We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in case-control studies. Graphical models represent assumptions about the conditional independencies among the variables. By including a node for the sampling indicator, assumptions about sampling processes can be made explicit. We demonstrate how to read off such graphs whether consistent estimation of the association between exposure and outcome is possible. Moreover, we give sufficient graphical conditions for testing and estimating the causal effect of exposure on outcome. The practical use is illustrated with a number of examples.
Relationship between two popular modeling frameworks of causal inference from observational data, namely, causal graphical model and potential outcome causal model is discussed. How some popular causal effect estimators found in applications of the potential outcome causal model, such as inverse probability of treatment weighted estimator and doubly robust estimator can be obtained by using the causal graphical model is shown. We confine to the simple case of binary outcome and treatment variables with discrete confounders and it is shown how to generalize results to cases of continuous variab…
When engagement with a randomized trial is driven by factors that affect the outcome or when trial engagement directly affects the outcome independent of treatment, the average treatment effect among trial participants is unlikely to generalize to a target population. In this paper, we use counterfactual and graphical causal models to examine under what conditions we can generalize causal inferences from a randomized trial to the target population of trial-eligible individuals. We offer an interpretation of generalizability analyses using the notion of a hypothetical intervention to "scale-up"…
Comparing the differences in outcomes (that is, in "dependent variables") between two subpopulations is often most informative when comparing outcomes only for individuals from the subpopulations who are similar according to "independent variables." The independent variables are generally known as "scores," as in propensity scores for matching or as in the probabilities predicted by statistical or machine-learned models, for example. If the outcomes are discrete, then some averaging is necessary to reduce the noise arising from the outcomes varying randomly over those discrete values in the ob…
Mediation analysis seeks to infer how much of the effect of an exposure on an outcome can be attributed to specific pathways via intermediate variables or mediators. This requires identification of so-called path-specific effects. These express how a change in exposure affects those intermediate variables (along certain pathways), and how the resulting changes in those variables in turn affect the outcome (along subsequent pathways). However, unlike identification of total effects, adjustment for confounding is insufficient for identification of path-specific effects because their magnitude is…
Several approaches to causal inference from observational studies have been proposed. Since the proposal of Rubin (1974) many works have developed a counterfactual approach to causality, statistically formalized by potential outcomes. Pearl (2000) has put forward a theory of structural causal models which gives an important role to graphical models and do not necessarily use potential outcomes. On the other hand, several authors have developed a dynamical approach in line with Granger (1969). We analyze prospective and retrospective causal questions and their different modalities. Following Da…
Causal inference is a science with multi-disciplinary evolution and applications. On the one hand, it measures effects of treatments in observational data based on experimental designs and rigorous statistical inference to draw causal statements. One of the most influential framework in quantifying causal effects is the potential outcomes framework. On the other hand, causal graphical models utilizes directed edges to represent causalities and encodes conditional independence relationships among variables in the graphs. A series of research has been done both in reading-off conditional indepen…
Unmeasured confounding and selection bias are often of concern in observational studies and may invalidate a causal analysis if not appropriately accounted for. Under outcome-dependent sampling, a latent factor that has causal effects on the treatment, outcome, and sample selection process may cause both unmeasured confounding and selection bias, rendering standard causal parameters unidentifiable without additional assumptions. Under an odds ratio model for the treatment effect, Li et al. 2022 established both proximal identification and estimation of causal effects by leveraging a pair of ne…
Graph-based causal discovery methods aim to capture conditional independencies consistent with the observed data and differentiate causal relationships from indirect or induced ones. Successful construction of graphical models of data depends on the assumption of causal sufficiency: that is, that all confounding variables are measured. When this assumption is not met, learned graphical structures may become arbitrarily incorrect and effects implied by such models may be wrongly attributed, carry the wrong magnitude, or mis-represent direction of correlation. Wide application of graphical model…
Graphical causal models led to the development of complete non-parametric identification theory in arbitrary structured systems, and general approaches to efficient inference. Nevertheless, graphical approaches to causal inference have not been embraced by the statistics and public health communities. In those communities causal assumptions are instead expressed in terms of potential outcomes, or responses to hypothetical interventions. Such interventions are generally conceptualized only on a limited set of variables, where the corresponding experiment could, in principle, be performed. By co…
Outcome-dependent sampling designs are common in many different scientific fields including epidemiology, ecology, and economics. As with all observational studies, such designs often suffer from unmeasured confounding, which generally precludes the nonparametric identification of causal effects. Nonparametric bounds can provide a way to narrow the range of possible values for a nonidentifiable causal effect without making additional untestable assumptions. The nonparametric bounds literature has almost exclusively focused on settings with random sampling, and the bounds have often been derive…
Unobserved confounding is a major hurdle for causal inference from observational data. Confounders---the variables that affect both the causes and the outcome---induce spurious non-causal correlations between the two. Wang & Blei (2018) lower this hurdle with "the blessings of multiple causes," where the correlation structure of multiple causes provides indirect evidence for unobserved confounding. They leverage these blessings with an algorithm, called the deconfounder, that uses probabilistic factor models to correct for the confounders. In this paper, we take a causal graphical view of the …
Educational systems have traditionally been evaluated using cross-sectional studies, namely, examining a pretest, posttest, and single intervention. Although this is a popular approach, it does not model valuable information such as confounding variables, feedback to students, and other real-world deviations of studies from ideal conditions. Moreover, learning inherently is a sequential process and should involve a sequence of interventions. In this paper, we propose various experimental and quasi-experimental designs for educational systems and quantify them using the graphical model and dire…
Respondent-driven sampling (RDS) is a chain-referral method for sampling members of a hidden or hard-to-reach population such as sex workers, homeless people, or drug users via their social network. Most methodological work on RDS has focused on inference of population means under the assumption that subjects' network degree determines their probability of being sampled. Criticism of existing estimators is usually focused on missing data: the underlying network is only partially observed, so it is difficult to determine correct sampling probabilities. In this paper, we show that data collected…
The Millennium Villages Project (MVP) is a ten-year integrated rural development project implemented in ten sub-Saharan African sites. At its conclusion we will conduct an evaluation of its causal effect on a variety of development outcomes, measured via household surveys in treatment and comparison areas. Outcomes are measured by six survey modules, with sample sizes for each demographic group determined by budget, logistics, and the group's vulnerability. We design a sampling plan that aims to reduce effort for survey enumerators and maximize precision for all outcomes. We propose two-stage …
Williams College（威廉姆斯学院）；Emory University（埃默里大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
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 …
We establish the inferential properties of the mean-difference estimator for the average treatment effect in randomized experiments where each unit in a population is randomized to one of two treatments and then units within treatment groups are randomly sampled. The properties of this estimator are well-understood in the experimental design scenario where first units are randomly sampled and then treatment is randomly assigned, but not for the aforementioned scenario where the sampling and treatment assignment stages are reversed. We find that the inferential properties of the mean-difference…
Strip-plot designs are very useful when the treatments have a factorial structure and the factors levels are hard-to-change. We develop a randomization-based theory of causal inference from such designs in a potential outcomes framework. For any treatment contrast, an unbiased estimator is proposed, an expression for its sampling variance is worked out, and a conservative estimator of the sampling variance is obtained. This conservative estimator has a nonnegative bias, and becomes unbiased under between-block additivity, a condition milder than Neymannian strict additivity. A minimaxity prope…
Gaussian graphical models have been used to study intrinsic dependence among several variables, but the Gaussianity assumption may be restrictive in many applications. A nonparanormal graphical model is a semiparametric generalization for continuous variables where it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations on each of them. We consider a Bayesian approach in the nonparanormal graphical model by putting priors on the unknown transformations through a random series based on B-splines where the coefficients are ordere…
Bayesian estimation of Gaussian graphical models has proven to be challenging because the conjugate prior distribution on the Gaussian precision matrix, the G-Wishart distribution, has a doubly intractable partition function. Recent developments provide a direct way to sample from the G-Wishart distribution, which allows for more efficient algorithms for model selection than previously possible. Still, estimating Gaussian graphical models with more than a handful of variables remains a nearly infeasible task. Here, we propose two novel algorithms that use the direct sampler to more efficiently…
Scalable and accurate identification of specific clinical outcomes has been enabled by machine-learning applied to electronic medical record (EMR) systems. The development of classification models requires the collection of a complete labeled data set, where true clinical outcomes are obtained by human expert manual review. For example, the development of natural language processing algorithms requires the abstraction of clinical text data to obtain outcome information necessary for training models. However, if the outcome is rare then simple random sampling results in very few cases and insuf…
We propose two model-free, permutation-based tests of independence between a pair of random variables. The tests can be applied to samples from any bivariate distribution: continuous, discrete or mixture of those, with light tails or heavy tails, \ldots The tests take advantage of the recent development of the global envelope tests in the context of spatial statistics. Apart from the broad applicability of the tests, their main benefit lies in the graphical interpretation of the test outcome: in case of rejection of the null hypothesis of independence, the combinations of quantiles in the two …
We consider the problem of learning the structure of a high dimensional precision matrix under sparsity assumptions. We propose to use a shrinkage prior, called the DL-graphical prior based on the Dirichlet-Laplace prior used for the Gaussian mean problem. A posterior sampling scheme based on Gibbs sampling is also provided along with theoretical guarantees of the method by obtaining the posterior convergence rate of the precision matrix.
We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcom…
Two-phase outcome dependent sampling (ODS) is widely used in many fields, especially when certain covariates are expensive and/or difficult to measure. For two-phase ODS, the conditional maximum likelihood (CML) method is very attractive because it can handle zero Phase 2 selection probabilities and avoids modeling the covariate distribution. However, most existing CML-based methods use only the Phase 2 sample and thus may be less efficient than other methods. We propose a general empirical likelihood method that uses CML augmented with additional information in the whole Phase 1 sample to imp…
Use copula to model dependency of variable extends multivariate gaussian assumption. In this paper we first empirically studied copula regression model with continous response. Both simulation study and real data study are given. Secondly we give a novel copula regression model with binary outcome, and we propose a score gradient estimation algorithms to fit the model. Both simulation study and real data study are given for our model and fitting algorithm.
This article * provides an overview of post-election audit sampling research and compares various approaches to calculating post-election audit sample sizes, focusing on risklimiting audits, * discusses fundamental concepts common to all risk-limiting post-election audits, presenting new margin error bounds, sampling weights and sampling probabilities that improve upon existing approaches and work for any size audit unit and for single or multi-winner election contests, * provides two new simple formulas for estimating post-election audit sample sizes in cases when detailed data, expertise, or…
In this work we handle with categorical (ordinal) variables and we focus on the (in)dependence relationship under the marginal, conditional and context-specific perspective. If the first two are well known, the last one concerns independencies holding only in a subspace of the outcome space. We take advantage from the Hierarchical Multinomial Marginal models and provide several original results about the representation of context-specific independencies through these models. By considering the graphical aspect, we take advantage from the chain graphical models. The resultant graphical model is…
Electronic health record (EHR) data are increasingly used for biomedical research, but these data have recognized data quality challenges. Data validation is necessary to use EHR data with confidence, but limited resources typically make complete data validation impossible. Using EHR data, we illustrate prospective, multi-wave, two-phase validation sampling to estimate the association between maternal weight gain during pregnancy and the risks of her child developing obesity or asthma. The optimal validation sampling design depends on the unknown efficient influence functions of regression coe…
Nonresponse is a common problem in survey sampling. Appropriate treatment can be challenging, especially when dealing with detailed breakdowns of totals. Often, the nearest neighbor imputation method is used to handle such incomplete multinomial data. In this article, we investigate the nearest neighbor ratio imputation estimator, in which auxiliary variables are used to identify the closest donor and the vector of proportions from the donor is applied to the total of the recipient to implement ratio imputation. To estimate the asymptotic variance, we first treat the nearest neighbor ratio imp…
We develop a novel full-Bayesian approach for multiple correlated precision matrices, called multiple Graphical Horseshoe (mGHS). The proposed approach relies on a novel multivariate shrinkage prior based on the Horseshoe prior that borrows strength and shares sparsity patterns across groups, improving posterior edge selection when the precision matrices are similar. On the other hand, there is no loss of performance when the groups are independent. Moreover, mGHS provides a similarity matrix estimate, useful for understanding network similarities across groups. We implement an efficient Metro…
