Population intervention causal effects based on stochastic interventions

Estimating the causal effect of an intervention on a population typically involves defining parameters in a nonparametric structural equation model (Pearl, 2000, Causality: Models, Reasoning, and Inference) in which the treatment or exposure is deterministically assigned in a static or dynamic way. We define a new causal parameter that takes into account the fact that intervention policies can result in stochastically assigned exposures. The statistical parameter that identifies the causal parameter of interest is established. Inverse probability of treatment weighting (IPTW), augmented IPTW (A-IPTW), and targeted maximum likelihood estimators (TMLE) are developed. A simulation study is performed to demonstrate the properties of these estimators, which include the double robustness of the A-IPTW and the TMLE. An application example using physical activity data is presented.     

Multiple imputation methods for treatment noncompliance and nonresponse in randomized clinical trials

Summary .  Randomized clinical trials are a powerful tool for investigating causal treatment effects, but in human trials there are oftentimes problems of noncompliance which standard analyses, such as the intention-to-treat or as-treated analysis, either ignore or incorporate in such a way that the resulting estimand is no longer a causal effect. One alternative to these analyses is the complier average causal effect (CACE) which estimates the average causal treatment effect among a subpopulation that would comply under any treatment assigned. We focus on the setting of a randomized clinical trial with crossover treatment noncompliance (e.g., control subjects could receive the intervention and intervention subjects could receive the control) and outcome nonresponse. In this article, we develop estimators for the CACE using multiple imputation methods, which have been successfully applied to a wide variety of missing data problems, but have not yet been applied to the potential outcomes setting of causal inference. Using simulated data we investigate the finite sample properties of these estimators as well as of competing procedures in a simple setting. Finally we illustrate our methods using a real randomized encouragement design study on the effectiveness of the influenza vaccine.     

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.     

Robust inference on the average treatment effect using the outcome highly adaptive lasso

Many estimators of the average effect of a treatment on an outcome require estimation of the propensity score, the outcome regression, or both. It is often beneficial to utilize flexible techniques, such as semiparametric regression or machine learning, to estimate these quantities. However, optimal estimation of these regressions does not necessarily lead to optimal estimation of the average treatment effect, particularly in settings with strong instrumental variables. A recent proposal addressed these issues via the outcome-adaptive lasso, a penalized regression technique for estimating the propensity score that seeks to minimize the impact of instrumental variables on treatment effect estimators. However, a notable limitation of this approach is that its application is restricted to parametric models. We propose a more flexible alternative that we call the outcome highly adaptive lasso. We discuss the large sample theory for this estimator and propose closed-form confidence intervals based on the proposed estimator. We show via simulation that our method offers benefits over several popular approaches.     

Point and interval estimation of baseline risk and treatment effect based on logistic model for observational studies

In observational studies with dichotomous outcome of a population, researchers usually report treatment effect alone, although both baseline risk and treatment effect are needed to evaluate the significance of the treatment effect to the population. In this article, we study point and interval estimates including confidence region of baseline risk and treatment effect based on logistic model, where baseline risk is the risk of outcome of the population under control treatment while treatment effect is measured by the risk difference between outcomes of the population under active versus control treatments. Using approximate normal distribution of the maximum‐likelihood (ML) estimate of the model parameters, we obtain an approximate joint distribution of the ML estimate of the baseline risk and the treatment effect. Using the approximate joint distribution, we obtain point estimate and confidence region of the baseline risk and the treatment effect as well as point estimate and confidence interval of the treatment effect when the ML estimate of the baseline risk falls into specified range. These interval estimates reflect nonnormality of the joint distribution of the ML estimate of the baseline risk and the treatment effect. The method can be easily implemented by using any software that generates normal distribution. The method can also be used to obtain point and interval estimates of baseline risk and any other measure of treatment effect such as risk ratio and the number needed to treat. The method can also be extended from logistic model to other models such as log‐linear model.     

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).     

Estimation and inference for the causal effect of receiving treatment on a multinomial outcome

Summary .  This article considers the analysis of two-arm randomized trials with noncompliance, which have a multinomial outcome. We first define the causal effect in these trials as some function of outcome distributions of compliers with and without treatment (e.g., the complier average causal effect, the measure of stochastic superiority of treatment over control for compliers), then estimate the causal effect with the likelihood method. Next, based on the likelihood-ratio (LR) statistic, we test those functions of or the equality of the outcome distributions of compliers with and without treatment. Although the corresponding LR statistic follows a chi-squared  (χ²)  distribution asymptotically when the true values of parameters are in the interior of the parameter space under the null, its asymptotic distribution is not  χ²  when the true values of parameters are on the boundary of the parameter space under the null. Therefore, we propose a bootstrap/double bootstrap version of a LR test for the causal effect in these trials. The methods are illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

Model misspecification and bias for inverse probability weighting estimators of average causal effects

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood lead levels.     

Interpretation of variation across marker loci as evidence of selection

Population structure and history have similar effects on the genetic diversity at all neutral loci. However, some marker loci may also have been strongly influenced by natural selection. Selection shapes genetic diversity in a locus-specific manner. If we could identify those loci that have responded to selection during the divergence of populations, then we may obtain better estimates of the parameters of population history by excluding these loci. Previous attempts were made to identify outlier loci from the distribution of sample statistics under neutral models of population structure and history. Unfortunately these methods depend on assumptions about population structure and history that usually cannot be verified. In this article, we define new population-specific parameters of population divergence and construct sample statistics that are estimators of these parameters. We then use the joint distribution of these estimators to identify outlier loci that may be subject to selection. We found that outlier loci are easier to recognize when this joint distribution is conditioned on the total number of allelic states represented in the pooled sample at each locus. This is so because the conditional distribution is less sensitive to the values of nuisance parameters.     

Estimating FST and kinship for arbitrary population structures

F_ST and kinship are key parameters often estimated in modern population genetics studies in order to quantitatively characterize structure and relatedness. Kinship matrices have also become a fundamental quantity used in genome-wide association studies and heritability estimation. The most frequently-used estimators of F_ST and kinship are method-of-moments estimators whose accuracies depend strongly on the existence of simple underlying forms of structure, such as the independent subpopulations model of non-overlapping, independently evolving subpopulations. However, modern data sets have revealed that these simple models of structure likely do not hold in many populations, including humans. In this work, we analyze the behavior of these estimators in the presence of arbitrarily-complex population structures, which results in an improved estimation framework specifically designed for arbitrary population structures. After generalizing the definition of F_ST to arbitrary population structures and establishing a framework for assessing bias and consistency of genome-wide estimators, we calculate the accuracy of existing F_ST and kinship estimators under arbitrary population structures, characterizing biases and estimation challenges unobserved under their originally-assumed models of structure. We then present our new approach, which consistently estimates kinship and F_ST when the minimum kinship value in the dataset is estimated consistently. We illustrate our results using simulated genotypes from an admixture model, constructing a one-dimensional geographic scenario that departs nontrivially from the independent subpopulations model. Our simulations reveal the potential for severe biases in estimates of existing approaches that are overcome by our new framework. This work may significantly improve future analyses that rely on accurate kinship and F_ST estimates.     

Identifiability and estimation of causal effects in randomized trials with noncompliance and completely nonignorable missing data

Summary .  In this article, we first study parameter identifiability in randomized clinical trials with noncompliance and missing outcomes. We show that under certain conditions the parameters of interest are identifiable even under different types of completely nonignorable missing data: that is, the missing mechanism depends on the outcome. We then derive their maximum likelihood and moment estimators and evaluate their finite-sample properties in simulation studies in terms of bias, efficiency, and robustness. Our sensitivity analysis shows that the assumed nonignorable missing-data model has an important impact on the estimated complier average causal effect (CACE) parameter. Our new method provides some new and useful alternative nonignorable missing-data models over the existing latent ignorable model, which guarantees parameter identifiability, for estimating the CACE in a randomized clinical trial with noncompliance and missing data.     

Improved confidence intervals for a difference of two cause-specific cumulative incidence functions estimated in the presence of competing risks and random censoring

A cause-specific cumulative incidence function (CIF) is the probability of failure from a specific cause as a function of time. In randomized trials, a difference of cause-specific CIFs (treatment minus control) represents a treatment effect. Cause-specific CIF in each intervention arm can be estimated based on the usual non-parametric Aalen–Johansen estimator which generalizes the Kaplan–Meier estimator of CIF in the presence of competing risks. Under random censoring, asymptotically valid Wald-type confidence intervals (CIs) for a difference of cause-specific CIFs at a specific time point can be constructed using one of the published variance estimators. Unfortunately, these intervals can suffer from substantial under-coverage when the outcome of interest is a rare event, as may be the case for example in the analysis of uncommon adverse events. We propose two new approximate interval estimators for a difference of cause-specific CIFs estimated in the presence of competing risks and random censoring. Theoretical analysis and simulations indicate that the new interval estimators are superior to the Wald CIs in the sense of avoiding substantial under-coverage with rare events, while being equivalent to the Wald CIs asymptotically. In the absence of censoring, one of the two proposed interval estimators reduces to the well-known Agresti–Caffo CI for a difference of two binomial parameters. The new methods can be easily implemented with any software package producing point and variance estimates for the Aalen–Johansen estimator, as illustrated in a real data example.     

Estimating marginal and incremental effects on health outcomes using flexible link and variance function models

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.     

Concordance indices with left-truncated and right-censored data

In the context of time-to-event analysis, a primary objective is to model the risk of experiencing a particular event in relation to a set of observed predictors. The Concordance Index (C-Index) is a statistic frequently used in practice to assess how well such models discriminate between various risk levels in a population. However, the properties of conventional C-Index estimators when applied to left-truncated time-to-event data have not been well studied, despite the fact that left-truncation is commonly encountered in observational studies. We show that the limiting values of the conventional C-Index estimators depend on the underlying distribution of truncation times, which is similar to the situation with right-censoring as discussed in Uno et al. (2011) [On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in Medicine 30(10), 1105–1117]. We develop a new C-Index estimator based on inverse probability weighting (IPW) that corrects for this limitation, and we generalize this estimator to settings with left-truncated and right-censored data. The proposed IPW estimators are highly robust to the underlying truncation distribution and often outperform the conventional methods in terms of bias, mean squared error, and coverage probability. We apply these estimators to evaluate a predictive survival model for mortality among patients with end-stage renal disease.     

Correction of confounding bias in non-randomized studies by appropriate weighting

In non-randomized studies, the assessment of a causal effect of treatment or exposure on outcome is hampered by possible confounding. Applying multiple regression models including the effects of treatment and covariates on outcome is the well-known classical approach to adjust for confounding. In recent years other approaches have been promoted. One of them is based on the propensity score and considers the effect of possible confounders on treatment as a relevant criterion for adjustment. Another proposal is based on using an instrumental variable. Here inference relies on a factor, the instrument, which affects treatment but is thought to be otherwise unrelated to outcome, so that it mimics randomization. Each of these approaches can basically be interpreted as a simple reweighting scheme, designed to address confounding. The procedures will be compared with respect to their fundamental properties, namely, which bias they aim to eliminate, which effect they aim to estimate, and which parameter is modelled. We will expand our overview of methods for analysis of non-randomized studies to methods for analysis of randomized controlled trials and show that analyses of both study types may target different effects and different parameters. The considerations will be illustrated using a breast cancer study with a so-called Comprehensive Cohort Study design, including a randomized controlled trial and a non-randomized study in the same patient population as sub-cohorts. This design offers ideal opportunities to discuss and illustrate the properties of the different approaches.     

Efficient nonparametric estimation of causal effects in randomized trials with noncompliance

Causal approaches based on the potential outcome framework providea useful tool for addressing noncompliance problems in randomizedtrials. We propose a new estimator of causal treatment effectsin randomized clinical trials with noncompliance. We use theempirical likelihood approach to construct a profile randomsieve likelihood and take into account the mixture structurein outcome distributions, so that our estimator is robust toparametric distribution assumptions and provides substantialfinite-sample efficiency gains over the standard instrumentalvariable estimator. Our estimator is asymptotically equivalentto the standard instrumental variable estimator, and it canbe applied to outcome variables with a continuous, ordinal orbinary scale. We apply our method to data from a randomizedtrial of an intervention to improve the treatment of depressionamong depressed elderly patients in primary care practices.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.     

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.     

Recent progress in polymorphism-based population genetic inference

The recent availability of whole-genome sequencing data affords tremendous power for statistical inference. With this, there has been great interest in the development of polymorphism-based approaches for the estimation of population genetic parameters. These approaches seek to estimate, for example, recently fixed or sweeping beneficial mutations, the rate of recurrent positive selection, the distribution of selection coefficients, and the demographic history of the population. Yet despite estimating similar parameters using similar data sets, results between methodologies are far from consistent. We here summarize the current state of the field, compare existing approaches, and attempt to reconcile emerging discrepancies. We also discuss the biases in selection estimators introduced by ignoring the demographic history of the population, discuss the biases in demographic estimators introduced by assuming neutrality, and highlight the important challenge to the field of achieving a true joint estimation procedure to circumvent these confounding effects.