SyncTwin: Transparent Treatment Effect Estimation under Temporal Confounding

1 INTRODUCTION . Estimating the causal individual treatment effect ( ITE ) on patient outcomes using observational data ( observational studies ) has become a promising alternative to clinical trials as large-scale electronic health records become increasingly available ( Booth & Tannock , 2014 ) . Figure 1 illustrates a common setting in medicine and it will be the focus of this work ( DiPietro , 2010 ) : an individual may start the treatment at some observed time ( black dashed line ) and we want to estimate the ITE on the outcomes over time after the treatment starts ( shaded area ) . The key limitation of observational studies is that treatment allocation is not randomised but typically influenced by prior measurable static covariates ( e.g . gender , ethnicity ) and temporal covariates ( e.g . all historical medical diagnosis and conditions , squares in Figure 1 ) . When the covariates also modulate the patient outcomes , they lead to the confounding bias in the direct estimation of the ITE ( Psaty et al. , 1999 ) . Although a plethora of methods overcome the confounding bias by adjusting for the static covariates ( Yoon et al. , 2018 ; Yao et al. , 2018 ; Louizos et al. , 2017 ; Shalit et al. , 2017 ; Li & Fu , 2017 ; Alaa & van der Schaar , 2017 ; Johansson et al. , 2016 ) , few existing works take advantage of the temporal covariates that are measured irregularly over time ( Figure 1 ) ( Bica et al. , 2020 ; Lim et al. , 2018 ; Schulam & Saria , 2017 ; Roy et al. , 2017 ) . Overcoming the confounding bias due to temporal covariates is especially important for medical research as clinical treatment decisions are often based on the temporal progression of a disease . Transparency is highly desirable in such a challenging problem . Although transparency is a general concept , we will focus on two specific aspects ( Arrieta et al. , 2020 ) . ( 1 ) Explainability : the method should estimate the ITE of any given individual ( the target individual ) based on a small subset of other individuals ( contributors ) whose amount of contribution can be quantified ( e.g using a weight between 0 and 1 ) . Although the estimate of different target individuals may depend on different contributors , the method can always shortlist the few contributors for the expert to understand the rationale for each estimate . ( 2 ) Trustworthiness : the method should identify the target individuals whose ITE can not be reliably estimated due to violation of assumptions , lack of data , or other failure modes . Being transparent about what the method can not do improves the overall trustworthiness because it guides the experts to only use the method when it is deemed reliable . Inspired by the well-established Synthetic Control method in Statistics and Econometrics ( Abadie et al. , 2010 ; Abadie , 2019 ) , we propose SyncTwin , a transparent ITE estimation method which deals with temporal confounding . Figure 2 A illustrates the schematics of SyncTwin . SyncTwin starts by encoding the irregularly-measured temporal covariates as representation vectors . For each treated target individual , SyncTwin selects and weights few contributors from the control group based on their representation vectors and the sparsity constraint . SyncTwin proceeds to construct a synthetic twin whose representation vector and outcomes are the weighted average of the contributors . Finally , the ITE is estimated as the difference between the target individual ’ s and the Synthetic Control ’ s outcomes after treatment . The difference in their outcomes before treatment indicates the quality of the synthetic twin and whether the model assumptions hold . If the target individual and synthetic twin do not match in pre-treatment outcomes , the estimated ITE should not be considered trustworthy . Transparency of SyncTwin . SyncTwin achieves explainability by selecting only a few contributors for each target individual . It achieves trustworthiness because it quantifies the confidence one should put into the estimated ITE as the difference between the target and synthetic pre-treatment outcomes . 2 PROBLEM SETTING . We consider a clinical observational study with N individuals indexed by i ∈ [ N ] = { 1 , . . . , N } . Let ai ∈ { 0 , 1 } be the treatment indicator with ai = 1 if i started to receive the treatment at some time and ai = 0 if i never initiated the treatment . 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 respectively . Denote N0 = |I0| and N1 = |I1| as the sizes of the groups . The time t = 0 is of special significance because it marks the initiation of the treatment ( black dashed line in Figure 1 ) . We call the period t < 0 the pre-treatment period and the period t ≥ 0 the treatment period ( shaded area in Figure 1 ) . Temporal covariates are observed during the pre-treatment period only and may influence the treatment decision and the outcome . Let Xi = [ xis ] s∈ [ Si ] be the sequence of covariates xis ∈ RD , which includes Si ∈ N observations taken at times t ∈ Ti = { tis } s∈ [ Si ] , where all tis ∈ R and tis < 0 . Note that xis may also include static covariates whose values are constant over time . To allow the covariates to be sampled at different frequencies , let mis ∈ { 0 , 1 } D be the masking vector with misd = 1 indicating the dth element in xis is sampled . The outcome of interest is observed both before and after the treatment . In many cases , the researchers are interested in the outcomes measured at regular time intervals ( e.g . the monthly average blood pressure ) . Hence , let T − = { −M , . . . , −1 } and T + = { 0 , . . . , H − 1 } be the observation times before and after treatment initiation . In this work , we focus on real-valued outcomes yit ∈ R observed at t ∈ T − ∪ T + . 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 ( Rubin , 2005 ) , 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 for yi ( 0 ) and y − i ( 0 ) . The individual treatment effect ( ITE ) is defined as τi = yi ( 1 ) − yi ( 0 ) ∈ RH . Under the consistency assumption ( discussed later in details ) , the factual outcome is observed yi ( ai ) = yi , which means for any i ∈ [ N ] only the unobserved counterfactual outcome yi ( 1− ai ) needs to be estimated in order to estimate the ITE . To simplify the notations , we focus on estimating the ITE for the treated , i.e . τ̂i = yi ( 1 ) − ŷi ( 0 ) for i ∈ I1 , though the same approach applies to the control i ∈ I0 and new units i /∈ [ N ] without loss of generality ( A.5 ) . SyncTwin relies on the following assumptions . ( 1 ) Consistency , also known as Stable Unit Treatment Value Assumption ( Rubin , 1980 ) : yit ( ai ) = yit , ∀i ∈ [ N ] , t ∈ T − ∪ T + . ( 2 ) No anticipation , also known as causal systems ( Abbring & Van den Berg , 2003 ; Dash , 2005 ) : yit = yit ( 1 ) = yit ( 0 ) , ∀t ∈ T − , i ∈ [ N ] . ( 3 ) Data generating model : the assumed directed acyclic graph is visualized in Figure 2 B ( Pearl , 2009 ) , where we introduce two variables ci ∈ RK and vi ∈ RU in addition to the previously defined ones . The latent variable ci is the common cause of yit ( 0 ) and xis , and it indirectly influences ai through xis . As we show later , SyncTwin tries to learn and construct a synthetic twin that has the same ci as the target . The variable vi is an unobserved confounder . Although SyncTwin , like all other ITE methods , works better without unobserved confounders ( i.e . vi = 0 , ∀i ∈ [ N ] ) , we develop a unique checking procedure in Equation ( 4 ) to validate if there exists vi 6= 0 . We also demonstrate that under certain favourable conditions , SyncTwin can overcome the impact of the vi . To establish the theoretical results , we further assume yit ( 0 ) follows a latent factor model with ci , vi as the latent “ factors ” ( Bai & Ng , 2008 ) : yit ( 0 ) = q > t ci + u > t vi + ξit , ∀t ∈ T − ∪ T + , ( 1 ) where qt ∈ RK , ut ∈ RU are weight vectors and ξit is the white noise . We require the weight vectors to have ||qt|| = 1 , ∀t ∈ T −∪T + ( Xu , 2017 ) , which does not reduce the expressiveness of the model . We further require the dimensionality of the latent factor to be smaller than the number of time steps before or after treatment , i.e . K < min ( M , H ) . Furthermore , let Q− = [ qt ] t∈T − ∈ RM×K and Q = [ qt ] t∈T + ∈ RH×K denote the matrices that stack all the weight vectors q ’ s before and after treatment as rows respectively . The latent factor model assumption may seem restrictive but as we show in Appendix A.4 it is applicable to many scenarios . In the simulation study ( 5.1 ) we further show SyncTwin performs well even when the data is not generated using model ( 1 ) but instead using a set of differential equations . We compare our assumptions with those used in the related works in Appendix A.3 . 3 RELATED WORK . 3.1 SYNTHETIC CONTROL . Similar to SyncTwin , Synthetic control ( SC ) ( Abadie , 2019 ) and its extensions ( Athey et al. , 2018 ; Amjad et al. , 2018 ) estimate ITE based on Synthetic Control outcomes . However , when applied to temporal confounding , SC will flatten the temporal covariates [ xis ] s∈ [ Si ] into a fixed-sized ( highdimensional ) vector xi and use it to construct the twin . As a result , SC does not allow the covariates to be variable-length or sampled at different frequencies ( otherwise xi ’ s dimensionality will vary across individuals ) . In contrast , SyncTwin can gracefully handle these irregularities because it constructs the twin using the representation vectors . Moreover , the covariates xi may contain observation noise and other sources of randomness that do not relate to the outcome or the treatment . Enforcing the target and the twin to have similar xi will inject these irrelevant noise to the twin , a situation we call over-match ( because it resembles over-fit ) . Over-match undermines ITE estimation as we show in the simulation study in Section 5.1 . Finally , SC assumes yit ( 0 ) = q > t xi + u > t vi + ξit , i.e . the flattened covariates xi linearly predicts yit ( 0 ) , which is a special case of our assumption ( 1 ) and unlikely to hold for many medical applications .

Summary: This paper provides an approach for treatment effect estimation when the observational data is longitudinal (with irregular time stamps) and consists of temporal confounding variables. The proposed method can be categorized under the matching methods, in which, in order to estimate the counterfactual outcomes, a subset of the subjects in the opposite treatment arm (i.e., contributors) is selected and weighted. The proposed method is designed such that it achieves explainability (by identifying a few contributors) and trustworthiness (by checking if the estimated outcome is reliable).