the ground-truth control. Figure Beguonstttt that the ground-truth control data is well predicted even when following a sine wave pattern.
12-wooks 1mg
12/eeaia Ing
12owees 2ng
12 weeks 2mg
goound mut comool syrtherr meen zonmi
graand tuth control
24 weets 1mg
systheric mean cannec
24-woeks:
00/c0m
20rcests 1i; goaund in rostel syrthake maan control
Jerec 2ng
arthiicmmee cantre
(wook)
Figure 4: Simulated crossover trial dataset (linear efficacy and sinusodial ground truth control) and synthetic control data generation.
The problem of estimating the nonlinear effect of dosage shown in Figure 3 cannot be adequately addressed with only two dose groups per arm. Therefore, Figure uses four dose groups per arm. Figure 5 shows the bad result when only two dose groups are used. It demonstrates that the syn- thetic control for the 12-week treatment group is pulled upward due to the nonlinear effect, and the 24-week treatment group shows unstable predictions with significant fluctuations. Increasing the diversity of dosages in the training data, i.c., using more dose groups per arm, can achieve more sta- ble and accurate counterfactual predictions. For instance, if hypothetical experiment is conducted where each patient receives slightly different dosages, as shown in Figure 6 the results improve even in more complex nonlinearities in efficacy.
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I--sots 1mg 2mg
1rsseel lng
Doweees 2igg
ground truc contol syrthert mann
grounc tuth concr
24 weets 1mg
systhetic maar
24-neeks 2mn
24rweaie 1ng
26 goond both contral systhete
gourd tuth contol sprthets meen contri
12
time
Figure 5: Simulated crossover trial dataset (nonlinear efficacy) and incorrect synthetic control data generation.
1e-ke 0.2mm
12-eee 1:37gg
2.0
asrthitic moon contril
gound brti contro
groane trth contro
synthibs moan carbel
1.0
95
cuccomo
palients mmze
wcrke Ging ground trit contrll
ueeic1 139C
syrthatic maan control
Figure 6: Simulated crossover trial dataset (nonlinear efficacy with diverse doses) and synthetic control data generation.
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D ADDITIONAL RESULTS OF LDL DATA EXPERIMENTS
In this experiment, we did not use dimensionality reduction. Thus, W becomes an identity matrix, making the EM algorithm of section B equivalent the standard GMM algorithm. Specifically, used one component for each of the treatment and control groups, which trivially means performing single Gaussian density estimation for each group. At this point, there is still an undetermined noise model: V I (see Appendix B to check how V affects the parameters of Eq. (19). and we manually substituted various a values obtain the counterfactual predictions given by Eq. (19). The results are shown in Fig. 7 The figure shows that very small very large sigma values resulted in high enror values, while values between 0.1 and yielded the lowest error values. Considering that we added noise with variance of 0.1 when simulating LDL data, it explains why the lowest error value terms of MAE was obtained around 0.1.
0.0
M46
0.00
30
30
30
10
10.
10
100
12' 307
Figure 7: Counterfactual prediction error (MAE) as function of σ² in the noise covariance model L = II
Figure ploots 15 different patients' examples obtained by applying the GMCG algorithm the LDL cholesterol data. Specifically, for total time length of T 30, the simulation was performed at t when statin medication was taken, and the factual data, ground truth counterfactual data, and counterfactual prediction obtained through the GMCG algorithm were plotted on each axis. There are two points observation here: the pretreatment data before 25 align exactly with the factual data, and the post-treatment data after 25 closely approximate the ground-truth counterfactual data.
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03
01-
-0.35
-100
13
04-
-03
-.0
o1 (00
00
0.
25
10
racall
1.5
1.3
62-
033
00
20
20
2.3
-2
-22
-3
-0.4
01.1
20
13
155
05-
Figure 8: Several plots derived from the results Po 0.1 and N 400 presented in Table T Each subplot represents the data of an individual patient. There are two key points to note here: First, the pre-treatment data before t = 25 matches exactly between the factual data and counterfactual prediction. Second, the counterfactual prediction after 25 closely approximates the given ground truth post-treatment data. This serves as further evidence of the excellent results shown Table
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-15
-1.
MORMAM 67)
25
30
1.3
02-
3.-
-1.0
0.2
03-
-1.0
-62
-25
0.3
Iettal
-03
-0.3
-1.5
20
coverrticual 271
114
1.5
0.2
20
Figure 9: Several plots derived from the results of Po 0.5 and N 1200 presented in Table 2 Each subplot represents the data of an individual patient. There are three key points to note here: First, these results were generated by a GMCG model trained without control data, generating control data. Second, the pre-treatment data before & 25 matches exactly between the factual data and counterfactual prediction. Third, the counterfactual prediction after 25 approximates the given ground truth post- treatment data. This serves as further evidence of the excellent results shown in T
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E SIMULATED TUMOR GROWTH DATA
Although the proposed algorithm is designed to generate synthetic control arms, it can also perform general causal inference, as defined in Section 3.3 This definition aligns with the problem defini- tions specified in the RMSN and CRN algorithms [Lim] 2018 Bica et al. 2020b). Therefore, in this section, we applied the algorithm to tumor growth data (Geng et al. 2017 which is commonly used for validating general causal inference algorithms. We used data from approximately 10,000 virtual patients and applied it to unbiased and biased data for T 10, 20, 30. The number of mixture com- ponents used in the GMCG model is K = 30. We compared the state-of-the afrt CRN model to the same data and obtained the results. Detailed information is provided the Appendix.
Table 3] shows the mean absolute error (MAE) obtained by varying å while keeping т fixed for data with T = 10 and T = 20. The error measure was averaged between = T + and t 7 Surprisingly, the GMM-based GMCG algorithm outperformed the RNN-based CRN regarding MAE. Results for unbiased data are also summarized in the Appendix.
(T,
(10,5) 5)
(20, 15)
8
2
3
5
2
3
5
GMCG
0.0436
0.0484
0.0520
0.0505
0.0497
0.0044
0.0051
0.0054
0.0054
0.0056
CRN
0.0587
0.0596
0.0627
0.0633
0.0644
0.0325
0.0283
0.0283
0.0292
0.0305
Table 3: Mean absolute errors (MAE) comparison for biased tumor growth data.
As the duration T' increases, there is a tendency for the error values to decrease. However, this cannot be solely attributed the more extended data being learned. Fundamentally, with longer durations, the tumor is exposed to more therapy sessions, and the tumor size has already reduced. Consequently, the ground truth counterfactual tumor size for other arbitrary therapy alternatives does not differ significantly from factual tumor size. While the GMCG results closely approximated the ground truth counterfactual data or factual data, the CRN's counterfactual predictions showed some differences from both of them.
How can GMMM-baed algorithm yield better results than an RNN-based adversarial network al- gorithm? Firstly, it is important to recall that GMM can act as excellent universal approximator as the number mixture components increases (Heaton 2018). Additionally, Problem defined in Section 3.2 is unsupervised learning problem, and GMM stably optimized through the EM algorithm (MeLachlan & Krishnan (007). The most crucial factor lies in the conditional proba- bility calculated through Equation (5). Even in regions with sparse training data, the conditional probability calculates a normalized probabilistic density distribution conditional on the counterfac- tual assumption, which creates robust characteristic against bias. This explains the successful generation of a synthetic control arm in the crossover trials example where training data did not exist.
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F ADDITIONAL MATERIALS FOR TUMOR DATA EXAMPLE
F.1 USED HYPERPARAMETERS
To train the CRN for our experiments, used the default hyperparameters. The exact values used are follows. These values fall within the optimized Appendix in the CRN paper (Bica al. 2020b).
The encoder's s used hyperparameters:
num_epochs = 100
hidden units 24,
br_size 12,
fc units 36,
learning_rate 0.01,
batch 128,
me. 0.9 }
The decoder's used hyperparameters:
num_epochs 100
{ imn_hidden_unin 12,
br_size: 18,
fc fc.hidden_mu units 36,
learning_rate rate 0.001,
batch_size 1024,
mn.keep-prob 0.9
where (br_size) is (balancing representation size) and (rnn (ce-kccp-prob) 1 (RNN dropout probabil- ity).
F.2
ADDITIONAL TABLE
(T, 7)
(10,5) 5)
(20, 15)
2
5
5
GMCG
0.0548
0.0636
0.0697
0.0694
0.0682
0.0057
0.0061
0.0071
0.0071
0.0073
CRN
0.0831
0.0795
0.0773
0.0788
0.0776
0.0429
0.0390
0.0348
0.0327
0.0296
Table 4: 20.
Mean absolute errors (MAE) comparison for unbiased tumor growth data for T = 10 and
(T; r)
(30, 15)
(30,25)
8
2
3
5
2
3
5
GMCG
0.0075 0.0179
0.0072
0.0077
0.0079
0.0081
0.0015
0.0017
0.0019
0.0019
0.0020
CRN
0.0226
0.0262
0.0293
0.0321
0.0128
0.0151
0.0167
0.0192
0.0209
Table Mean absolute errors (MAE) comparison for unbiased tumor growth data for T 30.
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G ASSUMPTIONS
G.1 STANDARD ASSUMPTIONS
This paper adopts Rubin's 's potential outcome framework for LDL cholesterol data analysis presented in Table I and is based on standard assumptions commonly found in many studies, such as consis- tency, overlap, and unconfoundedness. In addition, important new assumptions are discussed in Section 4.1, where we explore the concept of counterfactual reasoning. There two types of counterfactual reasoning: retrospective counterfactual reasoning, which examines the effects of tering past treatments at the current time, and prospective counterfactual reasoning, which forecasts the outcomes of choosing among various possible future treatments at the present moment.
Assumption 1: Consistency. If treatment a ( at (it) was administered to given patient, then
potential outcome for treatment a; is the same as the observed factual outcome: y/ (t-1)
((+ (1)
Assumption 2: Positivity. If p(xl:)) ao1:)) O, then p(a!)) a(0))(())) xx(:0) for all
(0)
Assumption 3: Sequential Strong Ignorability. (a TL ( (x(t)) for all (
G.2 ALTERNATIVE ASSUMPTIONS FOR POSITIVITY VIOLATION
When the positivity assumption is violated, as with the data used in Table an alternative assump- tion is needed. Generally, it can be assumed that the conditional expectation of the outcome variable sufficiently smooth function with respect the treatment variable. This continuity allows for the estimation of outcomes for treatment values that are not directly observed, i.e..
Assumption 4: Strong Continuity. Fy X; = 3r, aur = aer) is continuous and differentiable with respect to treatment to an appropriate order.
Alternatively, it can be assumed that the relationship between the outcome y. the counterfactual treatment acf. and the covariates follows a specific functional form (e.g.. linear or quadratic regression model). This assumption enables extrapolation to estimate causal effects:
Assumption 5: Functional Form of Outcome Model. EEyy X; Ty aet agl f(a, aut 0) where f is known functional form, and represents parameters to be estimated.
Specifically, the results in Figure I and the various examples in the Appendix were obtained based on this assumption.
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H PROOF OF GAUSSIAN MIXTURE COUNTERFACTUAL GENERATOR
Lemma 1. Let the random vector follow a mixture of multivariate normal distributions with the k-th mean my and the k-th covariance matrix Cy for -,..., ,K. Let A be a full-rank matrix and b be a translation vector. Then the random vector x defined by
x
AE + b
has mixture of multivariate normal distributions with the k-th mean
Amg b
(45)
and the k-th covariance matrix
AC
(46)
with the same proportions.
Proof. See Johnson et al. (0022 for the proof.
H.1 PROOF OF GMCG
Proof. Subtracting Eq. 6 from Eq. (
we obtain the following equation:
w (ser st).
Combining this with Eq.
13
write it follows:
æt + W(se -8)-))).. 8F).
Using Eq. (14) to simplify the expression and rearranging the equation with respect to E, we can organize the equation in the form of + AE:
X
Tt W(6 + +NE)
= D W + (WN)
+ AE
Using Eq.
45 and 46 of Lemma] I we finally obtain Eq. (19)
P& xx W(8. + +mm)), WNC
Xdf
25

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Published as a conference paper at ICLR 2025
I LIMITATION OF SCM IN CREATING A SYNTHETIC CONTROL ARM
Generating synthetic control arms in crossover trials challenging due to the unavailability con- trol group data after patients switch to active treatment. In standard RCT, control group data crucial for comparing outcomes over time. However, in crossover triaclyespecially those with an open-label extension phase patients initially assigned the control group often switch the active treatment after a certain period (e.g., after 12 weeks). This switch means that there are no remain- ing patients in the control group during the extended phase (e.g., up to 24 weeks). The absence of extended control data makes it difficult to assess the long-term efficacy and safety of the treatment, as there is no direct comparison group.
Abadie's s synthetic control method (SCM) attempts to approximate the treatment group's outcomes by creating a weighted combination of control units that closely resembles the geatment-assignee patient data before the intervention (Abadie & Gardeazabal] 2003 Abadie al. 2010). This method relies on the availability method becomes problematic in the context of crossover trials lacking extended control data. For example, in a trial with a 12-week RCT followed by 12-week open-lahel extension where all patients receive the active treatment, cannot calculate the weighted sum of the 24-week control outcomes needed to construct a synthetic control arm for the switched patients. This limitation prevents the direct application of Abadie's algorithm in such scenarios. In Zhou et al. (2024), SCM and üifference-inndifffeeeeee (Abadic) 2005| Sant' Anna & Zhao| (2020) methods successfully measured treatment effects in erossover trials with the availability of external controls; however, this approach is not feasible in studies where securing external controls challenging.
control data to compute the necessary weights. Applying Abadic's
24-week control data is available. Without this data,
26


Reviews:
[{"review_id": "BfutcBrhLU", "reviewer": "Reviewer_APKN", "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.", "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.\n\nWhat is $k$ -- the number of components in Gaussian mixture -- used in experiments? How do you find the parameters in Gaussian mixture?\n\nThe kernel matrix in equation (11) is unclear to me.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Introducing the Gaussian mixture counterfactual generator is relatively clear to the readers.\n\nThere 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.\n\nIt is unclear how to find the parameters $\\boldsymbol{\\mu}$ and $\\boldsymbol{\\Sigma}$ in Gaussian mixture.\n\nI am not convinced why does the proposed method tackle the difficult situation of scarce or no control group data.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oyT9zcwZxR", "reviewer": "Reviewer_9mXH", "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.", "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.\n\nIn 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?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Completed work: The article is clearly written, and the charts used for the presentation are visually appealing.\n\nClear motivation: I find the problem setting of the crossover trial is quite novel and worth thinking.\n\nWell-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.\n\nSufficient 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.\n\nFirst, 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.\n\nSecond, 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.\n\nThird, 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "quIiM89Jq4", "reviewer": "Reviewer_jKvu", "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.\n\n 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.", "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.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "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?\n\nOne 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.\n\nI 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.\n\nIn 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. \n\nIf 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DwKThxR2kQ", "reviewer": "Reviewer_zi3C", "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.\nSignificance: makes it difficult to assess long-term efficacy and safety of treatments in this setting.\nSolution: the authors propose a model (the Gaussian Mixture Counterfactual Generator) to generate the synthetic controls.", "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.\n\nAre there realistic placebo models that you could incorporate into your synthetic experiments?\n\nIs your non-linear dose response function a realistic one? Are there other functions like toxicity models that you could add?\n\nIn 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.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- a, to my knowledge, novel approach to a real problem.\n- technically sound , though perhaps missing important assumptions\n- 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.\n\nThe empirical results could be strengthened by including simulated but realistic placebo models. Perhaps adding toxicity models for dose responses.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)