Original: Causal Inference with Non-IID Data using Linear Graphical Models
Revised: Causal Inference with Non-IID Data using Linear Graphical Models

Original: Anonymous Author(s)

Original: AffiliationAddress

Original: Abstract
Revised: Abstract

Original: Traditional causal inference techniques assume data are independent and identicallydistributed (IID) and thus ignores interactions among units.
Revised: Traditional causal inference techniques assume data are independent and identically distributed (IID) and thus ignores interactions among units.

Original: However, a unit’streatment may affect another unit’s outcome (interference), a unit’s treatment maybe correlated with another unit’s outcome or a unit’s treatment and outcome may bespuriously correlated through another unit.
Revised: However, a unit’s treatment may affect another unit’s outcome (interference), a unit’s treatment may be correlated with another unit’s outcome, or a unit’s treatment and outcome may be spuriously correlated through another unit.

Original: To capture such nuances, we model thedata generating process using causal graphs and conduct a systematic analysis of thebias caused by different types of interactions when computing causal effects.
Revised: To capture such nuances, we model the data generating process using causal graphs and conduct a systematic analysis of the bias caused by different types of interactions when computing causal effects.

Original: Wederive theorems to detect and quantify the interaction bias , and derive conditionsunder which it is safe to ignore interactions.
Revised: We derive theorems to detect and quantify the interaction bias , and derive conditions under which it is safe to ignore interactions.

Original: Put differently, we present conditionsunder which causal effects can be computed with negligible bias by assuming thatsamples are IID.
Revised: Put differently, we present conditions under which causal effects can be computed with negligible bias by assuming that samples are IID.

Original: Furthermore, we develop a method to eliminate bias in caseswhere blindly assuming IID is expected to yield a significantly biased estimate.
Revised: Furthermore, we develop a method to eliminate bias in cases where blindly assuming IID is expected to yield a significantly biased estimate.

Original: Finally, we test the coverage and performance of our methods through simulations.
Revised: Finally, we test the coverage and performance of our methods through simulations.

Original:  Introduction
Revised:  Introduction

Original: Motivating Example Suppose we are interested in studying the effectiveness of Covid-19 vaccines.
Revised: Motivating Example Suppose we are interested in studying the effectiveness of Covid-19 vaccines.

Original: Specifically, we are interested in the causal effect of vaccine doses, V , on the severity of sickness S .
Revised: Specifically, we are interested in the causal effect of vaccine doses, V , on the severity of sickness S .

Original: A naive method would be building a causal model on V , S , and other related factors, and estimatingthe causal effect of V on S using available data.
Revised: A naive method would be building a causal model on V , S , and other related factors, and estimating the causal effect of V on S using available data.

Original: However, this method may result in biased estimationprimarily because traditional causal inference techniques assume that attributes of all units in thesample are independent and identically distributed (IID)[Rubin, 1978], which does not hold true inthe pandemic setting since units are not isolated from each other.
Revised: However, this method may result in biased estimation primarily because traditional causal inference techniques assume that attributes of all units in the sample are independent and identically distributed (IID)[Rubin, 1978], which does not hold true in the pandemic setting since units are not isolated from each other.

Original: We exemplify below few instancesof this problem that violate IID (Eyre et al.
Revised: We exemplify below few instances of this problem that violate IID (Eyre et al.

Original: Case 1: The vaccination V of a unit i , ( V i ) ,decreases their viral load, L i , which in turn decreases the transmission rate of the virus, and hencedecreases the severity of sickness S of another unit j , ( S j ) , who comes into contact with i .
Revised: Case 1: The vaccination V of a unit i , ( V i ) , decreases their viral load, L i , which in turn decreases the transmission rate of the virus, and hence decreases the severity of sickness S of another unit j , ( S j ) , who comes into contact with i .

Original: V icausally affects S j .
Revised: V i causally affects S j .

Original: Case 2: V i is affected by the area
Revised: Case 2: V i is affected by the area

Original: A that i lives in, and a contact j who lives invaccine deprived areas and areas with a higher incidence of Covid-19 infection is more likely to getsick.
Revised: A that i lives in, and a contact j who lives in vaccine deprived areas and areas with a higher incidence of Covid-19 infection is more likely to get sick.

Original: V i and S j are confounded.
Revised: V i and S j are confounded.

Original: Case 3: S i , determines whether or not i is quarantined and thusaffects whether i transmits the disease to another unit
Revised: Case 3: S i , determines whether or not i is quarantined and thus affects whether i transmits the disease to another unit

Original: j . S
Revised: j . S

Original: i causally affects S j .
Revised: i causally affects S j .

Original: Such interactions between units plague both observational and experimental studies.
Revised: Such interactions between units plague both observational and experimental studies.

Original: If the latter isperformed in a controlled environment where subjects are isolated from each other, the results wouldnot be valid for the target environment, where subject affect each other, and vice versa.
Revised: If the latter is performed in a controlled environment where subjects are isolated from each other, the results would not be valid for the target environment, where subjects affect each other, and vice versa.

Original: Modeling Interactions A line of existing work that analyzes interactions between units is interference[Cox, 1958].
Revised: Modeling Interactions A line of existing work that analyzes interactions between units is interference [Cox, 1958].

Original: Interference is the phenomenon in which treatment of unit i ( V i ) causally affects theoutcome, S j , of another unit j .
Revised: Interference is the phenomenon in which treatment of unit i ( V i ) causally affects the outcome, S j , of another unit j .

Original: In almost all existing literature this is interpreted as there existing acausal pathway from V i to S j .
Revised: In almost all existing literature this is interpreted as there existing a causal pathway from V i to S j .

Original: Case-1 above is a typical example.
Revised: Case-1 above is a typical example.

Original: Clearly, ignoring unit interactionswhile computing causal effects would result in a biased estimate.
Revised: Clearly, ignoring unit interactions while computing causal effects would result in a biased estimate.

Original: However, we note that interference is not the only type of interaction between units that can yield biased estimates.
Revised: However, we note that interference is not the only type of interaction between units that can yield biased estimates.

Original: For example, in
Revised: For example, in

Original: Case-2 V i and S j are confounded and V i is not a cause of S j .
Revised: Case-2 V i and S j are confounded and V i is not a cause of S j .

Original: Another example is an instance whereunit i ′ s treatment affects their own outcome through an attribute of unit j i.e., V i →
Revised: Another example is an instance where unit i ′ s treatment affects their own outcome through an attribute of unit j i.e., V i →

Original: W j → S i , forsome W j / ∈ { S j , V j } .
Revised: W j → S i , for some W j / ∈ { S j , V j } .

Original: In both these cases units interact with each other in a way that might bias theestimation of causal effects although they may not typically be classified as interference.
Revised: In both these cases units interact with each other in a way that might bias the estimation of causal effects although they may not typically be classified as interference.

Original: In spite ofthe prevalence of such interactions in applications related to health care, infectious diseases, socialnetworks and ad placements , they have not been systematically studied.
Revised: In spite of the prevalence of such interactions in applications related to health care, infectious diseases, social networks and ad placements, they have not been systematically studied.

Original: It is this deficiency that thispaper attempts to overcome.
Revised: It is this deficiency that this paper attempts to overcome.

Original: Questions addressed The scenario exemplified above raises several questions regarding the computation of causal effects given non-IID data.
Revised: Questions addressed The scenario exemplified above raises several questions regarding the computation of causal effects given non-IID data.

Original: How can we model different types of interactionsamong units in the population?
Revised: How can we model different types of interactions among units in the population?

Original: Under what conditions can we safely ignore unit interactions with theguarantee that assuming IID (and applying existing estimation techniques) will result in negligiblebias?
Revised: Under what conditions can we safely ignore unit interactions with the guarantee that assuming IID (and applying existing estimation techniques) will result in negligible bias?

Original: If assuming IID would yield a biased estimate, then how can we get rid of this bias?
Revised: If assuming IID would yield a biased estimate, then how can we get rid of this bias?

Original: Our contributions In this work we study causal inference in the presence of interactions amongsamples using linear models due to the convenience they offer with regard to path analysis.
Revised: Our contributions In this work we study causal inference in the presence of interactions among samples using linear models due to the convenience they offer with regard to path analysis.

Original: Wedevelop interaction models that portray different types of interactions among units and conduct asystematic analysis of the bias caused by different types of interactions.
Revised: We develop interaction models that portray different types of interactions among units and conduct a systematic analysis of the bias caused by different types of interactions.

Original: We derive theorems todetect and quantify the interaction bias.
Revised: We derive theorems to detect and quantify the interaction bias.

Original: We derive conditions under which it is safe to ignore unitinteractions when computing the average causal effects.
Revised: We derive conditions under which it is safe to ignore unit interactions when computing the average causal effects.

Original: Furthermore, we develop a method tocompute an unbiased estimate of causal effect in cases where blindly assuming IID is expected toyield a significant bias.
Revised: Furthermore, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.

Original: Finally, we corroborate our findings through simulation studies.
Revised: Finally, we corroborate our findings through simulation studies.

Revised: Summary of results in words Blindly assuming that data are IID when in fact they are not, can potentially bias the outcome of a research study.

Revised: Such bias can occur for the query: causal effect of treatment on outcome, when there is an open (not necessarily directed) path from the treatment of unit i to the outcome of unit j and/or to the outcome of unit i itself such that an intermediate node on the path belongs to unit j .

Original: Preliminaries
Revised: The formula in Theorem 1 quantifies the bias. Furthermore, only the two types of interaction structures previously mentioned can induce bias. In the presence of such bias inducing structures it is still possible to compute an unbiased estimate by selecting a subset of samples B such that no biasing paths exist in the interaction graph corresponding to samples in B (Theorem 2). More importantly, such a debiasing procedure does not require the selection of IID samples and may contain interactions among them.

Revised: Such a debiasing procedure can also be done in polynomial time (Algorithm 1).

Revised: Empirical analysis: We randomly generate interaction models and show that the bias can be huge if IID is wrongly assumed on non-IID data.

Revised: The debiasing method in this paper yields an unbiased estimate.

Revised: We further show that, as the number of IID samples increases, and as the strengths of bias structures decrease, the overall interaction bias decreases.

Revised:  Preliminaries

Original: Independent and Identically Distributed (IID)
Revised: Independent and Identically Distributed (IID)

Original: If X 1 , . .
Revised: If X 1 , . .

Original: , X n are independent and each has thesame marginal distribution with CDF F , we say that X 1 , . . .
Revised: , X n are independent and each has the same marginal distribution with CDF F , we say that X 1 , . . .

Original: X n are IID (independent and identicallydistributed)
Revised: X n are IID (independent and identically distributed)

Original: [Wasserman, 2013].
Revised: [Wasserman, 2013].

Original: For the sake of simplicity, we use X is IID to refer to all the units of
Revised: For the sake of simplicity, we use X is IID to refer to all the units of

Original: X , X 1 , . . .
Revised: X , X 1 , . . .

Original: , X n , being IID.
Revised: , X n , being IID.

Original: A dataset is IID if all variables in it are IID.
Revised: A dataset is IID if all variables in it are IID.

Original: Linear Causal Models A traditional linear causal model is also known as a linear structural causalmodel (SCM)
Revised: Linear Causal Models A traditional linear causal model is also known as a linear structural causal model (SCM)

Original: [Brito, 2004, Pearl, 2009, Chen and Pearl, 2014].
Revised: [Brito, 2004, Pearl, 2009, Chen and Pearl, 2014].

Original: The edge coefficients on the causal
Revised: The edge coefficients on the causal

Original: DAG represent direct effects.
Revised: DAG represent direct effects.

Original: An open path is collider-free, i.e., there are no head-to-head arrowson this path.
Revised: An open path is collider-free, i.e., there are no head-to-head arrows on this path.

Original: Note that if there exists an open path from W i to V j , it implies W i ̸⊥⊥ V j .
Revised: Note that if there exists an open path from W i to V j , it implies W i ̸⊥⊥ V j .

Original: The value‘ V al ( p ) ’ of an open path p in a linear model is defined as the product of the edge coefficients on p .
Revised: The value ‘ V al ( p ) ’ of an open path p in a linear model is defined as the product of the edge coefficients on p .

Original: The root of an open path p is defined as the variable on p that is the ancestor of all variables on p .
Revised: The root of an open path p is defined as the variable on p that is the ancestor of all variables on p .

Original: Average Causal Effects
Revised: Average Causal Effects

Original: In this work the query we are primarily interested in generalizing to thenon-IID case is the Average Causal Effect (ACE), also named as the Average Treatment Effect (ATE)[Rubin, 1977, Holland, 1988].
Revised: In this work the query we are primarily interested in generalizing to the non-IID case is the Average Causal Effect (ACE), also named as the Average Treatment Effect (ATE) [Rubin, 1977, Holland, 1988].

Original: For consistency, we use ACE to refer to both.
Revised: For consistency, we use ACE to refer to both.

Original: Given a causal model M , the average causal effect (ACE) of X = t vs X = c ( t and c are constants) on Y for k unitsis defined as ACE XY = 1 k (cid:80) i
Revised: Given a causal model M , the average causal effect (ACE) of X = t vs X = c ( t and c are constants) on Y for k units is defined as ACE XY = 1 k (cid:80) i

Original: ( Y i ( X i = t) − Y 
Revised: ( Y iX i = t − Y iX

Original: i( X i = c )) .
Revised: i = c ) .

Original: ACE is defined under the assumptionthat Y i depends  on factors of unit i and X i  Holland [1988].
Revised: ACE is defined under the assumption that Y i depends only on factors of unit i (including X i ) Holland [1988].

Original: Without loss of generality, we assumet = c + 1 1 .
Revised: Without loss of generality, we assume t = c + 1 1 .

Original: In linear models, ACE of X on Y can be identified as β Y X , the linear regression slopeof Y on X , if there is no backdoor (non-directed open paths) between X and Y [Pearl et al., 2016, Pearl, 2017].
Revised: In linear models, ACE of X on Y can be identified as β Y X , the linear regression slope of Y on X , if there is no backdoor (non-directed open paths) between X and Y [Pearl et al., 2016, Pearl, 2017].

Original:  Graphical Modeling of Interactions 
Revised:  Graphical Modeling of Interactions 3.1 Interaction models

Original: 3.1 Interaction models

Original: In this section, we define a graphical model derived from traditional causal models M = <
Revised: In this section, we define a graphical model derived from traditional causal models M = <

Original: G, S >(Pearl [2009], definition 7.1.1).
Revised: G, S > (Pearl [2009], definition 7.1.1).

Original: G is the causal graph and S is the set of structural equations ofvariables.
Revised: G is the causal graph and S is the set of structural equations of variables.

Original: We refer to the variables in a traditional causal model as generic variables .
Revised: We refer to the variables in a traditional causal model as generic variables .

Original: X, C, Y in
Revised: X, C, Y in

Original: Figure 1 are generic variables.
Revised: Figure 1 are generic variables.

Original: An explicit variable is similar to a generic variable except that itrepresents an attribute/event of one specific unit (or sample or individual).
Revised: An explicit variable is similar to a generic variable except that it represents an attribute/event of one specific unit (or sample or individual).

Original: For example, “treatment( X ) ” is a generic variable, and “the treatment of unit i ( X i ) ” is an explicit variable.
Revised: For example, “treatment ( X ) ” is a generic variable, and “the treatment of unit i ( X i ) ” is an explicit variable.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: X 1 = U X(1) Y 1 = X 1 + Y 2 + 3 C 1 + U Y(2) Y 2 = 2 X 2 − C 1 + X 3 + U Y(3) . . .
Revised: [Equation]

Original: Definition 1 (Interaction model M ∗ ( G ∗ , S ∗ ) ) .
Revised: Definition 1 (Interaction model M ∗ ( G ∗ , S ∗ ) ) .

Original: An interaction model, M ∗ ( G ∗ , S ∗ ) , is a causal modelwhere G ∗ is the interaction network and S ∗ is the set of structural equations defining the datagenerating process of the observed explicit variables.
Revised: An interaction model, M ∗ ( G ∗ , S ∗ ) , is a causal model where G ∗ is the interaction network and S ∗ is the set of structural equations defining the data generating process of the observed explicit variables.

Original: An interaction network, G ∗ , is a directedacyclic graph with each node representing an explicit variable and each directed edge A i
Revised: An interaction network, G ∗ , is a directed acyclic graph with each node representing an explicit variable and each directed edge A i

Original: → B jrepresenting A i causes B j .
Revised: → B j representing A i causes B j .

Original: An example of interaction model M ∗ ( G ∗ , S ∗ ) , is the interaction network, G ∗ , portrayed in Figure 2and the structural equations S ∗ (part of) specified beside it; U V i denotes the unobserved exogenouserror of an explicit variable V i .
Revised: An example of interaction model M ∗ ( G ∗ , S ∗ ) , is the interaction network, G ∗ , portrayed in Figureand the structural equations S ∗ (part of) specified beside it; U V i denotes the unobserved exogenous error of an explicit variable V i .

Original: Observe that interaction networks allow edges between explicitvariables of the same unit (e.g., X 1 → Y 1 ), as well as two distinct units (e.g., C 1 → Y 2 ).
Revised: Observe that interaction networks allow edges between explicit variables of the same unit (e.g., X 1 → Y 1 ), as well as two distinct units (e.g., C 1 → Y 2 ).

Original: We are now ready to define an isolated interaction model for an interaction model M ∗ .
Revised: We are now ready to define an isolated interaction model for an interaction model M ∗ .

Original: It is the “ideal”model constructed from M ∗ by eliminating all interactions between units.
Revised: It is the “ideal” model constructed from M ∗ by eliminating all interactions between units.

Original: Definition 2 (Isolated interaction model IM ∗ ( IG ∗ , IS ∗ ) ) .
Revised: Definition 2 (Isolated interaction model IM ∗ ( IG ∗ , IS ∗ ) ) .

Original: IM ∗ ( IG ∗ , IS ∗ ) is the Isolated interaction model with respect to an interaction model M ∗ ( G ∗ , S ∗ ) if it satisfies the following conditions:
Revised: IM ∗ ( IG ∗ , IS ∗ ) is the Isolated interaction model of an interaction model M ∗ ( G ∗ , S ∗ ) if IM satisfies the following conditions:

Original: IG ∗ =
Revised: IG ∗ =

Original: G ′ where G ′ is the graph obtained by removing from G ∗ all edges A i
Revised: G ′ where G ′ is the graph obtained by removing from G ∗ all edges A i

Original: → B j , i ̸
Revised: → B j , i ̸

Original: = j ,2.
Revised: = j , 2.

Original: IS ∗ =
Revised: IS ∗ =

Original: S ′ where S ′ is the set of equations obtained by removing from each equation
Revised: S ′ where S ′ is the set of equations obtained by removing from each equation

Original: X i = f
Revised: X i = f

Original: ( parents ( X i ))  in S ∗ all terms containing any Y j , ∀ j ̸
Revised: ( Pa ( X i )) 2 in S ∗ all terms containing any Y j , ∀ j ̸

Original: = i .
Revised: = i .

Original: For example, the interaction model M ∗ ( G ∗ , S ∗ ) has Figure 2 as G ∗ , and Equations (1-3) as part of
Revised: For example, the interaction model M ∗ ( G ∗ , S ∗ ) has Figure 2 as G ∗ , and Equations (1-3) as part of

Original: S ∗ .
Revised: S ∗ .

Original: The isolated model for M ∗ is denoted IM ∗ ( IG ∗ , IS ∗ ) .
Revised: The isolated model for M ∗ is denoted IM ∗ ( IG ∗ , IS ∗ ) .

Original: IG ∗ is given in Figure 3 below.
Revised: IG ∗ is given in Figure 3 below.

Original: And
Revised: And

Original: IS ∗ for Equations (1-3) are given by Equations (4-6).[Equation] [Equation]
Revised: IS ∗ for Equations (1-3) are given by Equations (4-6).

Original: 3.2 Symmetry Assumptions
Revised: 3.2 Symmetry Assumptions

Original: In real-world applications, we will have at our disposal limited (usually just one) observationscorresponding to a unit which in turn will make it hard to draw useful conclusions if the modelis completely arbitrary.
Revised: In real-world applications, we will have at our disposal limited (usually just one) observations corresponding to a unit which in turn will make it hard to draw useful conclusions if the model ( X [Figure] [Equation]

Original:  In traditional causal inference techniques this is not a problem since theyassume IID.
Revised: [Equation] is completely arbitrary. In traditional causal inference techniques this is not a problem since they assume IID, which is assuming for each variable, the distribution is the same and independent for all units.

Original: While we do not make strong assumptions such as IID, we need to make certain weakersymmetry restrictions (definitions 3, 4), in order to quantify bias and identify ACE .
Revised: While we do not make strong assumptions such as IID, we need to make certain weaker symmetry restrictions (definitions 3, 4), in order to quantify bias and identify ACE .

Revised: We only require some of the variables are IID instead of all.

Original: Definition 3 (Balanced interaction model M ∗ ( G ∗ , S ∗ ) ) .
Revised: Definition 3 (Balanced interaction model M ∗ ( G ∗ , S ∗ ) ) .

Original:  M ∗ is a balanced interaction model iftheisolated interaction model IM ∗of M ∗ has the same sub-model ( IM ∗ i ( IG ∗ i , IS ∗ i ) ) for every unit i .
Revised: Let M ∗ ( G ∗ , S ∗ ) be an interaction model with isolated model IM ∗ . M ∗ is a balanced interaction model if IM ∗ has the same unit-model ( IM ∗ i ( IG ∗ i , IS ∗ i ) ) for every unit i .

Original: Let G ∗ be the graph in Figure 2 and S ∗ be the set of equations (1-3) corresponding to M ∗ ( G ∗ , S ∗ ) .
Revised: Let G ∗ be the graph in Figure 2 and S ∗ be the set of equations (1-3) corresponding to M ∗ ( G ∗ , S ∗ ) .

Original: Let IG ∗ in Figure 3 be the graph and IS ∗ be the equations (4), (5) and (6) that correspond to IM ∗  .
Revised: IG ∗ in Figure 3 is the graph and IS ∗ are the equations (4), (5) and (6) that correspond to IM ∗ , which is the isolated model of M ∗ .

Original: The sub-graphs for unit 1 is different from unit 2 .
Revised: The unit-graph for unit 1 is different from unit 2 .

Original: Also, the structural equations for Y 1 and Y 2 of 
Revised: Also, the structural equations for Y 1 and Y 2 of the isolated interaction model (Equations (5) and (6)) are different. Hence, M ∗ is not a balanced interaction model.

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: X satisfies ASDC.

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original:  theisolated interaction model (Equations (5) and (6)) are different. Hence, M ∗ is not a balancedinteraction model. The interaction model M ∗ is balanced where G ∗ is the graph in Figure 4(a), and S ∗ is the set ofequations given in Figure 4(c).
Revised: For another example, the interaction model M ∗ is balanced where G ∗ is the graph in Figure 4(a), and S ∗ is the set of equations given in Figure 4(c).

Revised: Remark 1.

Original: A balanced interaction model M ∗ does not imply that data generated by it are IID.
Revised: Note that a balanced interaction model M ∗ does not imply that data generated by it are IID.

Revised: Being balanced only requires all units share the same causal relationships within each unit itself, but permits interactions and effects from other units.

Original: For example,the parents of explicit variables Y i and Y j , i ̸
Revised: For example, the parents of explicit variables Y i and Y j , i ̸

Original: =
Revised: =

Original: j can be different in G ∗ i.e., Y i can be caused by aset of variables S k corresponding to unit, k , and Y j can be caused by a distinct set of variables T k .
Revised: j can be different in G ∗ i.e., Y i can be caused by a set of variables S k corresponding to unit, k , and Y j can be caused by a distinct set of variables T k .

Original: However, for M ∗ to be balanced it is required that for all distinct units i and j , all Y i have the samerelationship with i ’s explicit variables as Y j with j ’s variables.
Revised: However, for M ∗ to be balanced it is required that for all distinct units i and j , all Y i have the same relationship with i ’s explicit variables as Y j with j ’s variables.

Original: We further note that if M ∗∗ is balanced then all the sub-models IM ∗ i ( IG ∗ i , IS ∗ i ) in definition 3 areidentical (with no edges between IM ∗ i and IM ∗ j ), and can be succinctly represented by a (single)causal model M ( G, S ) where G and S can be constructed from any IG ∗ i and IS ∗ i by replacingexplicit variables with generic variables.
Revised: We further note that if M ∗∗ is balanced then all the unit-models IM ∗ i ( IG ∗ i , IS ∗ i ) in definition 3 are identical (with no edges between IG ∗ i and IG ∗ j ), and can be succinctly represented by a (single) causal model M ( G, S ) where G and S can be constructed from any IG ∗ i and IS ∗ i by replacing explicit variables with generic variables.

Original: In addition to the  isolated components being the same, it would be helpful if we also have symmetricalassumptions on the underlying distributions of specific sets of variables.
Revised: In addition to the assumption that the isolated components being the same, it would be helpful if we also have symmetrical assumptions on the underlying distributions of specific sets of variables.

Original: For example, it is reasonableto assume all units’ treatments have the same distribution, i.e., for any treatment X
Revised: For example, it is reasonable to assume all units’ treatments have the same distribution, i.e., for any treatment X

Original: = x , all unitshave an equal chance of getting the treatment X
Revised: = x , all units have an equal chance of getting the treatment X

Original: = x .
Revised: = x .

Original: Definition 4 (Ancestral same-distribution condition (ASDC)) .
Revised: Definition 4 (Ancestral same-distribution condition (ASDC)) .

Original: In the interaction network G ∗ of abalanced interaction model, generic variable W to satisfies the ancestral same-distribution condition(ASDC) if for any unit i , 1) Pa ( W i ) satisfies ASDC, and 2)
Revised: In the interaction network G ∗ a balanced interaction model, generic variable W to satisfies the ancestral same-distribution condition (ASDC) if for all unit i , 1) Pa ( W i ) satisfies ASDC, and 2)

Original: Pa ( W i ) ⊆
Revised: Pa ( W i ) ⊆

Original: V ( i ) , and 3) for any differentunit j ̸
Revised: V ( i ) , and 3) for any different unit j ̸

Original: = i , Pa ( W i ) and Pa ( W j ) have the same set of generic variables, and their exogenous errors
Revised: = i , Pa ( W i ) and Pa ( W j ) have the same set of generic variables, and their exogenous errors

Original: U W i
Revised: U W i

Original: and U W j have the same distribution.
Revised: and U W j have the same distribution.

Original: (When i=j, the condition is automatically satisfied.)
Revised: (When i=j, the condition is automatically satisfied.)

Original: Pa ( X )denotes the parents of X in G ∗ .

Original: For example, in Figures 4(a) and 4(b), X satisfies ASDC in the former (assuming the condition onexogenous errors is satisfied) but not in the latter since in the latter Pa ( X 1 ) ̸ =
Revised: For example, in Figures 4(a) and 4(b), X satisfies ASDC in the former (assuming the condition on exogenous errors is satisfied) but not in the latter, since in the latter Pa ( X 1 ) ̸ =

Original: Pa ( X 2 ) .
Revised: Pa ( X 2 ) .

Original: ASDCimplies IID as stated in the following lemma.
Revised: ASDC implies IID as stated in the following lemma.

Original: Lemma 1.
Revised: Lemma 1.

Original: If W satisfies ASDC, then any two explicit variables W i and W j are IID (Independentand Identically Distributed.)
Revised: If W satisfies ASDC, then any two explicit variables W i and W j are IID (Independent and Identically Distributed.)

Original: Note that descendants of an ASDC variable need not be IID.
Revised: Remark 2. The descendants of an ASDC variable need not be IID.

Revised: For example, in Figure 4(a),

Original: For example, Y i and Y j that aredescendants of an ASDC variable W can have different sets of parents,thus making  Y  non-IID.
Revised: X satisfies ASDC, and Y i and Y j are descendants of X i and X j . Y i and Y j have different sets of parents, making their distributions different, so Y is non-IID.

Original: 3.3 Quantity of Interest: True Average Causal Effect (TACE)
Revised: 3.3 Quantity of Interest: True Average Causal Effect (TACE)

Original: We generalize traditional ACE to the non-IID setting.
Revised: We generalize traditional ACE to the non-IID setting.

Original: Examine the interactions depicted in Figure5(c).
Revised: Examine the interactions depicted in Figure 5(c).

Original: Unit i ’s
Revised: Unit i ’s

Original: treatment X i affects their outcome through unit j ’s outcome Y j .
Revised: treatment X i affects their outcome through unit j ’s outcome Y j .

Original: X i
Revised: X i

Original: → Y j → Y i is a“spurious” causal path.
Revised: → Y j → Y i is a “spurious” causal path.

Original: We are interested in computing the ACE of a unit’s treatment on their outcome,excluding the effects transmitted via spurious paths from their neighbors/contacts.
Revised: We are interested in computing the ACE of a unit’s treatment on their outcome, excluding the effects transmitted via spurious paths from their neighbors/contacts.

Original: In an experimentalsetting interactions can be eliminated by isolating all subjects.
Revised: In an experimental setting, interactions can be eliminated by isolating all subjects.

Original: Given non-IID observational data weare interested in computing the average causal effect of treatment on outcome as if all units wereisolated .
Revised: In an observational setting, where we are given the non-IID data, we are interested in computing the average causal effect of treatment on outcome as if all units were isolated .

Original: We present the formal definition below.
Revised: We present the formal definition below.

Original: Definition 5 (True Average Causal Effect ( TACE XY )) .
Revised: Definition 5 (True Average Causal Effect ( TACE XY )) .

Original: Let M ∗ be an interaction model.
Revised: Let M ∗ be an interaction model.

Original: Trueaverage causal effect of X= x + 1 vs X = x on Y , denoted as TACE XY , is defined as the 
Revised: True average causal effect of X on Y , denoted as TACE XY , is defined as the ACE of X on Y in the isolated interaction model IM ∗ corresponding to M ∗ .

Original: TACE is the non-IID version of ACE and is the same as ACE in a traditional causal model whereall samples are isolated.
Revised: TACE is the non-IID version of ACE and is the same as ACE in a traditional causal model where all samples are isolated.

Revised: Again, without loss of generality, we assume the difference between the treatment value and the outcome value is 1 , i.e. treatment is X

Revised: = c

Original: {++ 
+}
Revised: {++ 
+}

Revised: and the outcome is X = c .

Original:  Defining, Quantifying, Detecting and Removing Interaction Bias for TACE
Revised:  Defining, Quantifying, Detecting and Removing Interaction Bias for TACE

Original: Almost all machine learning algorithms including those that employ causal techniques assume thatdata are IID (Schölkopf [2022], section 3).
Revised: Almost all machine learning algorithms including those that employ causal techniques assume that data are IID (Schölkopf [2022], section 3).

Original: In other words, the theoretical and performance guaranteesof these algorithms are based on data being IID.
Revised: In other words, the theoretical and performance guarantees of these algorithms are based on data being IID.

Original: As such it would be useful to determine conditionsunder which an algorithm meant for IID data can be applied on non-IID data with the certainty thatthe resulting bias would be negligible.
Revised: As such it would be useful to determine conditions under which an algorithm meant for IID data can be applied on non-IID data with the certainty that the resulting bias would be negligible.

Original: We formally define interaction bias below.
Revised: We formally define interaction bias below.

Original: Definition 6 (Interaction bias) .
Revised: Definition 6 (Interaction bias) .

Original: Let balanced model M ∗ be the true model that generated the(available) non-IID dataset D .
Revised: Let balanced model M ∗ be the true model that generated the (available) non-IID dataset D .

Original: Let Q denote the query of interest and let Q ∗ be its true value.
Revised: Let Q denote the query of interest and let Q ∗ be its true value.

Original: Let Adenote an algorithm that outputs an unbiased estimate of Q given data that are IID and the causalgraph that generated the IID data.
Revised: Let A denote an algorithm that outputs an unbiased estimate of Q given data that are IID and the causal graph that generated the IID data.

Original: Let G † denote an approximate causal graph constructed underthe assumption that D is IID such that no assumption in G † is refuted by D .
Revised: Let G † denote an approximate causal graph constructed under the assumption that D is IID such that no assumption in G † is refuted by D .

Original: Let ˆ Q be the estimatecomputed by A using G † and D as input.
Revised: Let ˆ Q be the estimate computed by A using G † and D as input.

Original: Interaction bias is given by || Q ∗ − ˆ Q || .
Revised: Interaction bias is given by || Q ∗ − ˆ Q || .

Original: 4.1 Quantifying Bias
Revised: 4.1 Quantifying Bias

Original: We define the two main types of problematic graphical structures in a linear interaction network thatintroduces bias in the estimation of TACE .
Revised: We define the two main types of problematic graphical structures in a linear interaction network that introduces bias in the estimation of TACE .

Original: Definition 7 (Deflecting bias structure) .
Revised: Definition 7 (Deflecting bias structure) .

Original: A deflecting bias structure for TACE XY in an interactionnetwork G ∗ is an open path between X j and Y i forany i ̸
Revised: A deflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X j and Y i for i ̸

Original: = j .
Revised: = j .

Original: Deflecting bias structures are open paths from one unit to another unit.
Revised: Deflecting bias structures are open paths from one unit to another unit.

Original: For example, Figures 5(a)and 5(b) contain deflecting bias structures.
Revised: For example, Figures 5(a) and 5(b) contain deflecting bias structures.

Original: The interaction network in Figure 5(a) has a directedopen path between X j and Y i , and the interaction network in Figure 5(b) has a confounded open pathbetween X j and Y i .
Revised: The interaction network in Figure 5(a) has a directed open path between X j and Y i , and the interaction network in Figure 5(b) has a confounded open path between X j and Y i .

Original: Definition 8 (Reflecting bias structure) .
Revised: Definition 8 (Reflecting bias structure) .

Original: A reflecting bias structure for TACE XY in an interactionnetwork G ∗ is an open path between X i and Y i throughW j for some generic variable W, i ̸ = j.
Revised: A reflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X i and Y i through some explicit variable W j

Original: Reflecting bias structures are open paths that go from a unit through another unit and back to thesame unit.
Revised: Reflecting bias structures are open paths that go from a unit through another unit and back to the same unit.

Original: For example, Figures 5(c) and 5(d) contain a reflecting bias structure.
Revised: For example, Figures 5(c) and 5(d) contain a reflecting bias structure.

Original: In each of them,there is an open path from X i to Y i through Y j .
Revised: In each of them, there is an open path from X i to Y i through Y j .

Original: In some cases, there can be a deflecting bias structureembedded in a reflecting bias structure, as in Figures 5(c) and 5(d).
Revised: In some cases, there can be a deflecting bias structure embedded in a reflecting bias structure, as in Figures 5(c) and 5(d).

Original: However, this is not necessary.
Revised: However, this is not necessary.

Original: Figure 5(e) contains only a reflecting bias structure ( X i → C j → Y i ) but no deflecting bias structure.[Figure] [Equation]
Revised: Figure 5(e) contains only a reflecting bias structure ( X i → C j → Y i ) but no deflecting bias structure.

Original: [Figure]
Revised: [Figure]

Original: Theorem 1.
Revised: Theorem 1.

Original: Let M ∗ ( G ∗ , S ∗ ) be a balanced interaction model in which treatment variable X i andoutcome variable Y i are not confounded by any variable in V i , ∀ i .
Revised: Let M ∗ ( G ∗ , S ∗ ) be a balanced interaction model in which treatment variable X i and outcome variable Y i are not confounded by any variable in V i , ∀ i .

Original: Let D be the available datagenerated by M ∗ and let G † be the approximate graph constructed using D .
Revised: Let D be the available data generated by M ∗ and let G † be the approximate graph constructed using D .

Original: Let TACE XY beidentifiable in G † and be given by β Y X , the regression coefficient of Y on X .
Revised: Let TACE XY be identifiable in G † and be given by β Y X , the regression coefficient of Y on X .

Original: Let α denote the truevalue of TACE X,Y in M ∗ .
Revised: Let α denote the true value of TACE X,Y in M ∗ .

Original: If X satisfies ASDC then  interaction bias is given by,
Revised: If X satisfies ASDC then the interaction bias is given by,

Original: [Equation]
Revised: [Equation]

Original: [Table] where P [ iji ] is the set of reflecting bias structures between X i and Y i through any explicit variable
Revised: where P [ iji ] is the set of reflecting bias structures between X i and Y i through any explicit variable

Original: W j of unit j with i ̸
Revised: W j of unit j with i ̸

Original: = j , P [ ji ] is the set of deflecting bias structures between X j and Y i with i ̸
Revised: = j , P [ ji ] is the set of deflecting bias structures between X j and Y i with i ̸

Original: = j ,and R p is the root of path p .
Revised: = j , and R p is the root of path p .

Original: It follows from Theorem [-
-] that in a balanced interaction model in which no X i and Y i are confoundedby any variable in V i , the reflecting and deflecting structures are the only two structures that willbias the identification of TACE .
Revised: It follows from Theorem [-
-]1 that in a balanced interaction model in which no X i and Y i are confounded by any variable in V i , the reflecting and deflecting structures are the only two structures that will bias the identification of TACE .

Original: Note that although definition of interaction bias (Definition 6) on
Revised: Note that although definition of interaction bias (Definition 6) on

Original: TACE is for any unbiased estimator for ACE , we focus only on the ordinary least squares estimatorin this paper.
Revised: TACE is for any unbiased estimator for ACE , we focus only on the ordinary least squares estimator in this paper.

Original: This is because among the class of unbiased linear estimators, the OLS estimator hasthe minimum variance [Johnson et al., 2014].
Revised: This is because among the class of unbiased linear estimators, the OLS estimator has the minimum variance [Johnson et al., 2014].

Original: We exemplify theorem 1.
Revised: We exemplify theorem 1.

Original: [Figure]
Revised: [Figure]

Revised: [Figure]

Original: [Figure] [Figure] 
Revised: [Figure] [Figure] [Figure]

Original: Example 1.
Revised: Example 1.

Original: Figure 6 shows an example of an interaction model with 4 units where X 1 , . . .
Revised: Figure 6 shows an example of an interaction model with 4 units where X 1 , . . .

Original: , X 5 arethe treatments, and Y 1 , . . 
Revised: , X 5 are the treatments, and Y 1 , . . .

Original: , Y 5 the outcomes.
Revised: , Y 5 the outcomes.

Original: The numbers on the edges are the edge coefficients.
Revised: The numbers on the edges are the edge coefficients.

Original: Csatisfies ASDC, and C i for i
Revised: C satisfies ASDC, and C i for i

Original: = 1 , 2 , 3 , 5 are omitted from the graph for simplicity.
Revised: = 1 , 2 , 3 , 5 are omitted from the graph for simplicity.

Original: Suppose we want to estimate the ACE of X on Y as if the units were isolated: Input: the interactionnetwork G ∗ as shown in Figure 6 (no parameter i.e., S ∗ is not an input), Output: the TACE XY(should equal to 2 ).
Revised: Suppose we want to estimate the ACE of X on Y as if the units were isolated: Input: the interaction network G ∗ as shown in Figure 6 (no parameter i.e., S ∗ is not an input), Output: the TACE XY (should equal to 2 ).

Original: If we estimate ACE XY ignoring the connections between units, our estimator willbe ˆ β Y X , with Y = { Y 1 , . . 
Revised: If we estimate ACE XY ignoring the connections between units, our estimator will be ˆ β Y X , with Y = { Y 1 , . . .

Original: , Y 5 } and X =
Revised: , Y 5 } and X =

Original: { X 1 , . . .
Revised: { X 1 , . . .

Original: , X 5 } .
Revised: , X 5 } .

Original: This is because ignoring the connections,the graph becomes X i
Revised: This is because ignoring the connections, the graph becomes X i

Original: → Y i separated for i = 1 , . . .
Revised: → Y i separated for i = 1 , . . .

Original: , 5 , so is essentially X
Revised: , 5 , so is essentially X

Original: → Y Pearl [2009].
Revised: → Y Pearl [2009].

Original: However, by Theorem 1,
Revised: However, by Theorem 1,

Original: [Equation]
Revised: [Equation]

Original: Hence, the result is biased, and does not give us what we want.
Revised: Hence, the result is biased, and does not give us what we want.

Original: We show later in Theorem 2 how tocompute an unbiased estimate of TACE .
Revised: We show later in Theorem 2 how to compute an unbiased estimate of TACE .

Original: 4.2 Detecting Bias
Revised: 4.2 Detecting Bias

Original: In this section, we provide a graphical criterion resulting from Theorem 1, to detect interaction bias.
Revised: In this section, we provide a graphical criterion resulting from Theorem 1, to detect interaction bias.

Original: Corollary 1.
Revised: Corollary 1.

Original: Let M ∗∗ ( G ∗∗ , S ) be a balanced interaction model in which X satisfies ASDC and
Revised: Let M ∗∗ ( G ∗∗ , S ) be a balanced interaction model in which X satisfies ASDC and

Original: TACE is identified as β Y X =
Revised: TACE is identified as β Y X =

Original: α , then interaction bias exists iff G ∗∗ contains a reflecting or deflectingbias structure.
Revised: α in the approximate graph, then interaction bias exists iff G ∗∗ contains a reflecting or deflecting bias structure.

Original: For example, Figure 7 contains both reflecting and deflecting bias structures.
Revised: For example, Figure 7 contains both reflecting and deflecting bias structures.

Original: Figure 8 does notcontain any bias structure.
Revised: Figure 8 does not contain any bias structure.

Original: So Figure 7 has interaction bias and Figure 8 does not.
Revised: So Figure 7 has interaction bias and Figure 8 does not.

Revised: Note that the interactions in Figure 8 do not qualify as bias structures by Definitions 7 and 8.

Original: 4.3 Removing Bias
Revised: 4.3 Removing Bias

Original: Theorem 2 presents a technique for computing an unbiased estimate of TACE in cases where theorem1 predicts significant bias.
Revised: Theorem 2 presents a technique for computing an unbiased estimate of TACE in cases where theorem 1 predicts significant bias.

Original: It proceeds by applying linear regression on a set of samples S thatsatisfy the condition that no bias inducing structures exist between any two distinct units i and j .
Revised: It proceeds by applying linear regression on a set of samples B that satisfy the condition that no bias inducing structures exist between any two distinct units i and j .

Original: Inparticular, a subset of samples/units S is termed as a bias-free subset for TACE XY if no reflectingbias structures exist for any i ∈ S and no deflecting bias structures exist in G ∗ S where G ∗ S is the latentprojection of G ∗ on S (Definition 2.6.1, Pearl [2009]).
Revised: In particular, a subset of samples/units B is termed as a bias-free subset for TACE XY if no reflecting bias structures exist for any i ∈ S and no deflecting bias structures exist in G ∗ S where G ∗ S is the latent projection of G ∗ on B (Definition 2.6.1, Pearl [2009]).

Original: For example in figure 6, S comprises of units 2 & 5 and G ∗ S is X 2 → Y 2 X 5 → Y 5 .
Revised: For example in figure 6, B comprises of units 2 & 5 and G ∗ S is X 2 → Y 2 X 5 → Y 5 .

Original: However, S is not unique for a given interaction network.
Revised: However, B is not unique for a given interaction network.

Original: Another candidate for S is units 2 & 4 and the associated G ∗ S is X 2 → Y 2 X 4 → Y 4 .
Revised: Another candidate for B is units 2 & 4 and the associated G ∗ S is X 2 → Y 2 X 4 → Y 4 .

Original: An algorithmfor constructing S is presented in  the appendix.
Revised: An algorithm for constructing B is presented in Algorithm 1, with an example and discussion in the appendix.

Original: This algorithm starts by randomly initializing S witha sample.
Revised: This algorithm starts by randomly initializing B with a sample.

Original: Then it goes through the rest of the samples and adds a sample to S if its inclusion doesnot create bias structures in the resultant graph, G ∗ S . 
Revised: Then it goes through the rest of the samples and adds a sample to B if its inclusion does not create bias structures in the resultant graph, G ∗ S . Algorithm

Revised: Input: an interaction network G ∗ , iterations t

Revised: Output: the largest bias-free subset B selected

Original: Theorem 2.
Revised: 1: function F IND S UB ( G ∗ , t ) 2: B = ∅ 3: for i = 1 , . . . , t do 4: Units = randomly sorted list 1 , . .

Revised: . , n 5:

Revised: B = { Units [1] } (The index starts from 1) 6: for i = 2 , . .

Revised: . , n do 7: if Units [ i ] has no reflecting bias structure in G ∗ then 8: if Units [ i ] has no deflecting bias structure in G ∗ with an element in B then 9:

Revised: B = B ∪ { Unit [ i ] } 10:

Revised: B = B ∪ { B } 11: return Largest B in B

Revised: Theorem 2.

Original: Let G ∗ be an interaction network.
Revised: Let G ∗ be an interaction network.

Original: Given the conditions in Theorem 1 and ‘ S ’ a biasfree subset for G ∗ , TACE XY =
Revised: Given the conditions in Theorem 1 and ‘ B ’ a bias-free subset for G ∗ , TACE XY =

Original: E [ ˆ β Y X ] where the regression coefficient is calculated using onlysamples in set S .
Revised: E [ ˆ β Y X ] where the regression coefficient is calculated using only samples in set B .

Original: Note that bias-free subset of samples S used in Theorem 2 is not  IID .
Revised: Note that bias-free subset of samples B used in Theorem 2 is not always IID .

Original: While we insist that no reflectingor deflecting bias structures exist in G ∗ S , we do not restrict other forms of interactions among thesesamples.
Revised: While we insist that no reflecting or deflecting bias structures exist in G ∗ S , we do not restrict other forms of interactions among these samples.

Original: For example, G ∗ S can be X 1 → Y 1 ← C 2 → Y 2 ← X 2 where Y1 is caused by C 2 . Inthis case Y is not IID and hence S does not constitute an IID dataset.
Revised: For example, in Figure 8, Units {1, 2, 3} constitute a bias-free subset. In this case, Y is not IID ( Y 1 and Y 2 are dependent, Y 2 and Y 3 are dependent) and hence the bias-free subset is non-IID.

Original: Also note that to compute an unbiased estimate using Theorem 2, we have at our disposal a smallerset of samples; so the variance of estimation will be larger.
Revised: Also note that to compute an unbiased estimate using Theorem 2, we have at our disposal a smaller set of samples; so the variance of estimation will be larger.

Original: There is a trade off between ignoringinteraction (large bias, small variance), and using theorem 2 (no bias, large variance).
Revised: There is a trade off between ignoring interaction (large bias, small variance), and using theorem 2 (no bias, large variance).

Original: It remains futurework to quantify the variance of the estimator in Theorem 2 for different interaction models, but in
Revised: It remains future work to quantify the variance of the estimator in Theorem 2 for different interaction models, but in

Original: Section 5, we provide simulation results and case analysis study to empirically show its performance.
Revised: Section 5, we provide simulation results and case analysis study to empirically show its performance.

Original: Applicability of theorems 1 & 2 to real world problems: A natural question that arises at thisjuncture is whether we need an entire interaction network to apply these results to real world problems.
Revised: Applicability of theorems 1 & 2 to real world problems: A natural question that arises at this juncture is whether we need an entire interaction network to apply these results to real world problems.

Original: Theorem 1 quantifies bias and in doing so reveals to us if and how various factors such as samplesize and strength of connections (value of path coefficients) influence bias.
Revised: Theorem 1 quantifies bias and in doing so reveals to us if and how various factors such as sample size and strength of connections (value of path coefficients) influence bias.

Original: This in turn allows us touse available information about the problem from prior experience, domain knowledge or externalsources to determine if bias would be negligible or not.
Revised: This in turn allows us to use available information about the problem from prior experience, domain knowledge or external sources to determine if bias would be negligible or not.

Original: Specifically, bias is inversely proportionalto sample size; in fact the quadratic term n  ( n − 1) in thedenominator of deflecting bias showsthat it diminishes at a fast rate as sample size increases.
Revised: Specifically, bias becomes smaller as the number of bias-structure-free samples increases. In fact, if the numbers of deflecting and reflecting structures are fixed, the bias terms diminishes as n increases, indicated by the 1 /n for the reflecting bias term and 1 /n ( n − 1) for the deflecting bias term.

Original: It is also evident that if the values of pathcoefficients are high, V al ( p ) would be high and this will result in increased bias.
Revised: It is also evident that if the values of path coefficients are high, V al ( p ) would be high and this will result in increased bias.

Original: Finally, if theinteraction connections are sparse (fewer edges between units), the reduction in the total number ofpaths could potentially lower bias but more importantly the number of samples in the bias-free set Sused in theorem 2 will tend to be larger, which in turn will help in computing better quality estimates.
Revised: Finally, if the interaction connections are sparse (fewer edges between units), the reduction in the total number of paths could potentially lower bias but more importantly the number of samples in the bias-free set B used in theorem 2 will tend to be larger, which in turn will help in computing better quality estimates.

Original:  Experiments 
Revised:  Experiments 5.1 Simulations

Original: 5.1 Simulations

Original: Simulated Model We randomly generate balanced interaction network with n units (i.e., the samplesize is n ), with C i
Revised: Simulated Model We randomly generate balanced interaction network with n units (i.e., the sample size is n ), with C i

Original: → X i → Y i and X i
Revised: → X i → Y i and X i

Original: → M i for all i = 1 , . . . , n .
Revised: → M i for all i = 1 , . . . , n .

Original: For all ordered pairs of distinctunits i, j , we randomly add deflecting bias structures in the form of X
Revised: For all ordered pairs of distinct units i, j , we randomly add deflecting bias structures in the form of X

Original: i ← C i → Y j with probabilitydRate .
Revised: i ← C i → Y j with probability dRate .

Original: For all units i , we randomly add reflecting bias structures in the form of X i → M k → Y iwith probability rRate for a random k ̸
Revised: For all units i , we randomly add reflecting bias structures in the form of X i → M k → Y i with probability rRate for a random k ̸

Original: = i .
Revised: = i .

Original: Experiment: Bias of REG It follows from theorem 1 that larger sample sizes and smaller path valueson the bias structures result in smaller bias.
Revised: Experiment: Bias of REG It follows from theorem 1 that larger sample sizes and smaller path values on the bias structures result in smaller bias.

Original: We perform two simulations to show how bias varies as afunction of sample size and path values.
Revised: We perform two simulations to show how bias varies as a function of sample size and path values.

Original: We simulate data such that for each variable, the exogenouserror term follows a Gaussian distribution with mean 0 and standard deviation 1 .
Revised: We simulate data such that for each variable, the exogenous error term follows a Gaussian distribution with mean 0 and standard deviation 1 .

Original: For each set ofparameters, we randomly generate an interaction network, and simulate the data 10000 times.
Revised: For each set of parameters, we randomly generate an interaction network, and simulate the data 10000 times.

Original: Eachtime, we record the result from a naive regression of Y on X (REG).
Revised: Each time, we record the result from a naive regression of Y on X (REG).

Original: As a comparison, we also recordthe result from Theorem 2 (THM-2).
Revised: As a comparison, we also record the result from Theorem 2 (THM-2).

Original: We run the algorithm (provided in the appendix) to randomlyselect bias-free subsets for 10 times and select the largest subset.
Revised: We run the algorithm (provided in the appendix) to randomly select bias-free subsets for 10 times and select the largest subset.

Original: Simulation 1: X i → Y i ’s edge coefficient is 100, the edge coefficients of C i
Revised: Simulation 1: X i → Y i ’s edge coefficient is 100, the edge coefficients of C i

Original: → X i , X i →
Revised: → X i , X i →

Original: M j , M j → Y i are all set to 10 , the numbers of deflecting bias structures and additional reflectingbias structures are both 100 .
Revised: M j , M j → Y i are all set to 10 , the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .

Original: Simulation 2: Number of units n = 1000 , X i
Revised: Simulation 2: Number of units n = 1000 , X i

Original: → Y i ’s edge coefficient is 100, the numbers ofdeflecting bias structures and additional reflecting bias structures are both 100 .
Revised: → Y i ’s edge coefficient is 100, the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .

Original: The results are plottedin Figure 9.
Revised: The results are plotted in Figure 9.

Original: As seen in the plots, as n increases or the path values on the bias structures decreases(both with all other parameters fixed), β Y X from a naive regression approaches TACE .
Revised: As seen in the plots, as n increases or the path values on the bias structures decreases (both with all other parameters fixed), β Y X from a naive regression approaches TACE .

Original: Such resultscoincide with Theorem 1.
Revised: Such results coincide with Theorem 1.

Original: The β Y X computed by THM-2 is very close to TACE and the two linesalmost overlap.
Revised: The β Y X computed by THM-2 is very close to TACE and the two lines almost overlap.

Original: 5.2 Case Study
Revised: 5.2 Case Study

Original: Story We are interested in analyzing the effect of tutoring time on students’ grades.
Revised: Settings We are interested in analyzing the effect of tutoring time on students’ grades.

Original: In particular, wewish to compute the effect provided through the tutoring program only, but not through “side effects”from other units, such as learning from classmates, although such interactions are encouraged inthis scenario.
Revised: In particular, we wish to compute the effect provided through the tutoring program only, but not through “side effects” from other units, such as learning from classmates, although such interactions are encouraged in this scenario.

Original: For instance, unit i might help unit j understand the course materials better which inturn might improve j ’s grade.
Revised: For instance, unit i might help unit j understand the course materials better which in turn might improve j ’s grade.

Original: If unit i helped unit j improve their understanding and unit j statesthis in the peer review, then it would boost i ’s grade.
Revised: If unit i helped unit j improve their understanding and unit j states this in the peer review, then it would boost i ’s grade.

Original: To construct an interaction network and applyour results, we ask the students to fill out a survey including 1) their tutoring time, 2) their grade, 3)whom they helped, 4) who helped them, 5) peer review score.
Revised: To construct an interaction network and apply our results, we ask the students to fill out a survey including 1) their tutoring time, 2) their grade, 3) whom they helped, 4) who helped them, 5) peer review score.

Original: Construction of the Interaction Network Three generic variables are T (tutoring time in hours), U(understanding of course materials), and G (grade).
Revised: Construction of the Interaction Network Three generic variables are T (tutoring time in hours), U (understanding of course materials), and R (grade).

Original: For each unit i , T i → U i
Revised: For each unit i , T i → U i

Original: → G i .
Revised: → R i .

Original: In addition, ifi helped j , add U i → U j (deflecting bias structure).
Revised: In addition, if i helped j , add U i → U j (deflecting bias structure).

Original: If i first helped j and j mentioned this in thepeer review and thus boosted i ’s grade, add U i
Revised: If i first helped j and j mentioned this in the peer review and thus boosted i ’s grade, add U i

Original: → U j
Revised: → U j

Original: → G i (reflecting bias structure).
Revised: → R i (reflecting bias structure).

Original: We assumeno additional back-and-forth help happens.
Revised: We assume no additional back-and-forth help happens.

Original: Simulation Let there be 500 students, assume each student on average help 5 other students, andthe other student has a 0 .
Revised: Simulation Let there be 500 students, assume each student on average help 5 other students, and the other student has a 0 .

Original: 5 chance of helping back.
Revised: 5 chance of helping back.

Original: Let TACE = 2 , and the U i
Revised: Let TACE = 2 , and the U i

Original: → U j and U i → G jedges both have the value 2 .
Revised: → U j and U i → R j edges both have the value 2 .

Original: We randomly generate an interaction network and simulate data basedon these parameters.
Revised: We randomly generate an interaction network and simulate data based on these parameters.

Original: Results We apply THM-2 to select a bias-free subset, and compute β GT using data from that subset.
Revised: Results We apply THM-2 to select a bias-free subset, and compute β GT using data from that subset.

Original: We get the result 1 . 963 , with the size of the subset 72 .
Revised: We get the result 1 . 963 , with the size of the subset 72 .

Original: As a result, the effect of tutoring time onstudents’ grades not through other units is estimated to be 1 . 963 , which is close to the ground truth
Revised: As a result, the effect of tutoring time on students’ grades not through other units is estimated to be 1 . 963 , which is close to the ground truth

Original: TACE (2).
Revised: TACE (2).

Original: We further repeat the experiment 1000 times to show the distribution of the results.
Revised: We further repeat the experiment 1000 times to show the distribution of the results.

Original: Eachtime a random structure is generated and random data are simulated.
Revised: Each time a random structure is generated and random data are simulated.

Original: THM-2 is on average able toselect a bias-free subset of size 76 , and the average recovered TACE = 2 . 0002 .
Revised: THM-2 is on average able to select a bias-free subset of size 76 , and the average recovered TACE = 2 . 0002 .

Original: The result from
Revised: The result from

Original: REG had a significantly high bias with TACE averaging at 194 . 11 .
Revised: REG had a significantly high bias with TACE averaging at 194 . 11 .

Original: Also since every time the dataare regenerated, the model is different, and REG uses all the data, it has a larger variance.
Revised: Also since every time the data are regenerated, the model is different, and REG uses all the data, it has a larger variance.

Original: The twoplots below show the distribution of results from THM-2 and REG.
Revised: The two plots in Figure 10 show the distribution of results from THM

Original: The histograms of the results ofβ Y X computed by THM-2 and REG are as below.

Original:  Related Work
Revised:  Related Work

Original: One of the most studied concepts related to interactions among units is interference [Cox, 1958].
Revised: One of the most studied concepts related to interactions among units is interference [Cox, 1958].

Original: Majority of literature in empirical fields assume no-interference.
Revised: Majority of literature in empirical fields assume no-interference.

Original: In fact, SUTVA is a commonassumption in causal inference [Rubin, 1978].
Revised: In fact, SUTVA is a common assumption in causal inference [Rubin, 1978].

Original: Recent years have witnessed a rise in papers oninterference that employ graphical models.
Revised: Recent years have witnessed a rise in papers on interference that employ graphical models.

Original: These include Ogburn and VanderWeele [2014] that wasthe first to model interference using DAGs, Sherman and Shpitser [2018] that modeled interferenceusing chain graphs which permits modeling unknown interactions between units and Bhattacharyaet al.
Revised: These include Ogburn and VanderWeele [2014] that was the first to model interference using DAGs, Sherman and Shpitser [2018] that modeled interference using chain graphs which permits modeling unknown interactions between units and Bhattacharya et al.

Original: [2020]that proposed structure learning methods for chain graphs.
Revised: [2020]that proposed structure learning methods for chain graphs.

Original: These works rely on partialinterference which divides units into equal-sized blocks under the assumption that interactions occuronly within a block but not across different blocks.
Revised: These works rely on partial interference which divides units into equal-sized blocks under the assumption that interactions occur only within a block but not across different blocks.

Original: [Nabi et al., 2020] developed methods foridentification and estimation of multiple queries under conditions of interference and homophily, andapplied the results to the problem of ad-placements.
Revised: [Nabi et al., 2020] developed methods for identification and estimation of multiple queries under conditions of interference and homophily, and applied the results to the problem of ad-placements.

Original: Sobel [2006] was the first to notice the effect ofinterference in the housing mobility problem, and proposes causal estimands for this application.
Revised: Sobel [2006] was the first to notice the effect of interference in the housing mobility problem, and proposes causal estimands for this application.

Original: Aronow and Samii [2017], Sussman and Airoldi [2017] modeled general interference (withoutassuming partial inference) by constructing a function to define a unit’s exposure level on the numberof treated neighbors they have.
Revised: Aronow and Samii [2017], Sussman and Airoldi [2017] modeled general interference (without assuming partial inference) by constructing a function to define a unit’s exposure level on the number of treated neighbors they have.

Original: The methods are less restricted than partial interference methods, andallow units to be affected by any number of neighbors.
Revised: The methods are less restricted than partial interference methods, and allow units to be affected by any number of neighbors.

Original: However, they are limited to interference anddo not handle other forms of interactions.
Revised: However, they are limited to interference and do not handle other forms of interactions.

Original: Jagadeesan et al.
Revised: Jagadeesan et al.

Original: [2020] proposed a quasi-coloring methodto estimate direct effect under interference using experimental data.
Revised: [2020] proposed a quasi-coloring method to estimate direct effect under interference using experimental data.

Original: However, it does not easilygeneralize to observational studies.
Revised: However, it does not easily generalize to observational studies.

Original: Other papers along a similar direction include Fatemi and Zheleva[2020], which proposed experiment design to minimize interaction bias and selection bias at the sametime, and Liu and Hudgens [2014], which proposed a two-stage randomization design to minimizeinterference bias.
Revised: Other papers along a similar direction include Fatemi and Zheleva [2020], which proposed experiment design to minimize interaction bias and selection bias at the same time, and Liu and Hudgens [2014], which proposed a two-stage randomization design to minimize interference bias.

Original: Tchetgen Tchetgen et al.
Revised: Tchetgen Tchetgen et al.

Original: [2021] proposed a g-computation method, which is thefirst to model general interference using graphical models (chain-graphs), but requires the interferenceeffects to be symmetrical between units.
Revised: [2021] proposed a g-computation method, which is the first to model general interference using graphical models (chain-graphs), but requires the interference effects to be symmetrical between units.

Original: Sävje et al.
Revised: Sävje et al.

Original: [2021] and Hudgens and Halloran [2008] definedqueries similar to TACE, named EATE and PADE, respectively.
Revised: [2021] and Hudgens and Halloran [2008] defined queries similar to TACE, named EATE and PADE, respectively.

Original: These queries generalize traditional
Revised: These queries generalize traditional

Original: ACE to allow a unit’s outcome to be affected by treatments of other units.
Revised: ACE to allow a unit’s outcome to be affected by treatments of other units.

Original: However, they do not allowoutcomes to be affected by other units’ variables other than treatments.
Revised: However, they do not allow outcomes to be affected by other units’ variables other than treatments.

Original: Hudgens and Halloran [2008] defined six types of queries in the problems involving interference.
Revised: Hudgens and Halloran [2008] defined six types of queries in the problems involving interference.

Original: Work in interference that focuses on different queries/problems include a few as follows.
Revised: Work in interference that focuses on different queries/problems include a few as follows.

Original: VanderWeeleet al.
Revised: VanderWeele et al.

Original: [2012b] is the first to decompose the spillover effect (the effect of a unit’s treatment on another’soutcome (Quammen [2012])) to contagion and infectiousness effects using counterfactual mediationanalysis.
Revised: [2012b] is the first to decompose the spillover effect (the effect of a unit’s treatment on another’s outcome (Quammen [2012])) to contagion and infectiousness effects using counterfactual mediation analysis.

Original: Shpitser et al.
Revised: Shpitser et al.

Original: [2017] presented decomposition for units with unknown and symmetricalinteraction patterns and analyzed different interference paths.
Revised: [2017] presented decomposition for units with unknown and symmetrical interaction patterns and analyzed different interference paths.

Original: In linear models, the contagionand infectiousness effects reduce to the directed paths from X j to Y i .
Revised: In linear models, the contagion and infectiousness effects reduce to the directed paths from X j to Y i .

Original: Moreover, their work doesnot handle reflection bias.
Revised: Moreover, their work does not handle reflection bias.

Original: Hu et al.
Revised: Hu et al.

Original: [2021] was the first to define and provide estimands for theaverage indirect effect.
Revised: [2021] was the first to define and provide estimands for the average indirect effect.

Original: VanderWeele et al.
Revised: VanderWeele et al.

Original: [2014] developed methods for sensitivity analysis underinterference.
Revised: [2014] developed methods for sensitivity analysis under interference.

Original: Other types of interactions include the contagion effects, which are defined as a unit’soutcome affecting another unit’s outcome [VanderWeele and An, 2013].
Revised: Other types of interactions include the contagion effects, which are defined as a unit’s outcome affecting another unit’s outcome [VanderWeele and An, 2013].

Original: Work on this line usuallyused longitudinal data, including Burt [1987], Lyons [2011], VanderWeele et al.
Revised: Work on this line usually used longitudinal data, including Burt [1987], Lyons [2011], VanderWeele et al.

Original: [2012a].
Revised: [2012a].

Original: Homophilyeffects are defined as the behavior of connected units are similar
Revised: Homophily effects are defined as the behavior of connected units are similar

Original: [Jagadeesan et al., 2020].
Revised: [Jagadeesan et al., 2020].

Original: Work inthis line include McPherson et al.
Revised: Work in this line include McPherson et al.

Original: [2001], Jagadeesan et al.
Revised: [2001], Jagadeesan et al.

Original: The existing work above does notmodel interactions using graphical models.
Revised: The existing work above does not model interactions using graphical models.

Original: Work on relational causal models that also analyzes unit interactions include Maier et al.

Original: Arbour et al.

Original: [2016], Lee and Honavar [2016].

Original: Such work handles non-IID data in first or higherorder logic, while our work employs Bayesian networks (propositional logic), and focuses on makingresults such as those in Pearl [2009], applicable to non-IID data.

Original:  Conclusions
Revised:  Conclusions

Original: In this paper, we represent interactions among units using causal graphical models.
Revised: In this paper, we represent interactions among units using causal graphical models.

Original: We derivetheorems to quantify the interaction bias for average treatment effects in linear models.
Revised: We derive theorems to quantify the interaction bias for average treatment effects in linear models.

Original: We providesufficient and necessary graphical conditions to detect interaction bias.
Revised: We provide sufficient and necessary graphical conditions to detect interaction bias.

Original: Additionally, we developa method to compute an unbiased estimate of causal effect in cases where blindly assuming IIDis expected to yield a significant bias.
Revised: Additionally, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.

Original: Finally, we discuss the performance of our method throughsimulation studies.
Revised: Finally, we discuss the performance of our method through simulation studies.

Revised: Acknowledgments and Disclosure of Funding

Revised: We would like to thank anonymous reviewers for their comments, and Totte Harinen and Rumen Iliev for helpful discussions.

Revised: This research was supported in parts by grants from the National Science

Revised: Foundation [#IIS-2106908 and #2231798], Office of Naval Research [#N00014-21-1-2351], and

Revised: Toyota Research Institute of North America [#PO-000897].

Original: Checklist
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Original: [Yes] (c) Did you report error bars (e g., with respect to the random seed after running experiments multiple times)?[No]
Revised: [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

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Original:  with human subjects (a) Did you include the full text of instructions given to participants and screenshots, if 
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
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