Answering Complex Causal Queries With the Maximum Causal Set Effect

1 Introduction . Recent advances in machine learning technology have made it possible to non-parametrically estimate many parameters present in complex structural causal models ( SCMs ) . Specifically , such estimating technology has rapidly advanced for three major causal inference settings : the many causes setting , the many moderators setting , and the many mediators setting . All three settings represent a situation in which a particular causal query can be stated in terms of a large number of combinations of different variables . Specifically , a researcher could estimate a different treatment effect associated with each of the many different possible combinations of causes [ Imbens , 2000 , Wang and Blei , 2019 , Li et al. , 2019 , Wang et al. , 2018 , Zheng et al. , Forthcoming ] , a different conditional treatment effect for each of the many different combinations of moderators [ Green and Kern , 2012 , Athey and Imbens , 2016 , Grimmer et al. , 2017 , Wager and Athey , 2018 , Künzel et al. , 2019 ] , and a different mediated effect for each of the many different combinations of mediators [ Zhou and Yamamoto , 2020 , Daniel et al. , 2015 ] . Such causal queries are complex in the sense that they require summarizing the combined influence of a large number of causal variables . The main challenge for applied researchers in such settings is that standard causal inference algorithms are designed to provide a different estimate associated with each of the many causal variables rather than a single number summarizing the combined influence of all the causal variables together . Consider , for example , the setting of inferring the causal effect of actors on a film ’ s box office performance . Wang and Blei [ 2019 ] provide a framework for estimating the average treatment effect associated with every actor on a film ’ s performance . While certainly useful for making predictions ∗PHD Candidate , Interdisciplinary Doctoral Program in Statistics and Political Science , https : //zmarkovich.github.io/ 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . about which actors a director should cast , an economist studying the film industry might prefer a single number which summarizes the general importance of actors in general for a film ’ s box office success . As discussed in the next section , such settings are common in scientific research , suggesting the need for novel causal estimands to parameterize the predictions of such theories in the context of a particular SCM . Contribution The contribution of this paper is threefold . First , it introduces the notion of a complex causal query and argues that existing causal estimands are of limited utility to applied researchers in the face of such queries . Second , it defines a novel estimand – the Maximum Causal Set Effect ( MCSE ) – which can be used to provide an interpretable answer to such complex queries . Finally , the paper introduces an estimator for this estimand . The estimator is based on techniques proposed in the double Q-learning literature [ Hasselt , 2010 ] and is asymptotically consistent and conservative in finite samples under assumptions that are standard in the causal inference literature . It is also asymptotically normal and amenable to non-parametric bootstrap techniques , facilitating classical statistical inference about the MCSE . 2 Setting and Previous Work . 2.1 Problem Overview . Standard approaches to causal inference [ Pearl , 2009 ] typically begin with the researcher specifying an SCM and then defining a causal query which can be answered based on the assumed SCM . Under certain assumptions about the SCM , it may be possible to estimate the answer to that causal query using the conventional tools of statistical inference . The standard tools of causal inference are designed with settings in mind where the predictions of a scientific theory take the form of a simple causal query . Such queries are stated in terms of some low dimensional causal variable t and some outcome Y . For example , a question like how much does a medical procedure reduce the risk of disease , represents a simple causal query because it is defined in terms of a single unidimensional treatment . Such queries can be easily quantified using conventional statistical estimands because they are directly formulated in terms of a small number of theoretically motivated variables . This paper instead focuses on situations where a scientific theory makes diffuse predictions about the importance of a large number of causal variables , defying the stylization of simple causal queries . Such queries are common in scientific research . For example : • Genome Wide Association Studies ( GWAS ) – GWAS attempt to quantify the causal effect of a huge number of individual genotypes on the likelihood that some trait is expressed [ Stephens and Balding , 2009 , Visscher et al. , 2017 ] . • Personality – psychologists are often interested in the effect certain personality traits ( such as extraversion or neuroticism ) might have on life outcomes [ Pervin , 2003 ] , but such traits are only observed by the researcher as responses to a large number of survey questions . • Text – language is complex and multi-faceted and the causal effect of the wording of a document on a user ’ s response requires an assessment of the contribution of many different topics or words together [ Fong and Grimmer , 2016 , Egami et al. , 2018 , Fong and Grimmer , forthcoming ] . • Complex medical treatments – many medical treatments can not be reduced to a single low dimensional representation . For example , radiation exposure is observed as a high dimensional vector [ Nabi et al. , 2017 ] and medical researchers might also wish to understand the combined importance of many procedures using electronic medical records [ Gottesman et al. , 2013 ] . Such causal queries are complex because they require estimating the joint influence of many causal variables . The SCM undergirding such complex queries can take many forms . Three major examples are : ( a ) the many causes setting where the researcher wishes to understand the joint influence of many treatments ( b ) many moderators setting where the researcher wishes to understand how effect of a binary treatment varies based on many variables ( c ) the many mediators setting where the researcher wishes to model how a causal effect can be decomposed into many different pieces . These SCM ’ s are Note : Figure visualizes SCMs corresponding to complex causal queries in the case of ( a ) many causes , ( b ) many moderators , and ( c ) many mediators . ( a ) visualizes the case where treatment types t0 . . . tn each influence y described in Wang and Blei [ 2019 ] . ( b ) visualizes the case where the causal effect of t0 on y is directly modified by t1 . . . tn as described in VanderWeele and Robins [ 2007 ] . ( c ) visualizes the case where the effect of t0 on y is mediated by t1 . . . tn as described by Zhou and Yamamoto [ 2020 ] . visualized in Figure 1 in the form of directed acylical graphs ( DAGs ) . The unifying trait of a complex causal query is that it asks about the importance of many arrows present in each DAG . Techniques developed in the context of simple causal queries can not be readily used to answer complex ones . While the standard tools of causal inference can be used to estimate causal effects corresponding to every combination of causal variables in SCM ’ s like those visualized in Figure 1 , they do not provide applied researchers with a single unambiguous estimate with which to summarize the joint causal effect of many such variables . 2.2 Previous Work . The only existent proposal for addressing the challenge presented by complex causal queries in the machine learning literature is to dimension reduce the relevant causal variables and then focus on a simple causal query defined in terms of that latent trait [ Fong and Grimmer , 2016 , forthcoming , Nabi et al. , 2017 ] . This strategy has only been proposed in the many causes setting , but could also be extended to the many moderators or many mediators cases as well . Such a strategy is inherently reductive and risks understating the magnitude of causal effects because it disregards all variation in the treatment types that is not accounted for in the latent trait . Additionally such latent traits are often scale invariant and so may lack a scientifically meaningful interpretation . 2.3 Assumptions and Notation . We assume that the researcher observes a set of N independent ( ti , Yi , xi ) triplets where Yi is the outcome , and ti is a length K vector indicating the treatment type received by unit i , and xi is a length J vector representing a set of background covariates that causal effects should be adjusted for . Additionally , let T denote the support of the distribution of ti . We also assume that the researcher has knowledge of the population distribution of ti : g ( t ) . In many settings , the empirical distribution of ti will be the most logical choice , but other choices may be reasonable as well if the population distribution is known to the researcher , as might be the case when conducting survey research or if the treatment types were experimentally randomized . Finally , we assume that the researcher has specified some SCM and has specified a simple causal query , τ ( T ′ , T ′′ ) , which is defined in terms of two subsets : T ′ , T ′′ ⊆ T . In the many causes case , τ ( T ′ , T ′′ ) might take the form : τ ( T ′ , T ′′ ) ≡ E ( E ( Yi|do ( t ) ) |t ∈ T ′ ) − E ( E ( Yi|do ( t ) ) |t ∈ T ′′ ) where do ( · ) represents some causal intervention [ Pearl , 2009 ] . This estimand represents the average effect of receiving a set of treatments contained in T ′ rather than T ′′ . In the many moderators case on the other hand let the zeroth element of the treatment types vector recieved by unit i , ti,0 ∈ { 0 , 1 } , denote the level of some binary treatment received by unit i . Similarly , let the remaining elements of ti be denoted ti , −0 and indicate the level received by unit i on the many moderators . Then a possible choice for τ ( T ′ , T ′′ ) might be : τ ( T ′ , T ′′ ) ≡ E ( E ( Yi|do ( ti,0 = 1 ) ) − E ( Yi|do ( ti,0 = 0 ) ) |ti , −0 ∈ T ′ ) − E ( E ( Yi|do ( ti,0 = 1 ) ) − E ( Yi|do ( ti,0 = 0 ) ) |ti , −0 ∈ T ′′ ) which represents the difference in average treatment effects between units with moderators contained in T ′ rather than T ′′ . Defining such estimands in the many mediators case requires more cumbersome notation , but can be accomplished in terms of an average of different combinations of path specific effects [ Daniel et al. , 2015 , Zhou and Yamamoto , 2020 ] . 3 The Maximum Causal Set Effect . The challenge for applied researchers in the presence of such complex causal queries is that a different value of τ ( T ′ , T ′′ ) can be defined for every distinct pair of sets T ′ , T ′′ ⊆ T , leaving the analyst without a single unambiguous causal estimand to summarize their findings . In this section , we define a causal quantity of interest which overcomes this challenge by focusing on the contrast between two sets T Maxq and T Minq which maximize τ ( T ′ , T ′′ ) . To avoid choosing sets T Maxq and T Minq which correspond to unrepresentative edge cases , we require that the sets be of a researcher specified size : q . Formally , let the set of subsets of T such that the probability that ti is in T is at least q be defined as : Tq ≡ { T ′ ⊆ T : P ( ti ∈ T ′ ) ≥ q } where P ( ti ∈ T ′ ) = ∫ T g ( t ) 1 { t ∈ T } dt . We then define MCSEq as : MCSEq = max T ′ , T ′′∈Tq τ ( T ′ , T ′′ ) = τ ( T Maxq , T Minq ) We refer to T Maxq as the maximum causal set and T Minq as the minimum causal set . For many applications , the MCSE will have an intuitive and scientifically meaningful interpretation . In the actors example , it might be used to answer a question like what is the expected difference in box office performance between a film cast with one of the 10 % best performing casts rather than one of the bottom 10 % worst performing casts ? Similarly , in the genetics example , it might answer the question , what is the difference in the efficacy of some drug for patients with one of the top 10 % most treatment enhancing sets of genes rather than one of the bottom 10 % most treatment diminishing sets of genes ?

Summary: ### 1. Summary  The paper introduces a novel estimator for the multiple causes / moderators/ mediatiors scenario within causal inference processes.  The paper claims a threefold contribution, namely  (i) to introduce the notion of "complex causal queries"  (ii) to introduce a novel estimand (MCSE)  (iii) to introduce a related estimator