Causal Attribution Analysis for Continuous Outcomes
Authors: Shanshan Luo
Yu yixuan
Chunchen LIU
Feng Xie
Zhi Geng
Venue: ICML 2025 spotlightposter (2025)
Keywords: Attribution analysis
Causes of effects
Continuous outcome
Posterior causal estimands.
Primary area: general_machine_learning->causality

Abstract: Previous studies have extensively addressed the attribution problem for binary outcome variables. However, in many practical scenarios, the outcome variable is continuous, and simply binarizing it may result in information loss or biased conclusions. To address this issue, we propose a series of posterior causal estimands for retrospectively evaluating multiple correlated causes from a continuous outcome. These estimands include posterior intervention effects, posterior total causal effects, and posterior natural direct effects. Under assumptions of sequential ignorability, monotonicity, and perfect positive rank, we show that the posterior causal estimands of interest are identifiable and present the corresponding identification equations. We also provide a simple but effective estimation procedure and establish asymptotic properties of the proposed estimators. An artificial hypertension example and a real developmental toxicity dataset are employed to illustrate our method.

Causal Attribution Analysis for Continuous Outcomes
Shanshan Luo' Yixuan Yu?
Chunchen Liu? Feng Xie Zhi Geng' !
Abstract
uate whether the development of hypertension was caused by dietary habits, exercise routines, and physical character- istics. Additionally, such questions are also very common in developmental toxicity risk studies, where researchers in clinical trials aim to determine whether abnormal weight loss in pups is caused by potentially toxic agents, organ disease, or other risk factors.
Previous studies have extensively addressed the attribution problem for binary outcome variables. However, in many practical scenarios, the out- come variable is continuous, and simply binariz- ing it may result in information loss or biased conclusions. To address this issue, we propose a series of posterior causal estimands for retro- spectively evaluating multiple correlated causes from a continuous outcome. These estimands include posterior intervention effects, posterior total causal effects, and posterior natural direct effects. Under assumptions of sequential ignora- bility, monotonicity, and perfect positive rank, we show that the posterior causal estimands of inter- est are identifiable and present the corresponding identification equations. We also provide a simple but effective estimation procedure and establish asymptotic properties of the proposed estimators. An artificial hypertension example and real de- velopmental toxicity dataset are employed to il- lustrate our method.
To explain such retrospective studies, researchers need to use counterfactual scenarios that imagine what would have happened if certain conditions experienced previously had been different (Lu et al., 2023). For example, for a patient with high blood pressure who never exercises, it is inferred what his blood pressure would have been like there had been an intervention to make him exercise regularly. Simi- larly, for a pup exposed to a toxic reagent and abnormally thin, it is inferred how its body weight would have changed if it had received placebo instead. Pearl (2000) introduced three layer causal hierarchy, comprising association, in- tervention, and counterfactual levels. Retrospective causal analysis, which assesses the causes of observed effects, falls under the third level of this framework. The first two lev- els primarily involve predicting or evaluating the effects of interventions. In contrast, the third level focuses de- termining whether observed outcomes can be attributed to prior interventions or exposures (Pearl, 2015; Dawid et al., 2014; 2015).
1. Introduction
In social science (VanderWeele, 2012), health risk assess- ment (Khoury et al., 2004), legal contexts (Sanders et al., 2021), and explainable artificial intelligence (Galhotra et al., 2021), researchers are interested not only in assessing the
While randomized experiments and standard assumptions effectively address the first two levels of causation (Rosen- baum & Rubin, 1983; Pearl, 201 4)) they are not enough to address the challenges posed by the third level (Dawid & Musio, 2022). This limitation presents a significant chal- lenge for traditional causal inference methods when dealing with such retrospective analysis problems. To formally answer such questions, Pearl (1999) outlined three coun- terfactual definitions of causal relationships to capture the necessity or sufficiency of a cause for given binary effect, Additionally, Dawid al. (2014) introduced the probabil- ity of causation to infer the cause for given binary effect. When there are multiple potentially correlated causes, Lu et al. (2023) and Li et al. (2023) introduced posterior causal effects under observed post-treatment variables to retro- spectively deduce causes from a single effect and multiple effects, respectively. In many clinical trials, the outcome variables of interest may be continuous, such weight
effects of causes (Rosenbaum& & Rubin, 1983; Robins et al., 1994; Angrist et al., 1996; Bang & Robins, 2005; Ding et al.,
2011; Zhao et al., 2012; Jiang et al., 2016; Fan Li & Za- slavsky, 2018; Yang al., 2020), but also in inferring causes from specific outcomes (Pearl, 1995; Dawid, 2000; Tian & Pearl, 2000; Kuroki & Cai, 2011: Dawid & Musio, 2022; Pearl, 2022; Lu et al., 2023; Li et al., 2023). For instance, for hypertensive patients, researchers may retrospectively eval-
'School of Mathematics and Statistics, Beijing Technology and Business University, Fangshan District, Beijing, China 'Ling Co.Ltd, Alibaba Group, Hangzhou, China. Correspondence to: Feng Xic <fengxie @ btbu.cdu.cn.>.
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Causal Attribution for Continuous Outcomes
blood pressure, and income. However, most existing lit- erature primarily conducts attribution analysis for binary outcomes (Pearl, 1999; Dawid et al., 2014; Lu et al., 2023). Research on continuous outcome variables remains limited, and the formal definitions, identification expressions, and es- timation procedures for continuous outcomes require further exploration.
of changes in X, on the outcome Y,, thereby evaluating the likelihood of X; being the cause of event &..
Next, we consider the case with multiple causes X² (X1.... and a single outcome Y, where X is bi- nary vector of causes, and the causes may affect each other. Without loss of generality, we assume that the causes are arranged in topological order such that X₁ not a cause of X₁ for k < (. For example, X is sequence of observations ordered in time, or X consists of variables in a directed acyclic graph (DAG) where XR is not a descendant of X for ls < L. For generic sets of variables W and U, we use W. to denote the potential outcome of W that would have resulted if U were intervened to level u. In particular, if W = -Wi..... Ws), then W,, = (WW)...... ......,W... (W,).). We make the consistency assumption that connects observed variables to potential outcomes, i.e., W. = W if U = (Pearl, 2015). We further suppose the composition assump- tion holds in the sense that for any variable sets W. V and U, W - W, if U. = u (Pearl, 2015). The consistency as- sumption can be viewed as a special case of the composition assumption if V empty.
In this paper, we propose new framework for causal attri- bution with continuous outcomes. Unlike most prior work which focuses on binary responses, our method is tailored to continuous settings and enables retrospective analysis of how multiple causes contribute to an observed outcome. Our contributions are as follows: First, we extend the posterior attribution framework to continuous outcomes, which intro- duces new technical challenges and practical relevance, as binarizing outcomes often leads to information loss bias. Second, under sequential ignorability and the perfect posi- tive rank assumption (Heckman et al., 1997), we prove that individual treatment effects and posterior intervention ef- fects are identifiable. Third, assuming monotonicity among multiple causes (Lu et al., 2023; Li et al., 2023), we estab- lish the identifiability of all proposed estimands and provide explicit identification formulas. Fourth, we develop novel two-step estimation procedure: we recover individual-leve counterfactual mappings and treatment effects, and then estimate the remaining posterior causal estimands based on these mappings. Fifth, we also present simplified identifi- cation results under a known directed acyclic graph (DAG), and illustrate the application of our methods using an artifi- cial hypertension example. Proofs of all theoretical results provided in the Supplementary Materials.
To measure how likely Xx is a cause of the continuous effect given observed evidence (z: E)), extend the concept of posterior total causal effect (postTCE) defined by Lu et al. (2023) as follows:
PosT (X+ = Y Ex
= =E - Yxx=0 x,
where we use "x" to represent "X" = x² for notational simplicity. is important to note that this definition includes the event & defined by the observed outcome, and cannot simply be considered as a conditional average causal effect (CATE). As advocated by Lu et al. (2023), larger value of the posterior total causal effect indicates that the effect or outcome is more attributable to the cause Xg. The cause that produces the largest posterior total causal effect is usually considered the highest risk factor.
This paper is structured as follows. Section 2 presents the notation and definitions. Section 3 discusses the identifia- bility of the proposed posterior causal estimands. Section 4 outlines a two-step estimation method for the proposed esti- mands. In Section 5, we employ an artificial hypertension example to illustrate our proposed method. Finally, Section 6 concludes with a brief summary.
Similar to the direct causal effect considered by Pearl (2000), we define the posterior natural direct effect of X on Y given the observed evidence (z, E), which quantifies the et- feet of Xx on Y not mediated through intermediate variables. Let A& =(X,,.... -X Xx-1) and Dr = (Xx+1,.... Xp). Then X = (A,,Xi Da). and x = (ax, 2k, dx) denotes a value of X. Given the evidence (sr, E), the posterior natural direct effect (postNDE) of X₁ on Y is:
2. Notation and definitions
Assuming that we observe n independent and identically distributed samples from a superpopulation. We first con- sider the scenario with a single cause and a single outcome. For each unit i, let X₁ represent binary potential cause, where X, = indicates receiving treatment, and X; 0 in- dicates receiving control. Let Y; be the observed continuous outcome. Let Y and Xi denote the potential out- comes corresponding to X, = 0 and X, = 1, respectively. Many common measurements, such as weight, blood pres- sure, and income, are typically continuous, but Y, may fall within a specific interval of interest denoted as & Therefore, given the evidence (X,, E,), we aim to evaluate the effect
PostNDE(Xx + Y E)
= =E(YX,=L,D, - Yx-=o |r,E),
where (2 0) is the potential outcome under (A,, Xx) = (ax, 0). Throughout this paper, we use Dz(ak, 2x) and (De) )an interchangeably in the nested potential outcomes.
2

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Causal Attribution for Continuous Outcomes
The postNDE describes the effect observed when each indi- vidual in the subpopulation (xr, E) switches from Xx 0 to Xx= = 1, while keeping Da at its value when Xx = 0.
wy for all r S i 8. To identify the proposed estimands, make the following commonly used assumptions in pre- vious studies (Heckman et al., 1997; Pearl, 2000; 2014; Lu et al., 2023; Li et al., 2023).
Parallel to the natural indirect effect considered by Pearl (2000), we also define the posterior natural indirect effect (posINIE) of X on given the evidence (xr as follows:
Assumption 3.1 (Sequential ignorability). We consider the following assumptions:
PosINIE(Xx + Y E)
(i) there is no confounding between W. and W1:s-11 1, i.e., (W, [W))wm-- 1 W1: for all W1:s-1 and = 2, ................................ +
= B[Yx,- (x.=I.D((a..0) x 83.
1:
The PostNIE. quantifies, for each individual in the subpopu- lation (z, E), the effect observed when X₃ is set to Xx = 1, while all intermediate variables along the pathway from X to Y change from state (a(a 1) to state Du(ag, 0)-
(ii) the elements in ((W.), 1 are mutually inde- pendent for any given W1:pp
Through the definitions, we have that:
The independence condition in Assumption 3.1 can be re- laxed by introducing the baseline covariates, and we omit it for simplicity. The Assumption 3.1 implies that the potential outcome of each variable is independent of the prior variables in causal order. Under the Assumption 1.), if W. has a nonparametric causal structural model W. = m. (W (W::s-,,,, e.) with an unknown function m, (s)ea) and a error variable 6, TL W1:--1, then Assumption (ii holds naturally because Assumption 3.1 implies that 6g 1 C1:-1 for = 2.....P + 9. which further implies Assumption 3ii. Assumption 3.1 rules unobserved confounders between any two variables in W. However, each variable X₁ may still confound the relationship be- tween Y and X,, or between X, and X,, provided k: < &, Assumption 3.1 is frequently employed in causal inference with complex systems, including mediation analysis (Imai et al., 2010) and longitudinal data involving time-dependent confounders (Rohins, 2000).
P Y 8) PDES = Y E. + PXs = Y E.
Given the observed evidence E), in addition assessing the a posteriori total, direct and indirect effects of a particu- lar cause Xg. we need to consider assessing the synergistic effects of a joint intervention with all possible causes in an alternative state X = x². Indeed, synergistic effects are important in many applications. For example, having heart disease alone may have a limited effect on blood pressure, whereas the combination of unhealthy diet and heart disease may jointly contribute elevated blood pressure. Therefore, the posterior intervention causal effect (postICE) for another state X = æ'is defined as follows:
PostICE((2 1x, E (Y - Y Ex,
Assumption 3.2 (Monotonicity). For 2, .....PP. we assume that (W.)wi... S (W.) (W.)mi. whenever wis-1 1 w1:s-1 holds.
The individual treatment effect (ITE) for any pair (x',²*) can be defined as (TE(x',x') = Yur Yg-, representing the difference in potential outcomes for each individual under two different treatment conditions. Inferring ITEs presents a fundamental challenge because we can only ob- serve one potential outcome for each unit (Rosenbaum & Rubin, 1983).
Assumption 3.2 implies that each cause has non-negative effect subsequent causes. This assumption is commonly expressed in epidemiology as "no prevention", meaning that no individual is helped by exposure to a risk factor. To identify the posterior causal estimands, Lu et al. (2023) and Li et al. (2023) also introduce the same monotonicity assumption across multiple potentially correlated causes. The validity of monotonicity cannot be tested directly, but under Assumption 3.1, the monotonicity can be falsified by imposing testable restrictions on the observed data distribu- tion. For example, for any w I W1:--1, the following equality can be used to falsify the monotonicity assumption:
Table S3 in Section S1 of the Supplementary Material com- pares our estimands for continuous outcomes with those for binary outcomes in Lu et al. (2023), illustrating their notational alignment and practical relevance.
3. Identifiability of posterior causal estimands and required assumptions
pr(W, = 1 Wi:--1 = wis--1)
3.1. Assumptions required for identifiability
S pr(W,, = 1 Wi:s-1 w1:s-1)-
Define W = Y) and let Wrss denote a subvector (W,, Wr-1...., Ws) of W for r I s. Let wi: = (w....., 2 Wr:s = (Wr...., w.) denote that w/ S
Assumption 3.3 (Perfect positive rank). We assume that W Wpt1 = mp+1 (W1sps fpt1), or equivalently y
3

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Causal Attribution for Continuous Outcomes
Mp+1 (X, <p+1), where Ep+1 represents a scalar-valued er- variable. The unknown link function (X+)(X,.) is continuous and strictly increasing in Ep+1-
for some real-valued function mz+1(X) and positive func- tion a (X).
3.2. Identification equations of posterior causal estimands
For continuous outcome variables, Assumption 3.3 in our framework states that the individual-level outcome is a monotonic function of an unobserved latent variable fp+1- This implies that individuals preserve their relative ranks across different treatment conditions. Such rank- preseringg behavior commonly used to identify individual treatment effects or quantile treatment effects in counterfactual causal inference literature (Heckman et al., 1997; Chernozhukov & Hansen, 2005), but it is relatively novel assumption in retrospective attribution analysis. For the linear model,
Under Assumptions 3.1 and 3.3, we first consider the identifiability of ITEs and postICEs. Let Sy, denote the support of the potential outcome Ys, which can be iden- tified by Sy|x== under Assumption 3.1. The key to our identification strategy is to match the potential out- come Y, = mp+1 (I, 6p+1) i) with another potential outcome Y2 = mp+1 (z', cpt1) through a mapping Φ2-+2' (.), such that Y.' Qx- (Ya). This mapping (z)-z(() is termed counterfactual mapping (Vuong & Xu, 2017; Feng et al., 2020), because it allows us to find the counterfactual out- come Y from Y3 using the function Φx- (.), and vice versa.
== = og + +a + ... + Xp + Cp-1,
(2)
Assumption 3.3 holds naturally. Note that Assumption 3.3 imposes a condition only on the outcome variable Xp+1 = Y, while Assumption 3.2 concerns the structural relation- ship among multiple causes (X1...., Xx). Since these two assumptions relate to different variables, the former does not imply the latter.
Let "1 (x. be the inverse function of 1p+1 (2t, )). Ac- cording to Assumption 3.3, for any pair (X, = (z, y). we can uniquely represent the error term "
cp-1 -1 (1, v). although the specific form of m m -1 (z. v) is un-
known. Therefore, Yg is uniquely defined by Ox- (v) Mp+1 {z', mp+t (z, y)} for each E Sy,- Moreover, the counterfactual mapping (x-+r((-) is a continuous and strictly increasing function from Sy, onto Sy Thus, if we can identify c (v) for all y E Sy, and x # x', we can recover ITES for each individual.
Lu et al. (2023) considered the monotonicity assumption of the binary outcome with respect to multiple correlated causes X, denoted as Yx=r- & Yx=x for any x² 3 x. However, this monotonic relationship may not be applicable when the outcome variable is continuous. Specifically, the condition implies that increasing any component of the treat- ment vector, while keeping others fixed or increased, should not decrease the outcome. In the linear model (2), satisfying this monotonicity assumption requires all coefficients to be non- non-negative, i.e., aj 2 for all I....,P. P, which may be overly restrictive for continuous outcomes.
Lemma 3.4. Under Assumptions 3.1 and 3.3, for any y e Sy,, the counterfactual mapping (-) is identified by the continuous extension of
(2) = (Fo()))
Vy E Sy.
(3)
= PT s x and St is the inte- rior of Sy Moreover, the ITES of every individual in the population can be identified.
where =x)
The basic restriction in Assumption 3.3 is also referred to as the rank preservation or rank invariance (Heckman et al., 1997: Chernozhukov & Hansen, 2005; Vuong & Xu, 2017: Feng et al., 2020). If an individual with regular exercise and without heart disease (i.e., x = (1, 0)) has the lowest blood pressure in the subpopulation {(,,, Yi) X, z). then according to Assumption 3.3, an individual with exercise and heart disease (i.e., x' = (0, 1)) should also have the lowest blood pressure in the corresponding subpopulation {(X,,Y.) Y.): : X, = x'); and vice versa. The strict monotonic increase of can be changed to the strict monotonic decrease without affecting the subsequent discussion. For simplicity, we assume the strict monotonic increase. For any given error 67 Assumption 3.3 requires that the relative rank or quantile of Y2= = mp++11; (+1) be the same that of Yz =\t mp++1 1 (x', () for any A x'. A stronger version of Assumption 3.3 assumes that the error term 6p+1 is ad- ditive, that is, Y = mp+1 (X) + 1 for some real-valued function m (.). Moreover, the heteroscedasticity model also satisfies Assumption 3.3: Y = (X) + (X))p+11
Lemma 3.4 establishes the identifiability of the counterfac- tual mapping @a--ar (-) on Sy: constructively by matching the quantiles of Y, and Yge. We provide further intuition for Lemma 3.4. When we observe an individual in the sub- population {(x; = x. Y.)) with the highest blood pressure, we can recover the joint distribution of individual by iden- tifying the individual with the highest blood pressure in each observed subgroup {(X, = x', Y,)} for any x' + z. Given the identifiability of the counterfactual mapping and ITEs, the postICEs can also be identified using an inverse probability weighting expression (Horvitz & Thompson, 1952).
[[X x, pr = x, 2,
PostICE (Y, 1 E) = E
(6 (P)
where T(-) denotes the indicator function.
4

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Causal Attribution for Continuous Outcomes
Assumptions 3.1 and 3.3 establish the identifiability of ITES and postICES. However, they are not sufficient to ensure the identifiability of other posterior causal estimands (e.g.. PostNDEs and PostNIEs) when considering a specific cause Xx- This is due to the challenge posed by identifying expec- tations of the nested potential outcomes given the observed evidence. For example, the postNDE involves the condi- tional expectation of the world intervention potential outcome
Theorem 3.6 illustrates that with the additional monotonicity assumption 3.2, we can also identify postNDE, postNIE, and postTCE. In some cases, we may only want to conduct attribution analysis for continuous effects based on evidence from a subset of (X,E), while Lemma 3.4 and Theorem 3.6 are based on fully observed evidence (a, E). For some 8 < p, let (X',8) = (Xi,..... Xi, E) denote a subset of (X, E). We can summarize the remaining set X (X'' to obtain the expected results for the subset (X", EE based on Lemma 3.4 and Theorem 3.6. To simplify the exposition, we omit this part. We also refer to Corollary 1 in Lu et al. (2023) and Theorem 3 in Li et al. (2023) for parallel results.
which cannot be identified using Lemma 3.4 alone. Before formally identifying these ex- pectations, we provide another lemma for identifying the conditional probability of the counterfactual outcomes of the causes.
Lemma 3.5. Under Assumptions 3.1 and 3.2, given the observed evidence (ak, Fk, de, E), let d% = (1k+17---.. (%) and d2 = (I)...
3.3. Identification equations of posterior causal effects under causal networks
In this section, we consider the causal structure of observed variables (X, Y) represented by a directed acyclic graph (DAG) or network. We aim to present the simplified identi- fication expressions for the posterior causal estimands given known DAG, For k = 1.....P P, let Pag and Pay denote the sets of parents of Xx and Y in the graph, respectively. The joint probability distribution pr(X, Y) can be factor- ized as pr(X, Y) = II pr(X,, Pas) pr(Y Pay). We assume pr(x, y) > O for each (æ, y). For a given graph, the sequential ignorability assumption (Assumption 3.1) posits that there is no unobserved variable intervening be- tween any two or more nodes in the DAG. The monotonicity assumption 3.2 implies that each node X₃ has positive individual monotonic effect on its child nodes. Furthermore, Assumption 3.3 can be simplified such that the unknown link function Mp+1 (Pay, ( 1) only needs to be continuous with respect to Pay and strictly increasing in Ep+1- Specifi- cally, the continuous outcome variable only required to perfectly match the quantiles of potential outcomes under different realizations of its parent nodes Pay- We define simpler counterfactual mapping 6ps, -pa, (-) to character- ize the mapping relationship of the parent nodes of Y under different realizations pa, and pay The theoretic results pre- sented in Lemma 3.4 and (4) can be simplified for a given graph.
(i) For di.- 3 da, we have,
pr Dr(Daan 0) =dp ak. 1, - x²²) + (2x; Rosh
pr(xx 1 ak2 0, 2 + 23-1 pr(X, = ake 1, Tk+ f6-1)
where Ro.
(ii) For dg dig we have,
pr pr(Dx(ak:1) 1) = die ak. 0, ]
= -II==++1 (27 + (1 2xx (1 xz5 R1sh,
pr(X, = 0| Oks 1, # 2 x1-1)
where R1s
pr(X: 0| akj 0, Tk+1, 13-1)
Theorem 3.6. Under Assumptions 3.1-3.3, given the evi-
dence (ak, Tk, du, E), the PostNDE, PosINIE, and posiTCE of Xx on Y can be identified as follows:
(i) given zk for any 2 E {0, we have,
((Va)) 1x, = E(Y x,E), E(Yaz,Dv(ar.)) [x,E]
=d E(Yz;.4; |x, E)r( Da di [x),
Corollary 3.7. Suppose that the causal network of (X (X,,..., (X1,...,X,,,V X), and 3.3, for any # x' and y Sy,, let pay C and pal, C x', the counterfactual mapping &r-za' (.) can be reduced as follows:
is DAG. Then, under Assumptions 3.
(ii) given 2x = for any xt E {0, 13, we have,
ElY x, [x,]] = E(Y E,E),
E(Yaq,D((an.)) [2,E] = Ean F((zzga;) Ix, E)r (Da(( 1) = di x))
(x- -Op,, -pol, (v) = pal, (Fpa, (v)}, Vy S
where E(Y df xx E) can be identified by Lemma 3.4, and pr Da(ak, x1)) )))- = di œ) for 1/ E {0, 1} can be identified by Lemma 3.5.
where mpa, (v) = Pr(Y S y Pay = =pay)- Moreover the ITEs of every individual in the population can be identified.
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Causal Attribution for Continuous Outcomes
The following equation also holds:
4. Estimation
E (Y [x,E)= = EE pag, pay, E)
In this section, we propose a simple but effective method for estimating the counterfactual outcome mapping dn- (-) as well the posterior causal estimands. Let {(X,, Yt) :
=E I(Payy - pay, E)
pay -pa +py, {
pr(Pay = pay: E)
1, n be the independent and identically distributed samples generated according to Assumptions 3.1-3.3. Our estimation procedure consists of two steps: first, for each observation (X,, Y;) = (æ, y). we estimate the counterfac- tual mapping Øx-ts' (V) for x' * x using a simple estimator that minimizes convex population objective function and constructs pseudo samples of the counterfactual outcomes for all individuals. In the second step, nonparametri- cally estimate the posterior causal estimands based on the counterfactual mapping Pa--a' (-). For simplicity, let Sy, denote a compact interval [V M for any X = x, where -00 < v/2 < y < +oc. We also assume the compact sup- port [V'= y²) known. Otherwise, it can be estimated using the methods proposed in Korostelev & Tsybakov (2012). To establish the asymptotic properties of the estimators to be proposed in this section, make the following assump- tions.
and the postICE is identifiable.
Corollary 3.7 indicates that the counterfactual mapping de- pends only on Pay. the PostCCEs and ITEs can be calculated using the low-dimensional probabilities pr(Y S v pay)- Similarly, the identification equations of the conditional probability pr 0) = dž æ} can also be simplified by replacing Ros and Ris in Lemma 3.5 with Ris and Rigs
R = pr(XX = 0 pa,)/pr(Xx = 0 pa,); 1 = pr(X. =1 1 pa.)/pr(Xx. htr//p((X. = p)
(5)
where pag
(ax, 0, 21111-...... 2 and p C
(ak, 1, #kt11...... ,.g.:,...,,,..)........ The following Corollary 3.8 pro- vides a simplified identification equation of other posterior causal estimands for a given graph.
Assumption 4.1. (i) The function mp+1 (2. (p+1) is con- tinuously differentiable in error term 6p4-1 for any X = x; (ii) The probability density function of the error term 6p+1 is continuous; (iii) inf infgetyl yelyh Ia (y) > for any X- X, where Ir (v) - (F2(2))/0))))))))) /8y.
Suppose that the causal network of (Xi,...,Xp,. Y) is DAG. Under Assumptions 3. 3.1-3.3, given the observed evidence (0k, Lk, dx,&) E), the postNDE, postNIE, and postTCE of Xx on can be identified using the following equations:
Corollary 3.8.
Assumptions 4.1(1) and 4.i are regularity conditions and, together with Assumptions 3.1 and 3.3, ensure that the marginal distribution Fz()) is absolutely continuous with respect to the Lebesgue measure, and its probability density function gN (y) is also continuous for any X = x. Assumption 4. (iii) is introduced for the sake of simplic- ity in explanation. Trimming techniques can be employed to relax this assumption, but they may introduce technical complexities.
(i) when Ik= = for any x E {0,1),, we have,
E(Y | [ = E(ypa' par pay. E), D(Ysz,Da(an.0) [
Edgzos pay, pay, prr DD 0) = di x}
(ii) when Ik= = O, for any 1 E 10, we have,
We now aim to derive counterfactual outcome mapping (2- from two marginal distributions Fx(.) and (-) For a given E R. we define the objective function as follows: Pa. Pa-zr y) E {sign(Y v) X = z) x
xx [x = E(ypa- pay, E),
[r,]= = En E(Ypaz pay, De 1) = di x))
E(YY - x z'), where sign(u) = 2 x I[u > 0) The above objective function is motivated by the quantile regression method in Koenker & Bassett (1978) and Feng et al. (2020). Simple calculations reveal that the first-order and second-order conditions of this objective function are given by:
where E(Ypa; I pay,,) and E(ypa; pay: E) can be identified by Corollary 3.7 for pay C Fk,, dr) and par, C (ax. rt, di), and pr(Da (Ok,2") di x) can be identified by Lemma 3.5 and (5) for any at E 1).
Opz-Az (1:0) = 2 (t) [x)) = 0.
The theoretic results in this section very useful in prac- tice and can greatly simplify the computation, especially when the dimensionality of the causes is large. We suggest that practitioners first obtain a simple DAG, or a larger one but with the correct order of nodes, from expert knowledge or conditional independence tests.
Ot
8
Po- (1:0) (t 2 0.
0t2
The following lemma demonstrates that for any y E S the objective function pr-zz (::)) is uniquely minimized at
6

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Causal Attribution for Continuous Outcomes
the counterfactual outcome sz-z/(1). (v).
Table I: Results of posterior intervention causal effects
based on different evidence.
Lemma 4.2. Under Assumptions 3. 3.1.3.3, Pa- (:59) is continuously differentiable and weakly convex on R. Addi- tionally, Pa-rx' (: 4) is strictly convex Sy and uniquely minimized on IR at Ox-ra' (y) when y E SYy-
ParICEE( 1, Y 140)
(2).3a)
(x)
(x)
(0.0)
(0,1)
-1,00
(1,11
(x (0.0)
-12.43
-1.76
-18.19
(2).25) 10.
3.34
0,00
2.34
-5.43
( = (1.0)
1.75
-10.01
0.00
-15.75
The objective function Pa-rel (:)) in Lemma 4.2 has sev- eral important properties that support the validity of our es- timation procedure. It is continuously differentiable, which ensures that gradient-based optimization methods can be effectively applied. Moreover, the function Po-ra' (:y) is strictly convex in the interior of the support, which implies the uniqueness of the solution. Lemma 4.2 also provides the foundation for our nonparametric estimation of the counter- factual mappings and posterior causal estimands. For the th observational unit (X; = x, Y:), let
(x..--) (1.1)
12,05
5.57
10.86
0.00
are not preventive for heart disease HD. The various po- tential outcomes also satisfy the perfect rank assumption (Assumption 3.3). Specifically, sure Y as continuous outcome variable and ensured that, after binarization, its distribution matches that observed in Lu et al. (2023). The topological order in Figure 1 is X = -X1,,.. X2s) = (E, D, Hb, HD, CP). The joint probability of X and Y is calculated by substituting the con- ditional probabilities from Figure I into the following equa- tion: pr(E. D, Hb, HD, CP, BP) pr(EE) pr(D) pr(Hb D) (HD D) pr(CP Hb, HD) (r(PP HD). With- out loss of generality, we consider the event & = =[(Y> > 140), which indicates whether hypertension is present. The posterior causal estimands can be used to analyze the causes of hypertension for a patient based on the evidence (r, Y > 140).
modeled blood pres-
1 sY Yi) I(X, x)
Pa-e'' (t; (i)
xt
Σラ I(X; =))
E/= 1Y, - 11 x I(X, = x') I(X, x')
Hence, we can estimate the counterfactual outcome of the th unit by,
da-r (Yi) = arg min P2- (t; X, = x. (6) re)))
We first present the postICEs for hypertension, based on dif- ferent observed evidence. shown in Table 1. According to Corollary 3.7, we know that the postICEs are only related to the parent nodes of BP, namely X, (exercise) and X4 (heart disease). We find that the largest change in postICEs occurs with the evidence (#1, Zz E) = 1,, > 140) and the intervened treatment (x), = (0, Specifi- cally, for individuals who do not exercise, have heart dis- ease, and have high blood pressure, if they had exercised previously and did not have heart disease, i.e., receiving treatment x1) = (0, 0). their blood pressure would significantly decrease by 18. 19, that is, Post.CC((Y> I
By separately minimizing Pa-tx' (: for each unit i with (X; x, Yt), we can estimate the counterfactual out- come for an individual i under state X = x' by the coun- terfactual mapping Φ2-42' (Yi) in (6), and the individual treatment effect ITE((2,,' xx") can be estimated as follows: ÁITE,((''2"* xz) é (Y) Po-zz- (Yi). Moreover, other posterior causal estimands can also be constructed using similar moment estimators.
5. Example: risk factors for hypertension
X, = 1, = 1, > 140) & 18.199 Conversely, in the evidence (I1, E) = (0. 0. Y > 140), i.e., individu- als who exercise, do not have heart disease, and have high blood pressure, if they had not exercised previously and did not have heart disease, i.e., receiving (4'). x14)) = (1, O), their blood pressure would slightly increase by 1.75, that is, PostICE((2, | X1 = 0, X4 = 0, Y > 140) ~ 1.75.
In this section, we will use the example provided in Lu et al. (2023) to explain our proposed causal attribution estimands, and assess the posterior causal effects of risk factors on con- tinuous hypertension. Figure 1 presents the causal network and the corresponding conditional probabilities, where Ex- ercise (E), Diet (D), Heart Disease (HD), Heartburn (Hb), and Chest Pain (CP) are potential causes of hypertension. Let E- = denote no daily exercise, D = denote un- healthy diet, HD- 1 denote heart discase, Hbb = 1 denote heartburn, CP= = 1 denote chest pain, and BP denote con- tinuous blood pressure Y. We generate the data without unobserved confounders to satisfy Assumption 3.1. We also ensure that the relationships among the causes satisfy the monotonicity assumption (Assumption 3.2). For instance, lack of daily exercise (E = 1) and poor diet (D) = 1)
Since our data generation mechanism ensures the exact same observed data distribution as described in Lu al. (2023) after binarizing outcome Y. we directly use their posterior total causal effects in their Table I for comparative analysis. We use the symbol asterisks (+) to differentiate from the definitions provided in this paper. Specifically, let Yx,= be the binary event I(Yxa=ze > 140), we adopt the definitions of the binary outcomes from Lu et al. (2023), denoted as PN (Xx + E) and TCS (X + E
7

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Causal Attribution for Continuous Outcomes
E-1
D-1
0,75
Hb-1
Exercise
Diet (1)
llearbbum (N))
D-0
D=1
CP:1
IID+]
HDH
D-o
E-
0.30
Heart discase
Chesi pain (K)
HD
E-0
0.85
Hb-
HD
D-0
0.30
HP
HP 140
Blood pressure
HD:0
U(108, 148)
0.15
(Y)
HD
U(138.5, 148.5)
0.90
U(110, 150)
0.25
HD
E-i
U(138, 158)
Figure 1: A causal network representing hypertension and its risk factors, where U(a, b) denotes a uniform distribution on the interval [a, b].
Table 2: Results of marginal probabilities of necessity and posterior causal estimands based on the evidence {X 1,, 140).
table show very similar results in terms of sign and ordering. Specifically, HD, E, and D (denoted as X1, X2, and X4. respectively) all have non non-zero post values, indicating that they are risk factors for blood pressure. Among them, HD (i.c., X₂) has the largest postTCE, indicating that HD the most important risk factor for blood pressure. In addi- tion, the value of (X, = E) is also the largest, further confirming the importance of HD for HD (i.e., X4), it can be observed that the PostNDE is equal to the post of BP. On the other hand, the postNDE for E (i.c., X1) is smaller than the post for BP, implying that has direct and indirect causal effects on BP. The third to fifth rows of the table present the postNDE, POstNIE, and postTCE values for Hb and CP (denoted X3 and Xs. respectively), which are all equal to zero. This indicates that they are not risk factors for blood pressure. From the causal network in Figure 1, it can be observed that there is no causal path from Hb (i.c., X3) and CP (i.e., X4) to BP. However, PN (Xs = E) is greater than PN (X1 = E) and PN* (N2 = E) as shown in the first row.
X1
X2
X;
X;
PN = 0)
0.347
0,230
0,133
0,760
0,563
PostTCEE (X. 2 2, 140)
0.207
0.722
DOSINDE = |,,Y 140)
3,823
0
17.023
NoE/N = Y Y
6,805
4.561
risk factor. Notably,
TCE 22
10.628
4.561
17.023
I, Y 140). as follows: PN* + 8) pY 0 ak, Xx = 1, Y > 140),
TOst (Xx = E x, Y 140)
= =F(( x xr Y 140).
The above definitions can be identified using Lemma 1 and Theorem 1 in Lu et al. (2023). We do not consider the causal estimands postNDDE" (Xx = & I > 140) and postNIE* (X2 = E Y 140), because there is no literature to support the identifiability results for these two causal quantities after binarization.
We also present additional analysis conclusions under differ- ent evidence in Section S8 of the Supplementary Material. To assess the stability of the proposed estimation procedure in Section 4 of the Supplementary Material, we conduct sim- ulation studies by generating data according to the causal network depicted in Figure 1. The estimated results Table under different sample sizes were provided. The simu- lation results indicate negligible biases and small standard errors, particularly for large sample sizes. For detailed infor- mation on data generation procedures and simulation results,
Given the observed evidence {X = 140). Table 2 presents the results for each potential risk factor Xa with respect to posterior causal estimands. The first row of Table 2 displays the probabilities of necessity (X + E) computed after binarizing the outcome vari- able Y. while the second row presents postTCE" (X₁ + Y > 140). The third to fifth rows present the re- sults for postNDEs, PostNIEs, and PostTCEs considered in this paper. We find that the second and fifth rows of the
>
8

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Causal Attribution for Continuous Outcomes
please refer to Section S9 of the Supplementary Material. In Section S10 of the Supplementary Material, we also apply the proposed method to real dataset (NTP, 2023).
Beyond the path-specific effects, there are also several po- tential directions for future research. Firstly, in addition to continuous outcomes, the attribution analysis of continuous causes is common in practice and is an issue of great interest. Second, the problem of retrospective analysis when multiple causes and multiple continuous outcomes are involved in many medical diagnoses is also worth exploring (Li et al., 2023). Finally, it is also interesting to discuss the bounds of the proposed estimands or consider sensitivity analysis when the monotonicity assumption does not hold (Tian & Pearl, 2000; Dawid et al., 2024). These issues are beyond the scope of this paper and are left for future research.
6. Discussion
Dawid et al. (2014) pointed out that statistical inference about the causes of effects is particularly challenging and is difficult to justify even in randomized experiments. While some prior research has addressed attribution analysis prob- lems with binary outcomes (Pearl, 2000; Dawid et al., 2015; Dawid& & Musio, 2022; Li et al., 2023), evaluating the causes of continuous outcomes or effects remains underexplored. The identifiability of ITEs and postICEs with multiple po- tentially correlated causes is motivated by the identification of ITES in the case of a single cause (Heckman et al., 1997). However, even if ITEs are identified, it is insufficient for retrospective assessment of a specific cause of effects, as the monotonicity assumption 3.2 is still necessary. In practice, for estimation purposes, we can begin by recovering the counterfactual mappings between different potential out- comes as well as ITEs through quantile regression (Koenker & Bassett, 1978; Heckman al., 1997; Feng et al., 2020); then, we can utilize the identification expressions in Lemma 3.4 and Theorem 3.6 for nonparametric estimation.
A Acknowledgements
We appreciate the comments from anonymous reviewers, which greatly helped to improve the paper. The authors are supported by the National Natural Science Foundation of China (12401378, 62306019), the Beijing Key Laboratory of Applied Statistics and Digital Regulation, the BTBU Dig- ital Business Platform Project by BMEC, and the Research Foundation for Youth Scholars of Beijing Technology and Business University (RFYS2025).
Impact Statement
Our work is also related to recent studies that explain causal attribution using ideas from information theory and Shapley values. Specifically, Schamberg et al. (2020) uses informa- tion measures to separate direct and indirect effects, and Jung al. (2022) suggests method based on Shapley val- ues to measure the importance of different factors in causal models. These methods mainly focus average effects across the whole population and look at how outcomes would change under possible interventions. In contrast, we focus on explaining what caused a particular observed outcome. By conditioning on the observed treatments and outcome, we define a set of posterior causal estimands that allow for retrospective and individual-level explanations. This can be especially useful in applications where personal- ized decisions or responsibility need to be considered, such as in medicine or legal settings. Our method is complemen- tary to these prior approaches, offering a distinct and useful perspective on causal inference with continuous outcomes.
This paper proposes a method for causal attribution analy- sis in the context of continuous outcome variables, which has the potential to improve decision-making in fields such as medicine and policy analysis. By offering more ac- curate framework for identifying and estimating posterior intervention effects, our work may lead to more reliable conclusions in various practical scenarios. Ethical consider- ations include ensuring the robustness and fairness of causal conclusions, especially in domains affecting public health and social well-being. We believe that the methods devel- oped here could contribute to a deeper understanding of complex causal relationships in real-world data.
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Causal Attribution for Continuous Outcomes
Supplementary Material
The Supplementary Material contains proofs of all theoretical results, identifiability results under a causal network, simulation details of the proposed procedure, and a real data analysis.
S1. Comparison of posterior causal estimands
This section provides the definitions of posterior causal estimands under both continuous and binary outcomes. Let Y* denote binary outcome, and let = represent the potential outcome under intervention X = z. Following the framework of Lu et al. (2023), Table S3 clearly illustrates the notational correspondence between our proposed estimands and their binary counterparts. While the definitions PostNDE" and PostNIE" in Table S3 are conceptually consistent with their continuous counterparts, their identifiability in the binary outcome setting has not been formally established the existing literature.
Table S3: Definitions of posterior causal estimands under continuous and binary outcomes.
Estimand
Outcome Type
Definition
Continuous
PostTCE - 1
BB Yx, -D [x,E)
PostTCE
Binary
= =) 2,
E(Y - Vx x,=0 x, Y = =1)
Continuous
PostNDE(X; = Y [. [ x1 xx,=0 [x,E]
PostNDE
Binary
PostNDE (Xx; + 8 x,
= E( ((
Y =
I, Y = 1]
Continuous
PostNTE(X; + 2.8) = [(Xx Y X1=1, 0)
|x,
PostNIB
Binary
Post (X, & 2 Y 1) EVV x, Yx, DD
Ix,
Continuous
PostICE(Y, // x. B(Y Y x.E)
PostCE
Binary
PostICE' x, Y = B(y Y x, x==1)1)
Continuous
ITE(2', Yv Ya. ITE((","*) = VD - V2-
ITE
Binary
From a practical perspective, the choice of which estimand to use depends on the specific analytical goal. For examplc, given the observed evidence, PostTCE appropriate when the interest lies in evaluating the overall effect of a single cause, while PostNDE and PostNIE are useful when disentangling direct and indirect pathways of primary interest. PostICE may be preferred in settings involving multiple simultaneous interventions, where understanding the joint contribution of correlated causes is crucial. Finally, ITE provides individual-level contrast and is most relevant in personalized decision-making contexts, such as precision medicine targeted policy analysis.
S2. The proof of Lemma 3.4
Proof. For any I # x', we recall that the definitions of two marginal distributions (v) and (v) are F2(y) = pr(Y < y X = 2) and (y) pr(Y Sy X = x²). Let Fep- (t) = pr(cp+1 S t) denote the marginal distribution of the error term 1p+1. Let S.,,, be the support of the error term 1p41. Since the function mp41(1 (p+1) is strictly monotonic in 6p+1 under Assumption 3.3, for any Se Sptt (the interior of S,p+r); ), we have.
F F (T)
Fe' (2 Eg p 1 (I, T)
Hence, we have (p+1(1',7) S By the fact that the marginal distribution F (-) is continuous and strictly increasing at mp+1 (I', 7), we have,
mp41(x',7) Fz' (mpp2e2 r)-]
Let = (2+1(2, E &Sy and we have 7 = mpth (x. y). Then the above equation becomes
mp+1 (2', .) F7 (F)()))
12

---

Causal Attribution for Continuous Outcomes
which shows that P2-z2' (y) is identified S% (the interior of Sy, by
Φz (v) mp+a .7) Fer (Fz(v))
The function Φx-ra (y) is identified on Sy, by its continuous extension. The remaining results follow immediately.
S3. The proof of Lemma 3.5
Proof. Without loss of generality, let F=+1: = 1 = 0. Let Xja = (Xj....,X) and Tjat = (Zz...., Iz). We first simplify the expression of the conditional probability as follows:
pr (Dllax, [x) dz |
Ip) = d% D& (ax: xx)) = ]
Da xt.) dz D& (ax,, Inc = dx}
pr (ax, Ik) d
pr(Xx+1(ax: 21)) = 2it1........ Xp( 1, xxp-11 = zp. Xx+1 = Ik+1: Xplp:: 1k = Ip}
pr x (as, (Tk) = 0k+1, X, (1p Tkup = Tpp)
([Xu+ilaR,, xt) xht Xk+1 1 (ak, U&) xx+11 } x... pr X, (ap (p-- k kpp 1) = x 1k np}
pr( xf[X (0k, Fx)) = xx+1 x... x pr (. = Fpl
kk:p-1)
-II x -h.. ) = x= X, (ak; xh-s-1) ).}.
where the second-to- equation holds due to Assumption 3.1.
We first consider the case (i), namely (x)) (0, and di = (1k+1 , I ds
Under the
monotonicity assumption 3.2, we have,
Xr (ak.
O. xh (ak.
= 1, X, (ak: 1,21+118-1) = =1}
pr(Xx(ax, 0, 11 (o(kk 1, 1.441-111 = 11
pr (x, x (ak 2k:41: 1}
pr(X, (ax. 0, 27 1 1}
pr( (ak: 3 111-11111111 1]
pr(X, = 1 0, x x
pr(X, = axs 1, Ib+11a-1)
pr(Xs(ax, 0, 0, 0 Xs(ax, 1, 13 pr ( O, 1 11
X, 14 1}.
(r( 0, 2x 21 0 (a 1, 2x41:s-1) = 0} pr X, (ax. x 0, X, (ax: 11 0)
(x 1. = 0}
pr( X, (ak: 36+1:-1) = 0}
(r (ax: 1, 114118-11 0}
=
pr(X, O, 2k+1111))) = 1) =0
= pr Xa(an, 0, Xs 1, 1, =0)
0.
Therefore, we have,
pr( (ak, O, 0. x", X, Xx 1, Tb-11s = xz) (1 (1x)))= + (2x² 1os,
where
pr(X, 1 (kk 0, Ett x
Ros
pr(X, = ak. 1, 2x xz-1)
We next consider the case (ii), namely (aî. rK) (1,0) and de "..+,....,,p)) 1 di = (o ...,))...,.,))) Under the
13

---

Causal Attribution for Continuous Outcomes
monotonicity assumption 3.2, have,
pr(X, 1, 1.:101)11) = Xa( O, IR+1:8-1) = 0] pr(X, (ak: 1, (4,
0)
pr(X, (ax. 0, 02
pr(X, (ake 1, k
Xr (#k. 0, 12 0}
pr(X, 0k 1, 1 14118-11
pr(Xx ak 0, 2x++18-1)
pr(xx 1, 2 1 Xx(ax. 0,2R+1:0-1) = 0) = 1 p ( a2. 1, k 1:s-11 = 0 (a 0, = 0}.
pr X, (ak: 1, t k+1s--1111 = 1, 1x =1
(ax 1,
x, (a,, 0, 0,
pr(X, (ak. 0, 0,2
1}
pr(X, 0, 1)
( 0, =
1.
pr (ax, 1, ++ = 0 | (x 0, 0, =1}
= pr 1, 1, = (a O, 0,044111-1 =
O.
Therefore, we have,
pr( 1, 1, = x; X Xa(ax, 0, xk+1ls-1) xz) x + (1 - 2x²) xx, 2.)Ree.
where
R1s pr(X, ak 1, 2 2- 1)
pr(X, 0 ax 0, 2k+1 ..... x
S4. The proof of Theorem 3.6 Proof. When Fa = 1, we first consider EXY =2 De
E(Y D( ag 1, d2, Y EE} D'(an.st
Delan.1) (t.aa.)) 0x 1, 1, 1) = dz, Y E
E(Yan ag 1, De 1) dr, Y E E}
E(Yogaatdd ak. 1, Y E E
We next consider ((
De (k, 1, Y EE
ax. 1, d La E(Yoriiii av. 1, dr, (ax, 0) = di, Y e E)pr( 0) dī
=aazza E(Youaz 0k 1, E ( (a 0) = di. akk 1, dx
where the second equality holds due to Assumption 3.1 (ii) and Assumption 3.2. The proof's for æ 0 follow similar logic: for simplicity, omit them.
S5. The proof of Corollary 3.7
Proof. Note that because (X+1(X,-) is strictly monotone in Ep+1, for any T e S (the interior S.) and for any pa,, # pay, we have:
F Pep (T) Fpu; (mp+1 7)} Fpu, (mp+1 (pal,, 7)}.
The remaining proof follows a similar logic as Section S2; we omit for simplicity.
14

---

Causal Attribution for Continuous Outcomes
S6. The proof of Corollary 3.8
Proof. Given the Bayesian network, the distribution of each node depends only on its parent nodes. The proof is straightfor- ward, and we omit it for brevity.
S7. The proof of Lemma 4.2
Proof. First, we differentiate Pa-xz' (yar Un) with respect to Yr. Noting that
a y/ X = =x)( - F(iiin(Y - u) X x} = - 2pr(Y <リ X = x) = - (F)(
Øt
Y t X = x²)
=-(liign(Y (P(sign(Y-(1) X = ===xx =x'=====--- =1
2pr(Y <
X =x') = - (F,
where sugn(u) = 2 I[ > 0) 11- 1. It follows that
Ope-r (t: = (1) [2)) 0.
at
Fix yY E R. Note that the marginal distribution (-)(() is weakly increasing on R and strictly increasing on Sy Therefore, Pa-saa (<) is weakly and strictly convex on IR and Sy: Furthermore, under Assumption 3.3, we know that (Ux) = F7 (Fz)), For ye E " we have Fz (Va) FF (12) if and only if yz- dx-iz' (yer). Thus, V2' = Oa-zz' (yz)) uniquely solves the first-order condition of Pz- (t: yx) = 0.
0
S8. Additional analysis for hypertension example in Section 5
In this section, present additional analysis conclusions under different evidence. Given the observed evidence {X 11 > 140}. Table S4 presents the results for the posterior causal estimands with respect to each possible risk factor Xx. The first two rows of results are directly obtained from Lu et al. (2023) for comparison purposes. In the second row, we observe that postTTEE" (X1 + E I-) = 0.2, and postTCC" (Xa + E I-) = 0 for k = 2,...,5, 5. Our results regarding postTCE are presented in the fifth row, showing a significant increase in blood pressure by 10.483 units when suffering from heart disease, i.e., post (X, = Y () = 10.483. Our results differ from Lu et al. (2023) with respect to X2. This is because Lu et al. (2023) additionally requires the monotonicity assumption of the outcome variable for identifiability in the binary outcome case. This assumption ensures that given the evidence X4- = 0 (indicating no heart discase), X1 is the unique risk factor. However, the perfect rank assumption 3.3 introduced for the continuous variable does not guarantee this. It can also be observed that the probability of necessity is not zero for X2, X3. and Xs: postTCE" (X, = & -), postTCE (XA + Y -), and NOS (X& = Y 1--) are zero when X,- 0 for k = 1, 2, 5. Additionally, it can be observed from the fourth row that pOstNTE (X, = Y 1.) = 0 for all A = 1,..., 5, suggesting that all causes have no indirect effect on BP. From the causal network in Figure 1, since the evidence indicates no heart discase, i... X, 0, can be intuitively understood that X, blocks other nodes from transmitting effects to the outcome along the paths.
Table $4: Results of marginal probabilities of necessity and posterior causal estimands based the evidence {X 1, 1,0, 140).
X1
X₂
X3
X₃
X2
(N' + E)
0.347
0.230
0.133
#
0,563
post (X2 = El 1x, Y > 140)
0.200
0
0
Nost (XA Y x, 140)
2,000
0
10.483
0
postNIE (Xo = Y Y, > 140)
0
0
0
0
0
post (X2 = Y
Y,Y 140)
2.000
0
10.483
0
# Corresponding quantity is undefined.
For the observed evidence {X = (1, 1, Y> 140), where X₂ = 0 indicates a healthy diet, the values of post and postNDE are shown in Table $5. Comparing it with Table 2, we see that
TCE (X1 > 1,,0, 1), Y 140) 12.418 POstTCE (X1 Y| X 1,1,1,1,1). > 140 = 10.628,
15

---

Causal Attribution for Continuous Outcomes
Table S5: Results of marginal probabilities of necessity and posterior causal estimands based on the evidence {X (1,0, 1,0,1,11)), 140).
X1
X2
X3
X4
X
PostTCE" (X, + & 2, Y > 140)
0.449
0
0
0.722
0
postNDE/ (X: = Y x, 140)
3.424
0
0
16.856
0
postNIE (Xg= + Y 1Y|,Y,
a,Y Y 140)
8.994
0
0
0
post (Xa = Y J,, 140)
12.418
0
0
16.856
0
and
PostTCE = Y x =
> 140} = 0 < PCst (X2 = Y X = (1,1,1,,,,) > 140} = 4.561.
This is because changing X2 to X₂ in the evidence increases the possibility that X1 the cause and decreases the possibility that X₂ is the cause. Similar conclusions are also found Lu et (2023) the binary case.
S9. Simulation studies for hypertension example in Section 5
S9.1. Data generating details
In this section, provide additional simulation studies related to Section 5. To generate simulation data that satisfies Figure 1 in Lu et al. (2023), we consider the following data generation process.
(a) X1 is Bernoulli with pr(X1 1) = 0.3.
X2 is Bernoulli with pr(X2 1) 0.75.
(c) Let G3
Xx(2, = 0)), then G3 can take values on 00, 01, and 11 under monotonicity assumption
3.2,
pr(G3 00) = 0.15, pr(G3 01) = 0.65, pr(G3 = 11) = 0.2.
Here, Xg is generated follows:
X3 = 0 if G3 = 00,
X3 = G3 = 11,
X3 = G3 = 01, X2 = 0,
X3 0f G3 X1
(iv) Let G. = = -,,0,,1000.,,,,,.0..000000-1101-
= 1, 1)), then G4 can
take values on 0000, 0011,000,, 0111, and 1111 under monotonicity assumption 3.2,
pr(G. 0000) 0.25, pr(G, 0001) 0.10, pr(G. 0011) 0.20,
pr(G, 0101) 0.10, pr(G. 0111) 0.10, pr(G. 1111) 0.25.
16

---

Causal Attribution for Continuous Outcomes
Here, X₃ is generated follows:
X1 = G. = 0000,
X1 if G. 0001, X1 X2 1,
X4 0 if G4 0001, X1 X2 1,
X4 0 if G4 0001, X1 = 1, X2 0,
X4 if 44 = 0001, X1 = 0, X2 = 0,
X1 = 1 if G4 = 0011, X1 = 1,
X4 = 0 G4 = 0011, X = 0,
X4 if G4 0101, X2
X4 0 G4 0101, X2
X4 1 G4 = 0111, X1 = 1, X₂ 1,
X1 1 G4 = 0111, i = 0, 1,
X1 = 1 G4 = 0111, X = 1,X; = 0,
X4 0 G4 = 0111, 1 XI = X2 0,
X4 if G4 1111.
(d) Let G5 = -) 0, X4 =
1, = =)), then G5 can take values on 0000, 0001, 00110 0101, 0111, and 1111 under monotonicity assumption 3.2.
=
pr(Gs 0000) 0.10, pr(Gs 0001) 0.05, pr(G; 00110 0.45, pr(Gs = 0101) = 0.05, pr(Gs = 0111) = 0.25, pr(Gs = 1111) = 0.10.
Here, X5 is generated as follows:
X5 0 if G5 0000,
X5 if G5 0001, X3 1, X1 1,
X5 0 if G5 0001, X3 0, X4 1,
X5 if G5 0001, X3 = 1, X4 0,
X5 = if G5 = 0001, X3- = 0, X₁ = 0,
X5 if G5 0011, X3
X5 0 G5 0011, X3
X5 i G5 0101, X4 1,
X5 0 if G5 = 0101, X4 = O,
X5 = if G5 = 0111, X3 = 1, X4 = 1,
X5 = 1 G3 = 0111, X3 = 0, X1 = 1,
X5 if G5 0111, X3 1,X, 0,
X5 0 if G5 0111, X3 X4 0,
X5 if G5 1111.
The above data generation process will result in an observation distribution exactly identical to Lu et al. (2023) and will be utilized to compute the true values of posterior causal estimands considered in this paper.
$9.2. The simulation results for posterior causal estimands
In this section, we conducted simulation studies to assess the performance the proposed procedure in finite samples. We generated numerical examples corresponding to Figure of the main text using the method described in Section 19.1. Using data generated from 2000000 samples, we obtained the estimated results presented in Tables $4, and of the main text,
17

---

Causal Attribution for Continuous Outcomes
well the true values provided in Tables S6 and S7 of the Supplementary Material. Additionally, considered sample sizes of 1000, 2000, and 10000, and averaged the results over 500 repetitions. These simulation results demonstrate the stability of the proposed identification expressions. Specifically, we observed that the estimated values are close to the true values, and the standard errors are relatively small. As the sample size increases, both the bias and standard errors decrease.
Table S6 presents the true values and estimated results of the posterior intervention causal effect based different evidence.
(iI Table S7 presents the true values of the posterior natural direct and indirect causal effect based on different evidence.
(iii) Tables S8-S10 present the estimated results of the posterior direct effect based on different evidence.
(iv) Tables S11-S13 present the estimated results of the posterior indirect effect based on different evidence.
Table S6: Posterior intervention causal effect based on different pieces of evidence.
Post x, 140)
(#1,
(0,0)
(21, (0,1)
(21, (1,0)
(01,24) (1,1)
(z(,>%) (0,0)
0
-13.26
-2
-19
(x(,0)) (0,1)
3.50
2.25
True values
(xf.24) = (1,0)
-11.26
0
-17
(x1,=() [1,1)
12
5.25
10.5
(z),01) (0,0)
0.000 (0.000)
-13.252 (0.722)
-1.757 (1,210)
-18.955 (0. 7697
(x),=)) (0,
3.477 (0.317)
0.000 (0.000)
2.391 (0.699)
4.999 (0.432)
1000
(x),41) = (1,0)
1.561 (1.129)
-1.533 (1.496)
0.000 (0.000)
-17.224 (1.48))
(x1,-() 11
11.967 (0.435)
5.227 (0.518)
10.676 (0.835)
0,000 (0.000)
(z),=1) (0,0)
0.000 (0.000)
-13.224 (0.554)
-1.921 (0.928)
1189662 (0.602)
(1),=1) 1)
3.463 (0.216)
0.000 (0.000)
2.273 (0.495)
-4,977 (0.300)
2000
(xf,21) = (1,0)
1.808 (0.862)
-1.322 (1.031)
0.000 (0.000)
-17.057 (1.073)
(x1,-1) 1)
11.944 (0.296)
5.240 (0.361)
10.518 (0.631)
0,000 (0.000)
(1),=1) (0,0)
0.000 (0.0000)
13.266 (0.2636)
-1,950 (0.3588)
-190036 (0.2550)
(1) 1)
3.503 (0.0992)
0,000 (0.00100
2.291 (0.1965)
-5.005 (3.3)
10000
(x) = (1,0)
1.926 (0.3298)
-1.307 (0.4901)
0.000 (0.0000)
-17.076 (0.4652)
(1 (1,1)
11.995 (0. 1558)
5.261 (0.1576)
10.543 (0.2530)
0.000 (0.0000)
18

---

Causal Attribution for Continuous Outcomes
Table S7: True values of posterior natural direct effect and posterior natural indirect effect.
Evidence
Posterior natural direct effect
Posterior natural indirect effect
Case
(x,
X1
X₂
X3
X4
X5
X2
X2
X3
X,
Xs.
(0,0,0,0,00,.)
2.000
0.000
0.000
3.482
0.000
3.885
0.918
0.000
0.000
0.000
2
(0.0.0.0.1.E)
2.000
0.000
0.000
3.331
0.000
3.694
0.796
0.000
0.000
0.000
3
(0,0,0.1,0,E)
5.197
0.000
0.000
13,408
0.000
0.000
0,000
0.000
0,000
0.000
4
(0.0.0.1.1.E))
5.201
0.000
0.000
13.396
0.000
0.000
0.000
0.000
0.000
0.000
5
(0,0.1,0,0.E)
2,000
0.000
0.000
3.539
0.000
4.460
0.992
0.000
0.000
0.000
6
(0,0,1,0,1.8)
2.000
0.000
0.000
3.434
0.000
3.908
0.838
0.000
0.000
0.000
7
(0.0.1.1,0.8)
5,416
0.000
0.000
12.751
0.000
0.000
0.000
0.000
0.000
0.000
8
(0,0.1.11..)
5,171
0.000
0.000
13.486
0.000
0.000
0.000
0.000
0.000
0.000
9
(0.1,0,0.0.8)
2.000
0.000
0.000
3.455
0.000
5.655
0.000
0.000
0.000
0.000
10
(0.1,0.0.1..)
2,000
0.000
0.000
3.585
0.000
5,474
0.000
0.000
0,000
0.000
11
(0.1.0.1.0.g)
5.126
0.000
0.000
13.621
0.000
0.000
5,926
0.000
0.000
0.000
12
(0.1.0..I..))
5.291
0.000
0.000
13.127
0.000
0.000
5.950
0.000
0.000
0.000
13
(0.1,1000,))
2.000
0.000
0.000
3.547
0.000
5.664
0.000
0.000
0.000
0.000
14
(0.1,1,0.1.8))
2.000
0.000
0.000
3.483
0.000
5.407
0.000
0.000
0.000
0.000
15
(0.1.1.1,0.))
5,192
0.000
0.000
13.425
0.000
0.000
5,874
0.000
0.000
0.000
16
(0.1,111,1.8)
5.246
0.000
0.000
13.263
0.000
0.000
5.916
0.000
0.000
0.000
17
(1,0,0,0,0.))
2.000
0.000
0.000
10.490
0.000
0.000
4.602
0.000
0.000
0.000
18
(1.0.0.0...))
2.000
0.000
0.000
10.365
0.000
0.000
4.864
0.000
0.000
0.000
19
(1.0.0.1.0.))
3.342
0.000
0.000
16.955
0.000
9.184
0.000
0.000
0.000
0.000
20
(1.0.0.I1..))
3.304
0.000
0.000
17.188
0.000
9.477
0.000
0.000
0.000
0.000
21
(1.0.1,00..))
2.000
0.000
0.000
10.569
0.000
0.000
4.793
0.000
0.000
0.000
22
(1.0.10,1.))
2.000
0.000
0.000
10.645
0.000
0.000
3,802
0.000
0.000
0.000
23
(1.0.L.,0.E)
3,504
0.000
0.000
17.188
0.000
8.130
0.000
0.000
0.000
0.000
24
(1.0.1.1.1.8)
3.424
0.000
0.000
16.856
0.000
8.994
0.000
0.000
0.000
0.000
25
(1,1,0,0.0.))
2.000
0.000
0.000
10.363
0.000
0.000
0.000
0.000
0.000
0.000
26
(1.1,0.011.))
2,000
0.000
0.000
10.586
0.000
0.000
0.000
0.000
0.000
0.000
27
(1.1.0.10.E))
3.867
0.000
0.000
16.809
0.000
6.740
4.306
0.000
0.000
0.000
28
(1.1.0.1...))
3.799
0.000
0.000
17.045
0.000
6.692
4,251
0.000
0.000
0.000
29
(1,1,1,0,0.))
2.000
0.000
0.000
10.463
0.000
0.000
0.000
0.000
0.000
0.000
30
(1.1..0.1.))
2.000
0.000
0.000
10.483
0.000
0.000
0.000
0.000
0.000
0.000
31
(1111,08))
3,837
0.000
0.000
16.993
0.000
6.595
4.603
0.000
0.000
0.000
32
(1,1,I,1.))
3.823
0.000
0.000
17.023
0.000
6.805
4.561
0.000
0.000
0.000
S10. Real data analysis: risk factors for abnormal weight
In this section, we apply the proposed method to a real dataset from the developmental toxicology experiments conducted by the National Toxicology Program (NTP) (NTP, 2023). The primary objective of this study is to analyze whether mii((-chloro-2-propy) phosphate (TCPP) is risk factor for abnormal weight loss in B6C3F1/N mice, or if there are other causes that are the potential risks. In this experiment, total of 120 mice were randomly exposed six different dose levels of TCPP via dosed feed: O, 1250, 2500, 5000, 10000, 20000 ppm, for a duration of 3 months. Each pup's 's data includes gender (male/female), weekly body weights, organ weights, and whether organs exhibit pathology. In our analysis, let X1 represent the gender of the mice, where X, = 0 indicates female mice and X; = indicates male mice. Let X₂ denote the dose, where X₂ = 0 represents exposure to the low dose group including 0, 1250, and 2500 ppm, and X₂ = represents exposure to the high dose group including 5000, 10000, and 20000 ppm. Let X3 denote whether the liver or kidney exhibits pathology, where X3 three months as the outcome Y. In this analysis, we focus assessing the potential risk factors affecting the body weight of underweight mice, indicated by the event E = II(Y < 27).
indicates no pathology and X3 = 1 indicates pathology. We choose the body weight at the end of
We first use the R package "bnlearn" to construct a Bayesian network based on the collected data as shown in Figure S2. It is clear that dose X₂ indirectly affects body weight Y through the organ disease X3. Since both gender (i.e., X,) and dose (i.c., X2) can be considered as randomized trials, we can estimate the causal effect of X2 on Y using the difference in means estimator, i.e., E(Yx,=1 -o,= R 6.09; which implies that males are heavier. Similarly, we can estimate the causal
19

---

Causal Attribution for Continuous Outcomes
Table S8: The simulation results for the postNDE under a sample size of 1000.
Evidence
PostNDE(Xx + Y xr Y > 140)
Case
(I,E)
X1
X2
X3
X4
X5
1
(0.0,0,0,0.E)
1.561 (1.134)
0.000 (0.000)
0.000 (0.000)
3.489 (0.318)
0.000 (0.000)
2
(0,0,0,0,1,E)
1,561 (1.134)
0.000 (0.000)
0,000 (0.000)
3.489 (0.318)
0,000 (0.000)
3
(0,0,0,1,0,E)
5.226 (0.519)
0.000 (0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
4
(0,0,0,1,1.E)
5.226 (0.519)
0.000 (0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
5
(0,0,1,0.0.E)
1.561 (1.134)
0.000 (0.000)
0.000 (0.000)
3.489 (0.318)
0.000 (0.000)
6
(0,0,1.0.1,8)
1.561 (1.134)
0.000 (0.000)
0.000 (0.000)
3.489 (0.318)
0.000 (0.000)
7
(0.0,0,1.0.8)
5.226 (0.519)
0.000 (0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
8
(0.011,111,))
5.226 (0.519)
0.000 (0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
9
(0.100,0.0.8)
1,561 (1.134)
0.000 (0.000)
0.000 (0.000)
3,489 (0.318)
0.000 (0.000)
10
(0,1,0.0.1,8)
1,561 (1.134)
0.00000.0000
0.000 (0.000)
3,489 (0.318)
0.000 (0.000)
11
(0.1,0.1.0,8)
5,226 (0.519)
0.000
0.000 (0.000)
13,255 (0.722)
0.000 (0.000)
12
(0.1,0.111.E)
5.226 (0.519)
(0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
13
(0.1110.0..)
1.561 (1.134)
00.000000000
0.000 (0.000)
3.489 (0.318)
0.000 (0.000)
14
(0.1,10,1..))
1.561 (1.134)
0..00,0.0000
0.000 (0.000)
3.489 (0.318)
0.000 (0.000)
15
(0.11111008)
5.226 (0.519)
0.000
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
16
0011111,11.)
5.226 (0.519)
0.000 (0.000)
0.000 (0.000)
13.255 (0.722)
0.000 (0.000)
17
(1.000.000.8)
1,755 (1.194)
0..0000.0000
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
18
(1,0,0.0.1,8)
1.756
0.000
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
19
(1,0,0.100,8)
3,230 (0.903)
(0.000)
0.000 (0.000)
17.220 (1.479)
0.000 (0.000)
20
(1,0,0.1,1.E)
3,230 (0.903)
0.000 (0.000)
0.000 (0.000)
17.220 (1.479)
0.000 (0.000)
21
(1,0,1,000.8)
1,755 (1.194)
0.000 (0.000)
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
22
(1,0,1,0,1.8)
1.753 (1.193)
0.000 (0.000)
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
23
(1,0,1.1.0.8)
3.371 (1.153)
0.000 (0.000)
0.000 (0.000)
17.220 (1.479)
(0.000)
24
(1.0.1.1.1.))
3.365 (1.148)
0.000 (0.000)
0.000 (0.000)
17.217 (1.478)
0.000 (0.000)
25
(1,100.000.g)
1.755 (1.194)
0.000 (0.000)
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
26
(1,1,00011.))
1.804 (1.124)
0.000 (0.000)
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
27
(1,1,0.100,g)
3.732 (0.740)
0.000 (0.000)
(0.0000
17.220 (1.479)
0.000
28
(1.1,0,1,1.8)
3.732 (0.740)
0..00
(0.000)
17.220 (1.479)
0.000
29
(1,1,1,,0..))
1.755 (1.194)
(0.000)
(0.0000
10.641 (0.821)
0.000 (0.000)
30
(1,1,1,,118))
1,755 (1.194)
0..0000.0000
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
31
(1,111110,))
3.692 (0.667)
0.000 (0.000)
0.000 (0.000)
17,220 (1.479)
0.000 (0.000)
32
(L,1I,,11E))
3.692 (0.667)
0.000 (0.000)
0.000 (0.000)
17,220 (1.479)
0.000 (0.000)
effect of X2 on Y to be -3.353, i.e., Figure S2 also provides simple descriptive statistical analysis of these potential risk factors. We observe that as the toxin level increases, the occurrence rate of organ disease also increases. In addition, males showed higher variability in organ abnormalities compared to females, as demonstrated in previous studies (Bianco et al., 2023). These empirical findings align with monotonicity assumption 3.2, i.c., (0, 0) < {Xs(0, 1), X3 (1, < (1, 1). In this data analysis, we choose not to binarize the outcome and, therefore, do not report the results of the binarized posterior causal effects, as the outcome variable fails to satisfy the monotonicity assumption regarding the causes (Assumption 2(b) in Lu et al. (2023)).
3.333, which suggests that the higher dose lead to weight loss.
We now present the estimation results of PostICEs based on various observed evidence in Table S14. According to Corollary 3.7 and Figure S2 we know that the postICEs are only related to the parent nodes of body weight, namely X, (gender) and X3 (organ disease). Considering the evidence (X,,X3, E) = (0, 1, Y < 27), we observe PostICE(Y, 2, Y < 27) 10.51, which indicates that changing from male mice without organ disease (i.e., (41,*4) = (1, to female mice with organ disease results in the most significant weight loss, totaling 10.51. For the evidence (x1, 23, E) (0, (0,0,Y Y < 27), we observe PostICE(Y, Y < 27) = 1.42, which indicates that changing from female mice with organ disease (i.e., (a1, 2²/) = (0, 1)) to female mice without organ disease results in a slight decrease of about 1.42. In summary, for a given posterior evidence, changing from female mice (X1 = 0) to male mice (X1 = 1) results in an increase body weight,
20

---

Causal Attribution for Continuous Outcomes
Table S9: The simulation results for the postNDE under a sample size of 2000.
Evidence
PostNDE(Xx + Y xr Y > 140)
Case
(I,E)
X1
X2
X3
X4
X5
1
(0.0,0,0,0.E)
1.808 (0.863)
0.000 (0.000)
0.000 (0.000)
3.469 (0.217)
0.000 (0.000)
2
(0,0,0,0,1,E)
1,808 (0.863)
0.000 (0.000)
0,000 (0.000)
3.469 (0.217)
0,000 (0.000)
3
(0,0,0,1,0,E)
5.239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0.000 (0.000)
4
(0,0,0,1,1.E)
5.239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0.000 (0.000)
5
(0,0,1,0.0.E)
1.808 (0.863)
0.000 (0.000)
0.914 (0.)00)
3.469 (0.217)
0.000 (0.000)
6
(0,0,1.0.1,8)
1.808 (0.863)
0.000 (0.000)
0.000 (0.000)
3.469 (0.217)
0.000 (0.000)
7
(0.0,0,1.0.8)
5.239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0.000 (0.000)
8
(0.0,1,111,8)
5.239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0.000 (0.000)
9
(0.1,0,0.0.8)
1,808 (0.863)
0.000 (0.000)
0.000 (0.000)
3.469 (0.217)
0.000 (0.000)
10
(0,1,0.0.1,8)
1.808 (0.863)
0.000 (0.000)
0.000 (0.000)
3,469 (0.217)
0.000 (0.000)
11
(0.1,0.1.0,8)
5,239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0,000
12
(0.1,0.111.E)
5.239 (0.361)
0..000000000
0.000 (0.000)
13.226 (0.554)
0.000 (0.000)
13
(0.1110.0..)
1.808 (0.863)
0.000 (0.000)
0.000 (0.000)
3.469 (0.217)
0.000 (0.000)
14
(0.1,10,1.E)
1.808 (0.863)
0.000 (0.000)
0.000 (0.000)
3.469 (0.217)
15
(0.11111008)
5.239 (0.361)
(0.000)
0.000 (0.000)
13.226 (0.554)
16
0011111,11.)
5.239 (0.361)
0.000 (0.000)
0.000 (0.000)
13.226 (0.554)
0.000
17
(1.000.000.8)
1,920 (0.921)
0.000 (0.000)
0.000 (0.)00)
10.501 (0.625)
0.000
18
(1,0,0.0.1,8)
1.922 (0.922)
0.000 (0.000)
0.000 (0.000)
10.501 (0.625)
0.000
19
(1,0,0.100,8)
3.310 (0.622)
0..00
0.000 (0.000)
17.055 (1.073)
0.000
20
(1,0,0.1,1.E)
3.310 (0.622)
0.000 (0.000)
0.000 (0.000)
17.055 (1.073)
0.000 (0.000)
21
(1,0,1,000.8)
1.920 (0.921)
0.000 (0.000)
0.000 (0.)00)
10,501 (0.625)
0.000 (0.000)
22
(1,0,1,0,1.))
1.920 (0.921)
0.000 (0.000)
0.000 (0.000)
10.501 (0.625)
0.000 (0.000)
23
(1,0,1.100.8)
3.318 (0.724)
0.000 (0.000)
0.000 (0.000)
17.055 (1.073)
0.000 (0.000)
24
(1.0.1.1.1.8)
3.318 (0.724)
0.000 (0.000)
0.000 (0.000)
17.055 (1.073)
0.000 (0.000)
25
(1,100.000,g)
1.920 (0.921)
0.000 (0.000)
0.000 (0.000)
10.501 (0.625)
0.000 (0.000)
26
(1,1,00011.))
1.804 (1.124)
0.000 (0.000)
0.000 (0.000)
10.641 (0.821)
0.000 (0.000)
27
(1,1,0.100,g)
3.753 (0.523)
0.000 (0.000)
(0.0000)000)
17.055 (1.073)
0.000
28
(1.1,0,1,1.8)
3.753 (0.523)
0..0000.0000
(0.000)
17.055 (1.073)
0.000 (0.000)
29
(L.1,1,,0..))
1.920 (0.921)
0.0000000000
(0.0000
10.501 (0.625)
0.000 (0.000)
30
(1,1,10,118))
1.920 (0.921)
0..0000.0000
0.000 (0.000)
10.501 (0.625)
0.000 0.0000)0000) (0.000)
31
(1,111110,))
3.760 (0.461)
0.00000.0000
0.000 (0.000)
17.055 (1.073)
0.000 (0.000)
32
(L,1I,,))))
3.760 (0.461) 0.000 (0.000)
0.000 (0.000)
17.055 (1.073)
0.000 (0.000)
while changing from mice without organ disease (X3 = 0) to mice with organ disease (X3 = 1) results in a decrease in body weight in most cases.
Table $15 presents the estimation results of the posterior causal estimands for each potential risk factor. Given the evidence X2, E- 1,,, < 27), we find that the PostNDE value for gender Xı is the highest, indicating that gender might be the most important direct factor, followed by organ disease X3. The PostNDE value for toxin dose X₂ is O, which is consistent with the conclusions drawn from the Bayesian network in Figure S2. In addition, given the evidence (X1, X2, E) = 1, < 27). we observed that both gender X, and toxin dose X₂ have an indirect effect on the outcome, with the absolute value of PostNIE for gender X, being significantly greater than that for toxin dose X2, while organ disease X3 has no indirect effect, suggesting that gender may be the most important indirect factor influencing body weight among the three factors, followed by toxin dose X₂. However, in terms of the post organ disease X3 has the largest value, suggesting that organ disease X3 is the most important total risk factor for low body weight, followed by toxin dose X2, while gender X, has the least effect.
Under the evidence 0,, < 27) and (0, 1,,1,1, < 27), toxin dose X₂ exhibits the most significant PostNIE value for weight loss indicating that toxin dose X₂ is the most important indirect risk factor in these situations. In summary, for most evidence in Table $15, either gender X, or organ disease X3 has the highest absolute PostTCE value, especially X3.
21

---

Causal Attribution for Continuous Outcomes
Table S10: The simulation results for the postNDE under sample size of 10000.
Evidence
postNDE = Y
>>
140)
Case
(x,
X1
X2
X3
X4
X5
1
(0,000,0,0,8)
1.961(0.351)
0.000(0.000)
0.000
3.504(0.100
0.000(0.000)
2
(0,0,0.0.1,E)
1.961(0.351)
0.000(0.000)
0.000(0.000)
3.504(0.102)
0.000(0.000)
3
(0,0,0.1,0,8)
5.259(0.171)
0.000(0.000)
0.000(0.000)
13.259(0.261)
0.000(0.000)
4
(0,0,0.1,1.E)
5.259(0.171)
0.000(0.000)
0.000(0.000)
13.259(0.261)
0.000(0.000)
5
(0.0,1.0.0,8)
1.961(0.351)
0.000(0.000)
0.000(0.000)
3.504(0.102)
0.000(0.000)
6
(0.0,1,0,1.8)
1.961(0.351)
0.000(0.000)
0.000(0.000)
3.504(0.102)
0.000(0.000)
7
(0.0,0.1,0,8)
5.259(0.17))
0.000(0.000)
0.000(0.0000
(3.259(0.261)
0.000(0.000)
8
(0.0,1.1,118))
1.259(0.171)
0.000(0.000)
0.000
(3.259(0.261)
0.000(0.000)
9
(0,1,0.0.0,E)
1.961(0.351)
0.000(0.000)
0.000(0.0000
3.504(0.102)
0.000(0.000)
10
(0.1,0.0.11E)
1.961(0.351)
0.000(0.000)
0.000(0.000
3.504(0.102)
0.000(0.000)
11
(0.1,0.1,0,8)
5.259(0.171)
0.000(0.000)
0.000 (000)
(3.259(0.261)
0.000(0.000)
(0.1,0.1II.E)
5.259(0.171)
0.000(0.000))
0.000 (O0)
13.259(0.261)
0.000(0.000)
13
00.1110000..)
1.961(0.351)
0.000
0.000 (000)
3.504(0.102)
0.000
14
(0.1,1,0.1.8)
1.961(0.351)
0.000(0.000)
0.000(0.00M
3.504(0.102)
0.000
15
(0.111,100.8)
5.259(0.17))
0.000(0.000)
0.000(0.00M
13.259(0.261)
0.000
16
(0IIII11))))
1.259(0.171)
0.000(0.0000
0.000(0.000
13.259(0.261)
0.000(0.
17
(1.000.000.g)
1.983(0.382)
0.000(0.0000
0.000(0.000
(0.513(0.267)
0.000(0.000)
18
(1,0,0.0.1.))
1.983(0.382)
0.000(0.0000
0.000(0.000)
10.513(0.267)
0.000(0.000)
19
(1.0,0.1,0.8)
3.357(0.267)
0.000(0.000)
0.000(0.000)
17.050(0.466)
0.000(0.000)
20
(1,0,0.111.8)
3.357(0.267)
0.000(0.000)
0.000(0.000
17 77.050(0.466)
0.000(0.000)
21
(1,0,1,000.))
1.983(0.382)
0.000(0.000)
0.000(0.00M
10.513(0.267)
0.000(0.000)
22
(1,0,1,0,1.8)
1.983(
0.000(0.000)
0.000
10.513(0.267)
0.000(0.000)
23
(1.0.1,1,0.8)
3.364(0.322)
0.000(0.000)
0.000(0.000
17 77.05000.460
0.000(0.000)
24
(1,0,1,1,1.8)
3.364(0.322)
0.000(0.000)
0.000(O. 000
17 17.050(0.466)
0.000(0.000)
25
(111000000.)
1.983(0.382)
0.000(0.000)
0.000 000)
10.513
0.000(0.000)
26
(1,1,0.0.1.))
1.983(0.382)
0.000(0.000)
0.000
10.513(0.267)
0.000(0.000)
27
(1,1,0.1.0,g)
3.796(0.229)
0.000(0.000)
0.000(0.000)
1 (7.050(0.466)
0.000(0.000)
28
(1,1,0,1,1.8)
3.798(0.228)
0.000(0.000)
0.000(0.000)
17.045(0.466)
0.000(0.000)
29
(1,1,1,00,,g)
1.983(0.382)
0.000(0.000)
0.000(0.000)
10.513(0.267)
0.000(0.000)
30
(1,1,1,0,1.8)
1.978(0.381)
0.000(0.000)
0.000(0.000)
10.519(0.269)
0.000(0.000)
31
(1,1,1,10,.8)
3.798(0.222)
0.000(0.000)
0.000(0.000)
17.047(0.470)
0.000(0.000)
32
(1,1I,111))
3.790(0. 195)
0.000(0.000)
0.000(0.000)
17.076(0.465)
0.000(0.000)
suggesting that organ disease is the important risk factor leading weight loss.
22

---

Causal Attribution for Continuous Outcomes
Table $11: The simulation results for the postNIE under a sample size of 1000.
Evidence
PostNIE(Xx
2, Y
140)
Case
(x, 8)
X1
X2
X3
X4
X5
(0,0,0,0,0.E)
4.157 (1.094)
0.926 (0.186)
-0.001 (0.376)
0.000 (0.000)
0.000 (0.000)
(0,0,0.0.1.E)
4.159 (1.093)
0.926 (0.186)
-0.001 (0.376)
0.000 (0.000)
0.000 (0.000)
3
(0,0,0.1,0.E)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
4
(0,0,0,1,1.8))
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
5
(0.0,1.0.0.8)
4.058 (1.995)
0.903 (0.286)
0.000 (0.000)
0.0000000000
0.000 (0.000)
6
(0.0,1,0,1.))
4.082 (2.013)
0.901 (0.286)
0.000 (0.000)
000000)
0.000 (0.000)
7
(0.0,0.1.0.8)
0.000 (0.000)
0.000 (0.000)
(1.42(5)424)
(0.000)
0.000 (0.0000
8
(0.0,1,1,1..))
0.000 (0.000)
0.000 (0.000)
1.186(5.42))
0.000
0.000 (0.000)
9
(0.1,0,0,0.E)
1.58041.74))
0.000 (0.000)
-0.056 (0.429)
0.000
0.000 (0.000)
10
(0.1,0.0.1.8)
5.580 (1.742)
0.000 (0.000)
(0.469))
0.000
10.000
11
(0.1,0.1,0.8)
0.000 (0.000)
5,801 (1.560)
0.000 (0.000)
0..00
0.000 (0.000)
(0110.111..))
(0.0000))
5.801 (1.560)
0.000 (0.0000
0.000 (0.000)
00.00
13
(0.1,1,,00.))
5.707 (0.874)
0.000 (0.000)
0.000 (0.000)
0..000000000
0..00
14
(0.1,1.011.))
5.707 (0.874)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0..00
15
(0.1,1,100.8)
0.000 (0.0000
5.870 (1.056)
.0.094(1.87))
0.00000
0.000 (0.000))
16
(0.1,,,,,L))
0.000 (0.000)
5.870 (1.056)
(1.876)
(0.000)
0.000 (0.000)
17
(1.0,0.0.0..)
0..00
4.616 (1.206)
-0.266 (3.567)
00.000)
0.000 (0.0000
18
(1.0,0,0,1.))
0..0000
4.616 (1.206)
-0.246 (3.591)
00.000)
0,000 (0.000)
19
(1,0,0.100.))
0..7771.777
0.000 (0.000)
0.000 (0.000)
00.000)
0,000 (0.000)
20
(1.0,0,1,1.8)
9.322 (1.777)
0.000 (0.000)
0..00
0.0000)
0.000 (0.000)
21
(100,1,000.))
0.000 (0.000)
4.544 (2.096)
(0.0000
0.000
0.000 (0.000)
22
(1.0,1,0,1.8)
0.000 (0.000)
4.159 (2.725)
0.000 (0.0000
00.0000
0.000 (0.000)
23
(1.0,1,100.))
8.688 (4.183)
0.000 (0.000)
-1.352 (6.608)
0.000
0.000 (0.000)
24
(1,0,1.1,1.))
8.708 (4.163)
0.000 (0.000)
-1.345 (6.613)
0.000
00.000
25
(1,1,0,0,0.g)
0.000 (0.000)
0.000 (0.000)
-1.177 (5.164)
0..00
0..00
26
(1,1,0,0.1.))
0.000 (0.000)
0.000 (0.000)
-1.177 (5.164)
00.000)
(0.0000
27
(1,1,0.100.))
6.711 (1.876)
4.408 (2.022)
0.000 (0.000)
(0.000)
(0.000)
28
(1.1,0,111.))
6.711 (1.876)
4.408 (2.022)
0.000 (0.000)
00.000)
000000)
29
(1,1,10,00.))
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
000000)
(0.000)
30
(1.1,1,011.))
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
(0.000)))00)
31
(1,1111100))
6.902 (0.902)
4.582 (1.508)
0.043 (1.996)
0.000 (0.000)
0..00
32
(1111,1111)))
6.902 (0.902)
4.588 (1.503)
0.043 (1.996)
0..000000000
0.000 (0.000)
23

---

Causal Attribution for Continuous Outcomes
Table $12: The simulation results for the postNIE under a sample size of 2000.
Evidence
PostNIE(X:
2, Y
140)
Case
(x, 8)
X1
X2
X3
X4
X5
(0,0,0,0,0.E)
4.059 (0. (0.748)
0.921 (0.120)
-0.004 (0.256)
0.000 (0.000)
0.000 (0.000)
(0,0,0.0.1.E)
4.059 (0.748)
0.921 (0.120)
-0.004 (0.256)
0.000 (0.000)
0.000 (0.000)
3
(0,0,0.1,0.E)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
4
(0,0,0.1,1.8))
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
5
(0.0,1.0.0.8)
4.120 (1.418)
0.914 (0.200)
0.000 (0.000)
00,000)
0.000 (0.000)
6
(0.0,1,0,1.))
4.120 (1.418)
0.914 (0.200)
0.000 (0.000)
000000)
0.000 (0.000)
7
(0.0,0.1.0.8)
0.000 (0.000)
0.000 (0.000)
-0.576 (3.295)
(0.000)
0.000 (0.0000
8
(0.0,1,1,1..))
0,000 (0.000)
0.000 (0.000)
-0.576 (3.295)
0.000
0.000 (0.000)
9
(0.1,0,0,0.E)
5.531 (1.246)
0.000 (0.000)
-0,005 (0.271)
0.000
0.000 (0.000)
10
(0.1,0.0.1.8)
5.531 (1.246)
0.000 (0.000)
-0,005 (0.271)
0.000
10.000
11
(0.1,0.1,0.8)
0.000 (0.000)
5,808 (1.006)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
(0110.111..))
0.000 (0.000)
5.808 (1.006)
0.000 (0.000)
0.000 (0.000)
10.000
13
(0.1,1,,00.))
5..45300653))
0.000 (0.000)
0.000 (0.0000
0.000 (0.000)
0..00
14
(0.1,1.011.))
5..44
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0..00
15
(0.1,1,100.8)
0.000 (0.000)
5.887 (0.746)
0.041 (1.254)
0.000 (0.0000
0.000 (0.000))
16
(0.1,I,,,L))
0.000 (0.000)
5.887 (0.746)
0.041 (1.254)
0.000
0.000 (0.000)
17
(1.0,0.0.0..)
0.000 (0.000)
4.606 (0.828)
0.025 (2.427)
0.000 (0.000)
0.000 (0.0000
18
(1.0,0,0,1.))
0,000 (0.000)
4.606 (0.828)
0.025 (2.427)
0.000 (0.000)
0,000 (0.000)
19
(1,0,0.1.0.))
9.308 (1.114)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0,000 (0.000)
20
(1.0,0,1,1.8)
9.308 (1.114)
0.000 (0.000)
0.000 (0.000)
0.000
0.000 (0.000)
21
(100,1,000.))
0.000 (0.000)
4.406 (1.524)
(0,000)
0.000 (0.000)
0.000 (0.000)
22
(1.0,1.0,1.8)
0.000 (0.000)
4.296 (1.620)
0.000 (0.000)
00.0000
0.000 (0.000)
23
(1.0,1,100,))
9.227 (2.246)
0.000 (0.000)
-0.376 (3.501)
0.000
0.000 (0.000)
24
(1,0,1.1,1.))
9.227 (2.246)
0.000 (0.000)
-0.376 (3.501)
0.000
00.000
25
(1,1,0,0,0.g)
0.000 (0.000)
0.000 (0.000)
-0.546 (2.909)
0.000
0..00
26
(1,1,0,0.1.))
0.000 (0.000)
0.000 (0.000)
-0.546 (2.909)
0.000
(0.0000))00)
27
(1,1,0.100.))
6.819 (1.306)
4.437 (1.473)
0.000 (0.000)
0..00000000)
(0.000)
28
(1.1,0,111.))
6,819 (1.306)
4.437 (1.473)
0.000 (0.000)
00.000)
00.000)
29
(1,1,10,00.))
0.000 (0.000)
0.000 (0.000)
0.000
0.00000.0000
(0.000)
30
(1.1,1,011.))
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
(0.000)))00)
31
(1,1111100))
6.827 (0.693)
4.489 (1.034)
0.007 (1.395)
0.000 (0.000)
0.000
32
(1111,1111)))
6.827 (0.693)
4.489 (1.034)
0.007 (1.395)
0..00
0.000 (0.000)
24

---

Causal Attribution for Continuous Outcomes
Table S13: The simulation results for postNIE under sample size 10000.
Evidence
PostNIE(X; = Y xr, Y 140)
Case
(x,
X1
X2
X3
X4
X5
1
(0,0,0,0,0.E)
4.004 (0.331)
0.934 (0.057)
0.001 (0.122)
0.000 (0.000)
0.000 (0.000)
2
(0,0,0.0,1.E)
4.004 (0.331)
0.934 (0.057)
0.001 (0.122)
0.000 (0.000)
0.000 (0.000)
3
(0,0,0,1.0,8)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
4
(0,0,0,1,1.E)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
5
(0,0,1,0,0,8)
3,989 (0.590)
0,931 (0.088)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
6
(0.0,1,0,1.8)
3,989 (0.590)
0.931 (0.088)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
7
(0.0,0,1,0..)
0.000 (0.000)
0.000 (0.000)
-0.098 (1.409)
0.000 (0.000)
0.000 (0.000)
8
(0,0.1,1,1.))
0.000 (0.000)
0.000 (0.000)
-0.098 (1.409)
0.000 (0.000)
0.000 (0.000)
9
(0110.000.E)
5,458 (0.491)
0.000 (0.000)
-0.004 (0.256)
0.000 (0.000)
0.000 (0.000)
10
(0,100,0,1.8)
5.458 (0.491)
0.000 (0.000)
-0.004 (0.126)
0.000 (0.000)
0.000 (0.000)
11
(0,100,1,0.))
0.000 (0.000)
5.880 (0.466)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
12
(0.1...1.1.))
0.000 (0.000)
5.880 (0.466)
0.000
0.000 (0.000)
0.000 (0.000)
13
(0.1110.00.))
5.472 (0.259)
0.000 (0.000)
0.000
0.000 (0.000)
0.000 (0.000)
14
(0.1.110.1IE)
5,472 (0.259)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
15
(0.1,1,1,0.))
0.000 (0.000)
5.886 (0.336)
-0,005 (0.577)
0.000 (0.000)
0.000 (0_000)
16
(0.1.1,11I.))
0.000 (0.000)
5.886 (0.336)
-0.005 (0.577)
0.000 (0.000)
0.000 (0.000)
17
(1.0.0.0.0.))
0.000 (0.000)
4.664 (0.346)
-0.037 (1.038)
0.000 (0.000)
0.000 (0.000)
18
(1.0.0.0.1.))
0.000 (0.000)
4.664 (0.346)
-0033 (1.038)
0.000 (0.000)
0.000 (0.000)
19
(1.00.1,00E))
9.275 (0.501)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
20
(1.000.1I.E)
9.275 (0.501)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
21
(1.0.1,0,0.E)
0.000 (0.000)
4.640 (0.586)
0.000
0.000 (0.000)
0.000 (0.000)
22
(1,0,1,0I1.))
0.000 (0.000)
4.640 (0.586)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
23
(1.0.1,10.E)
9.220 (1.009)
0.000 (0.000)
-0.114 (1.414)
0.000 (0.000)
0.000 (0.000)
24
(1,0.1,1.1.))
9.220 (1.009)
0..0000
-0.114 (1,414)
0.000 (0.000)
0.000 (0.000)
25
(1,1.0,0,0.8)
0.000 (0.000)
00.0000
-0.069 (1.093)
0.000
0.000 (0.000)
26
(1.10.0.1.))
0.000 (0.000)
(0.000)
-0.069 (1.093)
0.000 (0.000)
0.000 (0.000)
27
(1,1.0,10.8))
6,800 (0.551)
4.541 (0.606)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
28
(1,100.11..)
6.793 (0.553)
4.532 (0.600)
0.000 (0.000)
0.000 (0.000)
0.000 (0.000)
29
(1,1,1,000.8)
0.000 (0.000)
0.000 (0.000)
0..000000000
0.000 (0.000)
0.000 (0.000)
30
(1,111,01..))
0.000 (0.000)
0.000 (0.000)
0.000
0.000 (0.000)
0.000 (0.000)
31
(1,11,,00.)
6.812 (0.282)
4.567 (0.451)
0.019 (0.577)
0.000 (0.000)
0.000 (0.000)
32
(1,L1,,1E))
6.816 (0.270)
4.620 (0.461)
0.008 (0.584)
0.000 (0.000)
0.000 (0.000)
X,=1
X,=1
0.5
0.5
x,-1
Gender (X,)
Dose (Xx)
X,-0,Xy00
Xy-0
0.03
X,1,
X111
0.87
Bodyweight (Y)
Organ (X+)
Figure S2: A causal network representing developmental toxicology experiments, including body weight and its potential risk factors.
25

---

Causal Attribution for Continuous Outcomes
Table S14: Results of postICEs based on different evidence.
PostICE(Y2 x.Y Y < 27)
(21, rz) = (0.
(I1, 23))
(0,
(#1, = (1.0)
(I1, Wa)
= (1,1)
(x1, azg) (0,0)
0.00
1.59
-7.10
-2.77
(x1, =2)) = (0,1)
-1.42
0.00
-5.50
-3.35
(x1, ang) = (1,0)
8.81
10.51
0.00
5.58
(ot, .zg) (1.1)
2.84
4.46
-8.90
0.00
Table S15: Results of posterior causal estimands based on the various evidence for the NTP dataset.
Evidence X = x
Posterior causal estimands
(0,0,0)
(1,0,0)
(0, 1,0)
(1, 1,0)
(0,0, 1)
(1,0, 1)
(0,
(1, 1)
PostNDE(X, Y
2, < 27)
8.83
7.00
8.83
7.00
10.00
8.31
10.00
8.12
PostNDE(X, = Y 2, Y < 27)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
PostNDE(X; = ix,Y 27)
-1.43
-8,90
-1.43
-8.90
-6.30
-5.51
-6.30
-5.51
PostNIE(XX = Y
<
-0.00
0.00
-5.18
0.00
0.00
-5.51
0.00
-5.0
Post.NIE(X, = Y Ir, 27)
-0.05
-7.58
0.00
0.00
0.00
0.00
-6.30
-4.87
Post.NIE(Xs = Y xx,Y
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
PostTCE(X: Y 27)
8.23
7.00
3,65
7.00
10.00
2.80
10.00
2.82
PostTTEE((2 Y <7)
-0.05
-7.58
0.00
0.00
0.00
0.00
-6.30
-4.87
PostTCE((X; = Y 27)
-1.43
-8.90
-1.43
-8.90
-6.30
-5.51
-6.30
-5.51
26


Reviews:
[{"review_id": "0ZZ6eNAgMF", "reviewer": "Reviewer_31BM", "summary": "The submission describe a method for counterfactual analysis for continuous outcomes.\nIdentifiability conditions and results are given. Moreover based on minimization of a certain loss the authors propose an estimator and derive some theoretical properties.\nA practical example using simulated data show the results of the proposed methods, moreover a real problem is developed.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "---\n\n**Q1.** *The paper is well written and clear; it can be followed well. The originality is somehow restricted to the extension of a previous approach to continuous outcomes.*\n\n**A1.** We sincerely thank the reviewer for the positive and encouraging feedback. We truly appreciate your kind recognition of the manuscript’s clarity, the soundness of the proposed methodology, and the overall quality of both the theoretical and empirical components.\n\nWhile our work builds upon the framework introduced by Lu et al. (2023), our aim is not simply to apply it to continuous outcomes, but to extend it in several meaningful directions. In particular, we  \n- formulate a new class of posterior causal attribution estimands suitable for continuous responses,  \n- establish their identification under structured assumptions tailored to the continuous setting, and  \n- design a two-step estimation strategy that facilitates practical implementation and retrospective interpretation.\n\nWe are grateful for your comments and will revise the manuscript to better highlight these contributions and more clearly position our work in relation to the existing literature.\n\n---", "claims_and_evidence": "Yes clear", "methods_and_evaluation_criteria": "Yes they make sense", "theoretical_claims": "I did not check proofs in the supplementary, the theoretical claims seems sounds.", "experimental_designs_or_analyses": "The experimental design and analysis seems sound and correct.\nIt mainly follows the experiments of Lu et al. 2023", "relation_to_broader_scientific_literature": "The submission extend the work of Lu et al. To the case of continuous outcomes..\nThe setting, definition and experiments are mainly taken from Lu et. al", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "The paper is well written and clear, it can be followed well.\nThe originality is somehow restricted to the extension of a previous approach to continuous outcomes.", "other_comments_or_suggestions": "None", "questions_for_authors": "No", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rCYd7ekTk7", "reviewer": "Reviewer_HUHT", "summary": "This paper addresses the causal attribution problem for continuous outcome variables, an interesting and realistic scenario compared to binary outcomes. It introduces a set of posterior causal estimands to retrospectively analyze causal attribution:  PostTCE, PostNDE, PostNIE, PostICE. Under assumptions of sequential ignorability, monotonicity, and perfect positive rank, the authors demonstrate the identifiability of these estimands, and propose an efficient two-step estimation method based on quantile matching. The theories are validated through hypertention dataset and toxicity risk dataset.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the positive evaluation and constructive suggestions, which have been very helpful in improving the clarity and quality of our manuscript.\n \n---\n  **Q1.** Discuss relation to essential references.\n\n**A1.** Thank you for pointing out the related work by Schamberg et al. (2020) and Jung et al. (2022). Both are highly relevant to our goal of understanding and quantifying causal contributions, and—like our study—they offer principled frameworks for attributing the effects of multiple variables. While we share a common objective, our approach differs in several important ways.\n\nFirst, our method adopts a retrospective **causal attribution perspective**, conditioning on the observed outcome $Y$ to assess how each treatment component contributed to that specific realization. In contrast, Schamberg et al. and Jung et al. take an **effects-of-causes** perspective, focusing on how interventions influence the distribution of outcomes. Second, we introduce posterior causal estimands, defined based on observed treatment–outcome pairs, which allow for individual-level attribution. In comparison, Schamberg et al. rely on population-level, information-theoretic measures, while Jung et al. use do-Shapley values to quantify feature contributions under hypothetical interventions.\n\nWe will cite both papers and discuss these connections in the revised manuscript.\n\n---\n  **Q2.** *How likely are Assumptions 3.2 and 3.3 to be satisfied in general for continuous outcomes?*\n\n**A2.** Thank you for raising this interesting question. We believe that in many practical applications with temporal structures, progressive interventions, or latent rank-based heterogeneity, Assumptions 3.2 and 3.3 are generally plausible, or at least serve as reasonable modeling approximations.\n\n- **Assumption 3.2 (Monotonicity)** concerns the relationship among the treatment variables $X_1, \\ldots, X_p$, rather than between treatments and the continuous outcome. Specifically, it assumes that earlier treatment components have non-negative effects on subsequent ones. This assumption is likely to be satisfied in sequentially administered or progressive treatments, where prior treatments increase the chance or intensity of receiving later ones. For example, in multi-stage educational or medical programs, earlier stages (e.g., early screening or primary education) often facilitate participation in later ones (e.g., advanced therapies or higher education).\n\n- **Assumption 3.3 (Perfect Positive Rank)** is more abstract but remains plausible in many applications. It assumes that the individual-level outcome $Y$ is determined by a stable, unobserved factor in a strictly increasing way. This holds approximately when treatment primarily shifts the overall level of the outcome without changing individuals' relative ranking. For example,  if a person has the lowest blood pressure under treatment, he is also expected to have the lowest blood pressure without treatment.  Such rank-preserving behavior is commonly assumed in structural equation models, linear models, and additive models such as\n\n  $$\n  Y = \\alpha_0 + \\alpha_1 X_1 + \\cdots + \\alpha_p X_p + \\epsilon, \\quad \\text{or} \\quad Y = f(X_1, \\ldots, X_p) + \\epsilon,\n  $$\n  where $\\epsilon$ captures unobserved heterogeneity. \n\n\nWe will incorporate the above discussions and examples in the revised manuscript.\n\n---\n **Q3.** *How difficult is it to justify or test these assumptions?*\n\n**A3.** Thank you for raising this insightful point. While Assumptions 3.2 and 3.3 are not statistically testable in the strict sense, their plausibility can be assessed through empirical strategies and domain expertise.\n\n- **Assumption 3.2** is **falsifiable** under Assumption 3.1 (sequential ignorability), by verifying inequality constraints on observed conditional probabilities. For example, in a setting with binary treatment variables $X_1$ and $X_2$, monotonicity implies: $\n  \\operatorname{pr}(X_2 = 1\\mid X_1 = 0) \\leq \\operatorname{pr}(X_2 = 1\\mid X_1 = 1). $\nIf we observe in the data that: $\n  \\operatorname{pr}(X_2 = 1\\mid X_1 = 0) = 0.7, \\operatorname{pr}(X_2 = 1\\mid X_1 = 1) = 0.5, $\n  then monotonicity is violated, offering empirical evidence against the assumption.\n\n- For **Assumption 3.3**, we first suggest assessing, in real data, whether a common unobserved rank variable exists across different treatment conditions. This can serve as an empirical check of the assumption’s plausibility. In addition, one may fit linear or additive models, examine residual patterns, and evaluate model fit using criteria such as AIC, BIC, or R$^2$. If these models can explain a substantial proportion of the outcome variation, this would provide indirect support for the reasonableness of the assumption.\n\n\nWe will incorporate these discussions in the revised manuscript.\n\n---\n  **Q4.** *Line 106: “PostDNE” → “PostNDE”*\n\n**A4.** Thank you for this careful observation. We will correct it in the revised manuscript.\n\n---", "claims_and_evidence": "The claims regarding the identifiability of posterior causal estimands seem to be sound under stated assumptions.", "methods_and_evaluation_criteria": "The proposed evaluation methods and datasets are relevant and standard for the issue being discussed.", "theoretical_claims": "I mainly checked the proofs of Theorem 3.6, but not for other corollaries. The part that I read is correct.", "experimental_designs_or_analyses": "Yes, I checked the experiments designs. It adequately satisfied the assumptions and the analysis is valid,.", "relation_to_broader_scientific_literature": "This paper mainly focus on extending attribution analysis from binary outcomes to continuous outcomes. It builds upon the line of work by Pearl (2000), Dawid et al. (2014), and especially recent studies by Lu et al. (2023) and Li et al. (2023).", "essential_references_not_discussed": "There are related work discussing causal attribution problems in the context of Directed Acyclic Graphs (DAG) [1] and using Shapley Values [2].  It might be interesting to discuss the relation between them and posterior estimands.\n\n[1] Schamberg, Gabriel, William Chapman, Shang-Ping Xie, and Todd P. Coleman. \"Direct and indirect effects—An information theoretic perspective.\" *Entropy* 22, no. 8 (2020): 854.\n\n[2] Jung, Yonghan, Shiva Kasiviswanathan, Jin Tian, Dominik Janzing, Patrick Blöbaum, and Elias Bareinboim. \"On measuring causal contributions via do-interventions.\" In *International Conference on Machine Learning*, pp. 10476-10501. PMLR, 2022.", "other_strengths_and_weaknesses": "Overall, I think the paper addresses an important and underexplored issue in causal inference for continuous outcomes. There are certain parts in the proof that I didn't follow because posterio causal estimands is not my expertise, and the proofs are largely built upon [Lu 2023] and [Li 2023] and quite dense, but based on my current understanding, the claims seem to be sound.", "other_comments_or_suggestions": "Typo: L106 on the right column: PostDNE -> PostNDE", "questions_for_authors": "Given that the assumptions are similar to to that of [Lu 2023] and [Li 2023] for binary outcomes, (i) ihow likely are assumptions 3.2 and 3.3 going to be satisfied in general for continuous outcomes? My worry is that the assumptions would be too strong. (ii) how difficult is it to be justified/tested out?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PV0ISVS1s6", "reviewer": "Reviewer_mnpa", "summary": "This paper focuses on the causal attribution analysis, which answers retrospective questions like \"given that an outcome has occurred, how can we figure out how much each potential cause contributed to it?\" Most existing literature on this are introduced with binary outcome variables, which is not the case in real-world analysis.\n\nTherefore, in this paper, authors define a new set of \"posterior causal estimands\" for continuous outcomes. These include: posterior total causal effect, posterior natural direct effect, posterior natural indirect effect, and posterior intervention causal effect. The authors develop identifiability of these measures under certain assumptions. Further, for estimation of these quantities, a two-step approach is proposed.\n\n---\n\n## update after rebuttal:\n\nI thank the authors for the response. I keep my score of acceptance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the constructive feedback and the positive recommendation. We appreciate your recognition of our contribution and your valuable suggestions, which we will address in the revised manuscript.\n\n---\n\n **Q1.** *The assumptions are strong, especially for Assumption 3.2 (Monotonicity), so that there are “no prevention” relations. Are these assumptions in any way testable?*\n\n**A1.** Thank you for this insightful question. We agree that Assumption 3.2 is strong and may not always hold in practice. As discussed in Section 3.2 of the manuscript, although this assumption is not confirmed in the statistical sense—that is, it cannot be verified solely from observed data—it can be rejected under Assumption 3.1 (sequential ignorability), by checking whether certain inequality constraints on observed conditional probabilities hold in the data.\n\nTo illustrate, consider the case with two binary treatment variables $X_1$ and $X_2$. The monotonicity assumption implies that if $X_1 = 1$, then $X_2$ must be 1; whereas if $X_1 = 0$, then $X_2$ can be either 0 or 1. This leads to the testable implication:\n $\\mathrm{pr}(X_2 = 1 \\mid X_1 = 0) \\leq \\mathrm{pr}(X_2 = 1 \\mid X_1 = 1).$ \nIf we observe in the data that $\\mathrm{pr}(X_2 = 1 \\mid X_1 = 0) = 0.7$ and $\\mathrm{pr}(X_2 = 1 \\mid X_1 = 1) = 0.5$, this inequality is violated, providing empirical evidence against monotonicity.\n\nWe will incorporate these testable implications and further concrete illustrations into the revised manuscript.\n\n---\n\n**Q2.** *Except for listing the four posterior causal effect estimands, a discussion and practical guide on which to use under different scenarios is expected.*\n\n**A2.** Thank you for the helpful suggestion. To clarify the roles of the proposed posterior causal effect estimands, we have added a summary table in the revised manuscript. A simplified version is provided below:\n\n| Estimand | Definition | Interpretation |\n|----------|------------|----------------|\n| **PostTCE** | $E(Y_{X_k=1} - Y_{X_k=0} \\mid x, \\mathcal{E})$ | Total effect of switching $X_k$ from 0 to 1, given treatments $x$ and observed event $\\mathcal{E}$. |\n| **PostNDE** | $E(Y_{X_k=1, D_k(a_k, 0)} - Y_{X_k=0} \\mid x, \\mathcal{E})$ | Direct effect of changing $X_k$ while holding mediator fixed at its value under the reference value $X_k = 0$. |\n| **PostNIE** | $E(Y_{X_k=1} - Y_{X_k=1, D_k(a_k, 0)} \\mid x, \\mathcal{E})$ | Indirect effect  induced by changes in the intermediate variables $D_k$. |\n| **PostICE** | $E(Y_{X=x'} - Y \\mid x, \\mathcal{E})$ | Effect of changing the entire treatment vector from $x$ to $x'$. |\n| **ITE** | $Y_{x'} - Y_{x^*}$ | Individual-level contrast between two treatment configurations. |\n\nWe have also revised the illustrative example. Suppose an individual has exposure profile $x = (\\text{E}, \\text{D}, \\text{Hb}, \\text{HD}, \\text{CP})$ and observed outcome $Y > 140$. Given the observed evidence, the posterior estimands enable retrospective attribution in the following way:\n\n- PostTCE assesses the overall effect of a single factor (e.g., lack of exercise) on blood pressure.  \n- PostNDE identifies the portion of the effect that is direct.  \n- PostNIE captures the indirect path through variables like heart disease.  \n- PostICE quantifies the joint impact of multiple exposures (e.g., poor diet and no exercise).  \n- ITE compares the individual's outcome under two hypothetical exposure profiles.  \n\nWe will include this practical guidance in the revised manuscript to assist interpretation and application.\n\n---\n\n**Q3.** *An analysis to show the connections between the estimands, and their counterparts in the binary outcome case, is expected.*\n\n**A3.** Thank you for your helpful comments. In the revised manuscript, we provide a unified set of definitions for the five posterior causal effect estimands under both continuous and binary outcomes, and clarify their applicability and identifiability.  \n\nSpecifically, let $ Y^*$ $ denote the binary outcome. Following the framework of Lu et al. (2023), we define the binary versions of the posterior causal estimands as follows:\n\n- PostTCE:  $\\operatorname{postTCE}^*(X_k \\Rightarrow\\mathcal{E} \\mid x,Y^* = 1) = E(Y^*_{X_k=1} - Y^*_{X_k=0} \\mid x, Y^* = 1)$ \n- PostNDE:  $\\operatorname{postNDE}^*(X_k \\Rightarrow\\mathcal{E} \\mid x,Y^* = 1) = E(Y^*_{X_k=1, D_k(a_k, 0)} - Y^*_{X_k=0} \\mid x, Y^* = 1) $\n- PostNIE:  $\\operatorname{postNIE}^*(X_k \\Rightarrow \\mathcal{E} \\mid x, Y^* = 1) = E(Y^*_{X_k=1} - Y^*_{X_k=1, D_k(a_k, 0)} \\mid x, Y^* = 1) $\n- PostICE: $\\operatorname{postICE}^*(Y^*_{x'} \\mid x, Y^* = 1) = E(Y^*_{x'} - Y^* \\mid x, Y^* = 1) $\n\nWhile the definitions of $\\operatorname{postNDE}^*$ and $\\operatorname{postNIE}^*$ are conceptually consistent with their continuous-outcome counterparts, their identifiability under the binary setting has not yet been fully established in the literature. \n\nWe will provide a unified presentation and discuss this point in the revised manuscript.\n\n---", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. I read the measures' formulations, the assumptions, and the theoretical claims about identifiability. I skimmed through the proofs. The results seem correct to me but I cannot guarantee.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "/", "essential_references_not_discussed": "/", "other_strengths_and_weaknesses": "Strengths:\n\n1. The problem studied is necessary and crucial. It addresses a clear gap by extending causal attribution analysis to continuous outcomes, which is quite novel.\n\n2. The technical development is well-grounded and rigorous. The paper is also well structured, with formulations of quantities built first, and then the identifiability guarantees under certain assumptions. \n\n3. The real-world examples on hypertension and abnormal weights are interesting. One of my questions is: when only observational data is available, do we have any metrics to evaluate the quality of the estimated quantities, in terms of accuracy to the true quantities from counterfactual experiments?\n\n\nWeaknesses:\n\n1. The assumptions are strong, especially for the Assumption 3.2 (Monotonicity) so that there are \"no prevention\" relations. Are these assumptions in any way testable?\n\n2. Except for listing such four posterior causal effect estimands, a discussion and practical guide on which to use under different scenarios is expected. On top of that, an analysis to showcase the connections between them, and also to their counterparts in the binary outcome case, are expected.", "other_comments_or_suggestions": "/", "questions_for_authors": "/", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ODRTKIB5Tu", "reviewer": "Reviewer_ZTX9", "summary": "Previous studies have focused on attribution problems for binary outcomes, but binarizing continuous outcomes can lead to information loss or bias. To address this, the study introduces posterior causal estimands for evaluating multiple correlated causes in continuous outcomes. These estimands include posterior intervention effects, posterior total causal effects, and posterior natural direct effects. Under assumptions like sequential ignorability, monotonicity, and perfect positive rank, the study establishes the identifiability of these estimands and derives corresponding identification equations. A simple yet effective estimation procedure is proposed, with theoretical guarantees on asymptotic properties. The method is demonstrated through an artificial hypertension example and a real developmental toxicity dataset.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "---\nWe sincerely thank the reviewer for the thoughtful feedback and strong accept recommendation. Your recognition of our contribution is very encouraging, and your suggestions are highly valuable for improving the paper.\n\n\n\n---\n**Q1: Path-Specific Effects and Pearl (2001)**   \n\n**A1:** Thank you for the interesting comments. Following your suggestion, we revisited Pearl (2001) and examined how path-specific effects can be incorporated into our framework.\n\n \nGiven a causal graph $G$ over $(X, Y)$ and a subgraph $l \\subseteq G$ representing the path(s) of interest, our goal is to isolate the effect of $X_k$ on $Y$ transmitted specifically through path $l$. Let $\\mathrm{Pa}_j$ denote the set of parent nodes of $X_j$, which we decompose as $\\mathrm{Pa}_j = \\mathrm{Pa}_j(l) \\cup \\mathrm{Pa}_j(\\bar{l})$, where $\\mathrm{Pa}_j(l)$ includes parents along path $l$, and $\\mathrm{Pa}_j(\\bar{l})$ includes parents outside of it. To activate only path $l$, we fix the non-path parents $\\mathrm{Pa}_j(\\bar{l})$ to their counterfactual values under the reference setting $X_k = 0.$   \n\n\nThe $l$-specific counterfactual outcome is defined as $Y^l_{X_k = x_k} = (X_{p+1})^l_{X_k = x_k}$, computed under the modified system with only path $l$ active. When $X_k = 0$, this reduces to the usual counterfactual: $Y^l_{X_k = 0} = Y_{X_k = 0}$, since all variables follow the reference path. \nWe define the posterior path-specific effect as  \n$$\n\\mathrm{postPSE}(X_k \\Rightarrow Y; x, \\mathcal{E}, l) = E\\left( Y^l_{X_k = 1} - Y_{X_k= 0} \\mid x, \\mathcal{E}\\right).\n$$\n\nThe second term is identifiable via Theorem 3.6. Identifying the first term requires a finer strategy that accounts for node-level structures along path $l$ and their interaction with the evidence $ \\mathcal{E} $. We leave this technical development for future work.\n\n\n---\n\n**Q2: Monotonicity with Continuous Outcomes**   \n\n**A2:**  Thank you for the insightful question. Our intention was not to suggest that monotonicity is inappropriate for continuous outcomes, but rather to note that a strong condition such as $Y_x\\leq Y_{x'}$ for all $x \\preceq x'$ may be restrictive in certain settings. To illustrate this, consider a linear model:\n$$Y = \\alpha_0 + \\alpha_1 X_1 + \\alpha_2 X_2 + \\cdots + \\alpha_p X_p + \\epsilon.$$\nIf we assume $Y_x \\leq Y_{x'}$ whenever $x \\preceq x'$, then this implies that increasing any component of exposure vairables $x$ (with others held fixed or increased) must not decrease $Y$. In a linear model, this requirement implies $\\alpha_j \\geq 0$ for all $j = 1, \\ldots, p$, which may limit model flexibility. \n\nWe hence adopt the perfect rank assumption as a more general alternative. It includes linear models and permits identification without requiring all coefficients to be non-negative.\n\nWe acknowledge that this point was not clearly explained and will revise the manuscript to clarify it. \n\n---\n\n**Q3: Relationship between Assumptions 3.2 and 3.3**   \n\n**A3:**  Thank you for this insightful question. Assumption 3.3 pertains solely to the outcome variable $Y$ (denoted as $W_{p+1}$), whereas Assumption 3.2 pertains to the treatment variables $W_2, \\ldots, W_p$. Since they apply to different variable sets, Assumption 3.3 does not imply Assumption 3.2. We will make this distinction clearer in the revised text.\n\n---\n\n**Q4: Required Assumptions for Theorem 3.6**   \n\n**A4:**  Thank you for this careful and insightful observation.  You are correct—our intention was to show that the identification expression of postNDE, postNIE, and postTCE in the *first part* of Theorem 3.6 relies only on Assumptions 3.1 and 3.2. However, identification of the counterfactual term $\\mathbb{E}(Y_{x_k^\\ast, d_k^\\ast} \\mid x, \\mathcal{E})$ in the *second part* of the theorem does require Assumption 3.3 via Lemma 3.4.\n\nWe appreciate your comments and will revise the statement of Theorem 3.6 to explicitly include Assumption 3.3.\n\n---\n\n**Q5: Clarification on Lemma 4.2**\n\n**A5:**  Thank you for your thoughtful question. The properties of the function $\\rho_{x\\to x'}(\\cdot;y)$ serve important theoretical roles:\n\n- **Continuous differentiability** ensures that gradient-based optimization is feasible and reliable.\n- **Weak convexity** guarantees the existence of at least one global minimizer.\n- **Strict convexity** (within the interior of the support $S_{Y_{x'}}$  guarantees uniqueness of the minimizer, which ensures that the counterfactual mapping $\\phi_{x \\to x'}(y)$ is well-defined and stable.\n\nTogether, these properties underpin the theoretical soundness of our estimation method and justify the nonparametric approach adopted. We will elaborate on this in the revised manuscript.\n\n---\n\n**Q6: On the Support of the Conditioning Set**   \n\n**A6:**  Thank you for the helpful comment!  Yes, we require $\\mathrm{pr}(X = x, \\mathcal{E}) \\neq 0$ to ensure that the conditional distributions are well-defined. We have made this assumption explicit in the revised manuscript for clarity.\n\n---", "claims_and_evidence": "Yes, the claims are very vlear.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I have checked the correctness of theoretical claims. They are all correct.", "experimental_designs_or_analyses": "It is advisable to include confidence intervals.", "relation_to_broader_scientific_literature": "As the authors indicate, their findings have broad applications in social science, health risk assessment, legal contexts, and explainable AI. They showed in Introduction section.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "This paper is well-structured. They address a naturally arising causal question that researchers have thus far not explored.", "other_comments_or_suggestions": "You have examined posterior natural direct and indirect effects for the variables (X_{k+1},...,X_p). Can you also identify the path‐specific effect, as described by Pearl (2001)? (May be future work)\n\nJudea Pearl. 2001. Direct and indirect effects. In Proceedings of the Seventeenth conference on Uncertainty in artificial intelligence (UAI'01). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 411–420.", "questions_for_authors": "1. In line 165, you wrote, “However, this monotonic relationship may not be applicable when the outcome variable is continuous.” However, I believe that the monotonicity assumption is indeed directly applicable to continuous outcomes. Are you suggesting that posterior causal estimands cannot be identified through this monotonicity assumption?\n\n2. Does Assumption 3.3 imply Assumption 3.2? It seems Assumption 3.3 imposes strict monotonicity, whereas Assumption 3.2 only requires non-strict monotonicity.\n\n3. Do you requires all Assumptions 3.1, 3.2, and 3.3 for identifying postNDE, postNIE, and postTCE? The statement Theorem 3.6 may look that postNDE, postNIE, and postTCE are idenrtified from Assumptions 3.1, 3.2.\n\n4. I would like additional clarification regarding Lemma 4.2. Specifically, what does each property guarantee?\n\n5. Do you presume that the probability of the evidence $pr(X=x,{\\cal E})\\ne 0$?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "No."}}]

Author rebuttal:

Meta-review: The paper studies the problem of causal attribution with continuous outcome variables. The authors introduce several posterior causal estimands and present conditions for identifiability, along with an effective estimation procedure. The problem addressed is novel, and the analysis is built on a solid theoretical foundation.
Decision: Accept (spotlight poster)