Causal mediation analysis seeks to investigate how the treatment effect of an exposure on outcomes is mediated through intermediate variables. Although many applications involve longitudinal data, the existing methods are not directly applicable to settings where the mediator and outcome are measured on sparse and irregular time grids. We extend the existing causal mediation framework from a functional data analysis perspective, viewing the sparse and irregular longitudinal data as realizations of underlying smooth stochastic processes. We define causal estimands of direct and indirect effects accordingly and provide corresponding identification assumptions. For estimation and inference, we employ a functional principal component analysis approach for dimension reduction and use the first few functional principal components instead of the whole trajectories in the structural equation models. We adopt the Bayesian paradigm to accurately quantify the uncertainties. The operating characteristics of the proposed methods are examined via simulations. We apply the proposed methods to a longitudinal data set from a wild baboon population in Kenya to investigate the causal relationships between early adversity, strength of social bonds between animals, and adult glucocorticoid hormone concentrations. We find that early adversity has a significant direct effect (a 9-14% increase) on females' glucocorticoid concentrations across adulthood, but find little evidence that these effects were mediated by weak social bonds.
A Bayesian semi-parametric approach to causal mediation for longitudinal mediators and time-to-event outcomes with application to a cardiovascular disease cohort study
In animal behavior studies, a common goal is to investigate the causal pathways between an exposure and outcome, and a mediator that lies in between. Causal mediation analysis provides a principled approach for such studies. Although many applications involve longitudinal data, the existing causal mediation models are not directly applicable to settings where the mediators are measured on irregular time grids. In this paper, we propose a causal mediation model that accommodates longitudinal mediators on arbitrary time grids and survival outcomes simultaneously. We take a functional data analys…
The paper considers mediation analysis with longitudinal data under latent growth curve models within a counterfactual framework. Estimators and their standard errors are derived for natural direct and indirect effects when the mediator, the outcome, and possibly also the exposure can be modeled by an underlying latent variable giving rise to a growth curve. Settings are also considered in which the exposure is instead fixed at a single point in time.
Causal mediation analysis for zero-inflated longitudinal data in the presence of treatment non-compliance and multiple mediators
University of Chicago（芝加哥大学）；Smeal College of Business, The Pennsylvania State University（宾夕法尼亚州立大学斯米尔商学院）；University of Florida（佛罗里达大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
Questions concerning mediated causal effects are of great interest in psychology, cognitive science, medicine, social science, public health, and many other disciplines. For instance, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker, Preacher, Tormala, & Petty, 2011). Standard parametric approaches to mediation analysis employ regression models, and either the "difference method" (Judd & Kenny, 1981), more common in epidemiology, or the "product method" (Baron & Kenny, 1986), more common in the social sciences. In this pa…
Mediation analysis is concerned with the decomposition of the total effect of an exposure on an outcome into the indirect effect through a given mediator, and the remaining direct effect. This is ideally done using longitudinal measurements of the mediator, as these capture the mediator process more finely. However, longitudinal measurements pose challenges for mediation analysis. This is because the mediators and outcomes measured at a given time-point can act as confounders for the association between mediators and outcomes at a later time-point; these confounders are themselves affected by …
A method using multiple imputation and bootstrap for dealing with missing data in mediation analysis is introduced and implemented in SAS. Through simulation studies, it is shown that the method performs well for both MCAR and MAR data without and with auxiliary variables. It is also shown that the method works equally well for MNAR data if auxiliary variables related to missingness are included. The application of the method is demonstrated through the analysis of a subset of data from the National Longitudinal Survey of Youth.
Doubly Robust Functional Representation Learning for Longitudinal Causal Inference with Irregular Histories
Academy of Life and Natural Sciences, Xi’an Jiaotong-Liverpool University（西交利物浦大学生命与自然科学学院）；Aerospace Information Research Institute, Chinese Academy of Sciences（中国科学院空天信息研究院）
We point out an instantiation of Simpson's paradox in Covid-19 case fatality rates (CFRs): comparing a large-scale study from China (17 Feb) with early reports from Italy (9 Mar), we find that CFRs are lower in Italy for every age group, but higher overall. This phenomenon is explained by a stark difference in case demographic between the two countries. Using this as a motivating example, we introduce basic concepts from mediation analysis and show how these can be used to quantify different direct and indirect effects when assuming a coarse-grained causal graph involving country, age, and cas…
The increase in the use of mobile and wearable devices now allows dense assessment of mediating processes over time. For example, a pharmacological intervention may have an effect on smoking cessation via reductions in momentary withdrawal symptoms. We define and identify the causal direct and indirect effects in terms of potential outcomes on the mean difference and odds ratio scales, and present a method for estimating and testing the indirect effect of a randomized treatment on a distal binary variable as mediated by the nonparametric trajectory of an intensively measured longitudinal varia…
Deciphering seasonal depression variations and interplays between weather changes, physical activity, and depression severity in real-world settings: Learnings from RADAR-MDD longitudinal mobile health study
With multiple potential mediators on the causal pathway from a treatment to an outcome, we consider the problem of decomposing the effects along multiple possible causal path(s) through each distinct mediator. Under Pearl's path-specific effects framework (Pearl, 2001; Avin et al., 2005), such fine-grained decompositions necessitate stringent assumptions, such as correctly specifying the causal structure among the mediators, and there being no unobserved confounding among the mediators. In contrast, interventional direct and indirect effects for multiple mediators (Vansteelandt and Daniel, 201…
Interventional effects have been proposed as a solution to the unidentifiability of natural (in)direct effects under mediator-outcome confounders affected by the exposure. Such confounders are an intrinsic characteristic of studies with time-varying exposures and mediators, yet the generalization of the interventional effect framework to the time-varying case has received little attention in the literature. We present an identification result for interventional effects in a general longitudinal data structure that allows flexibility in the specification of treatment-outcome, treatment-mediator…
The use of observational time series data to assess the impact of multi-time point interventions is becoming increasingly common as more health and activity data are collected and digitized via wearables, social media, and electronic health records. Such time series may involve hundreds or thousands of irregularly sampled observations. One common analysis approach is to simplify such time series by first discretizing them into sequences before applying a discrete-time estimation method that adjusts for time-dependent confounding. In certain settings, this discretization results in sequences wi…
Traditional mediation analysis typically examines the relations among an intervention, a time-invariant mediator, and a time-invariant outcome variable. Although there may be a direct effect of the intervention on the outcome, there is a need to understand the process by which the intervention affects the outcome (i.e. the indirect effect through the mediator). This indirect effect is frequently assumed to be time-invariant. With improvements in data collection technology, it is possible to obtain repeated assessments over time resulting in intensive longitudinal data. This calls for an extens…
Pollutant emissions from coal-burning power plants have been deemed to adversely impact ambient air quality and public health conditions. Despite the noticeable reduction in emissions and the improvement of air quality since the Clean Air Act (CAA) became the law, the public-health benefits from changes in emissions have not been widely evaluated yet. In terms of the chain of accountability (HEI Accountability Working Group, 2003), the link between pollutant emissions from the power plants (SO2) and public health conditions (respiratory diseases) accounting for changes in ambient air quality (…
Structural Nested Mean Models (SNMMs) are useful for causal inference of treatment effects in longitudinal observational studies. Most existing works assume that the data are collected at pre-fixed time points for all subjects, which, however, is restrictive in practice. To deal with irregularly spaced observations, we assume a class of continuous-time SNMMs and a martingale condition of no unmeasured confounding (NUC) to identify the causal parameters. We develop the first semiparametric efficiency theory and locally efficient estimators for continuous-time SNMMs. This task is non-trivial due…
In the statistical literature, a number of methods have been proposed to ensure valid inference about marginal effects of variables on a longitudinal outcome in settings with irregular monitoring times. However, the potential biases due to covariate-driven monitoring times and confounding have rarely been considered simultaneously, and never in a setting with an ordinal outcome and a continuous exposure. In this work, we propose and demonstrate a methodology for causal inference in such a setting, relying on a proportional odds model to study the effect of the exposure on the outcome. Irregula…
University of Cambridge（剑桥大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Columbia University（哥伦比亚大学）；The Alan Turing Institute（艾伦·图灵研究所）
Estimating counterfactual outcomes over time has the potential to unlock personalized healthcare by assisting decision-makers to answer ''what-iF'' questions. Existing causal inference approaches typically consider regular, discrete-time intervals between observations and treatment decisions and hence are unable to naturally model irregularly sampled data, which is the common setting in practice. To handle arbitrary observation patterns, we interpret the data as samples from an underlying continuous-time process and propose to model its latent trajectory explicitly using the mathematics of con…
Causal mediation analysis aims at disentangling a treatment effect into an indirect mechanism operating through an intermediate outcome or mediator, as well as the direct effect of the treatment on the outcome of interest. However, the evaluation of direct and indirect effects is frequently complicated by non-ignorable selection into the treatment and/or mediator, even after controlling for observables, as well as sample selection/outcome attrition. We propose a method for bounding direct and indirect effects in the presence of such complications using a method that is based on a sequence of l…
Outlying curves often occur in functional or longitudinal datasets, and can be very influential on parameter estimators and very hard to detect visually. In this article we introduce estimators of the mean and the principal components that are resistant to, and then can be used for detection of, outlying sample trajectories. The estimators are based on reduced-rank t-models and are specifically aimed at sparse and irregularly sampled functional data. The outlier-resistance properties of the estimators and their relative efficiency for noncontaminated data are studied theoretically and by simul…
Mediation analysis is appealing for its ability to improve understanding of the mechanistic drivers of causal effects, but real-world data complexities challenge its successful implementation, including: 1) the existence of post-exposure variables that also affect mediators and outcomes (thus, confounding the mediator-outcome relationship), that may also be 2) multivariate, and 3) the existence of multivariate mediators. Interventional direct and indirect effects (IDE/IIE) accommodate post-exposure variables that confound the mediator-outcome relationship, but currently, no estimator for IDE/I…
This paper considers canonical correlation analysis for two longitudinal variables that are possibly sampled at different time resolutions with irregular grids. We modeled trajectories of the multivariate variables using random effects and found the most correlated sets of linear combinations in the latent space. Our numerical simulations showed that the longitudinal canonical correlation analysis effectively recovers underlying correlation patterns between two high-dimensional longitudinal data sets. We applied the proposed LCCA to data from the Alzheimer's Disease Neuroimaging Initiative and…
We propose a distance between two realizations of a random process where for each realization only sparse and irregularly spaced measurements with additional measurement errors are available. Such data occur commonly in longitudinal studies and online trading data. A distance measure then makes it possible to apply distance-based analysis such as classification, clustering and multidimensional scaling for irregularly sampled longitudinal data. Once a suitable distance measure for sparsely sampled longitudinal trajectories has been found, we apply distance-based clustering methods to eBay onlin…
Longitudinal (panel) data provide the opportunity to examine temporal patterns of individuals, because measurements are collected on the same person at different, and often irregular, time points. The data is typically visualised using a "spaghetti plot", where a line plot is drawn for each individual. When overlaid in one plot, it can have the appearance of a bowl of spaghetti. With even a small number of subjects, these plots are too overloaded to be read easily. The interesting aspects of individual differences are lost in the noise. Longitudinal data is often modelled with a hierarchical l…
In many longitudinal studies, the covariate and response are often intermittently observed at irregular, mismatched and subject-specific times. How to deal with such data when covariate and response are observed asynchronously is an often raised problem. Bayesian Additive Regression Trees(BART) is a Bayesian non-Parametric approach which has been shown to be competitive with the best modern predictive methods such as random forest and boosted decision trees. The sum of trees structure combined with a Bayesian inferential framework provide a accurate and robust statistic method. BART variant so…
Repeated longitudinal measurements are commonly used to model long-term disease progression, and timing and number of assessments per patient may vary, leading to irregularly spaced and sparse data. Longitudinal trajectories may exhibit curvilinear patterns, in which mixed linear regression methods may fail to capture true trends in the data. We applied functional principal components analysis to model kidney disease progression via estimated glomerular filtration rate (eGFR) trajectories. In a cohort of 2641 participants with diabetes and up to 15 years of annual follow-up from the Chronic Re…
We consider the problem of learning predictive models from longitudinal data, consisting of irregularly repeated, sparse observations from a set of individuals over time. Such data often exhibit {\em longitudinal correlation} (LC) (correlations among observations for each individual over time), {\em cluster correlation} (CC) (correlations among individuals that have similar characteristics), or both. These correlations are often accounted for using {\em mixed effects models} that include {\em fixed effects} and {\em random effects}, where the fixed effects capture the regression parameters tha…
In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defective vision, or assessment of gait in patients with neurological disorders. Since the data collection is often costly and inconvenient, estimation of progression from sparse observations is of great interest for practitioners. From the statistical standpoint, such data is often analyzed in the context…
