Regaining power lost by non-compliance via full probability modelling

Non-compliance, or non-receipt of randomized intervention, is a common problem in randomized controlled trials. An intention-to-treat (ITT) analysis, which compares individuals as randomized, under-estimates the efficacy of the intervention and leads to a loss of power. We explore the possibility of regaining some of this power in a setting with all-or-nothing compliance, without making any assumptions about the comparability of compliers and non-compliers. Efficacy may be specified as the complier average causal effect (CACE), which is the difference in mean outcome among compliers. Compliance is only partially observed, but under an exclusion restriction assumption, the CACE may be estimated using maximum likelihood. In order to quantify the possible gain in power, we derive an expression for the asymptotic relative efficiency (ARE) of the CACE relative to the ITT effect with a Normally distributed outcome. Under the assumption of a common CACE across covariate strata, the CACE estimate is at least as powerful as ITT analysis. The inclusion of covariates that predict compliance enables an additional gain in power, which is investigated algebraically. Using data from three clinical trials, we obtain values of the ARE ranging up to 1.05 due to covariates alone, and 1.13 due to CACE modelling alone, corresponding to gains in power of up to 5 per cent. This implies that a large gain in power obtained using as-treated or per-protocol analyses is likely to be due to the strong and often implausible assumptions such analyses require to be valid.     

A causal model for longitudinal randomised trials with time‐dependent non‐compliance

In the presence of non‐compliance, conventional analysis by intention‐to‐treat provides an unbiased comparison of treatment policies but typically under‐estimates treatment efficacy. With all‐or‐nothing compliance, efficacy may be specified as the complier‐average causal effect (CACE), where compliers are those who receive intervention if and only if randomised to it. We extend the CACE approach to model longitudinal data with time‐dependent non‐compliance, focusing on the situation in which those randomised to control may receive treatment and allowing treatment effects to vary arbitrarily over time. Defining compliance type to be the time of surgical intervention if randomised to control, so that compliers are patients who would not have received treatment at all if they had been randomised to control, we construct a causal model for the multivariate outcome conditional on compliance type and randomised arm. This model is applied to the trial of alternative regimens for glue ear treatment evaluating surgical interventions in childhood ear disease, where outcomes are measured over five time points, and receipt of surgical intervention in the control arm may occur at any time. We fit the models using Markov chain Monte Carlo methods to obtain estimates of the CACE at successive times after receiving the intervention. In this trial, over a half of those randomised to control eventually receive intervention. We find that surgery is more beneficial than control at 6months, with a small but non‐significant beneficial effect at 12months. © 2015 The Authors. Statistics in Medicine Published by JohnWiley & Sons Ltd.     

Intention to treat,per protocol,as treated and instrumental variable estimators given non‐compliance and effect heterogeneity

We consider the behaviour of three approaches to efficacy estimation—using so‐called ‘as treated’ (AT), ‘per protocol’ (PP) and ‘instrumental variable’ (IV) analyses—and of the Intention to Treat estimator, in a two‐arm randomized treatment trial with a Normally distributed outcome when there is treatment effect heterogeneity and non‐random compliance with assigned treatment. Formulae are derived for the bias of estimators when used either to estimate average treatment effect (ACE) or to estimate complier average treatment effect (CACE) under several models for the relationship between compliance and potential outcomes. These enable the expected values of AT, PP and IV estimators to be ranked in relation to ACE, and show that AT and PP estimators are generally biased for both ACE and CACE even under homogeneity. However, we show that the difference between any pair of (AT, PP, IV) estimates can be used to estimate the correlation between the latent variable determining compliance behaviour and one potential outcome. In the absence of measures that predict compliance, bounds for ACE can only be set given strong assumptions. Regarding the Intention to Treat estimator, while this is ‘biased towards the null’ if viewed as a measure of CACE, we show that it is not always so in relation to ACE. Finally we discuss the behaviour of the estimators under weak and strong null hypotheses. Copyright © 2009 John Wiley & Sons, Ltd.     

On a model-based approach to estimating efficacy in clinical trials

Treatment efficacy is of primary importance in phase III clinical trials. Determining the true size of the treatment effect is often complicated by patient non-compliance with the regimen. This paper examines a model-based approach in the spirit of Efron and Feldman utilizing drug and placebo compliance information. One of the assumptioqs of this analysis is ‘comparability’ of drug and placebo compliance. Robustness in estimation of subgroup and population treatment effects when this assumption is violated is investigated in a simulation study. We find that even moderate non-comparability (for example, normalized compliance correlations of 0.4) may produce severely biased estimates. The bias is modulated by the strength of the relationship between compliance and the response on placebo.     

Handling Missing Data in Randomized Experiments with Noncompliance

Treatment noncompliance and missing outcomes at posttreatment assessments are common problems in field experiments in naturalistic settings. Although the two complications often occur simultaneously, statistical methods that address both complications have not been routinely considered in data analysis practice in the prevention research field. This paper shows that identification and estimation of causal treatment effects considering both noncompliance and missing outcomes can be relatively easily conducted under various missing data assumptions. We review a few assumptions on missing data in the presence of noncompliance, including the latent ignorability proposed by Frangakis and Rubin (Biometrika 86:365–379, 1999), and show how these assumptions can be used in the parametric complier average causal effect (CACE) estimation framework. As an easy way of sensitivity analysis, we propose the use of alternative missing data assumptions, which will provide a range of causal effect estimates. In this way, we are less likely to settle with a possibly biased causal effect estimate based on a single assumption. We demonstrate how alternative missing data assumptions affect identification of causal effects, focusing on the CACE. The data from the Johns Hopkins School Intervention Study (Ialongo et al., Am J Community Psychol 27:599–642, 1999) will be used as an example.     

Weighted estimation for confounded binary outcomes subject to misclassification

In the presence of confounding, the consistency assumption required for identification of causal effects may be violated due to misclassification of the outcome variable. We introduce an inverse probability weighted approach to rebalance covariates across treatment groups while mitigating the influence of differential misclassification bias. First, using a simplified example taken from an administrative health care dataset, we introduce the approach for estimation of the marginal causal odds ratio in a simple setting with the use of internal validation information. We then extend this to the presence of additional covariates and use simulated data to investigate the finite sample properties of the proposed weighted estimators. Estimation of the weights is done using logistic regression with misclassified outcomes, and a bootstrap approach is used for variance estimation.     

Causal inference in epidemiological studies with strong confounding

One of the identifiability assumptions of causal effects defined by marginal structural model (MSM) parameters is the experimental treatment assignment (ETA) assumption. Practical violations of this assumption frequently occur in data analysis when certain exposures are rarely observed within some strata of the population. The inverse probability of treatment weighted (IPTW) estimator is particularly sensitive to violations of this assumption; however, we demonstrate that this is a problem for all estimators of causal effects. This is due to the fact that the ETA assumption is about information (or lack thereof) in the data. A new class of causal models, causal models for realistic individualized exposure rules (CMRIER), is based on dynamic interventions. CMRIER generalize MSM, and their parameters remain fully identifiable from the observed data, even when the ETA assumption is violated, if the dynamic interventions are set to be realistic. Examples of such realistic interventions are provided. We argue that causal effects defined by CMRIER may be more appropriate in many situations, particularly those with policy considerations. Through simulation studies, we examine the performance of the IPTW estimator of the CMRIER parameters in contrast to that of the MSM parameters. We also apply the methodology to a real data analysis in air pollution epidemiology to illustrate the interpretation of the causal effects defined by CMRIER.     

Addressing positivity violations in causal effect estimation using Gaussian process priors

In observational studies, causal inference relies on several key identifying assumptions. One identifiability condition is the positivity assumption, which requires the probability of treatment be bounded away from 0 and 1. That is, for every covariate combination, it should be possible to observe both treated and control subjects the covariate distributions should overlap between treatment arms. If the positivity assumption is violated, population-level causal inference necessarily involves some extrapolation. Ideally, a greater amount of uncertainty about the causal effect estimate should be reflected in such situations. With that goal in mind, we construct a Gaussian process model for estimating treatment effects in the presence of practical violations of positivity. Advantages of our method include minimal distributional assumptions, a cohesive model for estimating treatment effects, and more uncertainty associated with areas in the covariate space where there is less overlap. We assess the performance of our approach with respect to bias and efficiency using simulation studies. The method is then applied to a study of critically ill female patients to examine the effect of undergoing right heart catheterization.     

Causal logistic models for non-compliance under randomized treatment with univariate binary response

We propose a method for estimating the marginal causal log-odds ratio for binary outcomes under treatment non-compliance in placebo-randomized trials. This estimation method is a marginal alternative to the causal logistic approach by Nagelkerke et al. (2000) that conditions on partially unknown compliance (that is, adherence to treatment) status, and also differs from previous approaches that estimate risk differences or ratios in subgroups defined by compliance status. The marginal causal method proposed in this paper is based on an extension of Robins' G-estimation approach for fitting linear or log-linear structural nested models to a logistic model. Comparing the marginal and conditional causal log-odds ratio estimates provides a way of assessing the magnitude of unmeasured confounding of the treatment effect due to treatment non-adherence. More specifically, we show through simulations that under weak confounding, the conditional and marginal procedures yield similar estimates, whereas under stronger confounding, they behave differently in terms of bias and confidence interval coverage. The parametric structures that represent such confounding are not identifiable. Hence, the proof of consistency of causal estimators and corresponding simulations are based on two different models that fully identify the causal effects being estimated. These models differ in the way that compliance is related to potential outcomes, and thus differ in the way that the causal effect is identified. The simulations also show that the proposed marginal causal estimation approach performs well in terms of bias under the different levels of confounding due to non-adherence and under different causal logistic models. We also provide results from the analyses of two data sets further showing how a comparison of the marginal and conditional estimators can help evaluate the magnitude of confounding due to non-adherence.     

Estimating treatment effects in randomized clinical trials with non-compliance: the impact of maternal smoking on birthweight

This paper assesses the causal impact of late-term (8th month) maternal smoking on birthweight using data from a randomized clinical trial, in which some women were encouraged not to smoke, while others were not. The estimation of treatment effects in this case is made difficult as a result of the presence of non-compliers, women who would not change their smoking status, regardless of the receipt of encouragement. Because these women are not at risk of changing treatment status, treatment effect distributions may be difficult to construct for them. Consequently, the paper focuses on obtaining the distribution of treatment impacts for the sub-set of compliers found in the data. Because compliance status is not observed for all subjects in the sample, a Bayesian finite mixture model is estimated that recovers the treatment effect parameters of interest. The complier average treatment effect implies that smokers give birth to infants weighing 348 g less than those of non-smokers, on average, although the 95% posterior density interval contains zero. The treatment effect is stronger for women who were moderate smokers prior to pregnancy, implying a birthweight difference of 430 g. However, the model predicts that only about 22% of the women in the sample were at risk of changing their smoking behaviour in response to encouragement to quit.     

A method for the analysis of repeated binary outcomes in randomized clinical trials with non-compliance

When analysing repeated binary data from randomized trials, the model-based approaches, such as generalized estimating equations, are frequently used. Such methods ignore compliance information and give the model-based intention-to-treat estimate of treatment effect. In this paper, the design-based (randomization-based) semi-parametric estimation procedure is given in the estimation of causal risk difference. The resulting risk difference estimator is interpreted as an extension of the instrumental variables estimator for a binary outcome which has the causal interpretation. Extension of the proposed method to stratified analysis is given for data from stratified randomization or meta-analysis. It yields a Mantel-Haenszel type risk difference estimator. As a special case of stratified analysis, the pattern mixture model which stratifies the data by pattern of missing data is performed. Application of the proposed method to a trial in which endpoints were the occurrences of fever over three courses is provided. The same ideas are applied to the causal risk ratio estimation.     

Mendelian randomization with a binary exposure variable: interpretation and presentation of causal estimates

Mendelian randomization uses genetic variants to make causal inferences about a modifiable exposure. Subject to a genetic variant satisfying the instrumental variable assumptions, an association between the variant and outcome implies a causal effect of the exposure on the outcome. Complications arise with a binary exposure that is a dichotomization of a continuous risk factor (for example, hypertension is a dichotomization of blood pressure). This can lead to violation of the exclusion restriction assumption: the genetic variant can influence the outcome via the continuous risk factor even if the binary exposure does not change. Provided the instrumental variable assumptions are satisfied for the underlying continuous risk factor, causal inferences for the binary exposure are valid for the continuous risk factor. Causal estimates for the binary exposure assume the causal effect is a stepwise function at the point of dichotomization. Even then, estimation requires further parametric assumptions. Under monotonicity, the causal estimate represents the average causal effect in ‘compliers’, individuals for whom the binary exposure would be present if they have the genetic variant and absent otherwise. Unlike in randomized trials, genetic compliers are unlikely to be a large or representative subgroup of the population. Under homogeneity, the causal effect of the exposure on the outcome is assumed constant in all individuals; rarely a plausible assumption. We here provide methods for causal estimation with a binary exposure (although subject to all the above caveats). Mendelian randomization investigations with a dichotomized binary exposure should be conceptualized in terms of an underlying continuous variable.     

Estimating treatment effects in randomized clinical trials in the presence of non-compliance

In clinical trials where patients are randomized between two treatment arms, not all patients comply with the treatment they were randomly assigned to. The reasons for (non)compliance may be associated with the outcome variable and thereby act as confounders. The standard way of analysing such trials is by the 'intention-to-treat' principle, which allows the use of permutation tests. Conclusions drawn from such tests do not depend on untested assumptions such as absence of confounding. However, this approach may yield biased estimators for the causal effects of treatments. We consider the estimation of such effects for clinical trials where non-compliers can be considered to have switched to the other trial arm. The most important example of this is the placebo-controlled clinical trial where no substantial placebo effects are anticipated. We consider the situation where the relationship between compliance, and thus treatment received, and outcome is influenced by unobserved confounders. The residual of the regression of the actual treatment indicator variable on the randomization arm indicator variable is shown to 'intercept' the effect of such confounders. Inclusion of this residual in a multivariate analysis, in conjunction with the treatment indicator variable, should thus adjust for confounding. Examples are given. In those examples, the results are similar to those obtained by more complex methods.     

On the use of propensity scores in principal causal effect estimation

We examine the practicality of propensity score methods for estimating causal treatment effects conditional on intermediate posttreatment outcomes (principal effects) in the context of randomized experiments. In particular, we focus on the sensitivity of principal causal effect estimates to violation of principal ignorability, which is the primary assumption that underlies the use of propensity score methods to estimate principal effects. Under principal ignorability (PI), principal strata membership is conditionally independent of the potential outcome under control given the pre‐treatment covariates; i.e. there are no differences in the potential outcomes under control across principal strata given the observed pretreatment covariates. Under this assumption, principal scores modeling principal strata membership can be estimated based solely on the observed covariates and used to predict strata membership and estimate principal effects. While this assumption underlies the use of propensity scores in this setting, sensitivity to violations of it has not been studied rigorously. In this paper, we explicitly define PI using the outcome model (although we do not actually use this outcome model in estimating principal scores) and systematically examine how deviations from the assumption affect estimates, including how the strength of association between principal stratum membership and covariates modifies the performance. We find that when PI is violated, very strong covariate predictors of stratum membership are needed to yield accurate estimates of principal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

Applying optimal model selection in principal stratification for causal inference

Noncompliance to treatment allocation is a key source of complication for causal inference. Efficacy estimation is likely to be compounded by the presence of noncompliance in both treatment arms of clinical trials where the intention‐to‐treat estimate provides a biased estimator for the true causal estimate even under homogeneous treatment effects assumption. Principal stratification method has been developed to address such posttreatment complications. The present work extends a principal stratification method that adjusts for noncompliance in two‐treatment arms trials by developing model selection for covariates predicting compliance to treatment in each arm. We apply the method to analyse data from the Esprit study, which was conducted to ascertain whether unopposed oestrogen (hormone replacement therapy) reduced the risk of further cardiac events in postmenopausal women who survive a first myocardial infarction. We adjust for noncompliance in both treatment arms under a Bayesian framework to produce causal risk ratio estimates for each principal stratum. For mild values of a sensitivity parameter and using separate predictors of compliance in each arm, principal stratification results suggested that compliance with hormone replacement therapy only would reduce the risk for death and myocardial reinfarction by about 47% and 25%, respectively, whereas compliance with either treatment would reduce the risk for death by 13% and reinfarction by 60% among the most compliant. However, the results were sensitive to the user‐defined sensitivity parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

ADJUSTING FOR NON-COMPLIANCE AND CONTAMINATION IN RANDOMIZED CLINICAL TRIALS

A method of analysis is presented for estimating the magnitude of a treatment effect among compliers in a clinical trial which is asymptotically unbiased and respects the randomization. The approach is valid even when compliers have a different baseline risk than non-compliers. Adjustments for contamination (use of the treatment by individuals in the control arm) are also developed. When the baseline failure rates in non-compliers and contaminators are the same as those who accept their allocated treatment, the method produces larger treatment effects than an ‘intent-to-treat’ analysis, but the confidence limits are also wider, and (even without this assumption) asymptotically the efficiencies are the same. In addition to providing a better estimate of the true effect of a treatment in compliers, the method also provides a more realistic confidence interval, which can be especially important for trials aimed at showing the equivalence of two treatments. In this case the intent-to-treat analysis can give unrealistically narrow confidence intervals if substantial numbers of patients elect to have the treatment they were not randomized to receive. © 1997 by John Wiley & Sons, Ltd. Stat. Med., Vol. 16, 1017–1029 (1997).     

Non-parametric bounds on treatment effects with non-compliance by covariate adjustment

Consider a clinical trial where subjects are randomized to two treatment arms but compliance to the assignment is not perfect. Concerning this problem, this paper derives non-parametric bounds on treatment effects by making use of the observed covariates information. The new bounds are narrower and more informative than the existing ones. In addition, a new non-parametric point estimation approach is proposed based on stratified analysis. Furthermore, to examine the accuracy of estimating the proposed bounds, we provide variance estimators for the proposed approach. The results of this paper can yield credible information on treatment effects, which will be useful for medical research and public health policy analysis.     

Estimation of causal mediation effects for a dichotomous outcome in multiple‐mediator models using the mediation formula

Mediators are intermediate variables in the causal pathway between an exposure and an outcome. Mediation analysis investigates the extent to which exposure effects occur through these variables, thus revealing causal mechanisms. In this paper, we consider the estimation of the mediation effect when the outcome is binary and multiple mediators of different types exist. We give a precise definition of the total mediation effect as well as decomposed mediation effects through individual or sets of mediators using the potential outcomes framework. We formulate a model of joint distribution (probit‐normal) using continuous latent variables for any binary mediators to account for correlations among multiple mediators. A mediation formula approach is proposed to estimate the total mediation effect and decomposed mediation effects based on this parametric model. Estimation of mediation effects through individual or subsets of mediators requires an assumption involving the joint distribution of multiple counterfactuals. We conduct a simulation study that demonstrates low bias of mediation effect estimators for two‐mediator models with various combinations of mediator types. The results also show that the power to detect a nonzero total mediation effect increases as the correlation coefficient between two mediators increases, whereas power for individual mediation effects reaches a maximum when the mediators are uncorrelated. We illustrate our approach by applying it to a retrospective cohort study of dental caries in adolescents with low and high socioeconomic status. Sensitivity analysis is performed to assess the robustness of conclusions regarding mediation effects when the assumption of no unmeasured mediator‐outcome confounders is violated. Copyright © 2013 John Wiley & Sons, Ltd.     

Weighted causal inference methods with mismeasured covariates and misclassified outcomes

Inverse probability weighting (IPW) estimation has been widely used in causal inference. Its validity relies on the important condition that the variables are precisely measured. This condition, however, is often violated, which distorts the IPW method and thus yields biased results. In this paper, we study the IPW estimation of average treatment effects for settings with mismeasured covariates and misclassified outcomes. We develop estimation methods to correct for measurement error and misclassification effects simultaneously. Our discussion covers a broad scope of treatment models, including typically assumed logistic regression models and general treatment assignment mechanisms. Satisfactory performance of the proposed methods is demonstrated by extensive numerical studies.     

Bayesian estimation of the average treatment effect on the treated using inverse weighting

We develop a Bayesian approach to estimate the average treatment effect on the treated in the presence of confounding. The approach builds on developments proposed by Saarela et al in the context of marginal structural models, using importance sampling weights to adjust for confounding and estimate a causal effect. The Bayesian bootstrap is adopted to approximate posterior distributions of interest and avoid the issue of feedback that arises in Bayesian causal estimation relying on a joint likelihood. We present results from simulation studies to estimate the average treatment effect on the treated, evaluating the impact of sample size and the strength of confounding on estimation. We illustrate our approach using the classic Right Heart Catheterization data set and find a negative causal effect of the exposure on 30-day survival, in accordance with previous analyses of these data. We also apply our approach to the data set of the National Center for Health Statistics Birth Data and obtain a negative effect of maternal smoking during pregnancy on birth weight.