ANOCE: ANALYSIS OF CAUSAL EFFECTS WITH MULTIPLE MEDIATORS VIA CONSTRAINED STRUCTURAL LEARNING

Abstract

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, medicine, genetics, and economics. 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. An analysis of causal effects that interprets the causal mechanism contributed through mediators is hence challenging but on demand. To the best of our knowledge, there are no feasible algorithms that give an exact decomposition of the indirect effect on the level of individual mediators, due to common interaction among mediators in the complex graph. In this paper, we establish a new statistical framework to comprehensively characterize causal effects with multiple mediators, namely, ANalysis Of Causal Effects (ANOCE), with a newly introduced definition of the mediator effect, under the linear structure equation model. We further propose a constrained causal structure learning method by incorporating a novel identification constraint that specifies the temporal causal relationship of variables. The proposed algorithm is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the spread of the coronavirus in Chinese major cities out of Hubei.

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. 2019). However, the current cutting-edge methods neglect the temporal causal relationship among variables, and thus cannot 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



A causal graph with parallel mediators.

A causal graph with interacted mediators.

Figure 1: Illustration of causal graphs with different types of mediators, where A is the exposure, {M 1 , • • • , M p } are mediators, and Y is the outcome of interest.

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 (

