ANOCE: Analysis of Causal Effects with Multiple Mediators via Constrained Structural Learning

1 INTRODUCTION . In the era of causal revolution , identifying the causal effect of an exposure on the outcome of interest is an important problem in many areas , such as epidemics ( Hernán , 2004 ) , medicine ( Hernán et al. , 2000 ) , education ( Card , 1999 ) , and economics ( Panizza & Presbitero , 2014 ) . Under a general causal graph , the exposure may have a direct effect on the outcome and also an indirect effect regulated by a set of mediators ( or intermediate variables ) . For instance , during the outbreak of Coronavirus disease 2019 ( COVID-19 ) , the Chinese government has taken extreme measures to stop the virus spreading such as locking Wuhan down on Jan 23rd , 2020 , followed by 12 other cities in Hubei , known as the “ 2020 Hubei lockdowns ” . This approach ( viewed as the exposure ) , directly blocked infected people leaving from Hubei ; and also stimulated various quarantine measures taken by cities outside of Hubei ( as the mediators ) , which further decreased the migration countrywide in China , and thus indirectly control the spread of COVID-19 . Quantifying the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread regulated by different cities outside Hubei is challenging but of great interest for the current COVID-19 crisis . An analysis of causal effects that interprets the causal mechanism contributed via individual mediators is thus very important . Many recent efforts have been made on studying causal effects that are regulated by mediators . Chakrabortty et al . ( 2018 ) specified the individual mediation effect in a sparse high-dimensional causal graphical model . However , the sum of marginal individual mediation effect is not equal to the effect of all mediators considered jointly ( i.e . the indirect effect ) due to the common interaction among mediators ( VanderWeele & Vansteelandt , 2014 ) . Here , ‘ interaction ’ means that there exists at least one mediator that is regulated by other mediator ( s ) ( see Figure 1b for illustration ) , in contrast to the simple ‘ parallel ’ case ( shown in Figure 1a ) . Vansteelandt & Daniel ( 2017 ) considered an exact decomposition of the indirect effect with a two-mediator setting based on the conditional densities of mediators , while there was no feasible algorithm provided to solve their proposed expressions yet . Therefore , a new framework with a computational friendly algorithm that gives an exact decomposition of the indirect effect on the level of individual mediators is desired under the complex causal network . To estimate the underlying causal network , structure learning algorithms of the directed acyclic graph ( DAG ) are widely used . Popular methods such as the PC algorithm ( Spirtes et al. , 2000 ) that uses conditional independence tests to examine the existence of edges between each pair of variables , require strong assumptions and thus have no guarantee in the finite sample regime . Recently , Zheng et al . ( 2018 ) opened up another class of causal discovery methods by directly formulating a pure optimization problem over real metrics with a novel characteristic of the acyclicity . Yu et al . ( 2019 ) further extended Zheng et al . ( 2018 ) ’ s work with a deep generative model , and showed better performance on the structure learning with weaker assumptions on the noise . See more follow-up works in Lachapelle et al . ( 2019 ) and Zhu & Chen ( 2019 ) . However , the current cutting-edge methods neglect the temporal causal relationship among variables , and thus can not appropriately represent the causal network with pre-specified exposure and outcome . In this paper , we consider establishing a new statistical framework to comprehensively characterize causal effects with multiple mediators , namely , ANalysis Of Causal Effects ( ANOCE ) , under the linear structure equation model ( LSEM ) . Specifically , we propose two causal effects on the level of individual mediators , the natural direct effect and the natural indirect effect for a mediator , denoted as DM and IM , respectively . Our proposed DM can be interpreted as the direct effect of a particular mediator on the outcome that is not regulated by other mediators , while the IM is the indirect effect of the mediator controlled by its descendant mediators . We prove that the DM is valid in the sense that it exactly decomposes the indirect effect of the exposure on the outcome , followed by an ANOCE table to explain different sources of causal effects . To bridge the cutting-edge graphical learning approaches with the temporal causal relationship of variables , we extend the variational auto-encoder ( VAE ) framework in Yu et al . ( 2019 ) with a novel identification constraint that specifies the topological order of the exposure and the outcome . The proposed constrained VAE algorithm is then used to estimate causal effects defined in our ANOCE table , named as ‘ ANOCE-CVAE ’ . Our contributions can be summarized in the following three aspects : • 1 ) . Conceptually , we define different sources of causal effects through mediators with a newly introduced definition of direct and indirect mediator effects , and give an exact decomposition of the indirect effect on the level of individual mediators , under the linear structure equation model . • 2 ) . Methodologically , we incorporate the background knowledge ( the temporal causal relationship among variables ) when using an optimization approach to the causal discovery . Such prior knowledge can be generalized for any measured variable and on the possible set of their parents . Our proposed constrained structural learning can be easily extended to other score-based algorithms . • 3 ) . Practically , extensive simulations are conducted to demonstrate the empirical validity of the proposed algorithm and its competitive performance among existing causal discovery algorithms . Our method is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread in China , by quantifying the individual effect for each city . 2 BACKGROUND AND RELATED WORK . 2.1 GRAPH TERMINOLOGY . Consider a graph G = ( X , E ) with a node set X and an edge set E. There is at most one edge between any pair of nodes . If there is an edge between Xi and Xj , then Xi and Xj are adjacent . A node Xi is said to be a parent of Xj if there is a directed edge from Xi to Xj . Let the set of all parents of node Xj in G as PAXj ( G ) . A path from Xi to Xj in G is a sequence of distinct vertices , π ≡ { a0 , a1 , · · · , aL } ⊂ V such that a0 = Xi , and aL = Xj . A directed path from Xi to Xj is a path between Xi and Xj where all edges are directed toward Xj . A directed cycle is formed by the directed path from Xi to Xj together with the directed edge Xj to Xi . A directed graph that does not contain directed cycles is called a directed acyclic graph ( DAG ) . A directed graph is acyclic if and only if it has a topological ordering . Suppose a DAG G = ( X , E ) that characterizes the causal relationship among |X| = d nodes , where X = [ X1 , X2 , · · · , Xd ] > represents a random vector and an edge Xi → Xj means that Xi is a direct cause of Xj . Let B = { bi , j } 1≤i≤d,1≤j≤d be a d × d matrix , where bi , j is the weight of the edge Xi → Xj ∈ E , and bi , j = 0 otherwise . Then , we say that G = ( X , B ) is a weighted DAG with the node set X and the weighted adjacency matrix B ( the edge set E is nested in B ) . 2.2 RELATED WORK . Our work connects to the literature of the causal graphical model . Pearl et al . ( 2009 ) provided a comprehensive review of recent advances in the analysis of causes and counterfactuals using ‘ dooperator ’ by graphical methods . Later , Maathuis et al . ( 2009 ) started to use an unknown DAG without hidden variables to estimate the causal effects from the high-dimensional observational data . Nandy et al . ( 2017 ) extended the work of Maathuis et al . ( 2009 ) with the linear structure equation model , followed by the individual mediation effect defined in Chakrabortty et al . ( 2018 ) . All of these models rely on the PC algorithm to search the Markov equivalence class of the partial DAG , and usually require strong assumptions due to the computational limit . Our ANOCE is established under the same causal structure of Chakrabortty et al . ( 2018 ) but without sparsity and normality assumptions . Wide literature on causal discovery can be summarized in three classes . The first type focuses on local conditional independence tests to find a causal skeleton and then determine the orientation of edges , such as the well-known PC algorithm ( Spirtes et al. , 2000 ; Kalisch & Bühlmann , 2007 ) . However , testing the conditional independence of continuous variables is not easy ( Shah & Peters , 2018 ) . The second class specifies properly functional causal models with additional assumptions on data distribution , including the ICA-LiNGAM ( Shimizu et al. , 2006 ) and the causal additive model ( CAM ) ( Bühlmann et al. , 2014 ) . The last class , the score-based method , includes the greedy equivalence search ( GES ) ( Chickering , 2002 ) and the fast GES ( fGES ) ( Ramsey et al. , 2017 ) that use for example Bayesian scores in searching a space of causal models . Recently , Zheng et al . ( 2018 ) opened up another track of score-based methods by constructing an optimization with an acyclicity constraint under the LSEM , i.e . the NOTEARS . A follow-up work using a VAE parameterized by a graph neural network that generalizes LSEM was proposed in Yu et al . ( 2019 ) with a more computational friendly constraint , namely DAG-GNN . Also see Zhu & Chen ( 2019 ) and Lachapelle et al . ( 2019 ) for other cutting-edge structural learning methods . The improvement of our ANOCE-CVAE over the state-of-the-arts is as follows . We consider a new constrained structural learning , by incorporating the background knowledge ( the temporal causal relationship among variables ) into the score-based algorithms . We formulated such prior information as the identification constraint and add it as the penalty term in the objective function for the causal discovery . In this paper , we typically extend the DAG-GNN for an illustration . Note that the proposed constraint is not limited to the DAG-GNN and can be easily extended to other score-based algorithms . 3 ANALYSIS OF CAUSAL EFFECTS . 3.1 STATISTICAL FRAMEWORK AND ASSUMPTIONS . Let A be the exposure/treatment , M = [ M1 , M2 , · · · , Mp ] > be mediators with dimension p , and Y be the outcome of interest . Suppose there exists a weighted DAG G = ( X , B ) that characterizes the causal relationship among X = [ A , M > , Y ] > , where the dimension of X is d = p + 2 . Let Y ∗ ( A = a , M = m ) be the potential outcome that would be observed after receiving treatment a and setting mediators as m , and M∗ ( A = a ) be the potential mediators that would be observed after receiving treatment a . As standard in the causal inference ( Rosenbaum & Rubin , 1983 ) , we assume that there is no unmeasured confounder : ( A1 ) the effect of the treatment A on the outcome Y is unconfounded , i.e . Y ∗ ( A = a , M = m ) ⊥ A , ∀a , m ; ( A2 ) the effect of the treatment A on the mediators M is unconfounded , i.e . M∗ ( A = a ) ⊥ A , ∀a ; ( A3 ) the effect of the mediators M on the outcome Y is unconfounded given the treatment A , i.e . Y ∗ ( A = a , M = m ) ⊥ M |A , ∀a , m . In addition , as standard in the graphical causal discovery , we also make the Markov condition , the faithfulness condition , causal sufficiency assumption , and the linear structural equation model ( LSEM ) such that X = [ A , M > , Y ] > characterized by the pair ( G , ) is generated by X = B > X + , ( 1 ) where is a random vector of jointly independent error variables . Denote all directed paths in G that start with the exposure A and end with the outcome Y as set { πAY ( G ) } . If there exists at least one directed path π∗ ∈ { πAY ( G ) } such that the length of π∗ is larger than 2 , we say there is an interaction among mediators , as shown in Figure 1b ; otherwise , we call mediators are ‘ parallel ’ as shown in Figure 1a . In this paper , we consider all possible causal structures with multiple mediators under assumptions ( A1-A3 ) . We next give the total effect ( TE ) , the natural direct effect that is not mediated by mediators ( DE ) , and the natural indirect effect that is regulated by mediators ( IE ) defined in Pearl et al . ( 2009 ) . Definition 3.1 ( Pearl et al. , 2009 ) TE = ∂E { Y |do ( A = a ) } /∂a = E { Y |do ( A = a+ 1 ) } − E { Y |do ( A = a ) } , DE = E { Y |do ( A = a+ 1 , M = m ( a ) ) } − E { Y |do ( A = a ) } , IE = E { Y |do ( A = a , M = m ( a+1 ) ) } − E { Y |do ( A = a ) } , where do ( A = a ) is a mathematical operator to simulate physical interventions that hold A constant as a while keeping the rest of the model unchanged , which corresponds to remove edges into A and replace A by the constant a in G. Here , m ( a ) is the value of M if setting do ( A = a ) , and m ( a+1 ) is the value of M if setting do ( A = a+ 1 ) . Refer to Pearl et al . ( 2009 ) for more details of ‘ do-operator ’ . Note that in the assumed linear model , the slope of the line is the same everywhere ; for convenience and simplicity , we use a and a+ 1 to present the change of the treatment of 1 in the definition .

Summary: The paper proposes a framework for the analysis of causal inference. Its main contribution is to decompose the indirect effect by teasing out the causal contribution of a set of mediators. In a series of experiments with simulated data the authors show that the proposed method, ANOCE, outperforms other comparison partners. The manuscript also contains an analysis of real-world data that describes the causal effects of the lockdown of cities in the Hubei province (China) to reduce the spread of COVID-19. An extensive supplementary file is also part of the submission and it includes additional experiments and technical proofs.