Visualization is a key aspect of communicating the results of any study aiming to estimate causal effects. In studies with time-to-event outcomes, the most popular visualization approach is depicting survival curves stratified by the variable of interest. This approach cannot be used when the variable of interest is continuous. Simple workarounds, such as categorizing the continuous covariate and plotting survival curves for each category, can result in misleading depictions of the main effects. Instead, we propose a new graphic, the survival area plot, to directly depict the survival probability over time and as a function of a continuous covariate simultaneously. This plot utilizes g-computation based on a suitable time-to-event model to obtain the relevant estimates. Through the use of g-computation, those estimates can be adjusted for confounding without additional effort, allowing a causal interpretation under the standard causal identifiability assumptions. If those assumptions are not met, the proposed plot may still be used to depict noncausal associations. We illustrate and compare the proposed graphics to simpler alternatives using data from a large German observational study investigating the effect of the Ankle Brachial Index on survival. To facilitate the usage of these plots, we additionally developed the contsurvplot R-package which includes all methods discussed in this paper.
Causal inference on multiple non-independent outcomes raises serious challenges, because multivariate techniques that properly account for the outcome's dependence structure need to be considered. We focus on the case of binary outcomes framing our discussion in the potential outcome approach to causal inference. We define causal effects of treatment on joint outcomes introducing the notion of product outcomes. We also discuss a decomposition of the causal effect on product outcomes into intrinsic and extrinsic causal effects, which respectively provide information on treatment effect on the i…
The literature for count modeling provides useful tools to conduct causal inference when outcomes take non-negative integer values. Applied to the potential outcomes framework, we link the Bayesian causal inference literature to statistical models for count data. We discuss the general architectural considerations for constructing the predictive posterior of the missing potential outcomes. Special considerations for estimating average treatment effects are discussed, some generalizing certain relationships and some not yet encountered in the causal inference literature.
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…
Assessing the causal effects of interventions on ordinal outcomes is an important objective of many educational and behavioral studies. Under the potential outcomes framework, we can define causal effects as comparisons between the potential outcomes under treatment and control. However, unfortunately, the average causal effect, often the parameter of interest, is difficult to interpret for ordinal outcomes. To address this challenge, we propose to use two causal parameters, which are defined as the probabilities that the treatment is beneficial and strictly beneficial for the experimental uni…
Many outcomes of interest in the social and health sciences, as well as in modern applications in computational social science and experimentation on social media platforms, are ordinal and do not have a meaningful scale. Causal analyses that leverage this type of data, termed ordinal non-numeric, require careful treatment, as much of the classical potential outcomes literature is concerned with estimation and hypothesis testing for outcomes whose relative magnitudes are well defined. Here, we propose a class of finite population causal estimands that depend on conditional distributions of the…
Unobserved confounding presents a major threat to causal inference from observational studies. Recently, several authors suggest that this problem may be overcome in a shared confounding setting where multiple treatments are independent given a common latent confounder. It has been shown that under a linear Gaussian model for the treatments, the causal effect is not identifiable without parametric assumptions on the outcome model. In this paper, we show that the causal effect is indeed identifiable if we assume a general binary choice model for the outcome with a non-probit link. Our identific…
Analyses of biomedical studies often necessitate modeling longitudinal causal effects. The current focus on personalized medicine and effect heterogeneity makes this task even more challenging. Towards this end, structural nested mean models (SNMMs) are fundamental tools for studying heterogeneous treatment effects in longitudinal studies. However, when outcomes are binary, current methods for estimating multiplicative and additive SNMM parameters suffer from variation dependence between the causal parameters and the non-causal nuisance parameters. This leads to a series of difficulties in int…
Instrumental variable methods are widely used for inferring the causal effect in the presence of unmeasured confounders. Existing instrumental variable methods for nonlinear outcome models require stringent identifiability conditions. This paper considers a flexible semi-parametric potential outcome model that allows for possibly invalid instruments. We propose new identifiability conditions to identify the causal parameters when the majority of the instrumental variables are valid. We devise a novel inference procedure for a new average structural function and the conditional average treatmen…
It is common that in multiarm randomized trials, the outcome of interest is "truncated by death," meaning that it is only observed or well defined conditioning on an intermediate outcome. In this case, in addition to pairwise contrasts, the joint inference for all treatment arms is also of interest. Under a monotonicity assumption we present methods for both pairwise and joint causal analyses of ordinal treatments and binary outcomes in presence of truncation by death. We illustrate via examples the appropriateness of our assumptions in different scientific contexts.
In this paper we propose a new template for empirical studies intended to assess causal effects: the outcome-wide longitudinal design. The approach is an extension of what is often done to assess the causal effects of a treatment or exposure using confounding control, but now, over numerous outcomes. We discuss the temporal and confounding control principles for such outcome-wide studies, metrics to evaluate robustness or sensitivity to potential unmeasured confounding for each outcome, and approaches to handle multiple testing. We argue that the outcome-wide longitudinal design has numerous a…
Noncompliance and missing data often occur in randomized trials, which complicate the inference of causal effects. When both noncompliance and missing data are present, previous papers proposed moment and maximum likelihood estimators for binary and normally distributed continuous outcomes under the latent ignorable missing data mechanism. However, the latent ignorable missing data mechanism may be violated in practice, because the missing data mechanism may depend directly on the missing outcome itself. Under noncompliance and an outcome-dependent nonignorable missing data mechanism, previous…
A parametric expression for causal natural direct and indirect effects is derived for the setting of a binary outcome with a binary mediator. The proposed effect decomposition does not require the outcome to be rare and generalizes the existing ones, allowing for interactions between both the exposure and the mediator and confounding covariates. Further, it outlines the relationship between the causal effects and the correspondent pathway-specific logistic regression parameters, in parallel with results derived under the rare outcome assumption. Formulae for standard errors, obtained via the d…
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…
It is common in medical studies that the outcome of interest is truncated by death, meaning that a subject has died before the outcome could be measured. In this case, restricted analysis among survivors may be subject to selection bias. Hence, it is of interest to estimate the survivor average causal effect, defined as the average causal effect among the subgroup consisting of subjects who would survive under either exposure. In this paper, we consider the identification and estimation problems of the survivor average causal effect. We propose to use a substitution variable in place of the la…
The term "interference" has been used to describe any setting in which one subject's exposure may affect another subject's outcome. We use causal diagrams to distinguish among three causal mechanisms that give rise to interference. The first causal mechanism by which interference can operate is a direct causal effect of one individual's treatment on another individual's outcome; we call this direct interference. Interference by contagion is present when one individual's outcome may affect the outcomes of other individuals with whom he comes into contact. Then giving treatment to the first indi…
In medical research, a scenario often entertained is randomized controlled $2^2$ factorial design with a binary outcome. By utilizing the concept of potential outcomes, Dasgupta et al. (2015) proposed a randomization-based causal inference framework, allowing flexible and simultaneous estimations and inferences of the factorial effects. However, a fundamental challenge that Dasgupta et al. (2015)'s proposed methodology faces is that the sampling variance of the randomization-based factorial effect estimator is unidentifiable, rendering the corresponding classic "Neymanian" variance estimator s…
Although the exposure can be randomly assigned in studies of mediation effects, any form of direct intervention on the mediator is often infeasible. As a result, unmeasured mediator-outcome confounding can seldom be ruled out. We propose semiparametric identification of natural direct and indirect effects in the presence of unmeasured mediator-outcome confounding by leveraging heteroskedasticity restrictions on the observed data law. For inference, we develop semiparametric estimators that remain consistent under partial misspecifications of the observed data model. We illustrate the proposed …
Causal mediation analysis is an important statistical method in social and medical studies, as it can provide insights about why an intervention works and inform the development of future interventions. Currently, most causal mediation methods focus on mediation effects defined on a mean scale. However, in health-risk studies, such as alcohol or risky sex, outcomes are typically count data and heavily skewed. Thus, mediation effects in these setting would be natural on a rate ratio scale, such as in Poisson and negative binomial regression methods. Existing methods also mainly rely on the assu…
What is the ideal regression (if any) for estimating average causal effects? We study this question in the setting of discrete covariates, deriving expressions for the finite-sample variance of various stratification estimators. This approach clarifies the fundamental statistical phenomena underlying many widely-cited results. Our exposition combines insights from three distinct methodological traditions for studying causal effect estimation: potential outcomes, causal diagrams, and structural models with additive errors.
Johns Hopkins Bloomberg School of Public Health（约翰霍普金斯大学布隆伯格公共卫生学院）；Boston University（波士顿大学）；Johns Hopkins Whiting School of Engineering（约翰霍普金斯大学惠廷工程学院）
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…
The goal of causal inference is to understand the outcome of alternative courses of action. However, all causal inference requires assumptions. Such assumptions can be more influential than in typical tasks for probabilistic modeling, and testing those assumptions is important to assess the validity of causal inference. We develop model criticism for Bayesian causal inference, building on the idea of posterior predictive checks to assess model fit. Our approach involves decomposing the problem, separately criticizing the model of treatment assignments and the model of outcomes. Conditioned on …
Inference for causal effects can benefit from the availability of an instrumental variable (IV) which, by definition, is associated with the given exposure, but not with the outcome of interest other than through a causal exposure effect. Estimation methods for instrumental variables are now well established for continuous outcomes, but much less so for dichotomous outcomes. In this article we review IV estimation of so-called conditional causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome conditional on the exposure level, instrumental variable and me…
A key challenge in causal inference from observational studies is the identification and estimation of causal effects in the presence of unmeasured confounding. In this paper, we introduce a novel approach for causal inference that leverages information in multiple outcomes to deal with unmeasured confounding. The key assumption in our approach is conditional independence among multiple outcomes. In contrast to existing proposals in the literature, the roles of multiple outcomes in our key identification assumption are symmetric, hence the name parallel outcomes. We show nonparametric identifi…
Methods for inferring average causal effects have traditionally relied on two key assumptions: (i) the intervention received by one unit cannot causally influence the outcome of another; and (ii) units can be organized into non-overlapping groups such that outcomes of units in separate groups are independent. In this paper, we develop new statistical methods for causal inference based on a single realization of a network of connected units for which neither assumption (i) nor (ii) holds. The proposed approach allows both for arbitrary forms of interference, whereby the outcome of a unit may de…
Researchers are often interested in predicting outcomes, conducting clustering analysis to detect distinct subgroups of their data, or computing causal treatment effects. Pathological data distributions that exhibit skewness and zero-inflation complicate these tasks - requiring highly flexible, data-adaptive modeling. In this paper, we present a fully nonparametric Bayesian generative model for continuous, zero-inflated outcomes that simultaneously predicts structural zeros, captures skewness, and clusters patients with similar joint data distributions. The flexibility of our approach yields p…
In Randomised Controlled Trials (RCT) with treatment non-compliance, instrumental variable approaches are used to estimate complier average causal effects. We extend these approaches to cost-effectiveness analyses, where methods need to recognise the correlation between cost and health outcomes. We propose a Bayesian full likelihood (BFL) approach, which jointly models the effects of random assignment on treatment received and the outcomes, and a three-stage least squares (3sls) method, which acknowledges the correlation between the endpoints, and the endogeneity of the treatment received. Thi…
In this work, we propose an approach for assessing sensitivity to unobserved confounding in studies with multiple outcomes. We demonstrate how prior knowledge unique to the multi-outcome setting can be leveraged to strengthen causal conclusions beyond what can be achieved from analyzing individual outcomes in isolation. We argue that it is often reasonable to make a shared confounding assumption, under which residual dependence amongst outcomes can be used to simplify and sharpen sensitivity analyses. We focus on a class of factor models for which we can bound the causal effects for all outcom…
Advances in wearables and digital technology now make it possible to deliver behavioral mobile health interventions to individuals in their everyday life. The micro-randomized trial (MRT) is increasingly used to provide data to inform the construction of these interventions. In an MRT, each individual is repeatedly randomized among multiple intervention options, often hundreds or even thousands of times, over the course of the trial. This work is motivated by multiple MRTs that have been conducted, or are currently in the field, in which the primary outcome is a longitudinal binary outcome. Th…
With continuous outcomes, the average causal effect is typically defined using a contrast of expected potential outcomes. However, in the presence of skewed outcome data, the expectation may no longer be meaningful. In practice the typical approach is to either "ignore or transform" - ignore the skewness altogether or transform the outcome to obtain a more symmetric distribution, although neither approach is entirely satisfactory. Alternatively the causal effect can be redefined as a contrast of median potential outcomes, yet discussion of confounding-adjustment methods to estimate this parame…
Causal Inference with Missing Exposures and Missing Outcomes
University of California Berkeley（加州大学伯克利分校）；University of California San Francisco（加州大学旧金山分校）；Infectious Diseases Research Collaboration（传染病研究协作组织）；Makerere University（马凯雷雷大学）
It is important to draw causal inference from observational studies, which, however, becomes challenging if the confounders have missing values. Generally, causal effects are not identifiable if the confounders are missing not at random. We propose a novel framework to nonparametrically identify causal effects with confounders subject to an outcome-independent missingness, that is, the missing data mechanism is independent of the outcome, given the treatment and possibly missing confounders. We then propose a nonparametric two-stage least squares estimator and a parametric estimator for causal…
Generalized Odds for Ordinal Outcomes: A Causal Inference Framework Adjusting for Covariates
Beijing Institute of Brain Disorders, Capital Medical University（首都医科大学北京脑重大疾病研究院）；Chinese Institutes for Medical Research（中国医学科学院）；Department of Neurology, Xuanwu Hospital, Capital Medical University（首都医科大学宣武医院神经内科）
This paper deals with the problem of evaluating the causal effect using observational data in the presence of an unobserved exposure/ outcome variable, when cause-effect relationships between variables can be described as a directed acyclic graph and the corresponding recursive factorization of a joint distribution. First, we propose identifiability criteria for causal effects when an unobserved exposure/outcome variable is considered to contain more than two categories. Next, when unmeasured variables exist between an unobserved outcome variable and its proxy variables, we provide the tightes…
In causal inference, principal stratification is a framework for dealing with a posttreatment intermediate variable between a treatment and an outcome, in which the principal strata are defined by the joint potential values of the intermediate variable. Because the principal strata are not fully observable, the causal effects within them, also known as the principal causal effects, are not identifiable without additional assumptions. Several previous empirical studies leveraged auxiliary variables to improve the inference of principal causal effects. We establish a general theory for identific…
We discuss the recent paper on "excursion effect" by T. Qian et al. (2020). We show that the methods presented have close relationships to others in the literature, in particular to a series of papers by Robins, Hernán and collaborators on analyzing observational studies as a series of randomized trials. There is also a close relationship to the history-restricted and the history-adjusted marginal structural models (MSM). Important differences and their methodological implications are clarified. We also demonstrate that the excursion effect can depend on the design and discuss its suitability …
