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We propose a doubly robust approach to characterizing treatment effect heterogeneity in observational studies. We utilize posterior distributions for both the propensity score and outcome regression models to provide valid inference on the conditional average treatment effect even when high-dimensional or nonparametric models are used. We show that our approach leads to conservative inference in finite samples or under model misspecification, and provides a consistent variance estimator when both models are correctly specified. In simulations, we illustrate the utility of these results in difficult settings such as high-dimensional covariate spaces or highly flexible models for the propensity score and outcome regression. Lastly, we analyze environmental exposure data from NHANES to identify how the effects of these exposures vary by subject-level characteristics.

We study causal inference under case-control and case-population sampling. For this purpose, we focus on the binary-outcome and binary-treatment case, where the parameters of interest are causal relative and attributable risk defined via the potential outcome framework. It is shown that strong ignorability is not always as powerful as it is under random sampling and that certain monotonicity assumptions yield comparable results in terms of sharp identified intervals. Specifically, the usual odds ratio is shown to be a sharp identified upper bound on causal relative risk under the monotone treatment response and monotone treatment selection assumptions. We then discuss averaging the conditional (log) odds ratio and propose an algorithm for semiparametrically efficient estimation when averaging is based only on the (conditional) distributions of the covariates that are identified in the data. We also offer algorithms for causal inference if the true population distribution of the covariates is desirable for aggregation. We show the usefulness of our approach by studying two empirical examples from social sciences: the benefit of attending private school for entering a prestigious university in Pakistan and the causal relationship between staying in school and getting involved with drug-trafficking gangs in Brazil.

Measurement error is a pervasive issue which renders the results of an analysis unreliable. The measurement error literature contains numerous correction techniques, which can be broadly divided into those which aim to produce exactly consistent estimators, and those which are only approximately consistent. While consistency is a desirable property, it is typically attained only under specific model assumptions. Two techniques, regression calibration and simulation extrapolation, are used frequently in a wide variety of parametric and semiparametric settings. However, in many settings these methods are only approximately consistent. We generalize these corrections, relaxing assumptions placed on replicate measurements. Under regularity conditions, the estimators are shown to be asymptotically normal, with a sandwich estimator for the asymptotic variance. Through simulation, we demonstrate the improved performance of the modified estimators, over the standard techniques, when these assumptions are violated. We motivate these corrections using the Framingham Heart Study, and apply the generalized techniques to an analysis of these data.

This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifiable in principle. Similar to no-free-lunch theorems for statistical inference, the present results clarify the inevitability of substantial assumptions for causal inference. An additional benefit of our topological approach is that it easily accommodates SCMs with infinitely many variables. We finally suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.

Instrumental variables allow for quantification of cause and effect relationships even in the absence of interventions. To achieve this, a number of causal assumptions must be met, the most important of which is the independence assumption, which states that the instrument and any confounding factor must be independent. However, if this independence condition is not met, can we still work with imperfect instrumental variables? Imperfect instruments can manifest themselves by violations of the instrumental inequalities that constrain the set of correlations in the scenario. In this paper, we establish a quantitative relationship between such violations of instrumental inequalities and the minimal amount of measurement dependence required to explain them. As a result, we provide adapted inequalities that are valid in the presence of a relaxed measurement dependence assumption in the instrumental scenario. This allows for the adaptation of existing and new lower bounds on the average causal effect for instrumental scenarios with binary outcomes. Finally, we discuss our findings in the context of quantum mechanics.

Proximal causal inference was recently proposed as a framework to identify causal effects from observational data in the presence of hidden confounders for which proxies are available. In this paper, we extend the proximal causal approach to settings where identification of causal effects hinges upon a set of mediators which unfortunately are not directly observed, however proxies of the hidden mediators are measured. Specifically, we establish (i) a new hidden front-door criterion which extends the classical front-door result to allow for hidden mediators for which proxies are available; (ii) We extend causal mediation analysis to identify direct and indirect causal effects under unconfoundedness conditions in a setting where the mediator in view is hidden, but error prone proxies of the latter are available. We view (i) and (ii) as important steps towards the practical application of front-door criteria and mediation analysis as mediators are almost always error prone and thus, the most one can hope for in practice is that our measurements are at best proxies of mediating mechanisms. Finally, we show that identification of certain causal effects remains possible even in settings where challenges in (i) and (ii) might co-exist.

During the COVID-19 pandemic, governments and public health authorities have used seroprevalence studies to estimate the proportion of persons within a given population who have antibodies to SARS-CoV-2. Seroprevalence is crucial for estimating quantities such as the infection fatality ratio, proportion of asymptomatic cases, and differences in infection rates across population subgroups. However, serologic assays are prone to false positives and negatives, and non-probability sampling methods may induce selection bias. In this paper, we consider nonparametric and parametric seroprevalence estimators that address both challenges by leveraging validation data and assuming equal probabilities of sample inclusion within covariate-defined strata. Both estimators are shown to be consistent and asymptotically normal, and consistent variance estimators are derived. Simulation studies are presented comparing the finite sample performance of the estimators over a range of assay characteristics and sampling scenarios. The methods are used to estimate SARS-CoV-2 seroprevalence in asymptomatic individuals in Belgium and North Carolina.

The presence of measurement error is a widespread issue which, when ignored, can render the results of an analysis unreliable. Numerous corrections for the effects of measurement error have been proposed and studied, often under the assumption of a normally distributed, additive measurement error model. One such correction is the simulation extrapolation method, which provides a flexible way of correcting for the effects of error in a wide variety of models, when the errors are approximately normally distributed. However, in many situations observed data are non-symmetric, heavy-tailed, or otherwise highly non-normal. In these settings, correction techniques relying on the assumption of normality are undesirable. We propose an extension to the simulation extrapolation method which is nonparametric in the sense that no specific distributional assumptions are required on the error terms. The technique is implemented when either validation data or replicate measurements are available, and it shares the general structure of the standard simulation extrapolation procedure, making it immediately accessible for those familiar with this technique.

This paper is concerned with data-driven unsupervised domain adaptation, where it is unknown in advance how the joint distribution changes across domains, i.e., what factors or modules of the data distribution remain invariant or change across domains. To develop an automated way of domain adaptation with multiple source domains, we propose to use a graphical model as a compact way to encode the change property of the joint distribution, which can be learned from data, and then view domain adaptation as a problem of Bayesian inference on the graphical models. Such a graphical model distinguishes between constant and varied modules of the distribution and specifies the properties of the changes across domains, which serves as prior knowledge of the changing modules for the purpose of deriving the posterior of the target variable $Y$ in the target domain. This provides an end-to-end framework of domain adaptation, in which additional knowledge about how the joint distribution changes, if available, can be directly incorporated to improve the graphical representation. We discuss how causality-based domain adaptation can be put under this umbrella. Experimental results on both synthetic and real data demonstrate the efficacy of the proposed framework for domain adaptation. The code is available at //github.com/mgong2/DA_Infer .

Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing research direction owing to the large amount of available data and low budget requirement, compared with randomized controlled trials. Embraced with the rapidly developed machine learning area, various causal effect estimation methods for observational data have sprung up. In this survey, we provide a comprehensive review of causal inference methods under the potential outcome framework, one of the well known causal inference framework. The methods are divided into two categories depending on whether they require all three assumptions of the potential outcome framework or not. For each category, both the traditional statistical methods and the recent machine learning enhanced methods are discussed and compared. The plausible applications of these methods are also presented, including the applications in advertising, recommendation, medicine and so on. Moreover, the commonly used benchmark datasets as well as the open-source codes are also summarized, which facilitate researchers and practitioners to explore, evaluate and apply the causal inference methods.