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).     

Identifiability of causal effects for binary variables with baseline data missing due to death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).     

Efficient and robust methods for causally interpretable meta-analysis: Transporting inferences from multiple randomized trials to a target population

We present methods for causally interpretable meta-analyses that combine information from multiple randomized trials to draw causal inferences for a target population of substantive interest. We consider identifiability conditions, derive implications of the conditions for the law of the observed data, and obtain identification results for transporting causal inferences from a collection of independent randomized trials to a new target population in which experimental data may not be available. We propose an estimator for the potential outcome mean in the target population under each treatment studied in the trials. The estimator uses covariate, treatment, and outcome data from the collection of trials, but only covariate data from the target population sample. We show that it is doubly robust in the sense that it is consistent and asymptotically normal when at least one of the models it relies on is correctly specified. We study the finite sample properties of the estimator in simulation studies and demonstrate its implementation using data from a multicenter randomized trial.     

The large sample bounds on the principal strata effect with application to a prostate cancer prevention trial

Issues of post-randomization selection bias and truncation-by-death can arise in randomized clinical trials; for example, in a cancer prevention trial, an outcome such as cancer severity is undefined for individuals who do not develop cancer. Restricting analysis to a subpopulation selected after randomization can give rise to biased outcome comparisons. One approach to deal with such issues is to consider the principal strata effect (PSE, or equally, the survivor average causal effect). PSE is defined as the effect of treatment on the outcome among the subpopulation that would have been selected under either treatment arm. Unfortunately, the PSE cannot generally be estimated without the identifying assumptions; however, the bounds can be derived using a deterministic causal model. In this paper, we propose a number of assumptions for deriving the bounds with narrow width. The assumptions and bounds, which differ from those introduced by Zhang and Rubin (2003), are illustrated using data from a randomized prostate cancer prevention trial.     

Coherent modeling of longitudinal causal effects on binary outcomes

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. Toward 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 noncausal nuisance parameters. This leads to a series of difficulties in interpretation, estimation, and computation. These difficulties have hindered the uptake of SNMMs in biomedical practice, where binary outcomes are very common. We solve the variation dependence problem for the binary multiplicative SNMM via a reparameterization of the noncausal nuisance parameters. Our novel nuisance parameters are variation independent of the causal parameters, and hence allow for coherent modeling of heterogeneous effects from longitudinal studies with binary outcomes. Our parameterization also provides a key building block for flexible doubly robust estimation of the causal parameters. Along the way, we prove that an additive SNMM with binary outcomes does not admit a variation independent parameterization, thereby justifying the restriction to multiplicative SNMMs.     

Variance estimation in clinical studies with interim sample size re-estimation

We consider clinical studies with a sample size re-estimation based on the unblinded variance estimation at some interim point of the study. Because the sample size is determined in such a flexible way, the usual variance estimator at the end of the trial is biased. We derive sharp bounds for this bias. These bounds have a quite simple form and can help for the decision if this bias is negligible for the actual study or if a correction should be done. An exact formula for the bias is also provided. We discuss possibilities to get rid of this bias or at least to reduce the bias substantially. For this purpose, we propose a certain additive correction of the bias. We see in an example that the significance level of the test can be controlled when this additive correction is used.     

A novel penalized inverse-variance weighted estimator for Mendelian randomization with applications to COVID-19 outcomes

Mendelian randomization utilizes genetic variants as instrumental variables (IVs) to estimate the causal effect of an exposure variable on an outcome of interest even in the presence of unmeasured confounders. However, the popular inverse-variance weighted (IVW) estimator could be biased in the presence of weak IVs, a common challenge in MR studies. In this article, we develop a novel penalized inverse-variance weighted (pIVW) estimator, which adjusts the original IVW estimator to account for the weak IV issue by using a penalization approach to prevent the denominator of the pIVW estimator from being close to zero. Moreover, we adjust the variance estimation of the pIVW estimator to account for the presence of balanced horizontal pleiotropy. We show that the recently proposed debiased IVW (dIVW) estimator is a special case of our proposed pIVW estimator. We further prove that the pIVW estimator has smaller bias and variance than the dIVW estimator under some regularity conditions. We also conduct extensive simulation studies to demonstrate the performance of the proposed pIVW estimator. Furthermore, we apply the pIVW estimator to estimate the causal effects of five obesity-related exposures on three coronavirus disease 2019 (COVID-19) outcomes. Notably, we find that hypertensive disease is associated with an increased risk of hospitalized COVID-19; and peripheral vascular disease and higher body mass index are associated with increased risks of COVID-19 infection, hospitalized COVID-19, and critically ill COVID-19.     

Inverse‐probability‐of‐treatment weighted estimation of causal parameters in the presence of error‐contaminated and time‐dependent confounders

Inverse‐probability‐of‐treatment weighted (IPTW) estimation has been widely used to consistently estimate the causal parameters in marginal structural models, with time‐dependent confounding effects adjusted for. Just like other causal inference methods, the validity of IPTW estimation typically requires the crucial condition that all variables are precisely measured. However, this condition, is often violated in practice due to various reasons. It has been well documented that ignoring measurement error often leads to biased inference results. In this paper, we consider the IPTW estimation of the causal parameters in marginal structural models in the presence of error‐contaminated and time‐dependent confounders. We explore several methods to correct for the effects of measurement error on the estimation of causal parameters. Numerical studies are reported to assess the finite sample performance of the proposed methods.     

Instrumental variable estimation of complier causal treatment effect with interval-censored data

Assessing causal treatment effect on a time-to-event outcome is of key interest in many scientific investigations. Instrumental variable (IV) is a useful tool to mitigate the impact of endogenous treatment selection to attain unbiased estimation of causal treatment effect. Existing development of IV methodology, however, has not attended to outcomes subject to interval censoring, which are ubiquitously present in studies with intermittent follow-up but are challenging to handle in terms of both theory and computation. In this work, we fill in this important gap by studying a general class of causal semiparametric transformation models with interval-censored data. We propose a nonparametric maximum likelihood estimator of the complier causal treatment effect. Moreover, we design a reliable and computationally stable expectation–maximization (EM) algorithm, which has a tractable objective function in the maximization step via the use of Poisson latent variables. The asymptotic properties of the proposed estimators, including the consistency, asymptotic normality, and semiparametric efficiency, are established with empirical process techniques. We conduct extensive simulation studies and an application to a colorectal cancer screening data set, showing satisfactory finite-sample performance of the proposed method as well as its prominent advantages over naive methods.     

Utilizing Propensity Scores to Estimate Causal Treatment Effects with Censored Time-Lagged Data

Observational studies frequently are conducted to compare long-term effects of treatments. Without randomization, patients receiving one treatment are not guaranteed to be prognostically comparable to those receiving another treatment. Furthermore, the response of interest may be right-censored because of incomplete follow-up. Statistical methods that do not account for censoring and confounding may lead to biased estimates. This article presents a method for estimating treatment effects in nonrandomized studies with right-censored responses. We review the assumptions required to estimate average causal effects and derive an estimator for comparing two treatments by applying inverse weights to the complete cases. The weights are determined according to the estimated probability of receiving treatment conditional on covariates and the estimated treatment-specific censoring distribution. By utilizing martingale representations, the estimator is shown to be asymptotically normal and an estimator for the asymptotic variance is derived. Simulation results are presented to evaluate the properties of the estimator. These methods are applied to an observational data set of acute coronary syndrome patients from Duke University Medical Center to estimate the effect of a treatment strategy on the mean 5-year medical cost.     

A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome

Targeted maximum likelihood estimation of a parameter of a data generating distribution, known to be an element of a semi-parametric model, involves constructing a parametric model through an initial density estimator with parameter ? representing an amount of fluctuation of the initial density estimator, where the score of this fluctuation model at ? = 0 equals the efficient influence curve/canonical gradient. The latter constraint can be satisfied by many parametric fluctuation models since it represents only a local constraint of its behavior at zero fluctuation. However, it is very important that the fluctuations stay within the semi-parametric model for the observed data distribution, even if the parameter can be defined on fluctuations that fall outside the assumed observed data model. In particular, in the context of sparse data, by which we mean situations where the Fisher information is low, a violation of this property can heavily affect the performance of the estimator. This paper presents a fluctuation approach that guarantees the fluctuated density estimator remains inside the bounds of the data model. We demonstrate this in the context of estimation of a causal effect of a binary treatment on a continuous outcome that is bounded. It results in a targeted maximum likelihood estimator that inherently respects known bounds, and consequently is more robust in sparse data situations than the targeted MLE using a naive fluctuation model. When an estimation procedure incorporates weights, observations having large weights relative to the rest heavily influence the point estimate and inflate the variance. Truncating these weights is a common approach to reducing the variance, but it can also introduce bias into the estimate. We present an alternative targeted maximum likelihood estimation (TMLE) approach that dampens the effect of these heavily weighted observations. As a substitution estimator, TMLE respects the global constraints of the observed data model. For example, when outcomes are binary, a fluctuation of an initial density estimate on the logit scale constrains predicted probabilities to be between 0 and 1. This inherent enforcement of bounds has been extended to continuous outcomes. Simulation study results indicate that this approach is on a par with, and many times superior to, fluctuating on the linear scale, and in particular is more robust when there is sparsity in the data.     

Statistical Machines for Trauma Hospital Outcomes Research: Application to the PRospective,Observational, Multi-Center Major Trauma Transfusion (PROMMTT) Study

Improving the treatment of trauma, a leading cause of death worldwide, is of great clinical and public health interest. This analysis introduces flexible statistical methods for estimating center-level effects on individual outcomes in the context of highly variable patient populations, such as those of the PRospective, Observational, Multi-center Major Trauma Transfusion study. Ten US level I trauma centers enrolled a total of 1,245 trauma patients who survived at least 30 minutes after admission and received at least one unit of red blood cells. Outcomes included death, multiple organ failure, substantial bleeding, and transfusion of blood products. The centers involved were classified as either large or small-volume based on the number of massive transfusion patients enrolled during the study period. We focused on estimation of parameters inspired by causal inference, specifically estimated impacts on patient outcomes related to the volume of the trauma hospital that treated them. We defined this association as the change in mean outcomes of interest that would be observed if, contrary to fact, subjects from large-volume sites were treated at small-volume sites (the effect of treatment among the treated). We estimated this parameter using three different methods, some of which use data-adaptive machine learning tools to derive the outcome models, minimizing residual confounding by reducing model misspecification. Differences between unadjusted and adjusted estimators sometimes differed dramatically, demonstrating the need to account for differences in patient characteristics in clinic comparisons. In addition, the estimators based on robust adjustment methods showed potential impacts of hospital volume. For instance, we estimated a survival benefit for patients who were treated at large-volume sites, which was not apparent in simpler, unadjusted comparisons. By removing arbitrary modeling decisions from the estimation process and concentrating on parameters that have more direct policy implications, these potentially automated approaches allow methodological standardization across similar comparativeness effectiveness studies.     

Statistical inference and power analysis for direct and spillover effects in two-stage randomized experiments

Two-stage randomized experiments become an increasingly popular experimental design for causal inference when the outcome of one unit may be affected by the treatment assignments of other units in the same cluster. In this paper, we provide a methodological framework for general tools of statistical inference and power analysis for two-stage randomized experiments. Under the randomization-based framework, we consider the estimation of a new direct effect of interest as well as the average direct and spillover effects studied in the literature. We provide unbiased estimators of these causal quantities and their conservative variance estimators in a general setting. Using these results, we then develop hypothesis testing procedures and derive sample size formulas. We theoretically compare the two-stage randomized design with the completely randomized and cluster randomized designs, which represent two limiting designs. Finally, we conduct simulation studies to evaluate the empirical performance of our sample size formulas. For empirical illustration, the proposed methodology is applied to the randomized evaluation of the Indian National Health Insurance Program. An open-source software package is available for implementing the proposed methodology.     

Solutions for surrogacy validation with longitudinal outcomes for a gene therapy

Valid surrogate endpoints S can be used as a substitute for a true outcome of interest T to measure treatment efficacy in a clinical trial. We propose a causal inference approach to validate a surrogate by incorporating longitudinal measurements of the true outcomes using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points. We extend these methods to accommodate a delayed-start treatment design where all patients eventually receive the treatment. Not all parameters are identified in the general setting. We apply a Bayesian approach for estimation and inference, utilizing more informative prior distributions for selected parameters. We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pretreatment covariates to improve identifiability. We examine the frequentist properties (bias of point and variance estimates, credible interval coverage) of a Bayesian imputation method. Our work is motivated by a clinical trial of a gene therapy where the functional outcomes are measured repeatedly throughout the trial.     

Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models

There is growing interest in conducting cluster randomized trials (CRTs). For simplicity in sample size calculation, the cluster sizes are assumed to be identical across all clusters. However, equal cluster sizes are not guaranteed in practice. Therefore, the relative efficiency (RE) of unequal versus equal cluster sizes has been investigated when testing the treatment effect. One of the most important approaches to analyze a set of correlated data is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which the “working correlation structure” is introduced and the association pattern depends on a vector of association parameters denoted by ρ. In this paper, we utilize GEE models to test the treatment effect in a two‐group comparison for continuous, binary, or count data in CRTs. The variances of the estimator of the treatment effect are derived for the different types of outcome. RE is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal cluster sizes. We discuss a commonly used structure in CRTs—exchangeable, and derive the simpler formula of RE with continuous, binary, and count outcomes. Finally, REs are investigated for several scenarios of cluster size distributions through simulation studies. We propose an adjusted sample size due to efficiency loss. Additionally, we also propose an optimal sample size estimation based on the GEE models under a fixed budget for known and unknown association parameter (ρ) in the working correlation structure within the cluster.     

A formal causal interpretation of the case-crossover design

The case-crossover design of Maclure is widely used in epidemiology and other fields to study causal effects of transient treatments on acute outcomes. However, its validity and causal interpretation have only been justified under informal conditions. Here, we place the design in a formal counterfactual framework for the first time. Doing so helps to clarify its assumptions and interpretation. In particular, when the treatment effect is nonnull, we identify a previously unnoticed bias arising from strong common causes of the outcome at different person-times. We analyze this bias and demonstrate its potential importance with simulations. We also use our derivation of the limit of the case-crossover estimator to analyze its sensitivity to treatment effect heterogeneity, a violation of one of the informal criteria for validity. The upshot of this work for practitioners is that, while the case-crossover design can be useful for testing the causal null hypothesis in the presence of baseline confounders, extra caution is warranted when using the case-crossover design for point estimation of causal effects.     

An application of collaborative targeted maximum likelihood estimation in causal inference and genomics

A concrete example of the collaborative double-robust targeted likelihood estimator (C-TMLE) introduced in a companion article in this issue is presented, and applied to the estimation of causal effects and variable importance parameters in genomic data. The focus is on non-parametric estimation in a point treatment data structure. Simulations illustrate the performance of C-TMLE relative to current competitors such as the augmented inverse probability of treatment weighted estimator that relies on an external non-collaborative estimator of the treatment mechanism, and inefficient estimation procedures including propensity score matching and standard inverse probability of treatment weighting. C-TMLE is also applied to the estimation of the covariate-adjusted marginal effect of individual HIV mutations on resistance to the anti-retroviral drug lopinavir. The influence curve of the C-TMLE is used to establish asymptotically valid statistical inference. The list of mutations found to have a statistically significant association with resistance is in excellent agreement with mutation scores provided by the Stanford HIVdb mutation scores database.     

On the estimation of average treatment effects with right-censored time to event outcome and competing risks

We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, censoring, and treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that these estimators are regular asymptotically linear estimators and estimate their variances based on estimates of their influence functions. In empirical studies, we assess the robustness of the estimators and the coverage of confidence intervals. The methods are further illustrated using data from a Danish registry study.     

An estimator for treatment comparisons among survivors in randomized trials

In clinical trials of advanced-stage disease it is often of interest to perform treatment comparisons for endpoints which are defined only for survivors. Examples include time on ventilation in ventilation studies, change in quality of life in health-related quality-of-life studies, and duration of response to therapy in therapeutic trials. Randomized treatment comparisons for these endpoints cannot be performed because the outcomes are only defined in the nonrandomly selected subgroup of survivors. We propose a new evaluation of the survivor average causal effect (SACE) for treatment comparisons of this nature. We provide an estimator of SACE in the presence of no unmeasured confounders, a nontestable assumption, which identifies SACE. We also outline a sensitivity analysis for exploring robustness of conclusions to deviations from this assumption. We apply this method to duration of ventilation in a clinical trial of acute respiratory distress syndrome.