Bias formulas for violations of proximal identification assumptions in a linear structural equation model
-
Raluca Cobzaru
,
Roy Welsch
Abstract
Causal inference from observational data often rests on the unverifiable assumption of no unmeasured confounding. Recently, Tchetgen Tchetgen and colleagues have introduced proximal inference to leverage negative control outcomes and exposures as proxies to adjust for bias from unmeasured confounding. However, some of the key assumptions that proximal inference relies on are themselves empirically untestable. In addition, the impact of violations of proximal inference assumptions on the bias of effect estimates is not well understood. In this article, we derive bias formulas for proximal inference estimators under a linear structural equation model. These results are a first step toward sensitivity analysis and quantitative bias analysis of proximal inference estimators. While limited to a particular family of data generating processes, our results may offer some more general insight into the behavior of proximal inference estimators.
1 Introduction
Causal inference using observational data often rests on the assumption of no unmeasured confounding. This assumption is not empirically verifiable, but sensitivity analysis methods [1–4] are available to assess robustness to possible violations. Alternatively, investigators might use methods such as instrumental variable analysis or difference-in-differences, which depend on different assumptions. Sensitivity analyses for violations of the assumptions required by these alternative methods are also available [5,6].
There has been recent interest in the use of negative control methods to detect and resolve confounder bias. A negative control outcome (NCO) is a variable known not to be causally affected by the treatment of interest, while a negative control exposure (NCE) is a variable known not to causally affect the outcome of interest [7]. Tchetgen Tchetgen et al. have developed a proximal inference framework [8–10], which uses NCE-NCO pairs sharing the same unmeasured confounders as the treatment–outcome relationship of interest as proxies to adjust for unmeasured confounding. Key assumptions of proximal inference include that the unmeasured confounders are associated with both the NCEs and NCOs (“
In this article, we characterize bias from violations of proximal inference assumptions in a linear structural equation model (LSEM). Our results build understanding of the sensitivity of proximal inference to assumption violations and enable “assumption-heavy” sensitivity analysis and quantitative bias analysis [11] tools for proximal inference. We hope that our results will also serve as a first step toward assumption-light sensitivity analysis.
As a motivating example, we consider the observational SUPPORT study estimating the effect of right heart catheterization (RHC) on 30 day survival in intensive care unit (ICU) patients. Initial analyses of these data [8] depended on the no unobserved confounding assumption and adjusted for a set of 71 baseline covariates. Despite extensive covariate adjustment, concern about unobserved confounding remains. Cui et al. [9] applied proximal inference to the SUPPORT data, using physiological measurements taken early in patients’ ICU stay as both NCEs and NCOs. These variables were noisy early measurements of indicators of evolving underlying health conditions and did not themselves influence treatment decisions or health outcomes, making them valid NCEs. They also preceded treatment, making them valid NCOs. However, violations of
The organization of the article is as follows. In Section 2, we review proximal inference and motivate the need for bias analysis given the assumptions of this framework. In Sections 3 and 4, we derive and numerically explore bias formulas in a setting with two-dimensional unobserved confounder
2 Proximal identification of the average treatment effect
2.1 Review of definitions and assumptions
We use the potential outcome framework [12] to define causal effects. Let
Let
Assumption 1
(Consistency)
In other words, the observed value of
Assumption 2
(Positivity)
Assumption 2 states that both exposure levels are observed at all levels of the observed covariates
Many analyses further make the assumption that there is no unobserved confounding, i.e., that observed covariates block all “backdoor” causal paths between treatment and outcome.
Assumption 3
(Exchangeability)
Under Assumptions 1–3, counterfactual mean
where
Exchangeability is a strong assumption that is empirically untestable, and much effort in causal inference research has been devoted to relaxing this assumption. Miao et al. [15] propose an alternative to Assumption 3 that allows identification of the counterfactual mean
As shown by Cui et al. [9], we consider a (potentially multidimensional) variable
Figure 1 contains directed acyclic graphs (DAGs) representing each of the proxy types included in
DAGs representing the three types of variables
In the study by Miao et al. [15], exchangeability is replaced with the following assumptions:
Assumption 4
(Treatment-inducing confounding proxy)
Assumption 5
(Outcome-inducing confounding proxy)
Assumption 6
(Latent unconfoundedness) If
Assumption 4 states that
Moreover, to be valid proxies, variables
Assumption 7
(
The
Finally, in addition to Assumptions 1–7, Miao et al. [10] introduce the following completeness conditions for the identification of
Assumption 8
(Completeness) For any
If
If
Assumption 8(a) can be interpreted as a requirement that the NCE
Completeness Assumption 8(a) has a simple interpretation in the case where confounders
In other words, proximal inference can account for unmeasured confounding if the number of categories of
2.2 Estimating the proximal g-formula via moment restriction
Miao et al. [15] introduce the notion of an outcome confounding bridge function, which transforms the NCO
for all values of
which means
Cui et al. [9] and Miao et al. [10] established the following proximal identification result for the outcome confounding bridge function that leverages the distribution of a NCE
Theorem 1
Suppose there exists an outcome confounding bridge function
almost surely. Then, under Assumptions 1, 2, 4–6, and 8(a),
almost surely.
Under Assumption 6, we have
Corollary 1.1
(Proximal g-formula) If (12) holds almost surely, then the counterfactual mean
and the ACE is identified by
Assuming the outcome confounding bridge function
We define the target parameter
for some vector function
we may choose a function
such that the dimension of
Then, for
As established by Miao et al. [15], the estimates
2.3 The need for bias analysis
We have so far collected a series of untestable Assumptions 4–8 that replace exchangeability and account for the effect of unmeasured confounders
3 Bias formulas for two-dimensional
U
In this section, we characterize the proximal inference estimator bias in an LSEM under scenarios in which each of
DAG encoding causal relationships among variables in (19) in which U-relevance Assumption 7 is violated.
DAG encoding the causal relationships among variables in (21) in which completeness 7(a) is violated.
In addition, for the settings of Figures 2 and 3, we compare the bias of the proximal estimator due to violations of Assumptions 7 and 8 to the bias of alternative estimators of the ACE which the analyst might implement under an incorrect unconfoundedness assumption. We consider
an unadjusted estimator (referred to as “unadj”), which assumes no unobserved confounding and estimates
an outcome regression estimator (referred to as “OR”), which adjusts for
3.1 Bias settings for two-dimensional
U
As outlined at the beginning of this section, we derive formulas for the proximal inference estimator bias under scenarios depicted in Figures 2 and 3, under an LSEM based on [10]. The notation
Figure 4 depicts the causal DAG corresponding to this LSEM. The dashed bidirectional arrow between
DAG encoding the causal relationships among variables in (17).
The NCE
If
The proof of this claim is in Appendix A.1. Therefore, like an analyst unaware of the additional assumption-violating component of
By Theorem 1, violating Assumption 8(a) leads to a potentially biased ACE estimate as the outcome confounding bridge function
3.2 Partial
U
-relevance for two-dimensional unobserved confounder
U
(as in Figure 2)
In this subsection, we consider the case where
where
From Theorem 2, we know that setup (19) violates Assumption 8(a). In addition, we do not have a derivation of the true outcome confounding bridge function, so a linear model might be misspecified. The following theorem provides a formula for this bias under a linear bridge function specification, which is still the specification an analyst unaware of
Theorem 3
If
The proof for Theorem 3 (as well as a more general formula for
3.3 Completeness violation: Association between negative controls through
U
=
(
U
1
,
U
2
)
(as in Figure 3)
As shown in Figure 3, for simplicity, we consider a scenario with no covariates
where
The aforementioned setup satisfies all assumptions except 8(a) (which is violated according to Theorem 2). Thus, solving for the parameters
Theorem 4
If
where
For
The more general formulas for arbitrary
One implication of Theorem 4 is that the proximal outcome regression bias
![Figure 5
Plots of the ACE estimate bias under DGP (21) and under violations of completeness Assumption 8(a), where
θ
u
1
μ
u
1
{\theta }_{{u}_{1}}{\mu }_{{u}_{1}}
and
θ
u
2
μ
u
2
{\theta }_{{u}_{2}}{\mu }_{{u}_{2}}
have the same sign as in Theorem 5(b). The completeness violation is imposed by setting both components of
U
U
to be associated with both negative controls and only including one NCE and one NCO, as per Theorem 2. Along the
x
x
-axis, we vary the strength of association between
U
2
{U}_{2}
and
Z
Z
and
W
W
, by varying
θ
u
2
{\theta }_{{u}_{2}}
and
μ
u
2
{\mu }_{{u}_{2}}
(set to be equal to each other,
θ
u
2
=
μ
u
2
{\theta }_{{u}_{2}}={\mu }_{{u}_{2}}
, so they always have positive product). The different panels correspond to different values of
α
u
1
{\alpha }_{{u}_{1}}
governing the strength of confounding by
U
1
{U}_{1}
. The figure shows that, as predicted by Theorem 5(b), the bias of the proximal outcome estimator (“PI”, solid line) can be greater or smaller than the bias of the unadjusted estimator (“Unadj”, dashed line —) and can be arbitrarily large under certain parameter settings. Moreover, the relationship of PI with the outcome adjusted estimator (“OR”, dotted line
⋯
\cdots \hspace{0.33em}
) varies with the strength of confounding
α
u
1
{\alpha }_{{u}_{1}}
across panels. All other parameters are fixed, with values:
α
0
=
γ
0
=
θ
0
=
μ
0
=
0
{\alpha }_{0}={\gamma }_{0}={\theta }_{0}={\mu }_{0}=0
,
θ
a
=
θ
u
1
=
1
{\theta }_{a}={\theta }_{{u}_{1}}=1
,
γ
u
1
=
1
{\gamma }_{{u}_{1}}=1
,
γ
u
a
1
=
1.5
{\gamma }_{{u}_{a}1}=1.5
,
μ
u
1
=
−
0.5
{\mu }_{{u}_{1}}=-0.5
,
γ
a
=
0.5
{\gamma }_{a}=0.5
, similar to the simulation in [15]. (a)
α
u
1
=
0.3
{\alpha }_{{u}_{1}}=0.3
, (b)
α
u
1
=
0.5
{\alpha }_{{u}_{1}}=0.5
, (c)
α
u
1
=
1
{\alpha }_{{u}_{1}}=1
, (d)
α
u
1
=
−
0.5
{\alpha }_{{u}_{1}}=-0.5
.](/document/doi/10.1515/jci-2023-0039/asset/graphic/j_jci-2023-0039_fig_005.jpg)
Plots of the ACE estimate bias under DGP (21) and under violations of completeness Assumption 8(a), where
Under the simplifying assumption that the unobserved confounder is not an effect modifier, i.e.,
Theorem 5
Assuming
If
If
3.4 Numerical experiments
We provide numerical examples based on the bias formulas derived earlier in this section to illustrate how the bias of different estimators (proximal and nonproximal) varies with different values of
3.4.1 Partial
U
-relevance for two-dimensional unobserved confounder
U
Figure 6 illustrates the change in absolute bias for the proximal and unadjusted estimators relative to the value of
Plots of the ACE estimate bias under DGP (19) and under violations of
3.4.2 Completeness violation: Association between negative controls through
U
=
(
U
1
,
U
2
)
Figures 5 and 7 illustrate the change in absolute bias for each of the three estimators relative to the value of
![Figure 7
Plots of the ACE estimate bias under DGP (21) and under violations of completeness Assumption 8(a), where
θ
u
1
μ
u
1
{\theta }_{{u}_{1}}{\mu }_{{u}_{1}}
and
θ
u
2
μ
u
2
{\theta }_{{u}_{2}}{\mu }_{{u}_{2}}
have the same sign as in Theorem 5(a). The completeness violation is imposed by setting both components of
U
U
to be associated with both negative controls and only including one NCE and one NCO, as per Theorem 2. Along the
x
x
-axis, we vary the strength of association between
U
2
{U}_{2}
and
Z
Z
and
W
W
, by varying
θ
u
2
{\theta }_{{u}_{2}}
and
μ
u
2
{\mu }_{{u}_{2}}
(set to be equal to each other,
θ
u
2
=
μ
u
2
{\theta }_{{u}_{2}}={\mu }_{{u}_{2}}
, so they always have positive product). The different panels correspond to different values of
α
u
1
{\alpha }_{{u}_{1}}
governing the strength of confounding by
U
1
{U}_{1}
. The figure shows that, as predicted by Theorem 5(a), the bias of the proximal outcome estimator (“PI”, solid line) is always less than the unadjusted bias (“Unadj”, dashed line —). However, the relationship of PI with the outcome adjusted estimator (“OR”, dotted line
⋯
\cdots \hspace{0.33em}
) varies with the strength of confounding
α
u
1
{\alpha }_{{u}_{1}}
across panels. All other parameters are fixed, with values:
α
0
=
γ
0
=
θ
0
=
μ
0
=
0
{\alpha }_{0}={\gamma }_{0}={\theta }_{0}={\mu }_{0}=0
,
θ
a
=
θ
u
1
=
1
{\theta }_{a}={\theta }_{{u}_{1}}=1
,
γ
u
1
=
1
{\gamma }_{{u}_{1}}=1
,
γ
u
a
1
=
1.5
{\gamma }_{{u}_{a}1}=1.5
,
μ
u
1
=
0.5
{\mu }_{{u}_{1}}=0.5
,
γ
a
=
0.5
{\gamma }_{a}=0.5
, similar to the simulation in [15]. (a)
α
u
1
=
0.3
{\alpha }_{{u}_{1}}=0.3
, (b)
α
u
1
=
0.5
{\alpha }_{{u}_{1}}=0.5
, (c)
α
u
1
=
1
{\alpha }_{{u}_{1}}=1
, (d)
α
u
1
=
−
0.5
{\alpha }_{{u}_{1}}=-0.5
.](/document/doi/10.1515/jci-2023-0039/asset/graphic/j_jci-2023-0039_fig_007.jpg)
Plots of the ACE estimate bias under DGP (21) and under violations of completeness Assumption 8(a), where
4 Bias formulas in arbitrary dimension with no confounder–treatment interaction
To tractably obtain bias formulas in the general case of multidimensional
We consider i.i.d. data generated by:
The following theorem provides a formula for the proximal outcome identification bias under a linear bridge function:
Theorem 6
Let
If
A proof of Theorem 6 (which also considers the case of general
Remark 1
If
Theorem 6 enables sensitivity analysis. Note that the terms
5 Illustration of sensitivity analysis on the SUPPORT data
In this section, we provide an illustrative sensitivity analysis of the proximal inference application in [8] using the Study to Understand Prognoses and Preferences for Outcomes and Risks of Treatment (SUPPORT) dataset. We do not aim to be prescriptive, but rather to provide an example of how one might apply the bias formulas we derived to explore the extent of likely bias in a proximal inference analysis. Importantly, if data were not generated from an LSEM, then clearly our bias formulas will be incorrect. Still, the bias ensuing from violations of proximal inference assumptions under the LSEM assumption may serve as an approximation to the actual bias of a proximal inference analysis even if the data were not generated by an LSEM. We hope bias formulas for more general data generating processes might be developed in future work.
The SUPPORT data comprise 5,735 individuals, of which 2,184 were treated by RHC and 3,551 belonged to the control group. Outcome variable
We assess the potential impact of assumption violations by evaluating bias formula (25) under draws from an assumed distribution on the dimension of
Draw
Set a subset of components of
Set proportion of violating
For each
We considered two approaches to drawing
Empirical correlation: Construct covariance matrix
Uncorrelated:
In this setup, we assume that
Draw the parameters in
The remaining parameters
Constraining
(26)Constraining
(27)To account for sampling variability of the covariance matrices plugged into the above formulas, we employ a bootstrapping strategy (Appendix D.2).
We would expect that settings with lower expected dimension
Sensitivity-adjusted confidence intervals of the average treatment effect, where the intervals are computed using
| Row # | Setup | Sensitivity-adjusted CIs for
|
|---|---|---|
| 1 | No bias-inducing
|
|
| 2 | No
|
|
| 3 |
|
|
| 4 |
|
|
| 5 |
|
|
| 6 |
|
|
| 7 |
|
|
| 8 |
|
|
| 9 |
|
|
In the presence of a rich adjustment set with significant correlation between
Due to the interconnectedness of biological systems, we believe that most unmeasured confounders related to patients’ pretreatment health status would be associated with both the covariates
6 Discussion
By deriving bias formulas for proximal inference estimators under violations of completeness and
We have also shown how our bias formulas enable assumption-heavy sensitivity analysis of proximal inference estimates. While (25) was derived under the strong assumptions that data were generated by an LSEM and
-
Funding information: Funding was provided by the MIT-IBM Watson AI Lab.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors state no conflict of interest.
-
Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors institutional review board or equivalent committee.
-
Informed consent: Informed consent was obtained from all individuals included in this study.
-
Data availability statement: These are secondary analyses that are de-identified and are publicly available. The datasets analysed during the current study are available from the corresponding author on reasonable request.
Appendix A Bridge function parameters for post-treatment NCE
A.1 Bridge functions derivation for one-dimensional unobserved
U
– case of no violations
We identify coefficients
Coefficients of
h
:
We have that
Since
Assigning values
Multiplying (A3) by
Since
Similarly, from (A4), we obtain
Coefficients of
q
:
We have that
Since
for each
Assigning values
As in the outcome bridge function case, it follows that the coefficients of 1 (the constant term),
B Proving violations of completeness Assumption 8(a)
We will prove that completeness Assumption 8(a) is violated under the DGP (17) with
For any values
Using
we obtain
Let us consider
We will prove that
Let
We have that
which implies
and
using the fact that
for any
C Bias computations
C.1 Computing the (asymptotic) bias obtained through method of moments estimator under setup (19)
We will compute the asymptotic bias obtained from the method of moments solver using bridge function
We define the moment restrictions
C.1.1 Case of
Corr
(
U
1
,
U
2
)
=
0
(used in main paper)
By using
Let
We obtain the estimated bridge function parameters
The estimated effect resulting from
which yields a bias equal to
We note that the expectations
cannot be computed in closed form but can be obtained numerically using software like Mathematica or Maple once we provide the values of
C.1.2 General case of
Corr
(
U
1
,
U
2
)
=
ν
Using
Proceeding similarly as in the case
C.2 Computing the (asymptotic) bias obtained through method of moments estimator under setup (21)
We will compute the asymptotic bias obtained from the method of moments solver using bridge function
We define the moment restrictions
C.2.1 Case of
Corr
(
U
1
,
U
2
)
=
0
(used in main paper):
Using
Let
We obtain the estimated bridge function parameters
The estimated effect resulting from
which yields a bias equal to
In the particular case
Similarly to Proof C.1, we note that the expectations
cannot be computed in closed form but can be obtained numerically using software like Mathematica or Maple once we provide the values of
C.2.2 General case of
Corr
(
U
1
,
U
2
)
=
ν
:
Using
Let
Proceeding similarly as in the case
where
This yields a bias equal to
As in the previous case, expectations
which help simplify the bias formula.
C.3 Computing the OR estimator bias under setup (19)
Let
we obtain a linear regression estimator bias equal to
In particular, for
C.4 Computing the OR estimator bias under setup (21)
Let
we obtain a linear regression estimator bias equal to
In particular, for
C.5 Comparison of proximal and unadjusted estimator biases under setup (21)
We begin by proving that both
Proof that
S
1
,
S
2
>
0
:
We have that
which implies
since
Similarly, if we consider
Thus,
Taking the ratio of magnitudes for the two biases, we have
Let
If
If
If
If
C.6 Computing the proximal estimator bias under
γ
a
u
=
0
and
h
(
W
,
A
,
X
)
=
b
0
+
b
a
A
+
b
x
T
X
+
b
w
T
W
We will compute the asymptotic bias obtained from the method of moments solver using bridge function
We define the moment restrictions
Using
Under assumption
Setting
where
The estimated effect resulting from
which yields a bias equal to
D Details for illustrative sensitivity analysis on real data
D.1 Extracting relationships between
U
-parameters from the data
We have that
where
as well as
The aforementioned equations show that parameterizing
Moreover, we have that
which imply
D.2 In-depth rationale for choice of sensitivity parameters
D.2.1 Choice of distribution for
p
=
dim
(
U
)
The rate of the Poisson distribution can be adjusted depending on the expected number of independent unobserved confounders. In this case, we assume the large number of observed covariates in
D.2.2 Drawing
ρ
– the base case
Construct covariance matrix
Draw elements
Rescale each
We operate under the assumption that the pairwise covariances
D.2.3 Choice of
θ
u
,
l
,
θ
u
,
r
,
μ
u
,
l
,
μ
u
,
l
In the absence of additional priors on each of the unobserved confounders
To inform our choice of
fit1: regress
fit2: regress
fit3: regress
fit4: regress
Assuming our LSEM, fit2 yields estimates for
In addition, the coefficients of
In fact, our experiments show that the distribution of biases does not change significantly for fixed
D.2.4 Bootstrapping strategy for setting
E
[
A
U
]
,
γ
u
We compute 500 bootstrap estimates of the covariance matrices and draw
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- Sharp bounds for causal effects based on Ding and VanderWeele's sensitivity parameters
- Detecting treatment interference under K-nearest-neighbors interference
- Bias formulas for violations of proximal identification assumptions in a linear structural equation model
- Current philosophical perspectives on drug approval in the real world
- Foundations of causal discovery on groups of variables
- Improved sensitivity bounds for mediation under unmeasured mediator–outcome confounding
- Potential outcomes and decision-theoretic foundations for statistical causality: Response to Richardson and Robins
- Quantifying the quality of configurational causal models
- Design-based RCT estimators and central limit theorems for baseline subgroup and related analyses
- An optimal transport approach to estimating causal effects via nonlinear difference-in-differences
- Estimation of network treatment effects with non-ignorable missing confounders
- Double machine learning and design in batch adaptive experiments
- The functional average treatment effect
- An approach to nonparametric inference on the causal dose–response function
- Review Article
- Comparison of open-source software for producing directed acyclic graphs
- Special Issue on Neyman (1923) and its influences on causal inference
- Optimal allocation of sample size for randomization-based inference from 2K factorial designs
- Direct, indirect, and interaction effects based on principal stratification with a binary mediator
- Interactive identification of individuals with positive treatment effect while controlling false discoveries
- Neyman meets causal machine learning: Experimental evaluation of individualized treatment rules
- From urn models to box models: Making Neyman's (1923) insights accessible
- Prospective and retrospective causal inferences based on the potential outcome framework
- Causal inference with textual data: A quasi-experimental design assessing the association between author metadata and acceptance among ICLR submissions from 2017 to 2022
- Some theoretical foundations for the design and analysis of randomized experiments
Artikel in diesem Heft
- Research Articles
- Evaluating Boolean relationships in Configurational Comparative Methods
- Doubly weighted M-estimation for nonrandom assignment and missing outcomes
- Regression(s) discontinuity: Using bootstrap aggregation to yield estimates of RD treatment effects
- Energy balancing of covariate distributions
- A phenomenological account for causality in terms of elementary actions
- Nonparametric estimation of conditional incremental effects
- Conditional generative adversarial networks for individualized causal mediation analysis
- Mediation analyses for the effect of antibodies in vaccination
- Sharp bounds for causal effects based on Ding and VanderWeele's sensitivity parameters
- Detecting treatment interference under K-nearest-neighbors interference
- Bias formulas for violations of proximal identification assumptions in a linear structural equation model
- Current philosophical perspectives on drug approval in the real world
- Foundations of causal discovery on groups of variables
- Improved sensitivity bounds for mediation under unmeasured mediator–outcome confounding
- Potential outcomes and decision-theoretic foundations for statistical causality: Response to Richardson and Robins
- Quantifying the quality of configurational causal models
- Design-based RCT estimators and central limit theorems for baseline subgroup and related analyses
- An optimal transport approach to estimating causal effects via nonlinear difference-in-differences
- Estimation of network treatment effects with non-ignorable missing confounders
- Double machine learning and design in batch adaptive experiments
- The functional average treatment effect
- An approach to nonparametric inference on the causal dose–response function
- Review Article
- Comparison of open-source software for producing directed acyclic graphs
- Special Issue on Neyman (1923) and its influences on causal inference
- Optimal allocation of sample size for randomization-based inference from 2K factorial designs
- Direct, indirect, and interaction effects based on principal stratification with a binary mediator
- Interactive identification of individuals with positive treatment effect while controlling false discoveries
- Neyman meets causal machine learning: Experimental evaluation of individualized treatment rules
- From urn models to box models: Making Neyman's (1923) insights accessible
- Prospective and retrospective causal inferences based on the potential outcome framework
- Causal inference with textual data: A quasi-experimental design assessing the association between author metadata and acceptance among ICLR submissions from 2017 to 2022
- Some theoretical foundations for the design and analysis of randomized experiments
