Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference

Thank you for the valuable feedback.

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For example, in Eq. 2, the subscript should be used to distinguish the orders of variables, specially, in the left-hand side of the equation.

In Eq. (2) included in Definition 3, $\mathbf{X}$ and $\mathbf{Y}$ are already defined as $(\mathbf{X}_1,\cdots ,\mathbf{X}_m)$ and $(\mathbf{Y}_0,\cdots ,\mathbf{Y}_m)$, respectively.

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(1) Definition 3 seems incorrect since the presentation 'Z is said to be a sequential adjustment set relative to (X,Y) in G if…' is difficult to understand. (2) How to determine a sequential adjustment set from G?

1. That is a standard and established way to express it. Similar expressions can be found in existing works [1, 2].
2. Section 4.1 and Section 4.2 address methods for constructing $\mathbf{Z}$, focusing on how to determine the sequential adjustment set.

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How to understand “the causal effect is given as an adjustment”?

It means that the causal effect (left-hand side of Eq. (13)) is expressed as covariate adjustment (right-hand side of Eq. (13)). For further explanation of the adjustment criterion, please refer to lines 73-75 of our paper or the existing works referenced in lines 17-27.

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The conclusion in Definition 4 is not readable

It would be greatly appreciated if you could specify which part you find "not readable," so that we can better address your comment.

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What type of causal graph satisfies the proposition 3? Can you provide a type to summarize this kind of causal graph?

In lines 142-154, we illustrate such an example immediately following Proposition 3, where mSBD is not satisfied, but the causal effect can still be expressed by SCA.

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Further, the causal graph should be given and the lack of extensive validation with real-world data sets might be seen as a limitation.Y

Our paper falls into the causal effect identification category in which the key task is to express the causal effect as a function of observational distribution using assumptions encoded in a causal graph. Therefore, we disagree that the presence of causal graphs is a limitation of our work.

As you noted, our work is theory-oriented, presenting a complete criterion for sequential covariate adjustment and introducing an algorithm for constructing a (minimal) sequential adjustment set. Our focus has been on clearly elucidating the theories and algorithms through appropriate examples. We believe that simulation studies or experimental validations would not enhance our theoretical work. However, to demonstrate the practical benefits of our theories, we will provide implementation code during the revision process.

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<Reference>

[1] Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669–710.

[2] Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press, New York. 2nd edition, 2009.

Thank you for the valuable feedback and detailed comments.

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For example, in Definition 3 the authors consider H_i and in Eq. (2) they use in the formula h_{j-1}. So, what is the definition of h_{-1} and of h_{0}? Note that according to the definition of H_i, for H_{-1}, the sets X^{(-1)} and Y^{(-1)} are undefined, and for H_{0}, the set X^{(0)} is undefined. It seems that the authors mean, in such cases H_i are empty. Also, some sets Y^{(i)} can be empty. These issues should be clarified.

Thank you for your feedback. We implicitly defined a set whose indices are outside their original range as an empty set. For example,

1. In Eq. (2), all the subscripted values whose indices are out of range (e.g., $\mathbf{x}\_0$, $\mathbf{z}\_0$ and $\mathbf{z}\_{m+1}$ are considered as an empty set.

2. In Eq. (16), it implies that we go over index j such that $1 \leq j \leq i-1$. Hence, if $i=1$ implies that there is no union, i.e., an empty set will be returned for the term.

We will explicitly state this in the revised version of the paper.

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The formula presented in Eq. (2) for sequential covariate adjustment is different than the formula in (Jung et al., 2020, Theorem 1) used to identify causal effect by mSBD adjustment. Could you show that they are equivalent?

Here is the proof of the equivalence, as derived in line 393 in the Appendix:

$\underbrace{P_\mathbf{x}(\mathbf{y}) = \sum_\mathbf{z} \prod\_{j=0}^n P(\mathbf{y}_j \mid \mathbf{x}^{(j)},\mathbf{z}^{(j)},\mathbf{y}^{(j-1)})\prod\_{k=1}^n P(\mathbf{z}_k \mid \mathbf{x}^{(k-1)},\mathbf{z}^{(k-1)},\mathbf{y}^{(k-1)})}\_{ \text{Jung et al.,2020, Thm1}}$

$\qquad \quad = \sum\_\mathbf{z} \prod\_{j=0}^n P(\mathbf{y}_j \mid \mathbf{x}_j,\mathbf{z}_j,\mathbf{h}\_{j-1})\prod\_{k=1}^n P(\mathbf{z}_k \mid \mathbf{h}\_{k-1})$

$\qquad \quad =\sum_\mathbf{z} \prod\_{j=0}^n P(\mathbf{y}_j \mid \mathbf{x}_j,\mathbf{z}_j,\mathbf{h}\_{j-1})P(\mathbf{z}\_{j+1} \mid \mathbf{h}_j)$

$\qquad \quad =\underbrace{\sum\_\mathbf{z} \prod\_{j=0}^n P(\mathbf{z}\_{j+1},\mathbf{y}_j \mid \mathbf{h}\_{j-1},\mathbf{x}_j,\mathbf{z}_j,) = P(\mathbf{y} \mid do(\mathbf{x}))}\_{\text{Ours Eq. (2)}}$

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Write explicitly how do you define sets H_i in Definition 5. In the same way as n Definition 3?

Yes. We will invoke the definition of $\mathbf{H}_i$ in Def. 3 to Def. 5.

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Why do you consider in Definition 3, 4, etc. in the sequence X = (X_1, . . . ,X_m) the elements X_i as sets? In fact every X_i denotes here a single variable.

This is because a set serves as a basic unit for graphical operators like $Pa$, $An$, and so on. Furthermore, by treating $X_i$ as a set, we can maintain consistency in handling $X_i$, $\mathbf{Y}_i$, and $\mathbf{Z}_i$ simultaneously.

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Eq. (7) is not correct, resp., it does not correspond to the definition given in Eq. (2) in Definition 3 as well as formula used in Eq. (6). Specifically, in the factor P(y2 | x1, x2, z1, z2) in Eq. (7) variable y1 in the conditioning set is missing. Note that in this example we have: Y1 = {Y1}.

Thank you for catching these typos. We will make the necessary corrections.

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then X ∪ Y ∪ H_{i−1} in (15) is equal to X ∪ Y ∪ Z^{(i)}. So why do you use H_{i−1} in (15)?

We use $\mathbf{H}\_{i-1}$ in (15) to main comprehensibility. Specifically, Equation (15) is read as follow -- the forbidden set is an union of (1) treatments, (2) outcomes, (3) predecessors ($\mathbf{H}\_{i-1}$), (4) dpcp, and (5) a descendent set.

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In Definition 7: (Sequential Adjustment Criterion) --> (Sequential Adjustment Criterion (SAC))

We will revise as suggested. Thanks.

We thank the reviewer for the time and valuable feedback!

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The work does seem to be a simple extension of AC to the sequential case.

Our proposed method is an extension of the AC, similar to how the sequential back-door adjustment extends the back-door adjustment. The principle of our method is to open the proper causal paths and block the proper non-causal paths. However, we disagree that the extension procedure is simple.

As the saying goes “The devil is in the details”, there are nontrivial challenges when extending the AC to the SAC:

1. Unlike the adjustment case, partitioning the variables is essential. Choosing a proper partition that preserves causal interpretation (e.g., $\mathbf{X}_i$ causes $\mathbf{Y}^{\geq i}$) is nontrivial.

2. There is a significant gap between conjecturing and actually witnessing the soundness and completeness of SAC. Specifically, the formal approach for witnessing the soundness and completeness of AC proposed by [1] is hardly generalizable to SAC cases. The original proof of AC by [1] uses the counterfactual network [2] graph to show the soundness and completeness of AC. Such a proof technique is hardly generalizable for sequential cases, since, by the nature of the counterfactual network, the number of treatment variables to be considered in the proof increases exponentially with the number of treatments. To circumvent such challenges, we devised a new proof technique that can be commonly applied to both AC and SAC.

3. One naive approach is to condition the previous set $\mathbf{H}_{i-1}$, and apply the AC with respect to $(X_i, \mathbf{Y}^{\geq i})$ in the graph where the previous set is conditioned on. However, the graph induced by conditioning the previous set may not be a DAG (with bidirected edges), even if the original causal graph is a DAG. This happens when the previous set contains colliders [3].

Circumventing such challenges while adhering to the principle—to open the proper causal paths and block the proper non-causal paths—is a nontrivial problem requiring formal treatments, as we have done in this paper.

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For an adjustment set Z where mSBD holds, SAC holds as well (corollary 1) but can SAC provide a smaller adjustment set?

Just to clarify, please note that mSBD and SAC determine whether a causal effect can be expressed as Sequential Covariate Adjustment (SCA) when $\mathbf{Z}$ is given, rather than constructing $\mathbf{Z}$ itself. Algorithm 1 (minSCA) is designed to provide a smaller (or equal) adjustment set such that no strict subset of it is a valid adjustment set.

For example, consider Figure 4a. Here, $\mathbf{Z} = (\emptyset, \{Z_a, Z_b, Z_c\})$ satisfies the mSBD criterion (and SAC) relative to $((X_1, X_2), (\emptyset, Y))$. However, as discussed in lines 301-308, Algorithm 1 (minSCA) yields $\mathbf{Z}^{min} = (\emptyset, \{Z_a\})$ and this set also satisfies the SAC. This example illustrates that the minSCA procedure can yield a smaller adjustment set.

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In the example of Figure 3, doesn't SAC collapse to be the same as AC?

In the example, the set $\mathbf{Z} = \\{ Z_a,Z_b \\}$ satisfies AC, and $\mathbf{Z} := (\\{Z_a,Z_b\\},\\{\\})$ satisfies SAC. In other words, the set $\mathbf{Z} = \\{ Z_a,Z_b \\}$ is a valid admissible & sequentially admissible set. However, this observation doesn't hold in general. For example, $\mathbf{Z} := (\\{\\},\\{\\})$ also satisfies the SAC in the same graph. However, the AC is not satisfied, because $\mathbf{Z}$ fails to block the path $Y_2 \to Z_b \to X_2$ in the proper back-door graph $\mathcal{G}_{\text{pbd}}^{\mathbf{X},\mathbf{Y}}$.

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Definition of proper causal paths is different from previous papers (eg Shipster 2010), would be good to give intuition here. The definition of proper causal path may not be strictly true.

Our definition is comparable with the definitions in the previous papers as follow. First, a causal path (equivalently, a directed path) is said to be proper if it does not intersect $\mathbf{X}$ except at the end point [1]. Equivalently, the path is proper if only its start node is in $\mathbf{X}$ [4]. These definitions match with our definition in line 78, which states that the path does not contain a node in $\mathbf{X} \setminus \\{ X \\}$. In other words, the path starting from $X$ is not overlapped with $\mathbf{X}$ if and only if the path doesn't contain a node in $\mathbf{X} \setminus \\{ X \\}$. Therefore, our definition is equivalent to the definitions in the previous papers.

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References for definitions in Section 3?

We presume that you are referring to Definition 3 (SCA). We will refer to it just before Definition 3, which we already referenced sequential covariate adjustment in lines 28-29 of the Introduction.

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H_i should be redefined in Definition 5 to make the definition complete, or refer back to where it is defined

Thank you for the suggestion. We first note that $\mathbf{H}_{i-1}$ has been defined in Def. 3. We will invoke this definition in Def. 5.

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<Reference>

[1] Shpitser, I., VanderWeele, T., and Robins, J. M. (2010). On the validity of covariate adjustment for estimating causal effects.

[2] Shpitser, I. and Pearl, J. (2008). Complete identification methods for the causal hierarchy. Journal of Machine Learning Research, 9:1941–1979.

[3] Richardson, T., & Spirtes, P. (2002). Ancestral graph Markov models. The Annals of Statistics, 30(4), 962-1030.

[4] van der Zander, B., Li ́skiewicz, M., and Textor, J. (2014). Constructing separators and adjustment sets in ancestral graphs. In Proceedings of the UAI 2014 Conference on Causal Inference: Learning and Prediction-Volume 1274, pages 11–24.

Thank you for sharing your thoughts and feedback with us.

Simulation studies are not provided, though it is fine for a theory-focused paper.

As you noted, our work is theory-oriented, presenting a complete criterion for sequential covariate adjustment and introducing an algorithm for constructing a (minimal) sequential adjustment set. Our focus has been on clearly elucidating the theories and algorithms through appropriate examples. We believe that simulation studies or experimental validations would not enhance our theoretical work. However, to demonstrate the practical benefits of our theories, we will provide implementation code during the revision process.

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The computational complexity

1. The proper sequential back-door graph (in Def. 6) can be constructed in $O(|\mathbf{V}|+|\mathbf{E}|)$ time [1].

2. Provided the partition of $\mathbf{Z}= (\mathbf{Z}_1,\cdots,\mathbf{Z}_m)$, checking if such $\mathbf{Z}$ satisfies the SAC in Def. 7 takes $O(m|\mathbf{V}| + m|\mathbf{E}|)$, since checking each d-separation takes $O(|\mathbf{V}| + |\mathbf{E}|)$.

3. Constructing $\mathbf{Z}^{an}_i$ in Theorem 3 takes $O(|\mathbf{V}| + |\mathbf{E}|)$ since finding the set dpcp, descendent and ancestor sets take $O(|\mathbf{V}| + |\mathbf{E}|)$.

4. As mentioned in line 286, finding closure can be done in $O(|\mathbf{V}| + |\mathbf{E}|)$.

5. Combining all of these analyses, the computational complexity of the proposed method is $O(m|\mathbf{V}| + m|\mathbf{E}|)$.

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<Reference>

[1] van der Zander, B., Li ́skiewicz, M., and Textor, J. (2014). Constructing separators and adjustment sets in ancestral graphs. In Proceedings of the UAI 2014 Conference on Causal Inference: Learning and Prediction-Volume 1274, pages 11–24.