Causal mediation analysis with a latent mediator

Health researchers are often interested in assessing the direct effect of a treatment or exposure on an outcome variable, as well as its indirect (or mediation) effect through an intermediate variable (or mediator). For an outcome following a nonlinear model, the mediation formula may be used to estimate causally interpretable mediation effects. This method, like others, assumes that the mediator is observed. However, as is common in structural equations modeling, we may wish to consider a latent (unobserved) mediator. We follow a potential outcomes framework and assume a generalized structural equations model (GSEM). We provide maximum‐likelihood estimation of GSEM parameters using an approximate Monte Carlo EM algorithm, coupled with a mediation formula approach to estimate natural direct and indirect effects. The method relies on an untestable sequential ignorability assumption; we assess robustness to this assumption by adapting a recently proposed method for sensitivity analysis. Simulation studies show good properties of the proposed estimators in plausible scenarios. Our method is applied to a study of the effect of mother education on occurrence of adolescent dental caries, in which we examine possible mediation through latent oral health behavior.     

Flexible maximum likelihood methods for bivariate proportional hazards models

This article presents methodology for multivariate proportional hazards (PH) regression models. The methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional PH models, along with assumptions about the association among lifetimes. Because the models are parametric, ordinary maximum likelihood can be applied; it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes, unlike existing semiparametric methods. A bivariate Clayton model (1978, Biometrika 65, 141-151) is used to illustrate the approach taken. Because a parametric assumption about association is made, efficiency and robustness comparisons are made between estimation based on the bivariate Clayton model and "working independence" methods that specify only marginal distributions for each lifetime variable.     

Natural Direct and Indirect Effects on the Exposed: Effect Decomposition under Weaker Assumptions

Summary We define natural direct and indirect effects on the exposed. We show that these allow for effect decomposition under weaker identification conditions than population natural direct and indirect effects. When no confounders of the mediator‐outcome association are affected by the exposure, identification is possible under essentially the same conditions as for controlled direct effects. Otherwise, identification is still possible with additional knowledge on a nonidentifiable selection‐bias function which measures the dependence of the mediator effect on the observed exposure within confounder levels, and which evaluates to zero in a large class of realistic data‐generating mechanisms. We argue that natural direct and indirect effects on the exposed are of intrinsic interest in various applications. We moreover show that they coincide with the corresponding population natural direct and indirect effects when the exposure is randomly assigned. In such settings, our results are thus also of relevance for assessing population natural direct and indirect effects in the presence of exposure‐induced mediator‐outcome confounding, which existing methodology has not been able to address.     

Marginal structural models for estimating principal stratum direct effects under the monotonicity assumption

In epidemiological and clinical research, investigators are frequently interested in estimating the direct effect of a treatment on an outcome that is not relayed by intermediate variables. In 2009, VanderWeele presented marginal structural models (MSMs) for estimating direct effects based on interventions on the mediator. This paper focuses on direct effects based on principal stratification, i.e. principal stratum direct effects (PSDEs), which are causal effects within latent subgroups of subjects where the mediator is constant, regardless of the exposure status. We propose MSMs for estimating PSDEs. We demonstrate that the PSDE can be estimated readily using MSMs under the monotonicity assumption.     

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).     

Time-dependent mediators in survival analysis: Modeling direct and indirect effects with the additive hazards model

We discuss causal mediation analyses for survival data and propose a new approach based on the additive hazards model. The emphasis is on a dynamic point of view, that is, understanding how the direct and indirect effects develop over time. Hence, importantly, we allow for a time varying mediator. To define direct and indirect effects in such a longitudinal survival setting we take an interventional approach (Didelez, 2018) where treatment is separated into one aspect affecting the mediator and a different aspect affecting survival. In general, this leads to a version of the nonparametric g-formula (Robins, 1986). In the present paper, we demonstrate that combining the g-formula with the additive hazards model and a sequential linear model for the mediator process results in simple and interpretable expressions for direct and indirect effects in terms of relative survival as well as cumulative hazards. Our results generalize and formalize the method of dynamic path analysis (Fosen, Ferkingstad, Borgan, & Aalen, 2006; Strohmaier et al., 2015). An application to data from a clinical trial on blood pressure medication is given.     

Generalized causal mediation analysis

The goal of mediation analysis is to assess direct and indirect effects of a treatment or exposure on an outcome. More generally, we may be interested in the context of a causal model as characterized by a directed acyclic graph (DAG), where mediation via a specific path from exposure to outcome may involve an arbitrary number of links (or "stages"). Methods for estimating mediation (or pathway) effects are available for a continuous outcome and a continuous mediator related via a linear model, while for a categorical outcome or categorical mediator, methods are usually limited to two-stage mediation. We present a method applicable to multiple stages of mediation and mixed variable types using generalized linear models. We define pathway effects using a potential outcomes framework and present a general formula that provides the effect of exposure through any specified pathway. Some pathway effects are nonidentifiable and their estimation requires an assumption regarding the correlation between counterfactuals. We provide a sensitivity analysis to assess the impact of this assumption. Confidence intervals for pathway effect estimates are obtained via a bootstrap method. The method is applied to a cohort study of dental caries in very low birth weight adolescents. A simulation study demonstrates low bias of pathway effect estimators and close-to-nominal coverage rates of confidence intervals. We also find low sensitivity to the counterfactual correlation in most scenarios.     

Robust tests for treatment effects based on censored recurrent event data observed over multiple periods

We derive semiparametric methods for estimating and testing treatment effects when censored recurrent event data are available over multiple periods. These methods are based on estimating functions motivated by a working "mixed-Poisson" assumption under which conditioning can eliminate subject-specific random effects. Robust pseudoscore test statistics are obtained via "sandwich" variance estimation. The relative efficiency of conditional versus marginal analyses is assessed analytically under a mixed time-homogeneous Poisson model. The robustness and empirical power of the semiparametric approach are assessed through simulation. Adaptations to handle recurrent events arising in crossover trials are described and these methods are applied to data from a two-period crossover trial of patients with bronchial asthma.     

A semiparametric joint model for cluster size and subunit-specific interval-censored outcomes

Clustered data frequently arise in biomedical studies, where observations, or subunits, measured within a cluster are associated. The cluster size is said to be informative, if the outcome variable is associated with the number of subunits in a cluster. In most existing work, the informative cluster size issue is handled by marginal approaches based on within-cluster resampling, or cluster-weighted generalized estimating equations. Although these approaches yield consistent estimation of the marginal models, they do not allow estimation of within-cluster associations and are generally inefficient. In this paper, we propose a semiparametric joint model for clustered interval-censored event time data with informative cluster size. We use a random effect to account for the association among event times of the same cluster as well as the association between event times and the cluster size. For estimation, we propose a sieve maximum likelihood approach and devise a computationally-efficient expectation-maximization algorithm for implementation. The estimators are shown to be strongly consistent, with the Euclidean components being asymptotically normal and achieving semiparametric efficiency. Extensive simulation studies are conducted to evaluate the finite-sample performance, efficiency and robustness of the proposed method. We also illustrate our method via application to a motivating periodontal disease dataset.     

Doubly robust multiple imputation using kernel‐based techniques

We consider the problem of estimating the marginal mean of an incompletely observed variable and develop a multiple imputation approach. Using fully observed predictors, we first establish two working models: one predicts the missing outcome variable, and the other predicts the probability of missingness. The predictive scores from the two models are used to measure the similarity between the incomplete and observed cases. Based on the predictive scores, we construct a set of kernel weights for the observed cases, with higher weights indicating more similarity. Missing data are imputed by sampling from the observed cases with probability proportional to their kernel weights. The proposed approach can produce reasonable estimates for the marginal mean and has a double robustness property, provided that one of the two working models is correctly specified. It also shows some robustness against misspecification of both models. We demonstrate these patterns in a simulation study. In a real‐data example, we analyze the total helicopter response time from injury in the Arizona emergency medical service data.     

Semiparametric estimation in copula models for bivariate sequential survival times

Sequentially observed survival times are of interest in many studies but there are difficulties in analyzing such data using nonparametric or semiparametric methods. First, when the duration of followup is limited and the times for a given individual are not independent, induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but robustness is a concern. We introduce a new approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. This provides more robust estimates and checks on the fit of parametric models. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.     

Joint semiparametric models for case-cohort designs

Two-phase studies such as case-cohort and nested case-control studies are widely used cost-effective sampling strategies. In the first phase, the observed failure/censoring time and inexpensive exposures are collected. In the second phase, a subgroup of subjects is selected for measurements of expensive exposures based on the information from the first phase. One challenging issue is how to utilize all the available information to conduct efficient regression analyses of the two-phase study data. This paper proposes a joint semiparametric modeling of the survival outcome and the expensive exposures. Specifically, we assume a class of semiparametric transformation models and a semiparametric density ratio model for the survival outcome and the expensive exposures, respectively. The class of semiparametric transformation models includes the proportional hazards model and the proportional odds model as special cases. The density ratio model is flexible in modeling multivariate mixed-type data. We develop efficient likelihood-based estimation and inference procedures and establish the large sample properties of the nonparametric maximum likelihood estimators. Extensive numerical studies reveal that the proposed methods perform well under practical settings. The proposed methods also appear to be reasonably robust under various model mis-specifications. An application to the National Wilms Tumor Study is provided.     

Semiparametric distributed lag quantile regression for modeling time-dependent exposure mixtures

Studying time-dependent exposure mixtures has gained increasing attentions in environmental health research. When a scalar outcome is of interest, distributed lag (DL) models have been employed to characterize the exposures effects distributed over time on the mean of final outcome. However, there is a methodological gap on investigating time-dependent exposure mixtures with different quantiles of outcome. In this paper, we introduce semiparametric partial-linear single-index (PLSI) DL quantile regression, which can describe the DL effects of time-dependent exposure mixtures on different quantiles of outcome and identify susceptible periods of exposures. We consider two time-dependent exposure settings: discrete and functional, when exposures are measured in a small number of time points and at dense time grids, respectively. Spline techniques are used to approximate the nonparametric DL function and single-index link function, and a profile estimation algorithm is proposed. Through extensive simulations, we demonstrate the performance and value of our proposed models and inference procedures. We further apply the proposed methods to study the effects of maternal exposures to ambient air pollutants of fine particulate and nitrogen dioxide on birth weight in New York University Children's Health and Environment Study (NYU CHES).     

Modeling spatial survival data using semiparametric frailty models

We propose a new class of semiparametric frailty models for spatially correlated survival data. Specifically, we extend the ordinary frailty models by allowing random effects accommodating spatial correlations to enter into the baseline hazard function multiplicatively. We prove identifiability of the models and give sufficient regularity conditions. We propose drawing inference based on a marginal rank likelihood. No parametric forms of the baseline hazard need to be assumed in this semiparametric approach. Monte Carlo simulations and the Laplace approach are used to tackle the intractable integral in the likelihood function. Different spatial covariance structures are explored in simulations and the proposed methods are applied to the East Boston Asthma Study to detect prognostic factors leading to childhood asthma.     

Residuals for proportional hazards models with interval-censored survival data

We develop diagnostic tools for use with proportional hazards models for interval-censored survival data. We propose counterparts to the Cox-Snell, Lagakos (or martingale), deviance, and Schoenfeld residuals. Many of the properties of these residuals carry over to the interval-censored case. In particular, the interval-censored versions of the Lagakos and Schoenfeld residuals may be derived as components of suitable score statistics. The Lagakos residuals may be used to check regression relationships, while the Schoenfeld residuals can help to detect nonproportional hazards in semiparametric models. The methods apply to parametric models and to the semiparametric model with discrete observation times.     

Estimating mean cost using auxiliary covariates

We study the estimation of mean medical cost when censoring is dependent and a large amount of auxiliary information is present. Under missing at random assumption, we propose semiparametric working models to obtain low-dimensional summarized scores. An estimator for the mean total cost can be derived nonparametrically conditional on the summarized scores. We show that when either the two working models for cost-survival process or the model for censoring distribution is correct, the estimator is consistent and asymptotically normal. Small-sample performance of the proposed method is evaluated via simulation studies. Finally, our approach is applied to analyze a real data set in health economics.     

A robust approach for electronic health record–based case-control studies with contaminated case pools

We consider analyses of case-control studies assembled from electronic health records (EHRs) where the pool of cases is contaminated by patients who are ineligible for the study. These ineligible patients, referred to as “false cases,” should be excluded from the analyses if known. However, the true outcome status of a patient in the case pool is unknown except in a subset whose size may be arbitrarily small compared to the entire pool. To effectively remove the influence of the false cases on estimating odds ratio parameters defined by a working association model of the logistic form, we propose a general strategy to adaptively impute the unknown case status without requiring a correct phenotyping model to help discern the true and false case statuses. Our method estimates the target parameters as the solution to a set of unbiased estimating equations constructed using all available data. It outperforms existing methods by achieving robustness to mismodeling the relationship between the outcome status and covariates of interest, as well as improved estimation efficiency. We further show that our estimator is root-n-consistent and asymptotically normal. Through extensive simulation studies and analysis of real EHR data, we demonstrate that our method has desirable robustness to possible misspecification of both the association and phenotyping models, along with statistical efficiency superior to the competitors.     

Interaction of static and extremely low frequency electric and magnetic fields with living systems: health effects and research needs

An international seminar was held June 4-6, 1997, on the biological effects and related health hazards of ambient or environmental static and extremely low frequency (ELF) electric and magnetic fields (0-300 Hz). It was cosponsored by the World Health Organization (WHO), the International Commission on Non-Ionizing Radiation Protection (ICNIRP), the German, Japanese, and Swiss governments. Speakers provided overviews of the scientific literature that were discussed by participants of the meeting. Subsequently, expert working groups formulated this report, which evaluates possible health effects from exposure to static and ELF electric and magnetic fields and identifies gaps in knowledge requiring more research to improve health risk assessments. The working groups concluded that, although health hazards exist from exposure to ELF fields at high field strengths, the literature does not establish that health hazards are associated with exposure to low-level fields, including environmental levels. Similarly, exposure to static electric fields at levels currently found in the living and working environment or acute exposure to static magnetic fields at flux densities below 2 T, were not found to have demonstrated adverse health consequences. However, reports of biological effects from low-level ELF-field exposure and chronic exposure to static magnetic fields were identified that need replication and further study for WHO to assess any possible health consequences. Ambient static electric fields have not been reported to cause any direct adverse health effects, and so no further research in this area was deemed necessary.     

Association Models for Clustered Data with Binary and Continuous Responses

Summary .  We consider analysis of clustered data with mixed bivariate responses, i.e., where each member of the cluster has a binary and a continuous outcome. We propose a new bivariate random effects model that induces associations among the binary outcomes within a cluster, among the continuous outcomes within a cluster, between a binary outcome and a continuous outcome from different subjects within a cluster, as well as the direct association between the binary and continuous outcomes within the same subject. For the ease of interpretations of the regression effects, the marginal model of the binary response probability integrated over the random effects preserves the logistic form and the marginal expectation of the continuous response preserves the linear form. We implement maximum likelihood estimation of our model parameters using standard software such as PROC NLMIXED of SAS . Our simulation study demonstrates the robustness of our method with respect to the misspecification of the regression model as well as the random effects model. We illustrate our methodology by analyzing a developmental toxicity study of ethylene glycol in mice.     

Inference from Stratified Samples: Study Design,Bias and Graphical Model Representations

Stratification is a widely used strategy in empirical research to improve efficiency of the sampling design. One concern of stratification is that ignoring it on analysis may bias the relationship between variables. A weighted analysis can only be carried out when sampling weights are known. When these are unknown, valid inference on the relationship between variables then depends on the ignorability of the design, which may be difficult to establish. Here, graphical representations of multivariate dependencies and independencies are used to find necessary conditions for ignorability of stratified sampling designs for inference on conditional and marginal relationships between variables.