MissDeepCausal: causal inference from incomplete data using deep latent variable models

1 INTRODUCTION . Many methods have been developed to estimate the causal effect of an intervention , such as the administration of a treatment , on an outcome such as survival , from observational data , i.e. , data that is potentially confounded by selection bias due to the absence of randomization . Classical ones include matching ( Iacus et al. , 2012 ) , inverse propensity weighting ( IPW , Horvitz & Thompson , 1952 ; Rosenbaum & Rubin , 1983 ) and doubly robust methods ( Robins et al. , 1994 ; Chernozhukov et al. , 2018 ; Wager & Athey , 2018 ; Athey et al. , 2019 ) . More recent proposals use deep learning methods that ensure balance of the population at the level of representation ( Johansson et al. , 2016 ; Shalit et al. , 2017 ) , infer the joint distribution of latent and observed confounders , the treatment and the outcome ( Louizos et al. , 2017 ) or predict the counterfactuals with GANs ( Yoon et al. , 2018 ) . For a detailed review of existing literature on treatment effect estimation we refer to Imbens ( 2004 ) , Lunceford & Davidian ( 2004 ) and Guo et al . ( 2019 ) . However , state-of-the-art methods still suffer from important shortcomings . In particular , they seldom consider the possibility that covariates have missing values , which is ubiquitous in many realworld situations ( Josse & Reiter , 2018 ) and has been widely discussed in different contexts ( Mayer et al. , 2019a ; van Buuren , 2018 ; Little & Rubin , 2002 ) . Although this question of missing attributes in the context of treatment effect estimation has been raised early in the development of causal inference ( Rosenbaum & Rubin , 1984 ) , there is still a lack of effective and consistent solutions addressing this problem , with a few notable exceptions such as Mattei & Mealli ( 2009 ) ; Seaman & White ( 2014 ) ; Yang et al . ( 2019 ) ; Kallus et al . ( 2018 ) which mainly focus on inverse propensity weighting ( IPW ) methods and Kuroki & Pearl ( 2014 ) who discuss identifiability of causal effects under measurement error or unobserved confounders . Recently , Mayer et al . ( 2019b ) , in addition to suggesting doubly robust estimators with missing data , classified the existing approaches into two families : the ones that adapt the causal inference assumptions to the missing values setting ( D ’ Agostino Jr & Rubin , 2000 ; Blake et al. , 2019 ) and the ones ( Mattei & Mealli , 2009 ; Seaman & White , 2014 ; Kallus et al. , 2018 ) that consider the classical machinery and missingness mechanisms assumptions ( Little & Rubin , 2002 ) . While the former are based on the assumption of unconfoundedness with missing values , which can be difficult to assess in practice , the latter have been developed under strong para- metric assumptions about the outcome , treatment and covariates models , in addition to relying on missing values hypotheses that can also be difficult to meet in practice ( Yang et al. , 2019 ) . To avoid relying on the hypothesis of unconfoundedness with missing values or being in the very parametric ( and linear ) framework of multiple imputation ( Mattei & Mealli , 2009 ; Seaman & White , 2014 ) and matrix factorization ( Kallus et al. , 2018 ) , we propose a new method for causal inference with missing data , which we call MissDeepCausal . MissDeepCausal is inspired by the work of Kallus et al . ( 2018 ) in the sense that we consider a model with latent confounders , and assume that we only have access to covariates with missing values that are noisy proxies of the true latent confounders . However , our approach generalizes and extends the work of Kallus et al . ( 2018 ) in different aspects : ( i ) instead of linear factor analysis models with missing values , we consider nonlinear versions using deep latent variable models ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) ; ( ii ) we rely on the missing at random ( MAR ) ( Rubin , 1976 ) assumption for the missing data mechanisms , and not on the stronger missing completely at random ( MCAR ) one ; ( iii ) we take into account the posterior distribution of the latent variables given observed data and not only their conditional expectation . This latter point allows us to define a multiple imputation strategy adapted to the latent confounders model , and to couple it with doubly robust treatment effect estimation ( Chernozhukov et al. , 2018 ) . In the remainder of this article we first introduce the problem framework and recall existing work for handling missing values in causal inference in Section 2 . We then introduce two variants of our MissDeepCausal approach in Section 3 . Finally we compare MissDeepCausal empirically with several state-of-the-art methods on simulated data in Section 4 . 2 SETTING , NOTATIONS AND RELATED WORKS . In this section we start by quickly reviewing the problem of causal inference from observational data without missing data . We consider the potential outcomes framework ( Rubin , 1974 ; Imbens & Rubin , 2015 ) where we have a sample of n independent and identically distributed ( i.i.d . ) observations ( Yi ( 0 ) , Yi ( 1 ) , Wi , Xi ) i=1 , ... , n with Wi ∈ { 0 , 1 } a binary treatment , Xi = ( Xi1 , . . . , Xip ) > ∈ Rp a vector of covariates , and ( Yi ( 0 ) , Yi ( 1 ) ) ∈ R2 the outcomes we would have observed had we assigned control or treatment to the i-th sample , respectively . The observed outcome for unit i , Yi ∈ R is defined as Yi , WiYi ( 1 ) + ( 1 − Wi ) Yi ( 0 ) . The individual causal effect of the treatment is τi , Yi ( 1 ) − Yi ( 0 ) and the average treatment effect ( ATE ) is defined as τ , E [ Yi ( 1 ) − Yi ( 0 ) ] = E [ τi ] . The ATE τ , i.e. , the link between W and Y , can be estimated from observational data by taking into account the confounding factors X , i.e. , the common causes of W and Y . A popular estimator of τ from observational data is the so-called doubly robust estimator : τ̂DR , 1 n n∑ i=1 µ̂1 ( Xi ) − µ̂0 ( Xi ) +Wi Yi − µ̂1 ( Xi ) ê ( Xi ) − ( 1−Wi ) Yi − µ̂0 ( Xi ) 1− ê ( Xi ) , ( 1 ) where µ̂w ( x ) are regression estimates of the conditional response surfaces µw ( x ) , E [ Y ( w ) |X = x ] , w ∈ { 0 , 1 } , and ê ( x ) is an estimate of the propensity score e ( x ) , P ( Wi = 1 |Xi = x ) ( Rosenbaum & Rubin , 1983 ; Imbens & Rubin , 2015 ) . Standard results state that if either ( µ̂0 , µ̂1 ) or ê is correctly specified , then τ̂DR is an unbiased estimator of τ ( Robins et al. , 1994 ; Chernozhukov et al. , 2018 ; Wager & Athey , 2018 ) under the following assumptions ( Rosenbaum & Rubin , 1983 ) : the ignorability or unconfoundedness assumption that states that all confounding factors are measured , i.e. , conditionally on X , the treatment assignment is independent of the potential outcomes : { Yi ( 1 ) , Yi ( 0 ) } ⊥ Wi |Xi , for all i ; ( 2 ) and the overlap assumption assuming the existence of some η > 0 such that η < e ( x ) < 1 − η , for all x ∈ X . We now consider an extension to account for possible missing entries in the covariates . For that purpose , we denote the missingness pattern of the i-th sample as Mi ∈ { 0 , 1 } p such that Mij = 0 if Xij is observed and Mij = 1 otherwise . The matrix of observed covariates can be written as X ? , X ( 1 −M ) + NA M , with the elementwise multiplication and 1 the matrix filled with 1 , so that X ? takes its value in the half discrete space X ? , ( R ∪ { NA } ) p. We model Mi as a random vector , and the possibility to infer causal effects with missing data now depends on additional assumptions on the joint law of ( Yi ( 0 ) , Yi ( 1 ) , Wi , Xi , Mi ) i=1 , ... , n. Methods for causal inference with missing covariates can be classified into two categories . Unconfoundedness with missing values . Rosenbaum & Rubin ( 1984 ) extend the unconfoundedness hypothesis ( 2 ) to missing values as { Yi ( 1 ) , Yi ( 0 ) } ⊥ Wi |X ? i , for all i . ( 3 ) This implies the assumption , illustrated in Figure 1 , that if a covariate is not observed , it is not a confounder . In particular , observations can have different confounders depending on their pattern of missing data . They define the generalized propensity score as : ∀x ? ∈ X ? , e ? ( x ? ) , P ( Wi = 1 |X ? i = x ? ) , ( 4 ) which is a balancing score under ( 3 ) . Consequently , an IPW estimator formed with estimators of e ? can be an unbiased estimator of the ATE with missing values . Nevertheless , this method relies both on the fact that the covariates X are the appropriate set of confounders , which can be questioned without missing data ( Kallus et al. , 2018 ) , and requires certain expert input and reasoning to verify that for each observation , treatment assignment and/or outcome values depend only on observed values of the confounders ( Blake et al. , 2019 ; Mayer et al. , 2019b ) . Note in particular , that it is not because the missing data in the covariates are completely at random ( MCAR ) , i.e. , M ⊥ X , that ( 3 ) is met . In practice , in addition , a difficulty with this approach is that estimating ( 4 ) requires fitting one model per pattern of missing values , which is unrealistic with classical tools ( Miettinen , 1985 ; D ’ Agostino Jr & Rubin , 2000 ; D ’ Agostino Jr et al. , 2001 ; Blake et al. , 2019 ) ; Mayer et al . ( 2019b ) address this problem using random forests adapted to covariates with missing values . Missingness mechanisms assumptions . Multiple imputation is one of the most powerful approaches to estimate parameters and their variance from an incomplete data ( Little & Rubin , 2002 ; van Buuren , 2018 ) . Seaman & White ( 2014 ) show that when assuming ( i ) identifiability of the ATE in the complete case , ( ii ) missing at random ( MAR ) values givenW and Y , ( iii ) correct specification of the propensity score with logistic regression and of the Gaussian distribution of covariates , then multiple imputation gives a consistent estimate for the ATE estimated with IPW . An extension to doubly robust estimation has been proposed by Mayer et al . ( 2019b ) . Instead of assuming that confounders are observed directly , Kallus et al . ( 2018 ) consider a more general model where observed covariates X are noisy and/or incomplete proxies of the true latent confounders Z . More specifically , they assume a low-rank model for the covariates and estimate the latent variables from the incomplete confounders using matrix completion methods ( Hastie et al. , 2015 ; Josse et al. , 2016 ) . Then , under the linear regression model Yi = Z T i α+ τWi + εi , ( 5 ) with random latent variables Z , missing values completely at random ( MCAR ) in X , unconfoundedness given Z , and some additional assumptions , they prove that regressing Y on Ẑ and W leads to a consistent ATE estimator . Both techniques , multiple imputation and matrix factorization , rely on parametric ( and linear ) frameworks .

Summary: This contribution considers deep latent-factor models for causal inference -inferring the effect of a treatment- in the presence of missing values. The core challenge is that of confounders not directly observed, and accessible only via noisy proxys, in particular in the missingness. The contributed method relies on using the latent-factor model for multiple imputation in doubly-robust causal treatment effect estimators. As a consequence, it requires the missing at random assumption to control for the impact of imputation. Given that the confounders are not directly assumed, a first approach estimates their effect via an estimate of P(Z|X*)  (probability of confounder given observed data), which is then plugged in the doubly robust estimator in a multiple imputation strategy. A second approach uses heuristically the estimated latent confounders as regressors of non interest in a linear-regression model. The approaches are empirically compared to other imputation strategies used as plugins in the doubly-robust estimator. The contributed approach show marked benefits when the problem is highly non-linear.