arXiv subjects
Publications and source records attributed to Razieh Nabi.
We are grateful to the discussants, Levis and Kennedy [2025], Luo and Geng [2025], Wang and van der Laan [2025], and Yang and Kim [2025], for their thoughtful comments on our paper (Nabi et al., 2025). In this rejoinder, we summarize our main contributions and respond to each discussion in turn.
Establishing cause-effect relationships from observational data often relies on untestable assumptions. It is crucial to know whether, and to what extent, the conclusions drawn from non-experimental studies are robust to potential unmeasured confounding. In this paper, we focus on the average causal effect (ACE) as our target of inference. We generalize the sensitivity analysis approach developed by Robins et al. (2000), Franks et al. (2020), and Zhou and Yao (2023). We use semiparametric theory to derive the non-parametric efficient influence function of the ACE, for fixed sensitivity parameters. We use this influence function to construct a one-step, split sample, truncated estimator of the ACE. Our estimator depends on semiparametric models for the distribution of the observed data; importantly, these models do not impose any restrictions on the values of sensitivity analysis parameters. We establish sufficient conditions ensuring that our estimator has root-n asymptotics. We use our methodology to evaluate the causal effect of smoking during pregnancy on birth weight. We also evaluate the performance of estimation procedure in a simulation study.
The identification theory for causal effects in directed acyclic graphs (DAGs) with hidden variables is well established, but methods for estimating and inferring functionals that extend beyond the g-formula remain underdeveloped. Previous studies have introduced semiparametric estimators for such functionals in a broad class of DAGs with hidden variables. While these estimators exhibit desirable statistical properties such as double robustness in certain cases, they also face significant limitations. Notably, they encounter substantial computational challenges, particularly involving density estimation and numerical integration for continuous variables, and their estimates may fall outside the parameter space of the target estimand. Additionally, the asymptotic properties of these estimators is underexplored, especially when integrating flexible statistical and machine learning models for nuisance functional estimations. This paper addresses these challenges by introducing novel one-step corrected plug-in and targeted minimum loss-based estimators of causal effects for a class of hidden variable DAGs that go beyond classical back-door and front-door criteria (known as the treatment primal fixability criterion in prior literature). These estimators leverage data-adaptive machine learning algorithms to minimize modeling assumptions while ensuring key statistical properties including double robustness, efficiency, boundedness within the target parameter space, and asymptotic linearity under $L^2(P)$-rate conditions for nuisance functional estimates that yield root-n consistent causal effect estimates. To ensure our estimation methods are accessible in practice, we provide the flexCausal package in R.
We develop a general framework for estimating function-valued parameters under equality or inequality constraints in infinite-dimensional statistical models. Such constrained learning problems are common across many areas of statistics and machine learning, where estimated parameters must satisfy structural requirements such as moment restrictions, policy benchmarks, calibration criteria, or fairness considerations. To address these problems, we characterize the solution as the minimizer of a penalized population risk using a Lagrange-type formulation, and analyze it through a statistical functional lens. Central to our approach is a constraint-specific path through the unconstrained parameter space that defines the constrained solutions. For a broad class of constraint-risk pairs, this path admits closed-form expressions and reveals how constraints shape optimal adjustments. When closed forms are unavailable, we derive recursive representations that support tractable estimation. Our results also suggest natural estimators of the constrained parameter, constructed by combining estimates of unconstrained components of the data-generating distribution. Thus, our procedure can be integrated with any statistical learning approach and implemented using standard software. We provide general conditions under which the resulting estimators achieve optimal risk and constraint satisfaction, and we demonstrate the flexibility and effectiveness of the proposed method through various examples, simulations, and real-data applications.
Real-world vaccine effectiveness has increasingly been studied using matching-based approaches, particularly in observational cohort studies following the target trial emulation framework. Although matching is appealing in its simplicity, it suffers important limitations in terms of clarity of the target estimand and the efficiency or precision with which is it estimated. Scientifically justified causal estimands of vaccine effectiveness may be difficult to define owing to the fact that vaccine uptake varies over calendar time when infection dynamics may also be rapidly changing. We propose a causal estimand of vaccine effectiveness that summarizes vaccine effectiveness over calendar time, similar to how vaccine efficacy is summarized in a randomized controlled trial. We describe the identification of our estimand, including its underlying assumptions, and propose simple-to-implement estimators based on two hazard regression models. We apply our proposed estimator in simulations and in a study to assess the effectiveness of the Pfizer-BioNTech COVID-19 vaccine to prevent infections with SARS-CoV2 in children 5-11 years old. In both settings, we find that our proposed estimator yields similar scientific inferences while providing significant efficiency gains over commonly used matching-based estimators.
It is often said that the fundamental problem of causal inference is a missing data problem -- the comparison of responses to two hypothetical treatment assignments is made difficult because for every experimental unit only one potential response is observed. In this paper, we consider the implications of the converse view: that missing data problems are a form of causal inference. We make explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had we (possibly contrary to fact) been able to observe them. Drawing analogies with causal inference, we show how identification assumptions in missing data can be encoded in terms of graphical models defined over counterfactual and observed variables. We review recent results in missing data identification from this viewpoint. In doing so, we note interesting similarities and differences between missing data and causal identification theories.
This paper introduces a framework for estimating fair optimal predictions using machine learning where the notion of fairness can be quantified using path-specific causal effects. We use a recently developed approach based on Lagrange multipliers for infinite-dimensional functional estimation to derive closed-form solutions for constrained optimization based on mean squared error and cross-entropy risk criteria. The theoretical forms of the solutions are analyzed in detail and described as nuanced adjustments to the unconstrained minimizer. This analysis highlights important trade-offs between risk minimization and achieving fairnes. The theoretical solutions are also used as the basis for construction of flexible semiparametric estimation strategies for these nuisance components. We describe the robustness properties of our estimators in terms of achieving the optimal constrained risk, as well as in terms of controlling the value of the constraint. We study via simulation the impact of using robust estimators of pathway-specific effects to validate our theory. This work advances the discourse on algorithmic fairness by integrating complex causal considerations into model training, thus providing strategies for implementing fair models in real-world applications.
Significant progress has been made in developing identification and estimation techniques for missing data problems where modeling assumptions can be described via a directed acyclic graph. The validity of results using such techniques rely on the assumptions encoded by the graph holding true; however, verification of these assumptions has not received sufficient attention in prior work. In this paper, we provide new insights on the testable implications of three broad classes of missing data graphical models, and design goodness-of-fit tests for them. The classes of models explored are: sequential missing-at-random and missing-not-at-random models which can be used for modeling longitudinal studies with dropout/censoring, and a no self-censoring model which can be applied to cross-sectional studies and surveys.
Conducting valid statistical analyses is challenging in the presence of missing-not-at-random (MNAR) data, where the missingness mechanism is dependent on the missing values themselves even conditioned on the observed data. Here, we consider a MNAR model that generalizes several prior popular MNAR models in two ways: first, it is less restrictive in terms of statistical independence assumptions imposed on the underlying joint data distribution, and second, it allows for all variables in the observed sample to have missing values. This MNAR model corresponds to a so-called criss-cross structure considered in the literature on graphical models of missing data that prevents nonparametric identification of the entire missing data model. Nonetheless, part of the complete-data distribution remains nonparametrically identifiable. By exploiting this fact and considering a rich class of exponential family distributions, we establish sufficient conditions for identification of the complete-data distribution as well as the entire missingness mechanism. We then propose methods for testing the independence restrictions encoded in such models using odds ratio as our parameter of interest. We adopt two semiparametric approaches for estimating the odds ratio parameter and establish the corresponding asymptotic theories: one involves maximizing a conditional likelihood with order statistics and the other uses estimating equations. The utility of our methods is illustrated via simulation studies.
Despite the growing interest in causal and statistical inference for settings with data dependence, few methods currently exist to account for missing data in dependent data settings; most classical missing data methods in statistics and causal inference treat data units as independent and identically distributed (i.i.d.). We develop a graphical modeling based framework for causal inference in the presence of entangled missingness, defined as missingness with data dependence. We distinguish three different types of entanglements that can occur, supported by real-world examples. We give sound and complete identification results for all three settings. We show that existing missing data models may be extended to cover entanglements arising from (1) target law dependence and (2) missingness process dependence, while those arising from (3) missingness interference require a novel approach. We demonstrate the use of our entangled missingness framework on synthetic data. Finally, we discuss how, subject to a certain reinterpretation of the variables in the model, our model for missingness interference extends missing data methods to novel missing data patterns in i.i.d. settings.
We implement Ananke: an object-oriented Python package for causal inference with graphical models. At the top of our inheritance structure is an easily extensible Graph class that provides an interface to several broadly useful graph-based algorithms and methods for visualization. We use best practices of object-oriented programming to implement subclasses of the Graph superclass that correspond to types of causal graphs that are popular in the current literature. This includes directed acyclic graphs for modeling causally sufficient systems, acyclic directed mixed graphs for modeling unmeasured confounding, and chain graphs for modeling data dependence and interference. Within these subclasses, we implement specialized algorithms for common statistical and causal modeling tasks, such as separation criteria for reading conditional independence, nonparametric identification, and parametric and semiparametric estimation of model parameters. Here, we present a broad overview of the package and example usage for a problem with unmeasured confounding. Up to date documentation is available at \url{https://ananke.readthedocs.io/en/latest/}.
Functional magnetic resonance imaging (fMRI) has become one of the most common imaging modalities for brain function analysis. Recently, graph neural networks (GNN) have been adopted for fMRI analysis with superior performance. Unfortunately, traditional functional brain networks are mainly constructed based on similarities among region of interests (ROI), which are noisy and agnostic to the downstream prediction tasks and can lead to inferior results for GNN-based models. To better adapt GNNs for fMRI analysis, we propose TBDS, an end-to-end framework based on \underline{T}ask-aware \underline{B}rain connectivity \underline{D}AG (short for Directed Acyclic Graph) \underline{S}tructure generation for fMRI analysis. The key component of TBDS is the brain network generator which adopts a DAG learning approach to transform the raw time-series into task-aware brain connectivities. Besides, we design an additional contrastive regularization to inject task-specific knowledge during the brain network generation process. Comprehensive experiments on two fMRI datasets, namely Adolescent Brain Cognitive Development (ABCD) and Philadelphia Neuroimaging Cohort (PNC) datasets demonstrate the efficacy of TBDS. In addition, the generated brain networks also highlight the prediction-related brain regions and thus provide unique interpretations of the prediction results. Our implementation will be published to https://github.com/yueyu1030/TBDS upon acceptance.
Identification theory for causal effects in causal models associated with hidden variable directed acyclic graphs (DAGs) is well studied. However, the corresponding algorithms are underused due to the complexity of estimating the identifying functionals they output. In this work, we bridge the gap between identification and estimation of population-level causal effects involving a single treatment and a single outcome. We derive influence function based estimators that exhibit double robustness for the identified effects in a large class of hidden variable DAGs where the treatment satisfies a simple graphical criterion; this class includes models yielding the adjustment and front-door functionals as special cases. We also provide necessary and sufficient conditions under which the statistical model of a hidden variable DAG is nonparametrically saturated and implies no equality constraints on the observed data distribution. Further, we derive an important class of hidden variable DAGs that imply observed data distributions observationally equivalent (up to equality constraints) to fully observed DAGs. In these classes of DAGs, we derive estimators that achieve the semiparametric efficiency bounds for the target of interest where the treatment satisfies our graphical criterion. Finally, we provide a sound and complete identification algorithm that directly yields a weight based estimation strategy for any identifiable effect in hidden variable causal models.
The front-door criterion can be used to identify and compute causal effects despite the existence of unmeasured confounders between a treatment and outcome. However, the key assumptions -- (i) the existence of a variable (or set of variables) that fully mediates the effect of the treatment on the outcome, and (ii) which simultaneously does not suffer from similar issues of confounding as the treatment-outcome pair -- are often deemed implausible. This paper explores the testability of these assumptions. We show that under mild conditions involving an auxiliary variable, the assumptions encoded in the front-door model (and simple extensions of it) may be tested via generalized equality constraints a.k.a Verma constraints. We propose two goodness-of-fit tests based on this observation, and evaluate the efficacy of our proposal on real and synthetic data. We also provide theoretical and empirical comparisons to instrumental variable approaches to handling unmeasured confounding.
Cause-effect relationships are typically evaluated by comparing outcome responses to binary treatment values, representing two arms of a hypothetical randomized controlled trial. However, in certain applications, treatments of interest are continuous and multidimensional. For example, understanding the causal relationship between severity of radiation therapy, summarized by a multidimensional vector of radiation exposure values and post-treatment side effects is a problem of clinical interest in radiation oncology. An appropriate strategy for making interpretable causal conclusions is to reduce the dimension of treatment. If individual elements of a multidimensional treatment vector weakly affect the outcome, but the overall relationship between treatment and outcome is strong, careless approaches to dimension reduction may not preserve this relationship. Further, methods developed for regression problems do not directly transfer to causal inference due to confounding complications. In this paper, we use semiparametric inference theory for structural models to give a general approach to causal sufficient dimension reduction of a multidimensional treatment such that the cause-effect relationship between treatment and outcome is preserved. We illustrate the utility of our proposals through simulations and a real data application in radiation oncology.
Recently there has been sustained interest in modifying prediction algorithms to satisfy fairness constraints. These constraints are typically complex nonlinear functionals of the observed data distribution. Focusing on the path-specific causal constraints proposed by Nabi and Shpitser (2018), we introduce new theoretical results and optimization techniques to make model training easier and more accurate. Specifically, we show how to reparameterize the observed data likelihood such that fairness constraints correspond directly to parameters that appear in the likelihood, transforming a complex constrained optimization objective into a simple optimization problem with box constraints. We also exploit methods from empirical likelihood theory in statistics to improve predictive performance by constraining baseline covariates, without requiring parametric models. We combine the merits of both proposals to optimize a hybrid reparameterized likelihood. The techniques presented here should be applicable more broadly to fair prediction proposals that impose constraints on predictive models.
In classical causal inference, inferring cause-effect relations from data relies on the assumption that units are independent and identically distributed. This assumption is violated in settings where units are related through a network of dependencies. An example of such a setting is ad placement in sponsored search advertising, where the clickability of a particular ad is potentially influenced by where it is placed and where other ads are placed on the search result page. In such scenarios, confounding arises due to not only the individual ad-level covariates but also the placements and covariates of other ads in the system. In this paper, we leverage the language of causal inference in the presence of interference to model interactions among the ads. Quantification of such interactions allows us to better understand the click behavior of users, which in turn impacts the revenue of the host search engine and enhances user satisfaction. We illustrate the utility of our formalization through experiments carried out on the ad placement system of the Bing search engine.
Missing data has the potential to affect analyses conducted in all fields of scientific study, including healthcare, economics, and the social sciences. Several approaches to unbiased inference in the presence of non-ignorable missingness rely on the specification of the target distribution and its missingness process as a probability distribution that factorizes with respect to a directed acyclic graph. In this paper, we address the longstanding question of the characterization of models that are identifiable within this class of missing data distributions. We provide the first completeness result in this field of study -- necessary and sufficient graphical conditions under which, the full data distribution can be recovered from the observed data distribution. We then simultaneously address issues that may arise due to the presence of both missing data and unmeasured confounding, by extending these graphical conditions and proofs of completeness, to settings where some variables are not just missing, but completely unobserved.