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1. Propensity Score Weighting for Causal Inference with Clustered Data

   https://doi.org/10.1515/jci-2017-0027  

   Yang, Shu (September 2018, Journal of Causal Inference)

   Abstract Propensity score weighting is a tool for causal inference to adjust for measured confounders in observational studies. In practice, data often present complex structures, such as clustering, which make propensity score modeling and estimation challenging. In addition, for clustered data, there may be unmeasured cluster-level covariates that are related to both the treatment assignment and outcome. When such unmeasured cluster-specific confounders exist and are omitted in the propensity score model, the subsequent propensity score adjustment may be biased. In this article, we propose a calibration technique for propensity score estimation under the latent ignorable treatment assignment mechanism, i. e., the treatment-outcome relationship is unconfounded given the observed covariates and the latent cluster-specific confounders. We impose novel balance constraints which imply exact balance of the observed confounders and the unobserved cluster-level confounders between the treatment groups. We show that the proposed calibrated propensity score weighting estimator is doubly robust in that it is consistent for the average treatment effect if either the propensity score model is correctly specified or the outcome follows a linear mixed effects model. Moreover, the proposed weighting method can be combined with sampling weights for an integrated solution to handle confounding and sampling designs for causal inference with clustered survey data. In simulation studies, we show that the proposed estimator is superior to other competitors. We estimate the effect of School Body Mass Index Screening on prevalence of overweight and obesity for elementary schools in Pennsylvania. 

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2. Identification of Linear Non-{G}aussian Latent Hierarchical Structure

   Xie, Feng; Huang, Biwei; Chen, Zhengming; He, Yangbo; Geng, Zhi; Zhang, Kun (July 2022, Proceedings of Machine Learning Research)

   Chaudhuri, Kamalika; Jegelka, Stefanie; Song, Le; Szepesvari, Csaba; Niu, Gang; Sabato, Sivan (Ed.)

   Traditional causal discovery methods mainly focus on estimating causal relations among measured variables, but in many real-world problems, such as questionnaire-based psychometric studies, measured variables are generated by latent variables that are causally related. Accordingly, this paper investigates the problem of discovering the hidden causal variables and estimating the causal structure, including both the causal relations among latent variables and those between latent and measured variables. We relax the frequently-used measurement assumption and allow the children of latent variables to be latent as well, and hence deal with a specific type of latent hierarchical causal structure. In particular, we define a minimal latent hierarchical structure and show that for linear non-Gaussian models with the minimal latent hierarchical structure, the whole structure is identifiable from only the measured variables. Moreover, we develop a principled method to identify the structure by testing for Generalized Independent Noise (GIN) conditions in specific ways. Experimental results on both synthetic and real-world data show the effectiveness of the proposed approach. 

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3. Causal Effect Estimation with Mixed Latent Confounders and Post-treatment Variables

   Zhu, Yaochen; Ma, Jing; Wu, Liang; Guo, Qi; Hong, Liangjie; Li, Jundong (April 2025, International Conference on Learning Representations)

   Causal inference from observational data has attracted considerable attention among researchers. One main obstacle is the handling of confounders. As direct measurement of confounders may not be feasible, recent methods seek to address the confounding bias via proxy variables, i.e., covariates postulated to be conducive to the inference of latent confounders. However, the selected proxies may scramble both confounders and post-treatment variables in practice, which risks biasing the estimation by controlling for variables affected by the treatment. In this paper, we systematically investigate the bias due to latent post-treatment variables, i.e., latent post-treatment bias, in causal effect estimation. Specifically, we first derive the bias when selected proxies scramble both latent confounders and post-treatment variables, which we demonstrate can be arbitrarily bad. We then propose a Confounder-identifiable VAE (CiVAE) to address the bias. Based on a mild assumption that the prior of latent variables that generate the proxy belongs to a general exponential family with at least one invertible sufficient statistic in the factorized part, CiVAE individually identifies latent confounders and latent post-treatment variables up to bijective transformations. We then prove that with individual identification, the intractable disentanglement problem of latent confounders and post-treatment variables can be transformed into a tractable independence test problem despite arbitrary dependence may exist among them. Finally, we prove that the true causal effects can be unbiasedly estimated with transformed confounders inferred by CiVAE. Experiments on both simulated and real-world datasets demonstrate significantly improved robustness of CiVAE. 

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4. Triad Constraints for Learning Causal Structure of Latent Variables

   Ruichu Cai, Feng Xie (December 2019, Advances in neural information processing systems)

   Learning causal structure from observational data has attracted much attention,and it is notoriously challenging to find the underlying structure in the presenceof confounders (hidden direct common causes of two variables). In this paper,by properly leveraging the non-Gaussianity of the data, we propose to estimatethe structure over latent variables with the so-called Triad constraints: we design a form of "pseudo-residual" from three variables, and show that when causal relations are linear and noise terms are non-Gaussian, the causal direction between the latent variables for the three observed variables is identifiable by checking a certain kind of independence relationship. In other words, the Triad constraints help us to locate latent confounders and determine the causal direction between them. This goes far beyond the Tetrad constraints and reveals more information about the underlying structure from non-Gaussian data. Finally, based on the Triad constraints, we develop a two-step algorithm to learn the causal structure corresponding to measurement models. Experimental results on both synthetic and real data demonstrate the effectiveness and reliability of our method. 

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5. Estimating Average Treatment Effects Utilizing Fractional Imputation when Confounders are Subject to Missingness

   https://doi.org/10.1515/jci-2019-0024  

   Corder, Nathan; Yang, Shu (January 2020, Journal of Causal Inference)

   null (Ed.)

   Abstract The problem of missingness in observational data is ubiquitous. When the confounders are missing at random, multiple imputation is commonly used; however, the method requires congeniality conditions for valid inferences, which may not be satisfied when estimating average causal treatment effects. Alternatively, fractional imputation, proposed by Kim 2011, has been implemented to handling missing values in regression context. In this article, we develop fractional imputation methods for estimating the average treatment effects with confounders missing at random. We show that the fractional imputation estimator of the average treatment effect is asymptotically normal, which permits a consistent variance estimate. Via simulation study, we compare fractional imputation’s accuracy and precision with that of multiple imputation. 

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