Paper: Causal Estimation of Exposure Shifts with Neural Networks: Evaluating the Health Benefits of Stricter Air Quality Standards in the US (ICLR-2024)

Reviewer comment (Theory):
I am not fully convinced about this causal estimand. In Section C's example, why $c$ is a better number than 0? Can the authors clarify?

Author response:
The application generally sets the sensible choice of shift value $c$. Using the example of Section C asked by the reviewer, $c=1.1$ asks the question of what would the average outcome be under a uniform 10% increase in the treatment; $c=0$ is a degenerate case in which everybody receives treatment $\tilde{A}=0$; $c=1$ corresponds simply to the observed average outcome without counterfactuals. It is generally the role of the scientist or policymaker to choose the shift value under consideration. For example, in the application solved in our paper, it is sensible to ask what the effect reducing or increasing by PM 2.5 air pollution by 10% would be (Fig. 2b). The cutoff shift, also used in the application, is another example (Fig. 2a) in which a maximum threshold is allowed. The relevant shifts, $8,9,10,11 \mu g/m^3$, analogous to $c$ with a different type of shift, are set by the US Environmental Protection Agency.

Reviewer comment (Theory):
Can the shift of treatment be stochastic?

Author response:
There is no problem with the shift being stochastic as our paper is currently formulated. Our key assumption is that we observe pairs $(A, \tilde{A})$, with the latter measurement potentially being a stochastic draw. This flexible setup allows us to address the application problem. We point out to the reviewer the clarity improvements that we made to Section 2 of the revision introducing the estimand and clarifying the reviewer's question.

Reviewer comment (Novelty):
Some key references on doubly robust estimator and efficiency of causal effect estimation are missing, for example, Robins's work.

Author response:
The revision has added various relevant works by Robin. However, do note that these previous works concern traditional treatment effect estimation and do not cover the case of estimating the effect of an exposure shift (SRF). Our paper addresses this problem building upon the targeted regularization framework (Nie et al., ICLR 2021; Shi et al., NeurIPS 2019). The SRF scenario is sufficiently different from traditional causal effect estimation, requiring additional theory and methods.

Reviewer comment (Writing):
Should $\mu(x,a)$ be (1) $\mu(X,\bar A)$?

Author response:
It should be $\mu(X, \tilde{A})$.

Reviewer comment (Writing):
There are typos, for example, in Section C, EIF should be ERF; some references are broken.

Author response:
We have fixed this typo and the broken reference in the revision. Thank you for your careful review.

Reviewer comment (Novelty):
The theoretical result is not particularly surprising given the existing literature on double-robustness and efficiency (although I do like the application side of this work).

Author response:
Thank you for your positive remarks highlighting the application’s strengths and the paper’s clarity.

As a point of clarification, our framework builds upon previous literature on targeted regularization (TR), particularly Nie et al. (ICLR 2021). Previous results in the TR literature and our theoretical results may not be entirely surprising from the point of view of the long tradition of double robust estimation in causal inference. Nonetheless, such theoretical results are required to ensure correctness in the novel framework of TR. Notice as well that compared to previous work in the TR literature, we address the problem of estimating the effect of an exposure shift (SRF). This scenario is sufficiently different that it requires additional theory and methods (e.g., see Diaz et al. (2021) and Diaz & Van der Laan (2012)). Our paper rigorously extends the TR literature to SRFs while solving an application of urgent importance, providing the first application of targeted regularization in a critical public health application.

Reviewer comment (Experiments):
I am in general curious about the reason for choosing neural networks to fit the propensities and the outcome model. How do they compare with, say, tree-based methods?

Author response:
1. Earlier research on TR by Nie et al. (ICLR 2021) and Shi et al. (NeurIPS 2019) gave experimental results suggesting that TR can outperform tree-based methods using the AIPW, TMLE, BART and causal forest frameworks. Tree-based methods are incompatible with TR since they are inherent to gradient-based methods. However, we believe that using neural networks or tree-based methods is ultimately a matter of convenience and the nature of the data, as with many decisions in machine learning. Tree methods are generally easier to train and are more resilient to overfitting, but neural networks can perform greatly when correctly specified and are compatible with other forms of learning and regularization. Multi-modal systems with neural network building blocks are gaining importance (text, images, time series).

Reviewer comment (Evaluation):
In many scenarios, the multiple shifts are being considered simultaneously, should there be adjustment for the multiplicity?

Author response:
2. Eq. (8) includes a normalization by $|\tilde{P}|$, the number of exposure shifts considered. Is this the type of normalization considered by the reviewer?

Reviewer comment (Writing):
In equation (1), should $\mu(x,a)$ be $\mu(X,\tilde{A})$?

Author response:
3. Yes, it should be $\mu(X, \tilde{A})$. We have fixed the missing citations; thank you.

Reviewer comment (Presentation):
There is a missing reference at the bottom of page 5.

Author response:
3. Yes, it should be $\mu(X, \tilde{A})$. We have fixed the missing citations; thank you.

Reviewer comment (Writing):
The problem of this paper was not well presented. For example, the key concept of exposure shift is very confusing. The notation $\tilde{A}$ is used first without definition in Section 2. How is the potential outcome framework defined under $\tilde{A}$? The equation (1) is also problematic as it should be $a\sim \tilde{p}(\tilde{A}|X)$.

Author response:
1. Section 2 in the original manuscript introduced $\tilde{A}$ as a variable to denote the shifted exposure, and examples were provided after eq (1). Nonetheless, the revision has improved and expanded Section 2, paragraph 3, to improve clarity. We hope the reviewer finds the updated version clearer.

Reviewer comment (Theory):
The assumptions of this work also need more justifications. It looks like all the causal identification assumptions are based on the original treatment $A$ except the positivity assumption. Since the efficient function also contains $\mu(X, $\tilde{A}$)$, would more assumptions on $\tilde{A}$ be needed like SUTVA?

Author response:
2. Theorem 2 in the original manuscript imposed the additional restrictions required on the class of admissible density ratio functions $w(x,a)=\tilde{p}(a|x)/p(a|x)$. To improve clarity in the revision, we modified Assumption 2.2 and its explanation to give more intuitive and direct conditions tied to the regularity of the density ratio $w$ used for SRF estimation. Note that we provide mathematical proofs of all statements, justifying their definition. Assumption 2.2 is now in line with previous related work, specifically, Diaz & van der Laan (2012).

Reviewer comment (Novelty):
The proposed method is not new compared with the semiparametric literature and causal inference, by considering double robustness and the density ratio of two propensities. Please find the references below and justify them.

- Yang, Shu, and Peng Ding. "Combining multiple observational data sources to estimate causal effects." Journal of the American Statistical Association (2019).  
- Kallus, Nathan, and Xiaojie Mao. "On the role of surrogates in the efficient estimation of treatment effects with limited outcome data." arXiv preprint arXiv:2003.12408 (2020).

Author response:
3. We reviewed the suggested citations, Shu & Ding (2019) and Kallus & Mao (2020). However, we did not find their immediate relevance to the problem addressed in our paper and the literature we are building upon. These works use auxiliary data (which we don’t use), and their aim is traditional treatment effect estimation comprising the average treatment effect (ATE) for binary treatments and the exposure-response function (ERF) for continuous treatments, as explained in Section 2 and Appendix C. Our paper addresses an application in which such traditional effect estimation is not informative, and estimating the effect of an exposure shift is necessitated instead. There are formulas involving density ratios in these works, but they have a different purpose and are defined in terms of other quantities, namely causal estimation with partial unmeasured confounding and causal inference with surrogate outcomes.

Reviewer comment (Theory):
The theoretical connections between Section 3 and Section 4 are weak. There are many theoretical works related to using neural network methods for nuisance function estimation. The authors may consider the following reference to complete the gap.

- Farrell, Max H., Tengyuan Liang, and Sanjog Misra. "Deep neural networks for estimation and inference." Econometrica 89.1 (2021): 181-213.

Author response:
4. The revision includes the recommended citation by Farrel et al. (2021) in Section 4 for useful background, as suggested. Notice that the architectures discussed there concern traditional causal effect estimation (ATE and ERFs): they cannot be used to estimate the ratio between generalized propensity scores ($w$ in our paper), nor do they allow for double robustness of SRFs, as our proposed method does. The SRF scenario requires additional theory and methods beyond traditional causal effect estimation (Diaz et al. 2021; Diaz & Van der Laan, 2012). The extended paragraph also indicates that we based our architecture on previous work in targeted regularization (Nie et al. 2021), focusing on the required extensions for density ratio estimation.

Reviewer comment (Theory):
Since the efficient influence function is derived, given Theorem 2, I am curious why not continue to get the asymptotic normality of the proposed effect? Specifically, the authors are using the asymptotic normal formula in their simulations. Or if the authors think it is challenging, why can you use this result directly in the simulation?

Author response:
5. The targeted regularization literature that our paper builds upon has yet to establish theoretical results and experiments concerning asymptotic normality. Notice that these works have been published at similar venues (Nie et al., ICLR 2021; Shi et al. NeurIPS 2019). We agree that future work in targeted regularization methods and theory should prioritize the investigation of these properties, but it is beyond the scope of our application paper.

Reviewer comment (Experiments):
Since the double robustness is one major advantage of the proposed method, it is better to conduct simulation studies to reflect this property.

Author response:
6. Similarly, our evaluation criterion aligns with the literature we build upon. As in our answer to 5, we agree that double robustness could be examined with specific experiments, but that would go beyond the scope of our application paper and would require a paper on itself to examine the properties of *all* targeted regularization methods that have been proposed.

Reviewer comment (Presentation):
Mics: The reference at the bottom of page 5 is missing.

Author response:
7. We have fixed the broken reference; thank you for identifying it.

Reviewer comment (Theory):
Is Assumption 2.2 sufficient? If $p(a|x) \propto 1/n$ for some $a$, then the results do not hold, I think $p(a|x)$ should be lower bounded by a positive constant independent of $n$.

Author response:
Additional regularity conditions stabilizing the density ratios were included as part of Theorem 2 (T2) since they weren’t required to present the key concepts. Specifically, assumption (ii) of T2 requires that the density ratio class $\mathcal{W}$ and its reciprocal have bounded $\infty$-norm. In the revision, we have reworded Assumption 2.2 to more clearly indicate the needed positivity assumptions on the density ratio function $w$. Assumption 2.2 is now in line with previous related work, specifically, Diaz & van der Laan (2012).

Reviewer comment (Writing):
In Theorem 2, what does $\to$ indicate? Convergence of non-random variables or convergence in probability?

Author response:
The arrow denotes convergence in probability (updated in the revision).

Reviewer comment (Theory):
In Theorem 2, what does $O(r_1(n))$ in $\|\hat{\mu} - \mu\|_\infty$ mean? Should it be $O_P$?

Author response:
Yes (updated in the revision).

Reviewer comment (Novelty):
Sugiyama et al. (2012) discussed the density-ratio estimation, and its interest differs from this study. Furthermore, it has been known that Eq. (8) can be used to estimate the propensity score. I think the citation may not be appropriate.

Author response:
To clarify, we are not estimating a propensity score as in traditional treatment effect estimation (see also our response to Q7 below). The SRF framework requires estimating the conditional density ratio between two generalized propensity scores (Diaz & Van der Laan, 2012; Diaz et al., 2021). We originally cited the review paper by Sugiyama et al. (2012) on density ratio estimation since it was informative while developing our methods, however, it is indeed not required as we are citing other related work using the same density ratio formula.

Reviewer comment (Novelty):
Does this study relate to automatic debiased learning proposed by Chernozhukov et al. (2022)?

Author response:
The work by Chernozhukov et al. (2022) does not provide any guarantees specific for estimating the effect of an exposure shift (SRF) as our method does. More broadly, there are connections between the double machine learning literature and doubly robust estimation with the EIF, which is used by the TR framework that our paper builds upon. However, these connections remain under-explored. As far as we know, these connections have not been discussed in the TR literature.

Reviewer comment (Theory):
"for some function $\eta:\mathcal{X}\times\mathcal{A}\to\mathbb{R}$" should be for "for some measurable function $\eta:\mathcal{X}\times\mathcal{A}\to\mathbb{R}$"?

Author response:
We intentionally avoided measure-theoretical language since the targeted regularization literature we build upon has proved the necessary formal results without resorting to it. We understand the advantages of additional formality, but we also sought to align our notation with the literature.

Reviewer comment (Theory):
Does the shift-response function is the same as the standard average treatment effect?

Author response:
SRFs are different from traditional treatment effects, comprising the average treatment effect (ATE) for binary treatment and the exposure-response function (ERF) for continuous treatments, as explained in Section 2 and Appendix C. SRFs allow for a different family of problems but require additional theory and methods (e.g., see Diaz et al. (2021) and Diaz & Van der Laan (2012)). We have improve the presentation of SRFs and examples in Section 2, paragraph 3, of the revision.

Reviewer comment (Novelty):
Some citations are missing.

Author response:
Thank you for your careful reading. We fixed typos and missing references in the revision.