Causal Inference with Non-IID Data using Linear Graphical Models
Authors: Chi Zhang
Karthika Mohan
Judea Pearl
Venue: NeurIPS 2022 Accept (2022)

Abstract: Traditional causal inference techniques assume data are independent and identically distributed (IID) and thus  ignores interactions among units. 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. 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. We derive theorems to detect and quantify the interaction bias, and derive conditions under which it is safe to ignore interactions. Put differently, we present conditions under which causal effects can be computed with negligible bias by assuming that samples are IID. Furthermore, we develop a method to eliminate bias in cases where blindly assuming IID is expected to yield a significantly biased estimate. Finally, we test the coverage and performance of our methods through simulations.

Causal Inference with Non-IID Data using Linear Graphical Models
Chi Zhang Department of CS UCLA
Karthika Mohan Department of EECS
Judea Pearl Department of CS UCLA California, USA, 90095 judea@cs www.acclcc.. edu
Oregon State University Oregon, USA, 97331 mohank@oregonstate edu
California, USA, 90095 zcccêcs. www.. edu
Abstract
Traditional causal inference techniques assume data are independent and identically distributed (IID) and thus ignores interactions among units. However, a unit's s treatment may affect another unit's outcome (interference), a unit's treatment may be correlated with another unit's outcome, or unit's 's treatment and outcome may be spuriously correlated through another unit. To capture such nuances, we model the data generating process using causal graphs and conduct systematic analysis of the bias caused by different types of interactions when computing causal effects. We derive theorems to detect and quantify the interaction bias, and derive conditions under which safe ignore interactions. Put differently, we present conditions under which causal effects can be computed with negligible bias by assuming that samples are IID. Furthermore, we develop a method to climinate bias cases where blindly assuming IID is expected to yield significantly biased estimate. Finally, we test the coverage and performance our methods through simulations.
1 Introduction
Motivating Example Suppose we are interested in studying the effectiveness of Covid- 19 vaccines. Specifically, we interested in the causal effect of vaccine doses, V. on the severity of sickness S. 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. However, this method may result in biased estimation primarily because traditional causal inference techniques assume that attributes of all units in the sample independent and identically distributed (IID)[Rubin 1978), which does not hold true in the pandemic setting since units are not isolated from each other. We exemplify below few instances of this problem that violate IID (Eyre al. (2022).
Case 1: The vaccination V of a unit i, (V.). decreases their viral load, I,,, which in turn decreases the transmission rate of the virus, and hence decreases the severity of sickness of another unit (S)), who comes into contact with i. V; causally affects Sj.
Case 2: is affected by the area A that lives in, and contact who lives in vaccine deprived areas and areas with higher incidence of Covid-19 infection is more likely to get sick. V, and Sj are confounded.
Case 3: S, determines whether or not i is quarantined and thus affects whether transmits the disease to another unit j. S, causally affects Sy-
Such interactions between units plague both observational and experimental studies. If the latter is performed in controlled environment where subjects isolated from each other, the results would not be valid for the target environment, where subjects affect each other, and vice versa.
Modeling Interactions A line of existing work that analyzes interactions between units is interference Interference is the phenomenon in which treatment of unit é (V)) causally affects the
Cox 1958
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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outcome, Sj, of another unit j. In almost all existing literature this is interpreted as there existing a causal pathway from V to Sj. Case-1 above is a typical example. Clearly, ignoring unit interactions while computing causal effects would result in a biased estimate. However, we note that interference not the only type of interaction between units that can yield biased estimates. For example, in Case-2 Vi and S are confounded and V; is not a cause of Sj. Another example is an instance where unit i's treatment affects their own outcome through an attribute of unit j i... V; + W, + S, for some W & {S,, V2}. 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. 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. It is this deficiency that this paper attempts to overcome.
Questions addressed The scenario exemplified above raises several questions regarding the com- putation of causal effects given non-Ill) data. How can we model different types of interactions among units in the population? 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? If assuming IID would yield a biased estimate, then how can we get rid of this bias?
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. 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. We derive theorems to detect and quantify the interaction bias. We derive conditions under which it is safe to ignore unit interactions when computing the average causal effects. Furthermore, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming III is expected to yield a significant bias. Finally, we corroborate our findings through simulation studies.
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. 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 the outcome of unit j and/or to the outcome of unit itself such that an intermediate node on the path belongs to unit j. The formula in Theorem T 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 debiasing procedure does not require the selection of IID samples and may contain interactions among them. Such a debiasing procedure can also be done in polynomial time (Algorithm D. Empirical analysis: We randomly generate interaction models and show that the bias can be huge if IID is wrongly assumed on non-lID data. The debiasing method in this paper yields unbiased estimate. We further show that, the number of bias-free samples increases, and as the strengths of bias structures decrease, the overall interaction bias decreases.
2 Preliminaries
Independent and Identically Distributed (IID) If X1 ... Xn, are independent and each has the same marginal distribution with CDF F, we say that X,,..., X, are IID (independent and identically distributed) [Wasserman] 2013). For the sake of simplicity, we use X is IID to refer to all the units of X, XI,..., Xu, being IID. A dataset is IID if all variables in it are IID.
Linear Causal Models A traditional linear causal model is also known as linear structural causal model (SCM) [Brito 2004 Pearl, 2209 Chen and Pearl 2014]. The edge coefficients on the causal DAG represent direct effects. An open path is collider-free i.c., there are no head-to-head arrows on this path. Note that if there exists an open path from W; to Vj. it implies W, M V,, The value 'V (p) of open path p in a linear model is defined the product of the edge coefficients on p. The root of an open path p is defined as the variable on p that is the ancestor of all variables on p.
Average Causal Effects In this work the query we are primarily interested in generalizing to the non-III case is the Average Causal Effect (ACE), also named as the Average Treatment Effect (ATE) Rubin 1977 Holland 1988). For consistency, we use ACE to refer to both. Given a causal model M, the average causal effect (ACE) of X = vs X = e (1 and are constants) on Y for k units is defined as ACExy = (Yx Yxx==). ACE is defined under the assumption that Y, depends only
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on factors of unit (including X;) Holland [1988]. Without loss of generality, we assume t t= c In linear models, AC'E of X on Y can be identified By x. the linear regression slope of Y X. if there is no backdoor (non-directed open paths) between X and Y [Pearl et al. 2016 Pearl 2017
3 Graphical Modeling of Interactions
3.1 Interaction models
In this section, we define graphical model derived from traditional causal models M G.S (Pearl [2009], definition 7.1.1). G' is the causal graph and S is the set of structural equations of variables. We refer the variables in traditional causal model as generic variables. X, C, in Figure | are generic variables. An explicit variable is similar to generic variable except that it represents an attribute/event of one specific unit (or sample or individual). For example, "treatment (X)" is generic variable, and "the treatment of unit (X;)" is an explicit variable.
Xi
3
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C1
C
C4
X1 Uxi
(1)
Y1 = X1 +2+ + +Uy
(2)
Y/
Y3
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Y22 2X2 -0 +1++3++ +Us
(3)
Figure 1: Tra- ditional DAG.
Figure Interaction network with units and explicit variables (X,, Y,, C. for = 1, 3,
Definition (Interaction model (G' ())). An interaction model, GG, (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. An interaction network, G is a directed acyclic graph with each node representing explicit variable and each directed edge A; - Bj representing A, causes By-
example of interaction model "G" ), the interaction network, G portrayed in Figure and the structural equations S" (part of) specified beside it; Uy, denotes the unobserved exogenous eтror of an explicit variable V,. Observe that interaction networks allow edges between explicit variables of the same unit (e.g., X1 + Yi). as well as two distinct units (c.g., C1 + Y2).
We are ready define isolated inferaction model for interaction model M is the "ideal" model constructed from M* by eliminating all interactions between units.
Definition (Isolated interaction model IM* (IG', (I"*,IS*) IM (IG", (IG*,IS*) is the Isolated interac- tion model of an interaction model M (G' ,S*) if TM satisfies the following conditions:
1. 1G = G where G' is the graph obtained by removing from G all edges A, + B i
2. 1S = 5' where St is the set of equations obtained by removing from each equation X, = ((Pa(X))))))))) Pin St all terms containing any Y,, Vj # i.
For example, the interaction model ("* has Figure 2 as G*, and Equations 13) as part of S*. The isolated model for M denoted 7M* (IG", TS* TG is given in Figure below. And
TS* for Equations
3 are given by Equations (4 6
3.2 Symmetry Assumptions
In real-world applications, we will have at our disposal limited (usually just onc) observations corresponding unit which in turn will make it hard to draw useful conclusions if the model completely arbitrary. 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
If & c 1, the ACE is multiplied by the constant (t c).
2 PP((X)) denotes the parents of X; in G*.
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XI
X2
X3
X4
C1
C
C3
Co
X1 Ux.
(4)
Y1 X1 + +3C1 + Uy,
(5)
Y1
Y2
Y3
Y4
1/2 2X + Ut2
(6)
Figure 3: Interaction network with 4 units.
all units. While do not make strong assumptions such IID, we need to make certain weaker symmetry restrictions (definitions) 3 in order to quantify bias and identify ACF. We only require some of the variables III instead of all.
Definition 3 (Balanced interaction model (M" Let (G" S*)) be an interaction model with isolated model TM M is a balanced interaction model if TM has the same unit-model (TM, (TG (S;) )) for every unit i.
Let G be the graph in Figure and S* be the set of equations IG in Figure Jis the graph and IS* are the equations (4). (5) and 6 that correspond to IM", which is the isolated model of M*. The unit-graph for unit is different from unit 2. Also, the structural equations for Y, and Y2 of the isolated interaction model (Equations (5 and (6) are different. Hence,
corresponding to ("" ,S)).
M is not a balanced interaction model.
X1
X
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C1 UC1
C2 Uc2
Ci
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X1 C1 + Ux1
X2 C2 + Ux2
Y1 2X - C1 + + Uyi
Yi
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Y2
Y2 2X2 - C2 + UY2
(a) X satisfies ASDC.
(b) X does not satisfy ASDC.
(c) Structural equations for
Figure 4: Two balanced interaction networks and the structural equations for (a)
For another example, the interaction model M is balanced where G* the graph in Figure a and S* is the set of equations given in Figure C
Remark 1. Note that a balanced interaction model M does not imply that data generated by it are IID. Being balanced only requires all units share the same causal relationships within each unit itself. but permits interactions and effects from other units. For example, the parents of explicit variables Y: and y. # j can be different G i.e., Y; can be caused by a set of variables Sş corresponding to unit, k, and Y, can be caused by distinct set of variables Tk- However, for M* to be balanced it is required that for all distinct units and j. all Y; have the same relationship with i's explicit variables as Yj with j 's variables.
We further note that if M is balanced then all the unit-models IM; (IG;, IS,) in definition 3 are identical (with edges between IG; and IGJ), and can be succinetly represented by (single) causal model ( S) where and S can be constructed from any IG; and IS; by replacing explicit variables with generic variables.
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. For example, it is reasonable to assume all units' treatments have the same distribution, i.e., for any treatment X = x, all units have an equal chance of getting the treatment X = x.
Definition (Ancestral same-distribution condition (ASDC)). In the interaction network G a balanced interaction model, generic variable W satisfies the ancestral same cume-distribution condition (ASDC) for all unit i, I) Pa(W.) satisfies ASDC, and 2) Pa(W.) S V(gy- and 3) for any different unit j + i, Pa(W.) and Pa(W,) have the same set of generic variables, and their exogenous errors Uw: and Uw, have the same distribution. (When i=j, the condition is automatically satisfied.)

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For example, in Figures 4 and b). X satisfies ASDC in the former (assuming the condition on exogenous errors is satished) but not in the latter, since in the latter Pa(Xi) * Pa(X2). ASDC implies IID as stated in the following lemma.
Lemma 1. If V satisfies ASDC, then any two explicit variables W; and W are IID (Independent and Identically Distributed.)
Remark 2. The descendants of an ASDC variable need not be IID. For example. in Figure Caa. X satisfies ASDC, and Y; and , are descendants of X; and X3. Y; and Y3 have different sets of parents, making their distributions different, Y is non-IIl)
3.3 Quantity of Interest: True Average Causal Effect (TACE)
We generalize traditional ACE to the non-lID setting. Examine the interactions depicted in Figure ct Unit i's treatment X; affects their outcome through unit j's outcome Yj. X; Y3 Y; is "ppriouss" causal path. We are interested in computing the ACE a unit's treatment on their outcome, excluding the effects transmitted via spurious paths from their neighhors/contacts In experimental setting, interactions can be eliminated by isolating all subjects. 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. We present the formal definition below.
Definition 5 (True Average Causal Effect (T (T)CExy))) Let M be an interaction model. True average causal effect of X on Y, denoted as T TACExY, is defined the ACE of X on Y in the isolated interaction model IM" corresponding to M
TACE is the non-IID version of ACE and is the same as ACE in a traditional causal model where all samples are isolated. Again, without loss of generality, we assume the difference between the treatment value and the outcome value is 1, i.e. treatment is X = c 1 and the outcome is X c.
Defining, Quantifying, Detecting and Removing Interaction Bias for TACE
Almost all machine learning algorithms including those that employ causal techniques assume that data are IID (Schölkopf) [2022). section 3). In other words, the theoretical and performance guarantees of these algorithms are based data being IID. As such it would be useful to determine conditions under which an algorithm meant for IID data can be applied on non-III data with the certainty that the resulting bias would be negligible. We formally define interaction bias below.
Definition 6 (Interaction bias). Let balanced model M be the true model that generated the (available) non- dataset D. Let Q denote the query of interest and let Q' be its true value. Let A denote algorithm that outputs unbiased estimate Q given data that are IID and the causal graph that generated the HID data. Let G denote an approximate causal graph constructed under the assumption that D is IID such that no assumption in Gt is refuted by D. Let Q be the estimate computed by A using G and D as input. Interaction bias is given by I|Q" - Q11-
4.1 Quantifying Bias
We define the two main types of problematic graphical structures linear interaction network that introduces bias in the estimation of T A.E.
Definition 7 (Deflecting bias structure). A deflecting bias structure for T TACExY in an interaction network G an open path between X, and Y, foro + j.
Deflecting bias structures are open paths from one unit to another unit. For example, Figures 5 a and 5 b contain deflecting bias structures. The interaction network in Figure 5 has a directed open path between X, and Y,, and the interaction network in Figure 5 b) has a confounded open path between X, and Y,
Definition 8 (Reflecting bias structure). A reflecting bias structure for TACEXY in an interaction network G* is an open path between X; and Y, through some explicit variable W, with + j.
Reflecting bias structures are open paths that go from a unit through another unit and back to the same unit. For example, Figures and d) contain a reflecting bias structure. In each of them,
5

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there is an open path from X; to Y through Yj. In some cases, there can be deflecting bias structure embedded in reflecting bias structure, as in Figures O and However, this is not necessary.
Figure 5 e contains only reflecting bias structure (X; Cj but no deflecting bias structure.
X
Xj
X
X
X
X
X
Xj
C
C
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Uj
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Y;
Y,
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Y,
Y
(a) Deflecting bias: (b) Deflecting hias: (c) Reflecting bias: (d) Reflecting bias: (e) Reflecting bias Xj causes Yj through X, and y, are con- X, causes Y; through X; and Y; have con- only. founding path.
path.
founded.
Wy(= Y3 here).
Figure 5: Two main types interaction bias
Theorem 1. Let (G" 8*) be balanced interaction model in which treatment variable X; and outcome variable Y; are not confounded by any variable in Vú, Vi. Let D be the available data generated by M and let G be the approximate graph constructed using D. Let T Exx be identifiable in G and be given by By x, the regression coefficient of Y on X. Let a denote the true value T ACAECE in M If X satisfies ASDC then the interaction bias is given by.
E E E| - ISISH PEP
Val(p)
Σ Σ ISISA PEP
Val(p)
xx
where P(ija] is the of reflecting bias structures between X; and Y; through any explicit variable Wi of unit with + j. P[ji] is the set of deflecting bias structures between X, and Y; with i + j. and Rg is the root of path
follows from Theorem that in a balanced interaction model in which no X; and Y, are confounded by any variable in V,, the reflecting and deflecting structures the only two structures that will bias the identification
TACE. Note that although definition of interaction bias (Definition 6 on TACE is for any unbiased estimator for ACF, focus only the ordinary least squares estimator in this paper. This is because among the class of unbiased linear estimators, the OLS estimator has the minimum variance [Johnson al. 2014].
We exemplify theorem]
X1
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Y33
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Figure Interaction net- work with 3 units. (Other A, B. C variables including AI, Bı.... omitted)
Figure 6: Interaction network with 4 units. The numbers represent edge coefficients. (C1, C2, Ca, C's are omitted)
Figure 8: Interaction net- work with units. (C's omitted)
Example Figure shows an example of an interaction model with 4 units where XI...., Xs are the treatments, and Y .I...., Y, the outcomes. The numbers on the edges the edge coefficients. C satisfies ASDC, and C,
1,2,3, 3, 5 are omitted from the graph for simplicity.
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 (no parameter i.e., S* not an input), Output: the TACExY (should equal to 2). If we estimate ACExv ignoring the connections between units, our estimator will be By with Y .Y....., Ys) and X {X1.... ..)................... This is because ignoring the connections,
Xs}.

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the graph becomes X; + Y, separated for i = 1,....5, 5, so is essentially X
+ Y Pearl 20099)
However, by Theorem !
0.3 0.4
1
1
1 0.6
0.20 0.6-0.20
1
18vx 2|
0.5
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5
20
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x
20
#0.
Hence, the result is biased, and does not give us what we want. We show later in Theorem 2 how to compute an unbiased estimate of AC
4.2 Detecting Bias
In this section, we provide graphical criterion resulting from Theorem | to detect interaction bias.
Corollary 1. Let M (G* .S) be balanced interaction model in which X satisfies ASDC and TACE identiiied as xy = a in the approximate graph, then interaction bias exists iff G contains reflecting deflecting bias structure.
For example, Figure 7 contains both reflecting and deflecting bias structures. Figure g does not contain any bias structure. So Figure 7 has interaction bias and Figure 8 does not. Note that the interactions in Figure do not qualify as bias structures by Definitions and
4.3 Removing Bias
Theorem 2 presents technique for computing unbiased estimate of TACE in cases where theorem 1 predicts significant bias. It proceeds by applying linear regression on a set of samples B that satisfy the condition that bias inducing structures exist between any particular, a subset of samples/units B is termed as a bias-free subset for T ExACExy if no reflecting bias structures exist for any E S and deflecting bias structures exist in G where 3 is the latent projection of G* on B (Definition 2.6.1, Pearl 2009). For example in figure 6 B comprises of units 2 & 5 and G is X2 > Y2 X5 Ys- However, Another candidate for B is units 2 & 4 and the associated G is X2 + Y2 X₂ + Yı. An algorithm for constructing B is presented in Algorithm with an example and discussion in the appendix. This algorithm starts by randomly initializing B with sample. Then it goes through the rest of the samples and adds sample to its inclusion does not create bias structures in the resultant graph, G's-
distinct units and j. In
not unique for a given interaction network.
Algorithm 1 Select bias-free subset B from an interaction network G and return the largest subset from iterations
Input: interaction network G iterations
Output: the largest bias-free subset selected from iterations
1: function FINDSUB(G",
2:
B
3:
for = 1,...,t . do
4:
Units randomly sorted list L,...,
5:
B = nitss (The indices for Units start from 1) ........ .. 2......... do
6:
for i = 2,
7:
if Units(i [i] has no reflecting bias structure in G" then
8:
if U niis has no deflecting bias structure in G* with an element in B then
9:
B B B (i)
B B (B}
11:
return Largest B B
Theorem 2. Let G be an interaction network. Given the conditions in Theorem 7 and 'B' bias-free subset for G*, TACEXY E[Syx] where the regression coefficient is calculated using only samples in set B.
Note that bias-free subset of samples B used in Theorem is not always IID. While we insist that no reflecting or deflecting bias structures exist in G3,, do not restrict other forms of interactions

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among these samples. For example, in Figure 8 Units {1. 2. 3) constitute a bias-free subset. In this case, Y is not IID (Y₁ and Y2 are dependent, 22 and Y3 are dependent) and hence the bias-free subset is non-III.
Also note that compute an unbiased estimate using Theorem 2 have at our disposal smaller set of samples; so the variance of estimation will be larger. There is a trade off between ignoring interaction (large bias, small variance), and using theorem (no bias, large variance). remains future work to quantify the variance of the estimator in Theorem 2 for different interaction models, but in Section we provide simulation results and case analysis study to empirically show its performance.
Applicability of theorems & 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. Theorem IT 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. This in turn allows to use available information about the problem from prior experience, domain knowledge or external sources to determine if bias would be negligible or not. Specifically, bias becomes smaller as the number of bias-structure-Fi free samples increases. In fact, if the numbers of deflecting and reflecting structures are fixed, the bias terms diminishes as increases, indicated by the 1/n for the reflecting bias term and 1 1/n(n 1) for the deflecting bias term. It is also evident that if the values of path coefficients high, Val (p) would be high and this will result in increased bias. 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.
Experiments
5.1 Simulations
Simulated Model We randomly generate balanced interaction network with n units (i.e., the sample size is n), with C + X; + Yú and X; - M; for all = I,..., n. For all ordered pairs of distinct units j, we randomly add deflecting bias structures in the form of X, + C₁ - Y, with probability dRate. For all units i, randomly add reflecting bias structures in the form of X; + Mr + Y. with probability Rate for a random *
Bias of REG It follows from theorem I that larger sample sizes and smaller path values on the bias structures result in smaller bias. We perform two simulations to show how bias varies as a function of sample size and path values. We simulate data such that for each variable, the exogenous error term follows a Gaussian distribution with mean and standard deviation 1. For each set of parameters, we randomly generate an interaction network, and simulate the data 10000 times. Each time, we record the result from a naive regression of Y on X (REG). As comparison, we also record the result from Theorem 2 (THM-2). We run the algorithm (provided in the appendix) to randomly select bias-free free subsets for 10 times and select the largest subset.
Experiment:
Simulation 1: X; - Y''s edge coefficient is 100, the edge coefficients of C. + X,, My + Y; are all set to 10, the numbers of deflecting bias structures and additional reflecting bias structures are both 100.
Simulation 2: Number of units n = 1000, X; + Y,'s edge coefficient is 100, the numbers of deflecting bias structures and additional reflecting bias structures are both 100. The results are plotted in Figure 9 As seen in the plots, as increases or the path values on the bias structures decreases (both with all other parameters fixed), fy from naive regression approaches TACE. Such results coincide with Theorem The Byx computed by THM-2 is very close to TACE and the two lines almost overlap.
5.2 Case Study
Settings We are interested in analyzing the effect of tutoring time on students' grades. 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. For instance, unit i might help unit understand the course materials better which in turn might improve j's grade. 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. To construct interaction network and apply
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180 2002 Mod 1000 Bid
500
Figure 9: Left: x vs. number of units na, Right: By.x vs. path value on the bias structures. TACE 100.
435
200
3.2
-2400
3000
Figure 10: Left: estimated T TACE distribution from THM-2. Right: estimated 7 TACE distribution from REG.
our results, we ask the students to out survey including 1) their tutoring time, 2) their grade, 3) whom they helped, 4) who helped them, 5) peer review score.
Construction of the Interaction Network Three generic variables are I (tutoring time in hours). U/ (understanding of course materials), and R (grade). For each unit i, T; - U, - RG In addition, if i helped j, add U, - U, (deflecting bias structure). If first helped j and j mentioned this in the peer review and thus boosted i's grade, add U; U, - R; (reflecting bias structure). We assume no additional back-and and-forrthh help happens.
Simulation Let there be 500 students, assume each student on average help 5 other students, and the other student has 0.5 chance of helping back. Let 7 TACE 2, and the U; - Uj and U; - R₁ edges both have the value 2. We randomly generate an interaction network and simulate data based on these parameters.
Results We apply THM-2 to select a bias-free subset, and compute 3gr using data from that subset. We get the result 1.963, with the size of the subset 72. As result, the effect of tutoring time students' grades not through other units is estimated to be 1.963, which is close to the ground truth TACE (2). We further repeat the experiment 1000 times to show the distribution of the results. Each time a random structure is generated and random data are simulated. THM-2 is on average able to select bias-free subset of size 76, and the average recovered TACE 2.0002. The result from REG had a significantly high bias with TACE averaging at 194.11. Also since every time the data are regenerated, the model is different, and REG uses all the data, has a larger variance. The two plots in Figure [0 show the distribution of results from THM-2 and REG. The histograms of the results of By x computed by THM-2 and REG are shown in Figure I0
Related Work
One of the most studied concepts related interactions among units interference Cox 1958 Majority of literature in empirical fields assume no-interference. In fact, SUTVA is a common assumption in causal inference [Rubin] 1978). Recent years have witnessed rise in papers on interference that employ graphical models. These include Ogburn and Vander Weele [204]] that was first to model interference using DAGs, [herman and Shpitser [2018] that modeled interference using chain graphs which permits modeling unknown interactions between units and [Bhattacharyl al. that proposed structure learning methods for chain graphs. These works rely on partial interference which divides units into equal-sized blocks under the assumption that interactions occur

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only within a block but not across different blocks. 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 dl-placceents
(006) was the first to notice the effect of interference in the housing mobility problem, and proposes causal estimands for this application.
Aronow and Samii 2017]. Sussman and Airoldi [2017] modeled general interference (without assuming partial inference) by constructing a function define a unit's exposure level on the number of treated neighbors they have. The methods are less restricted than partial interference methods, and allow units be affected by any number of neighbors. However, they are limited to interference and do not handle other forms of interactions.
Dagadeesan al. 20200 proposed quasi- method to estimate direct effect under interference using experimental data. However, it does not easily generalize to observational studies. Other papers along similar direction include Faatemi and Zheleva 2020), which proposed experiment design to minimize interaction bias and selection bias at the same time, Liu Liu and Hudgens [2014), which proposed a two-stage randomization design minimize interference bias. [chetgen Tchetgen et al.) [202]] 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. Sävje et al. [022]] and Hudgens and Halloran (2008) defined queries similar to TACE, named EATE and PADE, respectively. These queries generalize traditional ACE allow a unit's outcome to be affected by treatments of other units. However, they do not allow outcomes to be affected by other units' variables other than treatments.
Hudgens and Halloran 2008 defined six types of queries in the problems involving interference. Work in interference that focuses different queries/problems include few follows. VanderWeels ct al. 2012bb 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. Shpitser et al. [2117] presented decomposition for units with unknown and symmetrical interaction patterns and analyzed different interference paths. In linear models, the contagion and infectiousness effects reduce to the directed paths from X₁ to Yg. Moreover, their work does not handle reflection bias. Hu et al. [2021] was the first define and provide estimands for the average indirect effect. Vander Weele et al. [2014] developed methods for sensitivity analysis under interference.
Other types of interactions include the contagion effects, which defined a unit's outcome affecting another unit's outcome VanderWeele and An [2013]. Work on this line usually used longitudinal data, including [Burt] [1987] Lyons] [2011]. VanderWeele al.] [2012a]. Homophily effects are defined as the behavior of connected units are similar (Jagadeesan et al. 2020). Work in
this line include McPherson et al. (2001) Jagadeesan et al. [2020). The existing work above does
model interactions using graphical models.
7 Conclusions
In this paper, we represent interactions among units using causal graphical models. We derive theorems to quantify the interaction bias for average treatment effects in linear models. We provide sufficient and necessary graphical conditions to detect interaction bias. Additionally, we develop method to compute an unbiased estimate of causal effect in cases where blindly assuming IID expected to yield significant bias. Finally, we discuss the performance of our method through simulation studies.
Acknowledgments and Disclosure of Funding
We would like to thank anonymous reviewers for their comments, and Totte Harinen and Rumen Iliev for helpful discussions. This research was supported in parts by grants from the National Science Foundation [#I[S-2106908 and #2231798), Office of Naval Research [#N00014-21-1-2351 and Toyota Research Institute of North America [#PO-000897].
10

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Fredrik Sävje, Peter M Aronow, and Michael G Hudgens. Average treatment effects in the presence of unknown interference. The Annals of Statistics, 49(2):673-701, 2021.
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Ilya Shpitser, Eric Tchetgen Tchetgen, and Ryan Andrews. Modeling interference via symmetric treatment decomposition. arXiv preprint 170.001700.01050. 2017.
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12


Reviews:
[{"review_id": "hWaO6wDGhxo", "reviewer": "Reviewer_wxC8", "summary": "This paper studies causal inference from non-iid data when there are interactions among units, in the case of linear causal relationships. In particular, it analyzes different types of interactions, detects and quantifies the corresponding interaction bias, and derives conditions under which it is safe to ignore interactions and estimate the causal effect with methods that are designed for iid data.", "questions": "I would like to suggest the authors improve their writing. For experimental studies, it would be more convincing if the authors could consider different types of interactions that are considered in previous sections.", "limitations": "The authors did not discuss the limitations and potential negative societal impact of their work", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nStrengths:\nThe studied problem is very interesting and meaningful because, in many real-world problems, the iid assumption is hard to hold. As far as I know, this work is original.\n\nThe authors show theoretical results on detecting and quantifying interaction bias, as well as a technique for removing the bias.\n\nWeakness:\nThis paper does not seem easy to follow; there are too many definitions, and the writing also needs improvement. \n\nFor the notation Xi, sometimes i is written as a subscript, but sometimes it is not.\n\nThe experimental studies do not seem enough.\n\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "bicGO11SowT", "reviewer": "Reviewer_bAan", "summary": "The goal of the paper is to study non typical forms of interference that cause iid violation and bias, and how these affect Average Causal Treatment Effects estimates. Moreover the authors propose set of conditions which when they hold true, violation of iid assumption when using an algorithm meant for iid will result in negligible bias of the estimated ATE.\n", "questions": "\nWhile the paper's ultimate goal seems very relevant for real datasets, i would like to ask the authors some questions about some points:\n1. In Figure 5 I do not understand why figures 5c-5e do not contain a deflecting path as well. According to the open path definition (no collider), in all these graphs there is \nan open path from X_j to Y_i.\n\n2. Does the proposed method assume that the user almost needs to know the full graph, and if so, don't we then already know if there is interference or not?\n\n3. Could you please clarify what is meant by \"no parameter i.e., S* is not an input\" in line 207?", "limitations": "No limitations are discussed by the authors. I do not see any limitation with respect to societal impact for this work.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n\nThe paper is well written  and with gradual introduction of the terminology needed for the reader. Also it addresses a very interesting statistical problem, relevant for many fields.\n\nWeaknesses:\n\nThe interaction bias formula contains the variance of the root path node R_p. This means that the authors assume that R_p must be known in a real dataset.\nI would like the authors to elaborate how realistic is this requirement, given that in many use cases the main problem is to identify the root cause of a target. \n\nI would like to see application on some real-dataset if possible. Also the comparison with Regression in the simulated graph is the absolute minimum. Could the authors include comparisons of their method with some state of the art ATE method?\n\nSome comments about structure: \n- Captions of Figure 5, 6 and 7 cannot be read, as they are very close to each other. Please, add appropriate spacing.\n- line 307, \"some polots above' not 'below'.\n- Minor typos: drop 'to' line 134, l.115 is -> are\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Ma92q6kiD7e", "reviewer": "Reviewer_xxrS", "summary": "The paper studies the causal inference problem with interference under a linear model. It assumes known interference structure, and present conditions under which the bias due to ignoring interference is negligible. It also presents a linear regression method on a set of samples that satisfies the condition that no bias-inducing structures exist between any two distinct units.\n", "questions": "Is it possible to illustrate the main idea, and under what conditions you can identify/remove the bias, without resorting to these new terminologies you introduce? \n", "limitations": "The linearity assumptions seem rather strong and do they ever hold in practice (for the whole network)?\n\nAlso the same for the other assumptions. More discussions on when one may be able to detect and remove the bias in a real context would be very helpful.", "rating": 3, "confidence": 1, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I am not aware of many previous papers tackling this problem using the language/perspective that the authors use, so that seems something novel. On the other hand, my feeling is that the authors introduce too many terminologies so that the main idea is buried behind these notations.\n\nThe authors motivate the discussion with an example of COVID-19 but finish with a simulation (!) on student grades. Why not make it consistent? This is a simulation anyway.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept