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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. Soft calibration for selection bias problems under mixed-effects models

   https://doi.org/10.1093/biomet/asad016  

   Gao, Chenyin; Yang, Shu; Kim, Jae Kwang (March 2023, Biometrika)

   Abstract Calibration weighting has been widely used to correct selection biases in nonprobability sampling, missing data and causal inference. The main idea is to calibrate the biased sample to the benchmark by adjusting the subject weights. However, hard calibration can produce enormous weights when an exact calibration is enforced on a large set of extraneous covariates. This article proposes a soft calibration scheme, where the outcome and the selection indicator follow mixed-effect models. The scheme imposes an exact calibration on the fixed effects and an approximate calibration on the random effects. On the one hand, our soft calibration has an intrinsic connection with best linear unbiased prediction, which results in a more efficient estimation compared to hard calibration. On the other hand, soft calibration weighting estimation can be envisioned as penalized propensity score weight estimation, with the penalty term motivated by the mixed-effect structure. The asymptotic distribution and a valid variance estimator are derived for soft calibration. We demonstrate the superiority of the proposed estimator over other competitors in simulation studies and using a real-world data application on the effect of BMI screening on childhood obesity. 

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3. Augmented balancing weights as linear regression

   https://doi.org/10.1093/jrsssb/qkaf019  

   Bruns-Smith, David; Dukes, Oliver; Feller, Avi; Ogburn, Elizabeth L (April 2025, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning. These popular doubly robust estimators combine outcome modelling with balancing weights—weights that achieve covariate balance directly instead of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine those of the original outcome model with those from unpenalized ordinary least-squares (OLS). Under certain choices of regularization parameters, the augmented estimator in fact collapses to the OLS estimator alone. We then extend these results to specific outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression—implying a novel analysis of undersmoothing. When the weighting model is instead lasso-penalized, we demonstrate a familiar ‘double selection’ property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights. 

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4. An Information Theoretic Approach to Prevalence Estimation and Missing Data

   Hossjer, Ola; Diaz-Pachon, Daniel; Zhao, Chen; Rao, J Sunil (October 2023, IEEE transactions on information theory)

   Many data sources, including tracking social behav- ior to election polling to testing studies for understanding disease spread, are subject to sampling bias whose implications are not fully yet understood. In this paper we study estimation of a given feature (such as disease, or behavior at social media platforms) from biased samples, treating non-respondent individuals as missing data. Prevalence of the feature among sampled individuals has an upward bias under the assumption of individuals’ willingness to be sampled. This can be viewed as a regression model with symptoms as covariates and the feature as outcome. It is assumed that the outcome is unknown at the time of sampling, and therefore the missingness mechanism only depends on the covariates. We show that data, in spite of this, is missing at random only when the sizes of symptom classes in the population are known; otherwise data is missing not at random. With an information theoretic viewpoint, we show that sampling bias corresponds to external information due to individuals in the population knowing their covariates, and we quantify this external information by active information. The reduction in prevalence, when sampling bias is adjusted for, similarly translates into active information due to bias correction, with opposite sign to active information due to testing bias. We develop unified results that show that prevalence and active information estimates are asymptotically normal under all missing data mechanisms, when testing errors are absent and present respectively. The asymptotic behavior of the estimators is illustrated through simulations. 

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5. Causal discovery in the presence of missing data

   Ruibo Tu, Cheng Zhang (April 2019, Proceedings of Machine Learning Research)

   Missing data are ubiquitous in many domain such as healthcare. When these data entries are not missing completely at random, the (conditional) independence relations in the observed data may be different from those in the complete data generated by the underlying causal process.Consequently, simply applying existing causal discovery methods to the observed data may lead to wrong conclusions. In this paper, we aim at developing a causal discovery method to recover the underlying causal structure from observed data that are missing under different mechanisms, including missing completely at random (MCAR),missing at random (MAR), and missing not at random (MNAR). With missingness mechanisms represented by missingness graphs (m-graphs),we analyze conditions under which additional correction is needed to derive conditional independence/dependence relations in the complete data. Based on our analysis, we propose Miss-ing Value PC (MVPC), which extends the PC algorithm to incorporate additional corrections.Our proposed MVPC is shown in theory to give asymptotically correct results even on data that are MAR or MNAR. Experimental results on both synthetic data and real healthcare applications illustrate that the proposed algorithm is able to find correct causal relations even in the general case of MNAR. 

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