SyncTwin: Treatment Effect Estimation with Longitudinal Outcomes

1 Introduction . Estimating the causal individual treatment effect ( ITE ) using observational data has become increasingly common in the medical literature due to the popularization of electronic health records ( EHR ) . The EHR is a longitudinal collection of records : it contains repeated measurements of a patient ’ s health condition over irregular time intervals . Although the treatments may also vary over time , many studies consider a point treatment , where the treatment allocation is performed at some observed time and stays fixed during the study [ 12 , 44 , 34 , 5 , 51 ] . Point treatment is widely applicable to problems involving one-off treatments ( e.g . surgical operations ) or treatments that do not change frequently ( e.g . long term medication for chronic disease ) . We will refer to the setting above as Longitudinal and Irregularly sampled data with Point treatment setting , or LIP . The LIP setting will be the focus of this work ( Figure 1 A ) . The LIP setting is different from the conventional settings in causal inference . This is because we observe the pre-treatment outcome yt over time leading to the treatment allocation . These pretreatment outcomes may unveil the inherent temporal structure in the outcome time series ( e.g . trend and seasonality ) , leading to better ITE estimation over time . In contrast , the conventional settings 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . only consider the outcomes after treatment initiation [ 54 , 52 , 36 , 43 , 6 ] . This is true even for works that deal with dynamic treatment allocations [ 40 , 35 , 11 ] . A naive way to incorporate pre-treatment outcomes into the standard ITE methods is to treat them as additional covariates . This approach is viable but inadequate : the pre-treatment outcomes are arguably much more closely linked to the outcomes after treatment than the covariates — they hence deserve special considerations , e.g . modifying the architecture or loss function to reflect their importance . Without such modification , the ITE methods will not incorporate our prior belief on the importance of the pre-treatment outcomes , which may lead to worse performance . To bridge this gap , we propose SyncTwin , a novel ITE estimation method tailored for the LIP setting . We assume that the temporal outcomes are generated by individualized latent factors and timevarying latent trends . This assumption is similar to the “ factor model ” assumption commonly used in Econometrics [ 1 ] and allows us to achieve the right balance between the parametric assumption and modeling flexibility . Figure 1 B illustrates the schematics of SyncTwin using an example with two treatment options ( treated and control ) . SyncTwin first uncovers the individualized latent factors using representation learning . For a target individual , SyncTwin selects and weights a few contributors based on their latent factors and a sparsity constraint . It proceeds to construct a synthetic twin whose temporal outcomes are the weighted average of the contributors . Finally , the ITE is estimated as the difference between the target individual and the synthetic twin ’ s outcomes . Unlike conventional methods , SyncTwin does not use the learned latent factors to directly predict the outcomes ; instead it uses these variables to find the contributors and their importance weights so as to construct the synthetic twin . This approach brings two bonus features ; both are important to medical applications . ( 1 ) We can calculate an individualized estimation error bound based on the actual and the synthetic outcomes before treatment . In practice , the clinician can accept the recommended treatment when the error bound is below a threshold and resort to expert knowledge otherwise . ( 2 ) We can identify the most important contributors to an estimate as the ones who receive the highest weights . The clinician can further examine these contributors ( e.g . whether they are indeed similar to the target individual ) to validate and interpret the estimate . Contributions . ( 1 ) We develop SyncTwin to leverage the temporal structure in the outcomes for better ITE estimation . ( 2 ) We provide theoretical justifications for each step involved in SyncTwin and prove an individualized error bound that can be used to identify untrustworthy estimates . ( 3 ) In addition to extensive simulations , we conduct an observational study using real EHR data and successfully reproduced the findings of a randomized controlled clinical trial with SyncTwin . 2 Problem setting . We consider an observational study with N individuals indexed by i ∈ [ N ] = { 1 , . . . , N } . Let ai ∈ { 0 , 1 } be the treatment indicator with ai = 1 if i received the treatment at some time and ai = 0 otherwise1 . We realign the time steps such that all treatments were initiated at time t = 0 . Let I1 = { i ∈ [ N ] | ai = 1 } and I0 = { i ∈ [ N ] | ai = 0 } be the set of the treated and the control individuals respectively . Denote N0 = |I0| and N1 = |I1| as the sizes of the groups . Let Xi = [ xis ] s∈ [ Si ] be the temporal covariates consisting of a sequence of observations xis ∈ RD . Let Mi = [ mis ] s∈ [ Si ] be the sequence of binary masking vectors mis ∈ { 0 , 1 } D , where [ mis ] d = 1 indicates the dth element of xis is measured and [ mis ] d = 0 otherwise . The entire sequence Xi contains Si ∈ N observations taken before treatment at times Ti = [ tis ] s∈ [ Si ] , tis < 0 . The maximum sequence length S = maxi ( Si ) . The outcome yit ∈ R is observed at time t ∈ T − ∪ T + . Let T − = { −M , . . . , −1 } and T + = { 0 , . . . , H − 1 } be the observation times before and after treatment allocation . We arrange the outcomes after treatment into a H-dimensional vector denoted as yi = [ yit ] t∈T + ∈ RH . Similarly define pre-treatment outcome vector y−i = [ yit ] t∈T − ∈ RM . Using the potential outcome framework [ 41 ] , let yit ( ai ) ∈ R denote the potential outcome at time t in a world where i received the treatment as indicated by ai . Let yi ( 1 ) = [ yit ( 1 ) ] t∈T + ∈ RH , and y−i ( 1 ) = [ yit ( 1 ) ] t∈T − ∈ RM . Similarly , let yi ( 0 ) = [ yit ( 0 ) ] t∈T + and y − i ( 0 ) = [ yit ( 0 ) ] t∈T − . The individual treatment effect ( ITE ) is defined as τi = yi ( 1 ) − yi ( 0 ) ∈ RH . 3 ITE estimation via SyncTwin . SyncTwin estimates the ITE by predicting both potential outcomes yi ( 1 ) and yi ( 0 ) and take the difference . Note that for an individual in the dataset i ∈ [ N ] , it is only necessary to predict the counterfactual outcome yi ( 1− ai ) because the factual outcome yi ( ai ) = yi is observed ( under the consistency assumption to be discussed later ) . Without loss of generality , we will use estimating yi ( 0 ) as an example in the following sections . One can estimate yi ( 1 ) using the same method . 3.1 Assumptions . SyncTwin relies on three assumptions . ( 1 ) Stable Unit Treatment Value Assumption [ 41 ] : yit ( ai ) = yit , ∀i ∈ [ N ] , t ∈ T − ∪ T + . ( 2 ) No anticipation , also known as causal systems [ 4 ] : yit = yit ( 1 ) = yit ( 0 ) , ∀t ∈ T − , i ∈ [ N ] . ( 3 ) Data generating process ( DGP ) . The DGP assumption involves two parts . First , the causal directed acyclic graph ( DAG ) is specified in Figure 2 [ 37 ] , which involves a latent factor ci ∈ RK . Secondly , we assume the potential outcomes have a parametric form : yit ( 0 ) = q > t ci + ξit , ∀t ∈ T − ∪ T + , ( 1 ) where qt ∈ RK , K < min ( M , H ) is a weight vector and ξit is the white noise . Equation 1 is commonly referred to as a latent factor model with ci as the individualized latent factor and qt as the time-varying latent trend [ 10 ] . Without loss of generality , we require the weight vectors ||qt|| = 1 , ∀t ∈ T − ∪ T + [ 50 ] . Discussion . SyncTwin differs from the nonparametric ITE estimation methods [ 27 ] because it assumes the parametric form in Equation 1 . In essence , it means that we can fully separate the time effect from the individual effect . We would like to highlight two aspects of this assumption . First , the latent factor ci does not change over time . It provides a constant link between the pre-treatment outcomes and the post-treatment outcomes . Secondly , the time-varying latent trend qt is common to all individuals . Together , they ensure that any linear combination of different individuals ’ outcomes will automatically preserve qt , i.e . ∑ i biyit ( 0 ) = q > t ∑ i bici + ∑ i biξit , for any weights bi ∈ R , ∀i ∈ [ N ] . For this reason , SyncTwin bypasses qt and directly finds the weights bi ’ s to issue counterfactual predictions . As we will show later , doing so allows us to derive various theoretical results to inform the design of SyncTwin . It also leads to a checking procedure to validate estimation quality and improvements in interpretability . 1SyncTwin can model more than two treatment options , but we focus on binary treatments for illustration . The DAG in Figure 2 is different from the one commonly used in the static setting [ 41 ] . We assume the latent factor ci captures the physiological factors that impact both the outcome and the covariates . Here , the covariates xis is a “ confounder ” in a general sense because it opens up a backdoor path from treatment to outcome , i.e . ai → xis → ci → yis [ 37 ] . Based on the backdoor criterion [ 37 ] , adjusting for the covariates xis is sufficient to identify the treatment effect from observational data . We compare our assumptions with those used in the related works in Appendix A.2 . In particular , we show that our assumption is weaker than the one used by Synthetic Control , a widely-applied method in Econometrics [ 2 ] . In Appendix A.5 , we further demonstrate the plausibility of our assumptions in real-world scenarios . In the simulation study ( Section 5.1 ) we show experimentally that SyncTwin performs well even when the data is not generated from the assumed DGP exactly but instead from a set of differential equations . Finally , and perhaps most importantly , we show that based on our assumptions , SyncTwin is able to reproduce the findings of a large-scale randomized controlled trial using observational data ( Section 5.2 ) .

Summary: The paper introduces a latent variable model (SyncTwin) for estimating time-varying individualized (point-) treatment effects (ITE) given temporal irregularly sampled pre-treatment covariates and outcomes. Theoretical results justifying the proposed approach include upper bound errors on the learned representations and counterfactual predictions. Experimental results on synthetic and real-world datasets show SyncTwin outperforms baselines per mean absolute error (MAE) on ITE and recovering RCT findings, respectively.