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Plots of N { ( 0 , 0 ) T , I 2 } I ( Δ MSE j ≥ 0 ) (which appears in the prior for ( β j , γ j ) T under BSSCE) as a function of βj and ηj. The density of the prior is given by the color: white indicates a prior density of zero, and blue colors indicate a smaller prior density than red colors. The top left plot has σ Y | A , X − j 2 = 10 , σ X j | X − j , A 2 = 0.25 , σ A | X − j 2 = 10 , and σ X j | X − j 2 = 1 . The other three plots double one of these parameters while leaving the other parameters unchanged: the top right plot has σ Y | A , X − j 2 = 20 , the bottom left plot has σ A | X − j 2 = 20 , and the bottom right plot has σ X j | X − j , A 2 = 0.5 . For this figure, n = 250, σ 2 = 1 , and τ j 2 = 1 .
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Unbiased estimation of causal effects with observational data requires adjustment for confounding variables that are related to both the outcome and treatment assignment. Standard variable selection techniques aim to maximize predictive ability of the outcome model, but they ignore covariate associations with treatment and may not adjust for import...
Citations
... In recent years, several Bayesian linear models have been proposed for genetic studies using the SNP data, including Bayesian sparse linear mixed models, Bayesian spike-and-slab regression models, and Bayesian variable selection models [7]. These models have been used to identify genetic variants associated with complex traits, predict the traits using the SNP data, and identify genetic pathways involved in disease pathogenesis. ...
... Still, it is highly desirable to try to adjust for confounding in our statistical models to the best of our ability. This is termed (perhaps somewhat unfortunately) causal inference in the literature [1,2,3,4,5,6,7,8,9,10]. This is also the approach that we will follow here, under the disclaimer that whether actual causality can be inferred remains a subject of interpretation and conjecture specific to the situation being studied. ...
... In this paper, we take inspiration from the approach of Koch et al. [10], who proposed a bi-level spike and slab prior for causal effect estimation in high dimensional problems (i.e. when number of predictors is larger than the number of observations). They considered a data-driven adaptive approach to propose their prior which reduces the variance of the causal estimate. ...
... We use these studies to compare our method with three other approaches. From now on, for the sake of illustration, we use the following acronyms: RBCE for robust Bayesian causal estimation (our method); SSCE for spike and slab causal estimation [10]; BSSCE for bi-level spike and slab causal estimation [10]; and BSSL for Bayesian spike and slab LASSO [20]. ...
Causal effect estimation is a critical task in statistical learning that aims to find the causal effect on subjects by identifying causal links between a number of predictor (or, explanatory) variables and the outcome of a treatment. In a regressional framework, we assign a treatment and outcome model to estimate the average causal effect. Additionally, for high dimensional regression problems, variable selection methods are also used to find a subset of predictor variables that maximises the predictive performance of the underlying model for better estimation of the causal effect. In this paper, we propose a different approach. We focus on the variable selection aspects of high dimensional causal estimation problem. We suggest a cautious Bayesian group LASSO (least absolute shrinkage and selection operator) framework for variable selection using prior sensitivity analysis. We argue that in some cases, abstaining from selecting (or, rejecting) a predictor is beneficial and we should gather more information to obtain a more decisive result. We also show that for problems with very limited information, expert elicited variable selection can give us a more stable causal effect estimation as it avoids overfitting. Lastly, we carry a comparative study with synthetic dataset and show the applicability of our method in real-life situations.
... In the literature, variable selection for causal inference (e.g., [14][15][16]) or measurement error correction (e.g., [17][18][19][20][21][22]) are discussed under various settings. However, in the concurrent presence of both features, limited work has been carried out to estimate ATE except for [23]. ...
In causal inference, the estimation of the average treatment effect is often of interest. For example, in cancer research, an interesting question is to assess the effects of the chemotherapy treatment on cancer, with the information of gene expressions taken into account. Two crucial challenges in this analysis involve addressing measurement error in gene expressions and handling noninformative gene expressions. While analytical methods have been developed to address those challenges, no user-friendly computational software packages seem to be available to implement those methods. To close this gap, we develop an R package, called AteMeVs, to estimate the average treatment effect using the inverse-probability-weighting estimation method to handle data with both measurement error and spurious variables. This developed package accommodates the method proposed by Yi and Chen (2023) as a special case, and further extends its application to a broader scope. The usage of the developed R package is illustrated by applying it to analyze a cancer dataset with information of gene expressions.
... In this paper we take inspiration from the approach of Koch et al. [11], who proposed a bi-level spike and slab prior for causal effect estimation. They considered a data-driven adaptive approach to propose their prior which reduces the variance of the causal estimate. ...
... We present our analyses in Table 1 and Table 2. For the sake of clarity we use the following accronyms: RBCE for robust Bayesian causal estimation (our method); SSCE for spike and slab causal estimation [11]; BSSCE for bi-level spike and slab causal estimation [11]; and BSSL for Bayesian spike and slab lasso [15]. As it can be seen from both the tables, SSCE and BSSCE are formulated for problems where p ≤ n and therefore we do not have any results for n < 50. ...
... We present our analyses in Table 1 and Table 2. For the sake of clarity we use the following accronyms: RBCE for robust Bayesian causal estimation (our method); SSCE for spike and slab causal estimation [11]; BSSCE for bi-level spike and slab causal estimation [11]; and BSSL for Bayesian spike and slab lasso [15]. As it can be seen from both the tables, SSCE and BSSCE are formulated for problems where p ≤ n and therefore we do not have any results for n < 50. ...
Causal inference concerns finding the treatment effect on subjects along with causal links between the variables and the outcome. However, the underlying heterogeneity between subjects makes the problem practically unsolvable. Additionally, we often need to find a subset of explanatory variables to understand the treatment effect. Currently, variable selection methods tend to maximise the predictive performance of the underlying model, and unfortunately, under limited data, the predictive performance is hard to assess, leading to harmful consequences. To address these issues, in this paper, we consider a robust Bayesian analysis which accounts for abstention in selecting explanatory variables in the high dimensional regression model. To achieve that, we consider a set of spike and slab priors through prior elicitation to obtain a set of posteriors for both the treatment and outcome model. We are specifically interested in the sensitivity of the treatment effect in high dimensional causal inference as well as identifying confounder variables. However, confounder selection can be deceptive in this setting, especially when a predictor is strongly associated with either the treatment or the outcome. To avoid that we apply a post-hoc selection scheme, attaining a smaller set of confounders as well as separate sets of variables which are only related to treatment or outcome model. Finally, we illustrate our method to show its applicability.
... In this paper we take inspiration from the approach of Koch et al. [8], who proposed a bi-level spike and slab prior for causal effect estimation. They considered a data-driven adaptive approach to propose their prior which reduces the variance of the causal estimate. ...
... We present our analyses in Table 1 and Table 2. For the sake of clarity we use the following accronyms: RBCE for robust Bayesian causal estimation (our method); SSCE for spike and slab causal estimation [8]; BSSCE for bi-level spike and slab causal estimation [8]; and BSSL for Bayesian spike and slab lasso [17]. As it can be seen from both the tables, SSCE and BSSCE are formulated for problems where p ≤ n and therefore we do not have any results for n < 50. ...
... We present our analyses in Table 1 and Table 2. For the sake of clarity we use the following accronyms: RBCE for robust Bayesian causal estimation (our method); SSCE for spike and slab causal estimation [8]; BSSCE for bi-level spike and slab causal estimation [8]; and BSSL for Bayesian spike and slab lasso [17]. As it can be seen from both the tables, SSCE and BSSCE are formulated for problems where p ≤ n and therefore we do not have any results for n < 50. ...
Causal inference using observational data is an important aspect in many fields such as epidemiology, social science, economics, etc. In particular, our goal is to find the treatment effect on the subjects along with the causal links between the variables and the outcome. However, estimation for such problems are extremely difficult as the treatment effects may vary from subject to subject and modelling the underlying heterogeneity explicitly makes the problem practically unsolvable. Another issue we often face is the dimensionality of the problem and we need to find a subset of explanatory variables to initiate the treatment. However, currently variable selection methods tend to maximise the predictive performance of the outcome model only. This can be problematic in the case of limited information. As the consequence of mistreatment can be harmful. So, in this paper, we suggest a general framework with robust Bayesian analysis which accounts for abstention in deciding an explanatory variable in the high dimensional regression model. To achieve that, we consider a set of spike and slab priors through prior elicitation to obtain robust estimates for both the treatment and outcome model. We are specifically interested in the sensitivity of the treatment effect in the high dimensional causal inference as well as the identifying the confounder variables by means of variable selection. However, indicator based confounder selection can be deceptive in some cases. Especially, when the predictor is strongly associated with either the treatment or the outcome. This increases the posterior expectation of the selection indicators. To avoid that we apply a post-hoc selection scheme which successfully remove negligible non-zero effects from the model attaining a smaller set of confounders. Finally, we illustrate our result using synthetic dataset.
... Ertefaie et al. 7 developed a penalized objective function by employing both the outcome and treatment models to do a variable selection. Assuming Spike and slab priors for covariate coefficients, Koch et al. 8 explored a Bayesian method to estimate causal effects with outcome and treatment models simultaneously employed. Ghosh et al. 9 proposed the "multiply impute, then select" approach by employing the lasso method. ...
... Let X * Ii (k, ψ) denote the subvector of X * i (k, ψ) corresponding to X * Ii , generated from Step 1. Using the selected treatment model (8) with X Ii replaced by X * Ii (k, ψ), we calculate the fitted value π i (k, ψ) ≜ g(X * Ii (k, ψ), Z Ii ; γ I ). Then we obtain an estimate, say, τ(k, ψ), of τ 0 using (2) with π i replaced by π i (k, ψ), and calculate ...
... When V is equal to Σ −1 , the covariance matrix of γ I is no greater than the covariance variance of γ I in the Loewner order, where γ I is the subvector of the SIMEX estimator γ corresponding to γ I . Theorem 3.1(a) establishes the asymptotic distribution for the estimators for the effects corresponding to important pre-treatment variables in model (3), or equivalently, for the estimators of the parameters for the selected treatment model (8). Theorem 3.1(b) ensures the oracle property in the sense of Fan and Li 23 for the variable selection procedure for building the final treatment model (8). ...
In the framework of causal inference, the inverse probability weighting estimation method and its variants have been commonly employed to estimate the average treatment effect. Such methods, however, are challenged by the presence of irrelevant pre-treatment variables and measurement error. Ignoring these features and naively applying the usual inverse probability weighting estimation procedures may typically yield biased inference results. In this article, we develop an inference method for estimating the average treatment effect with those features taken into account. We establish theoretical properties for the resulting estimator and carry out numerical studies to assess the finite sample performance of the proposed estimator.
... Citing previous simulation studies (Austin, Grootendorst and Anderson, 2007;Brookhart et al., 2006), Austin concluded that "there were merits to including only the potential confounders or the true confounders in the propensity score model". The idea that only controlling for the "true confounders" is sufficient, if not superior, is commonplace in practice (Glymour, Weuve and Chen, 2008) and methodological development (Ertefaie, Asgharian and Stephens, 2017;Shortreed and Ertefaie, 2017;Koch et al., 2020). Although this is sometimes referred to as the "common cause principle", we think a more accurate name is perhaps "conjunction heuristic", as this principle is often applied by merely testing conditional independence (see Section 3.3). ...
Confounder selection is perhaps the most important step in the design of observational studies. A number of criteria, often with different objectives and approaches, have been proposed, and their validity and practical value have been debated in the literature. Here, we provide a unified review of these criteria and the assumptions behind them. We list several objectives that confounder selection methods aim to achieve and discuss the amount of structural knowledge required by different approaches. Finally, we discuss limitations of the existing approaches and implications for practitioners.
... Drawing a causal conclusion generally requires the adjustment of sufficient covariates to satisfy the unconfoundedness assumption. Because omitting important confounders can bias estimation, researchers have been advocating the inclusion of all available covariates into the PS model [10,11]. However, with the ability to collect high-dimensional covariates rapidly improving, this 'all inclusive' approach may fail because (1) the inclusion of many unnecessary covariates could reduce efficiency and even induce bias [10][11][12]; and (2) it may not be possible to fit the PS model with high-dimensional covariates for reasons such as singularity [13]. ...
... Because omitting important confounders can bias estimation, researchers have been advocating the inclusion of all available covariates into the PS model [10,11]. However, with the ability to collect high-dimensional covariates rapidly improving, this 'all inclusive' approach may fail because (1) the inclusion of many unnecessary covariates could reduce efficiency and even induce bias [10][11][12]; and (2) it may not be possible to fit the PS model with high-dimensional covariates for reasons such as singularity [13]. For example, genome-wide SNP [14] or DNA methylation data [15] have been regarded as potential confounders or surrogates for unmeasured confounders when PS methods are used to estimate causal effects. ...
... For example, Chernozhukov et al. [22] proposed the double/debiased ML method (DML), which uses methods such as the random forest and neural network to fit the PS and the outcome model. Furthermore, considerable studies have been developed within the Bayesian paradigm [11,12,[23][24][25][26][27][28]. For example, Bayesian model averaging type methods [23,24] or a DR estimator proposed by Antonelli et al. [27]. ...
In recent work, researchers have paid considerable attention to the estimation of causal effects in observational studies with a large number of covariates, which makes the unconfoundedness assumption plausible. In this paper, we review propensity score (PS) methods developed in high-dimensional settings and broadly group them into model-based methods that extend models for prediction to causal inference and balance-based methods that combine covariate balancing constraints. We conducted systematic simulation experiments to evaluate these two types of methods, and studied whether the use of balancing constraints further improved estimation performance. Our comparison methods were post-double-selection (PDS), double-index PS (DiPS), outcome-adaptive LASSO (OAL), group LASSO and doubly robust estimation (GLiDeR), high-dimensional covariate balancing PS (hdCBPS), regularized calibrated estimators (RCAL) and approximate residual balancing method (balanceHD). For the four model-based methods, simulation studies showed that GLiDeR was the most stable approach, with high estimation accuracy and precision, followed by PDS, OAL and DiPS. For balance-based methods, hdCBPS performed similarly to GLiDeR in terms of accuracy, and outperformed balanceHD and RCAL. These findings imply that PS methods do not benefit appreciably from covariate balancing constraints in high-dimensional settings. In conclusion, we recommend the preferential use of GLiDeR and hdCBPS approaches for estimating causal effects in high-dimensional settings; however, further studies on the construction of valid confidence intervals are required.
... 7 To improve large-scale propensity score analyses in healthcare databases, several papers have proposed using data-adaptive algorithms to help identify confounding factors and exclude variables that harm the properties of effect estimates (e.g., instrumental variables). [14][15][16][17][18][19][20][21][22][23][24][25] In healthcare database studies, however, outcome events are often rare, and it can be difficult to accurately characterize the joint correlation structure between a high-dimensional set of variables and the outcome to distinguish between instruments, confounders, and other variable types. Various approaches rely on tuning parameters and assumptions that hold in different cases, and it can be unclear if a given approach adequately adjusts for all measured confounders. ...
The propensity score has become a standard tool to control for large numbers of variables in healthcare database studies. However, little has been written on the challenge of comparing large-scale propensity score analyses that use different methods for confounder selection and adjustment. In these settings, balance diagnostics are useful but do not inform researchers on which variables balance should be assessed or quantify the impact of residual covariate imbalance on bias. Here, we propose a framework to supplement balance diagnostics when comparing large-scale propensity score analyses. Instead of focusing on results from any single analysis, we suggest conducting and reporting results for many analytic choices and using both balance diagnostics and synthetically generated control studies to screen analyses that show signals of bias caused by measured confounding. To generate synthetic datasets, the framework does not require simulating the outcome-generating process. In healthcare database studies, outcome events are often rare, making it difficult to identify and model all predictors of the outcome to simulate a confounding structure closely resembling the given study. Therefore, the framework uses a model for treatment assignment to divide the comparator population into pseudo-treatment groups where covariate differences resemble those in the study cohort. The partially simulated datasets have a confounding structure approximating the study population under the null (synthetic negative control studies). The framework is used to screen analyses that likely violate partial exchangeability due to lack of control for measured confounding. We illustrate the framework using simulations and an empirical example.
... Across all these methods, one critical issue is how to infuse the theory and features of TCM into methodological components. Despite relatively high degree of standardization when using acupuncture and Chinese patent medicine, objective complex factors continue to present challenges in assessing clinical effects, including outcome measurements that are applicable to TCM features (Wang and Huang, 2019;Zhang et al., 2021), effect estimation in rare event setting (Greenland et al., 2016), study designs involving patient preferences (Angell, 1984;Rücker, 1989;Cameron et al., 2018), repeated measures in TCM setting (Albert, 1999;Twisk, 2004;Goetgeluk and Vansteelandt, 2008;Wang et al., 2012), causal inference of treatment effects in complex design and data settings (Zigler and Dominici, 2014;Koch et al., 2020), and decomposition of effects in complex intervention setting. Continuing efforts are warranted to address these issues. ...
