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.     

An extended general location model for causal inferences from data subject to noncompliance and missing values

Noncompliance is a common problem in experiments involving randomized assignment of treatments, and standard analyses based on intention-to-treat or treatment received have limitations. An attractive alternative is to estimate the Complier-Average Causal Effect (CACE), which is the average treatment effect for the subpopulation of subjects who would comply under either treatment (Angrist, Imbens, and Rubin, 1996, Journal of American Statistical Association 91, 444-472). We propose an extended general location model to estimate the CACE from data with noncompliance and missing data in the outcome and in baseline covariates. Models for both continuous and categorical outcomes and ignorable and latent ignorable (Frangakis and Rubin, 1999, Biometrika 86, 365-379) missing-data mechanisms are developed. Inferences for the models are based on the EM algorithm and Bayesian MCMC methods. We present results from simulations that investigate sensitivity to model assumptions and the influence of missing-data mechanism. We also apply the method to the data from a job search intervention for unemployed workers.     

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.     

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.     

Likelihood methods for treatment noncompliance and subsequent nonresponse in randomized trials

While several new methods that account for noncompliance or missing data in randomized trials have been proposed, the dual effects of noncompliance and nonresponse are rarely dealt with simultaneously. We construct a maximum likelihood estimator (MLE) of the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. The EM algorithm is used for parameter estimation. Our likelihood procedure relies on a latent compliance state covariate that describes the behavior of a subject under all possible treatment assignments and characterizes the missing data mechanism as in Frangakis and Rubin (1999, Biometrika 86, 365-379). Using simulated data, we show that the MLE for normal outcomes compares favorably to the method-of-moments (MOM) and the standard intention-to-treat (ITT) estimators under (1) both normal and non-normal data, and (2) departures from the latent ignorability and compound exclusion restriction assumptions. We illustrate methods using data from a trial to compare the efficacy of two antipsychotics for adults with refractory schizophrenia.     

Bounds on the average causal effects in randomized trials with noncompliance by covariate adjustment

In randomized trials with noncompliance, causal effects cannot be identified without strong assumptions. Therefore, several authors have considered bounds on the causal effects. Applying an idea of VanderWeele ( 2008 ), Chiba ( 2009 ) gave bounds on the average causal effects in randomized trials with noncompliance using the information on the randomized assignment, the treatment received and the outcome under monotonicity assumptions about covariates. But he did not consider any observed covariates. If there are some observed covariates such as age, gender, and race in a trial, we propose new bounds using the observed covariate information under some monotonicity assumptions similar to those of VanderWeele and Chiba. And we compare the three bounds in a real example.     

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.     

Generalizing causal inferences from individuals in randomized trials to all trial‐eligible individuals

We consider methods for causal inference in randomized trials nested within cohorts of trial‐eligible individuals, including those who are not randomized. We show how baseline covariate data from the entire cohort, and treatment and outcome data only from randomized individuals, can be used to identify potential (counterfactual) outcome means and average treatment effects in the target population of all eligible individuals. We review identifiability conditions, propose estimators, and assess the estimators' finite‐sample performance in simulation studies. As an illustration, we apply the estimators in a trial nested within a cohort of trial‐eligible individuals to compare coronary artery bypass grafting surgery plus medical therapy vs. medical therapy alone for chronic coronary artery disease.     

Analyzing a randomized trial on breast self-examination with noncompliance and missing outcomes

Recently, instrumental variables methods have been used to address non-compliance in randomized experiments. Complicating such analyses is often the presence of missing data. The standard model for missing data, missing at random (MAR), has some unattractive features in this context. In this paper we compare MAR-based estimates of the complier average causal effect (CACE) with an estimator based on an alternative, nonignorable model for the missing data process, developed by Frangakis and Rubin (1999, Biometrika, 86, 365-379). We also introduce a new missing data model that, like the Frangakis-Rubin model, is specially suited for models with instrumental variables, but makes different substantive assumptions. We analyze these issues in the context of a randomized trial of breast self-examination (BSE). In the study two methods of teaching BSE, consisting of either mailed information about BSE (the standard treatment) or the attendance of a course involving theoretical and practical sessions (the new treatment), were compared with the aim of assessing whether teaching programs could increase BSE practice and improve examination skills. The study was affected by the two sources of bias mentioned above: only 55% of women assigned to receive the new treatment complied with their assignment and 35% of the women did not respond to the post-test questionnaire. Comparing the causal estimand of the new treatment using the MAR, Frangakis-Rubin, and our new approach, the results suggest that for these data the MAR assumption appears least plausible, and that the new model appears most plausible among the three choices.     

Inference for the effect of treatment on survival probability in randomized trials with noncompliance and administrative censoring

In many clinical studies with a survival outcome, administrative censoring occurs when follow-up ends at a prespecified date and many subjects are still alive. An additional complication in some trials is that there is noncompliance with the assigned treatment. For this setting, we study the estimation of the causal effect of treatment on survival probability up to a given time point among those subjects who would comply with the assignment to both treatment and control. We first discuss the standard instrumental variable (IV) method for survival outcomes and parametric maximum likelihood methods, and then develop an efficient plug-in nonparametric empirical maximum likelihood estimation (PNEMLE) approach. The PNEMLE method does not make any assumptions on outcome distributions, and makes use of the mixture structure in the data to gain efficiency over the standard IV method. Theoretical results of the PNEMLE are derived and the method is illustrated by an analysis of data from a breast cancer screening trial. From our limited mortality analysis with administrative censoring times 10 years into the follow-up, we find a significant benefit of screening is present after 4 years (at the 5% level) and this persists at 10 years follow-up.     

Stratified analysis in randomized trials with noncompliance

This article develops methods for stratified analyses of additive or multiplicative causal effect on binary outcomes in randomized trials with noncompliance. The methods are based on a weighted estimating function for an unbiased estimating function under randomization in each stratum. When known weights are used, the derived estimator is a natural extension of the instrumental variable estimator for stratified analyses, and test-based confidence limits are solutions of a quadratic equation in the causal parameter. Optimal weights that maximize asymptotic efficiency incorporate variability in compliance aspects across strata. An assessment based on asymptotic relative efficiency shows that a substantial enhancement in efficiency can be gained by using optimal weights instead of conventional ones, which do not incorporate the variability in compliance aspects across strata. Application to a field trial for coronary heart disease is provided.     

Methods for clustered encouragement design studies with noncompliance and missing data

Encouragement design studies are particularly useful for estimating the effect of an intervention that cannot itself be randomly administered to some and not to others. They require a randomly selected group receive extra encouragement to undertake the treatment of interest, where the encouragement typically takes the form of additional information or incentives. We consider a "clustered encouragement design" (CED), where the randomization is at the level of the clusters (e.g. physicians), but the compliance with assignment is at the level of the units (e.g. patients) within clusters. Noncompliance and missing data are particular problems in encouragement design studies, where encouragement to take the treatment, rather than the treatment itself, is randomized. The motivating study looks at whether computer-based care suggestions can improve patient outcomes in veterans with chronic heart failure. Since physician adherence has been inadequate, the original study focused on methods to improve physician adherence, although an equally important question is whether physician adherence improves patient outcomes. Here, we reanalyze the data to determine the effect of physician adherence on patient outcomes. We propose causal inference methodology for the effect of a treatment versus a control in a randomized CED study with all-or-none compliance at the unit level. These methods extend the current approaches to account for nonignorable missing data and use an alternative approach to inference using multiple imputation methods, which have been successfully applied to a wide variety of missing data problems and have recently been applied to the potential outcomes framework of causal inference (Taylor and Zhou, 2009b).     

A causal proportional hazards estimator for the effect of treatment actually received in a randomized trial with all-or-nothing compliance

Survival data from randomized trials are most often analyzed in a proportional hazards (PH) framework that follows the intention-to-treat (ITT) principle. When not all the patients on the experimental arm actually receive the assigned treatment, the ITT-estimator mixes its effect on treatment compliers with its absence of effect on noncompliers. The structural accelerated failure time (SAFT) models of Robins and Tsiatis are designed to consistently estimate causal effects on the treated, without direct assumptions about the compliance selection mechanism. The traditional PH-model, however, has not yet led to such causal interpretation. In this article, we examine a PH-model of treatment effect on the treated subgroup. While potential treatment compliance is unobserved in the control arm, we derive an estimating equation for the Compliers PROPortional Hazards Effect of Treatment (C-PROPHET). The jackknife is used for bias correction and variance estimation. The method is applied to data from a recently finished clinical trial in cancer patients with liver metastases.     

Sensitivity analyses comparing outcomes only existing in a subset selected post-randomization, conditional on covariates, with application to HIV vaccine trials

In many experiments, researchers would like to compare between treatments and outcome that only exists in a subset of participants selected after randomization. For example, in preventive HIV vaccine efficacy trials it is of interest to determine whether randomization to vaccine causes lower HIV viral load, a quantity that only exists in participants who acquire HIV. To make a causal comparison and account for potential selection bias we propose a sensitivity analysis following the principal stratification framework set forth by Frangakis and Rubin (2002, Biometrics58, 21-29). Our goal is to assess the average causal effect of treatment assignment on viral load at a given baseline covariate level in the always infected principal stratum (those who would have been infected whether they had been assigned to vaccine or placebo). We assume stable unit treatment values (SUTVA), randomization, and that subjects randomized to the vaccine arm who became infected would also have become infected if randomized to the placebo arm (monotonicity). It is not known which of those subjects infected in the placebo arm are in the always infected principal stratum, but this can be modeled conditional on covariates, the observed viral load, and a specified sensitivity parameter. Under parametric regression models for viral load, we obtain maximum likelihood estimates of the average causal effect conditional on covariates and the sensitivity parameter. We apply our methods to the world's first phase III HIV vaccine trial.     

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model—a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an “expected” power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.     

Compliance Mixture Modelling with a Zero‐Effect Complier Class and Missing Data

Summary Randomized experiments are the gold standard for evaluating proposed treatments. The intent to treat estimand measures the effect of treatment assignment, but not the effect of treatment if subjects take treatments to which they are not assigned. The desire to estimate the efficacy of the treatment in this case has been the impetus for a substantial literature on compliance over the last 15 years. In papers dealing with this issue, it is typically assumed there are different types of subjects, for example, those who will follow treatment assignment (compliers), and those who will always take a particular treatment irrespective of treatment assignment. The estimands of primary interest are the complier proportion and the complier average treatment effect (CACE). To estimate CACE, researchers have used various methods, for example, instrumental variables and parametric mixture models, treating compliers as a single class. However, it is often unreasonable to believe all compliers will be affected. This article therefore treats compliers as a mixture of two types, those belonging to a zero‐effect class, others to an effect class. Second, in most experiments, some subjects drop out or simply do not report the value of the outcome variable, and the failure to take into account missing data can lead to biased estimates of treatment effects. Recent work on compliance in randomized experiments has addressed this issue by assuming missing data are missing at random or latently ignorable. We extend this work to the case where compliers are a mixture of types and also examine alternative types of nonignorable missing data assumptions.     

A matching procedure for sequential experiments that iteratively learns which covariates improve power

We propose a dynamic allocation procedure that increases power and efficiency when measuring an average treatment effect in sequential randomized trials exploiting some subjects' previous assessed responses. Subjects arrive sequentially and are either randomized or paired to a previously randomized subject and administered the alternate treatment. The pairing is made via a dynamic matching criterion that iteratively learns which specific covariates are important to the response. We develop estimators for the average treatment effect as well as an exact test. We illustrate our method's increase in efficiency and power over other allocation procedures in both simulated scenarios and a clinical trial dataset. An R package “SeqExpMatch ” for use by practitioners is available on CRAN .     

Assessing intervention effects in a randomized trial within a social network

Studies of social networks provide unique opportunities to assess the causal effects of interventions that may impact more of the population than just those intervened on directly. Such effects are sometimes called peer or spillover effects, and may exist in the presence of interference, that is, when one individual's treatment affects another individual's outcome. Randomization-based inference (RI) methods provide a theoretical basis for causal inference in randomized studies, even in the presence of interference. In this article, we consider RI of the intervention effect in the eX-FLU trial, a randomized study designed to assess the effect of a social distancing intervention on influenza-like-illness transmission in a connected network of college students. The approach considered enables inference about the effect of the social distancing intervention on the per-contact probability of influenza-like-illness transmission in the observed network. The methods allow for interference between connected individuals and for heterogeneous treatment effects. The proposed methods are evaluated empirically via simulation studies, and then applied to data from the eX-FLU trial.     

Analysis of times to repeated events in two-arm randomized trials with noncompliance and dependent censoring

This article develops randomization-based methods for times to repeated events in two-arm randomized trials with noncompliance and dependent censoring. Structural accelerated failure time models are assumed to capture causal effects on repeated event times and dependent censoring time, but the dependence structure among repeated event times and dependent censoring time is unspecified. Artificial censoring techniques to accommodate nonrandom noncompliance and dependent censoring are proposed. Estimation of the acceleration parameters are based on rank-based estimating functions. A simulation study is conducted to evaluate the performance of the developed methods. An illustration of the methods using data from an acute myeloid leukemia trial is provided.     

Principal stratification in causal inference

Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable tinder each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate. such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance. and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to forrmulate estimands based on principal stratification and principal causal effects and show their superiority.