Clustered encouragement designs with individual noncompliance: bayesian inference with randomization,and application to advance directive forms

In many studies comparing a new 'target treatment' with a control target treatment, the received treatment does not always agree with assigned treatment-that is, the compliance is imperfect. An obvious example arises when ethical or practical constraints prevent even the randomized assignment of receipt of the new target treatment but allow the randomized assignment of the encouragement to receive this treatment. In fact, many randomized experiments where compliance is not enforced by the experimenter (e.g. with non-blinded assignment) may be more accurately thought of as randomized encouragement designs. Moreover, often the assignment of encouragement is at the level of clusters (e.g. doctors) where the compliance with the assignment varies across the units (e.g. patients) within clusters. We refer to such studies as 'clustered encouragement designs' (CEDs) and they arise relatively frequently (e.g. Sommer and Zeger, 1991; McDonald et al., 1992; Dexter et al., 1998) Here, we propose Bayesian methodology for causal inference for the effect of the new target treatment versus the control target treatment in the randomized CED with all-or-none compliance at the unit level, which generalizes the approach of Hirano et al. (2000) in important and surprisingly subtle ways, to account for the clustering, which is necessary for statistical validity. We illustrate our methods using data from a recent study exploring the role of physician consulting in increasing patients' completion of Advance Directive forms.     

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.     

Assessing the effect of an influenza vaccine in an encouragement design

Many randomized experiments suffer from noncompliance. Some of these experiments, so-called encouragement designs, can be expected to have especially large amounts of noncompliance, because encouragement to take the treatment rather than the treatment itself is randomly assigned to individuals. We present an extended framework for the analysis of data from such experiments with a binary treatment, binary encouragement, and background covariates. There are two key features of this framework: we use an instrumental variables approach to link intention-to-treat effects to treatment effects and we adopt a Bayesian approach for inference and sensitivity analysis. This framework is illustrated in a medical example concerning the effects of inoculation for influenza. In this example, the analyses suggest that positive estimates of the intention-to-treat effect need not be due to the treatment itself, but rather to the encouragement to take the treatment: the intention-to-treat effect for the subpopulation who would be inoculated whether or not encouraged is estimated to be approximately as large as the intention-to-treat effect for the subpopulation whose inoculation status would agree with their (randomized) encouragement status whether or not encouraged. Thus, our methods suggest that global intention-to-treat estimates, although often regarded as conservative, can be too coarse and even misleading when taken as summarizing the evidence in the data for the effects of treatments.     

Local average treatment effects estimation via substantive model compatible multiple imputation

Nonadherence to assigned treatment is common in randomized controlled trials (RCTs). Recently, there has been increased interest in estimating causal effects of treatment received, for example, the so‐called local average treatment effect (LATE). Instrumental variables (IV) methods can be used for identification, with estimation proceeding either via fully parametric mixture models or two‐stage least squares (TSLS). TSLS is popular but can be problematic for binary outcomes where the estimand of interest is a causal odds ratio. Mixture models are rarely used in practice, perhaps because of their perceived complexity and need for specialist software. Here, we propose using multiple imputation (MI) to impute the latent compliance class appearing in the mixture models. Since such models include an interaction term between the latent compliance class and randomized treatment, we use “substantive model compatible” MI (SMC MIC), which can additionally handle missing data in outcomes and other variables in the model, before fitting the mixture models via maximum likelihood to the MI data sets and combining results via Rubin's rules. We use simulations to compare the performance of SMC MIC to existing approaches and also illustrate the methods by reanalyzing an RCT in UK primary health. We show that SMC MIC can be more efficient than full Bayesian estimation when auxiliary variables are incorporated, and is superior to two‐stage methods, especially for binary outcomes.     

Causal Structure of Brain Physiology after Brain Injury from Subarachnoid Hemorrhage

High frequency physiologic data are routinely generated for intensive care patients. While massive amounts of data make it difficult for clinicians to extract meaningful signals, these data could provide insight into the state of critically ill patients and guide interventions. We develop uniquely customized computational methods to uncover the causal structure within systemic and brain physiologic measures recorded in a neurological intensive care unit after subarachnoid hemorrhage. While the data have many missing values, poor signal-to-noise ratio, and are composed from a heterogeneous patient population, our advanced imputation and causal inference techniques enable physiologic models to be learned for individuals. Our analyses confirm that complex physiologic relationships including demand and supply of oxygen underlie brain oxygen measurements and that mechanisms for brain swelling early after injury may differ from those that develop in a delayed fashion. These inference methods will enable wider use of ICU data to understand patient physiology.     

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.     

Quantifying the impact of fixed effects modeling of clusters in multiple imputation for cluster randomized trials

In cluster randomized trials (CRTs), identifiable clusters rather than individuals are randomized to study groups. Resulting data often consist of a small number of clusters with correlated observations within a treatment group. Missing data often present a problem in the analysis of such trials, and multiple imputation (MI) has been used to create complete data sets, enabling subsequent analysis with well-established analysis methods for CRTs. We discuss strategies for accounting for clustering when multiply imputing a missing continuous outcome, focusing on estimation of the variance of group means as used in an adjusted t-test or ANOVA. These analysis procedures are congenial to (can be derived from) a mixed effects imputation model; however, this imputation procedure is not yet available in commercial statistical software. An alternative approach that is readily available and has been used in recent studies is to include fixed effects for cluster, but the impact of using this convenient method has not been studied. We show that under this imputation model the MI variance estimator is positively biased and that smaller intraclass correlations (ICCs) lead to larger overestimation of the MI variance. Analytical expressions for the bias of the variance estimator are derived in the case of data missing completely at random, and cases in which data are missing at random are illustrated through simulation. Finally, various imputation methods are applied to data from the Detroit Middle School Asthma Project, a recent school-based CRT, and differences in inference are compared.     

Nested Markov compliance class model in the presence of time-varying noncompliance

Summary .  We consider a Markov structure for partially unobserved time-varying compliance classes in the Imbens–Rubin (1997, The Annals of Statistics 25, 305–327) compliance model framework. The context is a longitudinal randomized intervention study where subjects are randomized once at baseline, outcomes and patient adherence are measured at multiple follow-ups, and patient adherence to their randomized treatment could vary over time. We propose a nested latent compliance class model where we use time-invariant subject-specific compliance principal strata to summarize longitudinal trends of subject-specific time-varying compliance patterns. The principal strata are formed using Markov models that relate current compliance behavior to compliance history. Treatment effects are estimated as intent-to-treat effects within the compliance principal strata.     

Polydesigns and causal inference

In an increasingly common class of studies, the goal is to evaluate causal effects of treatments that are only partially controlled by the investigator. In such studies there are two conflicting features: (1) a model on the full cohort design and data can identify the causal effects of interest, but can be sensitive to extreme regions of that design's data, where model specification can have more impact; and (2) models on a reduced design (i.e., a subset of the full data), for example, conditional likelihood on matched subsets of data, can avoid such sensitivity, but do not generally identify the causal effects. We propose a framework to assess how inference is sensitive to designs by exploring combinations of both the full and reduced designs. We show that using such a "polydesign" framework generates a rich class of methods that can identify causal effects and that can also be more robust to model specification than methods using only the full design. We discuss implementation of polydesign methods, and provide an illustration in the evaluation of a needle exchange program.     

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.     

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.     

Rational Design and Adaptive Management of Combination Therapies for Hepatitis C Virus Infection

Recent discoveries of direct acting antivirals against Hepatitis C virus (HCV) have raised hopes of effective treatment via combination therapies. Yet rapid evolution and high diversity of HCV populations, combined with the reality of suboptimal treatment adherence, make drug resistance a clinical and public health concern. We develop a general model incorporating viral dynamics and pharmacokinetics/ pharmacodynamics to assess how suboptimal adherence affects resistance development and clinical outcomes. We derive design principles and adaptive treatment strategies, identifying a high-risk period when missing doses is particularly risky for de novo resistance, and quantifying the number of additional doses needed to compensate when doses are missed. Using data from large-scale resistance assays, we demonstrate that the risk of resistance can be reduced substantially by applying these principles to a combination therapy of daclatasvir and asunaprevir. By providing a mechanistic framework to link patient characteristics to the risk of resistance, these findings show the potential of rational treatment design.     

Meta-analysis of studies with missing data

Summary .  Consider a meta-analysis of studies with varying proportions of patient-level missing data, and assume that each primary study has made certain missing data adjustments so that the reported estimates of treatment effect size and variance are valid. These estimates of treatment effects can be combined across studies by standard meta-analytic methods, employing a random-effects model to account for heterogeneity across studies. However, we note that a meta-analysis based on the standard random-effects model will lead to biased estimates when the attrition rates of primary studies depend on the size of the underlying study-level treatment effect. Perhaps ignorable within each study, these types of missing data are in fact not ignorable in a meta-analysis. We propose three methods to correct the bias resulting from such missing data in a meta-analysis: reweighting the DerSimonian–Laird estimate by the completion rate; incorporating the completion rate into a Bayesian random-effects model; and inference based on a Bayesian shared-parameter model that includes the completion rate. We illustrate these methods through a meta-analysis of 16 published randomized trials that examined combined pharmacotherapy and psychological treatment for depression.     

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.     

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.     

On small‐sample inference in group randomized trials with binary outcomes and cluster‐level covariates

Group randomized trials (GRTs) randomize groups, or clusters, of people to intervention or control arms. To test for the effectiveness of the intervention when subject‐level outcomes are binary, and while fitting a marginal model that adjusts for cluster‐level covariates and utilizes a logistic link, we develop a pseudo‐Wald statistic to improve inference. Alternative Wald statistics could employ bias‐corrected empirical sandwich standard error estimates, which have received limited attention in the GRT literature despite their broad utility and applicability in our settings of interest. The test could also be carried out using popular approaches based upon cluster‐level summary outcomes. A simulation study covering a variety of realistic GRT settings is used to compare the accuracy of these methods in terms of producing nominal test sizes. Tests based upon the pseudo‐Wald statistic and a cluster‐level summary approach utilizing the natural log of observed cluster‐level odds worked best. Due to weighting, some popular cluster‐level summary approaches were found to lead to invalid inference in many settings. Finally, although use of bias‐corrected empirical sandwich standard error estimates did not consistently result in nominal sizes, they did work well, thus supporting the applicability of marginal models in GRT settings.     

Statistical power and sample size requirements for three level hierarchical cluster randomized trials

SUMMARY: Cluster randomized clinical trials (cluster-RCT), where the community entities serve as clusters, often yield data with three hierarchy levels. For example, interventions are randomly assigned to the clusters (level three unit). Health care professionals (level two unit) within the same cluster are trained with the randomly assigned intervention to provide care to subjects (level one unit). In this study, we derived a closed form power function and formulae for sample size determination required to detect an intervention effect on outcomes at the subject's level. In doing so, we used a test statistic based on maximum likelihood estimates from a mixed-effects linear regression model for three level data. A simulation study follows and verifies that theoretical power estimates based on the derived formulae are nearly identical to empirical estimates based on simulated data. Recommendations at the design stage of a cluster-RCT are discussed.     

Semiparametric estimation of treatment effect in a pretest-posttest study

Inference on treatment effects in a pretest-posttest study is a routine objective in medicine, public health, and other fields. A number of approaches have been advocated. We take a semiparametric perspective, making no assumptions about the distributions of baseline and posttest responses. By representing the situation in terms of counterfactual random variables, we exploit recent developments in the literature on missing data and causal inference, to derive the class of all consistent treatment effect estimators, identify the most efficient such estimator, and outline strategies for implementation of estimators that may improve on popular methods. We demonstrate the methods and their properties via simulation and by application to a data set from an HIV clinical trial.