Statistical inference for data-adaptive doubly robust estimators with survival outcomes

The consistency of doubly robust estimators relies on the consistent estimation of at least one of two nuisance regression parameters. In moderate-to-large dimensions, the use of flexible data-adaptive regression estimators may aid in achieving this consistency. However, n^1/2-consistency of doubly robust estimators is not guaranteed if one of the nuisance estimators is inconsistent. In this paper, we present a doubly robust estimator for survival analysis with the novel property that it converges to a Gaussian variable at an n^1/2-rate for a large class of data-adaptive estimators of the nuisance parameters, under the only assumption that at least one of them is consistently estimated at an n^1/4-rate. This result is achieved through the adaptation of recent ideas in semiparametric inference, which amount to (i) Gaussianizing (ie, making asymptotically linear) a drift term that arises in the asymptotic analysis of the doubly robust estimator and (ii) using cross-fitting to avoid entropy conditions on the nuisance estimators. We present the formula of the asymptotic variance of the estimator, which allows for the computation of doubly robust confidence intervals and p values. We illustrate the finite-sample properties of the estimator in simulation studies and demonstrate its use in a phase III clinical trial for estimating the effect of a novel therapy for the treatment of human epidermal growth factor receptor 2 (HER2)–positive breast cancer.     

The use of propensity scores and observational data to estimate randomized controlled trial generalizability bias

Although randomized controlled trials are considered the ‘gold standard’ for clinical studies, the use of exclusion criteria may impact the external validity of the results. It is unknown whether estimators of effect size are biased by excluding a portion of the target population from enrollment. We propose to use observational data to estimate the bias due to enrollment restrictions, which we term generalizability bias. In this paper, we introduce a class of estimators for the generalizability bias and use simulation to study its properties in the presence of non‐constant treatment effects. We find the surprising result that our estimators can be unbiased for the true generalizability bias even when all potentially confounding variables are not measured. In addition, our proposed doubly robust estimator performs well even for mis‐specified models. Copyright © 2013 John Wiley & Sons, Ltd.     

Doubly robust generalized estimating equations for longitudinal data

A popular method for analysing repeated‐measures data is generalized estimating equations (GEE). When response data are missing at random (MAR), two modifications of GEE use inverse‐probability weighting and imputation. The weighted GEE (WGEE) method involves weighting observations by their inverse probability of being observed, according to some assumed missingness model. Imputation methods involve filling in missing observations with values predicted by an assumed imputation model. WGEE are consistent when the data are MAR and the dropout model is correctly specified. Imputation methods are consistent when the data are MAR and the imputation model is correctly specified. Recently, doubly robust (DR) methods have been developed. These involve both a model for probability of missingness and an imputation model for the expectation of each missing observation, and are consistent when either is correct. We describe DR GEE, and illustrate their use on simulated data. We also analyse the INITIO randomized clinical trial of HIV therapy allowing for MAR dropout. Copyright © 2009 John Wiley & Sons, Ltd.     

Causal inference with noisy data: Bias analysis and estimation approaches to simultaneously addressing missingness and misclassification in binary outcomes

Causal inference has been widely conducted in various fields and many methods have been proposed for different settings. However, for noisy data with both mismeasurements and missing observations, those methods often break down. In this paper, we consider a problem that binary outcomes are subject to both missingness and misclassification, when the interest is in estimation of the average treatment effects (ATE). We examine the asymptotic biases caused by ignoring missingness and/or misclassification and establish the intrinsic connections between missingness effects and misclassification effects on the estimation of ATE. We develop valid weighted estimation methods to simultaneously correct for missingness and misclassification effects. To provide protection against model misspecification, we further propose a doubly robust correction method which yields consistent estimators when either the treatment model or the outcome model is misspecified. Simulation studies are conducted to assess the performance of the proposed methods. An application to smoking cessation data is reported to illustrate the use of the proposed methods.     

Identification of the optimal treatment regimen in the presence of missing covariates

Covariates associated with treatment-effect heterogeneity can potentially be used to make personalized treatment recommendations towards best clinical outcomes. Methods for treatment-selection rule development that directly maximize treatment-selection benefits have attracted much interest in recent years, due to the robustness of these methods to outcome modeling. In practice, the task of treatment-selection rule development can be further complicated by missingness in data. Here, we consider the identification of optimal treatment-selection rules for a binary disease outcome when measurements of an important covariate from study participants are partly missing. Under the missing at random assumption, we develop a robust estimator of treatment-selection rules under the direct-optimization paradigm. This estimator targets the maximum selection benefits to the population under correct specification of at least one mechanism from each of the two sets—missing data or conditional covariate distribution, and treatment assignment or disease outcome model. We evaluate and compare performance of the proposed estimator with alternative direct-optimization estimators through extensive simulation studies. We demonstrate the application of the proposed method through a real data example from an Alzheimer's disease study for developing covariate combinations to guide the treatment of Alzheimer's disease.     

Bias due to missing exposure data using complete-case analysis in the proportional hazards regression model

We studied bias due to missing exposure data in the proportional hazards regression model when using complete-case analysis (CCA). Eleven missing data scenarios were considered: one with missing completely at random (MCAR), four missing at random (MAR), and six non-ignorable missingness scenarios, with a variety of hazard ratios, censoring fractions, missingness fractions and sample sizes. When missingness was MCAR or dependent only on the exposure, there was negligible bias (2-3 per cent) that was similar to the difference between the estimate in the full data set with no missing data and the true parameter. In contrast, substantial bias occurred when missingness was dependent on outcome or both outcome and exposure. For models with hazard ratio of 3.5, a sample size of 400, 20 per cent censoring and 40 per cent missing data, the relative bias for the hazard ratio ranged between 7 per cent and 64 per cent. We observed important differences in the direction and magnitude of biases under the various missing data mechanisms. For example, in scenarios where missingness was associated with longer or shorter follow-up, the biases were notably different, although both mechanisms are MAR. The hazard ratio was underestimated (with larger bias) when missingness was associated with longer follow-up and overestimated (with smaller bias) when associated with shorter follow-up. If it is known that missingness is associated with a less frequently observed outcome or with both the outcome and exposure, CCA may result in an invalid inference and other methods for handling missing data should be considered.     

A sensitivity analysis for missing outcomes due to truncation by death under the matched‐pairs design

The matched‐pairs design enables researchers to efficiently infer causal effects from randomized experiments. In this paper, we exploit the key feature of the matched‐pairs design and develop a sensitivity analysis for missing outcomes due to truncation by death, in which the outcomes of interest (e.g., quality of life measures) are not even well defined for some units (e.g., deceased patients). Our key idea is that if 2 nearly identical observations are paired prior to the randomization of the treatment, the missingness of one unit's outcome is informative about the potential missingness of the other unit's outcome under an alternative treatment condition. We consider the average treatment effect among always‐observed pairs (ATOP) whose units exhibit no missing outcome regardless of their treatment status. The naive estimator based on available pairs is unbiased for the ATOP if 2 units of the same pair are identical in terms of their missingness patterns. The proposed sensitivity analysis characterizes how the bounds of the ATOP widen as the degree of the within‐pair similarity decreases. We further extend the methodology to the matched‐pairs design in observational studies. Our simulation studies show that informative bounds can be obtained under some scenarios when the proportion of missing data is not too large. The proposed methodology is also applied to the randomized evaluation of the Mexican universal health insurance program. An open‐source software package is available for implementing the proposed research.     

Robust versus consistent variance estimators in marginal structural Cox models

In survival analyses, inverse‐probability‐of‐treatment (IPT) and inverse‐probability‐of‐censoring (IPC) weighted estimators of parameters in marginal structural Cox models are often used to estimate treatment effects in the presence of time‐dependent confounding and censoring. In most applications, a robust variance estimator of the IPT and IPC weighted estimator is calculated leading to conservative confidence intervals. This estimator assumes that the weights are known rather than estimated from the data. Although a consistent estimator of the asymptotic variance of the IPT and IPC weighted estimator is generally available, applications and thus information on the performance of the consistent estimator are lacking. Reasons might be a cumbersome implementation in statistical software, which is further complicated by missing details on the variance formula. In this paper, we therefore provide a detailed derivation of the variance of the asymptotic distribution of the IPT and IPC weighted estimator and explicitly state the necessary terms to calculate a consistent estimator of this variance. We compare the performance of the robust and consistent variance estimators in an application based on routine health care data and in a simulation study. The simulation reveals no substantial differences between the 2 estimators in medium and large data sets with no unmeasured confounding, but the consistent variance estimator performs poorly in small samples or under unmeasured confounding, if the number of confounders is large. We thus conclude that the robust estimator is more appropriate for all practical purposes.     

Model misspecification and robustness in causal inference: comparing matching with doubly robust estimation

In this paper, we compare the robustness properties of a matching estimator with a doubly robust estimator. We describe the robustness properties of matching and subclassification estimators by showing how misspecification of the propensity score model can result in the consistent estimation of an average causal effect. The propensity scores are covariate scores, which are a class of functions that removes bias due to all observed covariates. When matching on a parametric model (e.g., a propensity or a prognostic score), the matching estimator is robust to model misspecifications if the misspecified model belongs to the class of covariate scores. The implication is that there are multiple possibilities for the matching estimator in contrast to the doubly robust estimator in which the researcher has two chances to make reliable inference. In simulations, we compare the finite sample properties of the matching estimator with a simple inverse probability weighting estimator and a doubly robust estimator. For the misspecifications in our study, the mean square error of the matching estimator is smaller than the mean square error of both the simple inverse probability weighting estimator and the doubly robust estimators.     

Propensity score estimation with missing values using a multiple imputation missingness pattern (MIMP) approach

Propensity scores have been used widely as a bias reduction method to estimate the treatment effect in nonrandomized studies. Since many covariates are generally included in the model for estimating the propensity scores, the proportion of subjects with at least one missing covariate could be large. While many methods have been proposed for propensity score‐based estimation in the presence of missing covariates, little has been published comparing the performance of these methods. In this article we propose a novel method called multiple imputation missingness pattern (MIMP) and compare it with the naive estimator (ignoring propensity score) and three commonly used methods of handling missing covariates in propensity score‐based estimation (separate estimation of propensity scores within each pattern of missing data, multiple imputation and discarding missing data) under different mechanisms of missing data and degree of correlation among covariates. Simulation shows that all adjusted estimators are much less biased than the naive estimator. Under certain conditions MIMP provides benefits (smaller bias and mean‐squared error) compared with existing alternatives. Copyright © 2009 John Wiley & Sons, Ltd.     

Doubly robust estimation and causal inference for recurrent event data

Many longitudinal databases record the occurrence of recurrent events over time. In this article, we propose a new method to estimate the average causal effect of a binary treatment for recurrent event data in the presence of confounders. We propose a doubly robust semiparametric estimator based on a weighted version of the Nelson-Aalen estimator and a conditional regression estimator under an assumed semiparametric multiplicative rate model for recurrent event data. We show that the proposed doubly robust estimator is consistent and asymptotically normal. In addition, a model diagnostic plot of residuals is presented to assess the adequacy of our proposed semiparametric model. We then evaluate the finite sample behavior of the proposed estimators under a number of simulation scenarios. Finally, we illustrate the proposed methodology via a database of circus artist injuries.     

Allowing for uncertainty due to missing continuous outcome data in pairwise and network meta‐analysis

Missing outcome data are commonly encountered in randomized controlled trials and hence may need to be addressed in a meta‐analysis of multiple trials. A common and simple approach to deal with missing data is to restrict analysis to individuals for whom the outcome was obtained (complete case analysis). However, estimated treatment effects from complete case analyses are potentially biased if informative missing data are ignored. We develop methods for estimating meta‐analytic summary treatment effects for continuous outcomes in the presence of missing data for some of the individuals within the trials. We build on a method previously developed for binary outcomes, which quantifies the degree of departure from a missing at random assumption via the informative missingness odds ratio. Our new model quantifies the degree of departure from missing at random using either an informative missingness difference of means or an informative missingness ratio of means, both of which relate the mean value of the missing outcome data to that of the observed data. We propose estimating the treatment effects, adjusted for informative missingness, and their standard errors by a Taylor series approximation and by a Monte Carlo method. We apply the methodology to examples of both pairwise and network meta‐analysis with multi‐arm trials. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Bias adjustment in analysing longitudinal data with informative missingness.

The recent biostatistical literature contains a number of methods for handling the bias caused by 'informative censoring', which refers to drop-out from a longitudinal study after a number of visits scheduled at predetermined intervals. The same or related methods can be extended to situations where the missing pattern is intermittent. The pattern of missingness is often assumed to be related to the outcome through random effects which represent unmeasured individual characteristics such as health awareness. To date there is only limited experience with applying the methods for informative censoring in practice, mostly because of complicated modelling and difficult computations. In this paper, we propose an estimation method based on grouping the data. The proposed estimator is asymptotically unbiased in various situations under informative missingness. Several existing methods are reviewed and compared in simulation studies. We apply the methods to data from the Wisconsin Diabetes Registry Project, a longitudinal study tracking glycaemic control and acute and chronic complications from the diagnosis of type I diabetes.     

A Bayesian framework to account for uncertainty due to missing binary outcome data in pairwise meta‐analysis

Missing outcome data are a common threat to the validity of the results from randomised controlled trials (RCTs), which, if not analysed appropriately, can lead to misleading treatment effect estimates. Studies with missing outcome data also threaten the validity of any meta‐analysis that includes them. A conceptually simple Bayesian framework is proposed, to account for uncertainty due to missing binary outcome data in meta‐analysis. A pattern‐mixture model is fitted, which allows the incorporation of prior information on a parameter describing the missingness mechanism. We describe several alternative parameterisations, with the simplest being a prior on the probability of an event in the missing individuals. We describe a series of structural assumptions that can be made concerning the missingness parameters. We use some artificial data scenarios to demonstrate the ability of the model to produce a bias‐adjusted estimate of treatment effect that accounts for uncertainty. A meta‐analysis of haloperidol versus placebo for schizophrenia is used to illustrate the model. We end with a discussion of elicitation of priors, issues with poor reporting and potential extensions of the framework. Our framework allows one to make the best use of evidence produced from RCTs with missing outcome data in a meta‐analysis, accounts for any uncertainty induced by missing data and fits easily into a wider evidence synthesis framework for medical decision making. © 2015 The Authors. Statistics in MedicinePublished by John Wiley & Sons Ltd.     

Weighted Kaplan–Meier estimators for two‐stage treatment regimes

In two‐stage randomization designs, patients are randomized to one of the initial treatments, and at the end of the first stage, they are randomized to one of the second stage treatments depending on the outcome of the initial treatment. Statistical inference for survival data from these trials uses methods such as marginal mean models and weighted risk set estimates. In this article, we propose two forms of weighted Kaplan–Meier (WKM) estimators based on inverse‐probability weighting—one with fixed weights and the other with time‐dependent weights. We compare their properties with that of the standard Kaplan–Meier (SKM) estimator, marginal mean model‐based (MM) estimator and weighted risk set (WRS) estimator. Simulation study reveals that both forms of weighted Kaplan–Meier estimators are asymptotically unbiased, and provide coverage rates similar to that of MM and WRS estimators. The SKM estimator, however, is biased when the second randomization rates are not the same for the responders and non‐responders to initial treatment. The methods described are demonstrated by applying to a leukemia data set. Copyright © 2010 John Wiley & Sons, Ltd.     

Multiple imputation methods for bivariate outcomes in cluster randomised trials

Missing observations are common in cluster randomised trials. The problem is exacerbated when modelling bivariate outcomes jointly, as the proportion of complete cases is often considerably smaller than the proportion having either of the outcomes fully observed. Approaches taken to handling such missing data include the following: complete case analysis, single‐level multiple imputation that ignores the clustering, multiple imputation with a fixed effect for each cluster and multilevel multiple imputation. We contrasted the alternative approaches to handling missing data in a cost‐effectiveness analysis that uses data from a cluster randomised trial to evaluate an exercise intervention for care home residents. We then conducted a simulation study to assess the performance of these approaches on bivariate continuous outcomes, in terms of confidence interval coverage and empirical bias in the estimated treatment effects. Missing‐at‐random clustered data scenarios were simulated following a full‐factorial design. Across all the missing data mechanisms considered, the multiple imputation methods provided estimators with negligible bias, while complete case analysis resulted in biased treatment effect estimates in scenarios where the randomised treatment arm was associated with missingness. Confidence interval coverage was generally in excess of nominal levels (up to 99.8%) following fixed‐effects multiple imputation and too low following single‐level multiple imputation. Multilevel multiple imputation led to coverage levels of approximately 95% throughout. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Sensitivity analysis for the estimation of rates of change with non-ignorable drop-out: an application to a randomized clinical trial of the vitamin D3

The vitamin D(3) trial was a repeated measures randomized clinical trial for secondary hyperparathyroidism in haemodialysis patients where the efficacy of the vitamin D(3) infusions for suppressing the secretion of parathyroid hormone (PTH) was compared among four dose groups over 12 weeks. In this trial, patients terminated the study before the scheduled end of the study due to their elevated serum calcium (Ca) level, that is, the administration of the vitamin D(3) was expected to cause hypercalcaemia as an adverse event. In this setting of monotone missingness, there is a potential for bias in estimation of mean rates of decline in PTH for each treatment group using the standard methods such as the generalized estimating equations (GEE) which ignore the observed past Ca histories. We estimated the treatment-group-specific mean rates of decline in PTH by the inverse probability of censoring weighted (IPCW) methods which account for the observed past histories of time-dependent factors that are both a predictor of drop-out and are correlated with the outcomes. The IPCW estimator can be viewed as an extension of the GEE estimator that allows for the data to be MAR but not MCAR. With missing data, it is rarely appropriate to analyse the data solely under the assumption that the missing data process is ignorable, because the assumption of ignorable missingness cannot be guaranteed to hold and is untestable from the observed data. We proposed a sensitivity analysis that examines how inference about the IPCW estimates of the treatment-group-specific mean rates of decline in PTH changes as we vary the non-ignorable selection bias parameter over a range of plausible values.     

Non‐ignorable missingness in logistic regression

Nonresponses and missing data are common in observational studies. Ignoring or inadequately handling missing data may lead to biased parameter estimation, incorrect standard errors and, as a consequence, incorrect statistical inference and conclusions. We present a strategy for modelling non‐ignorable missingness where the probability of nonresponse depends on the outcome. Using a simple case of logistic regression, we quantify the bias in regression estimates and show the observed likelihood is non‐identifiable under non‐ignorable missing data mechanism. We then adopt a selection model factorisation of the joint distribution as the basis for a sensitivity analysis to study changes in estimated parameters and the robustness of study conclusions against different assumptions. A Bayesian framework for model estimation is used as it provides a flexible approach for incorporating different missing data assumptions and conducting sensitivity analysis. Using simulated data, we explore the performance of the Bayesian selection model in correcting for bias in a logistic regression. We then implement our strategy using survey data from the 45 and Up Study to investigate factors associated with worsening health from the baseline to follow‐up survey. Our findings have practical implications for the use of the 45 and Up Study data to answer important research questions relating to health and quality‐of‐life. Copyright © 2017 John Wiley & Sons, Ltd.     

ANALYSIS OF SEMI-PARAMETRIC REGRESSION MODELS WITH NON-IGNORABLE NON-RESPONSE

In this article we consider the problem of making inferences about the parameter β₀ indexing the conditional mean of an outcome given a vector of regressors when a subset of the variables (outcome or covariates) are missing for some study subjects and the probability of non-response depends upon both observed and unobserved data values, that is, non-response is non-ignorable. We propose a new class of inverse probability of censoring weighted estimators that are consistent and asymptotically normal (CAN) for estimating β₀ when the non-response probabilities can be parametrically modelled and a CAN estimator exists. The proposed estimators do not require full specification of the likelihood and their computation does not require numerical integration. We show that the asymptotic variance of the optimal estimator in our class attains the semi-parametric variance bound for the model. In some models, no CAN estimator of β₀ exists. We provide a general algorithm for determining when CAN estimators of β₀ exist. Our results follow after specializing a general representation described in the article for the efficient score and the influence function of regular, asymptotically linear estimators in an arbitrary semi-parametric model with non-ignorable non-response in which the probability of observing complete data is bounded away from zero and the non-response probabilities can be parametrically modelled. © 1997 by John Wiley & Sons, Ltd.     

Finite sample variance estimation for optimal dynamic treatment regimes of survival outcomes

Deriving valid confidence intervals for complex estimators is a challenging task in practice. Estimators of dynamic weighted survival modeling (DWSurv), a method to estimate an optimal dynamic treatment regime of censored outcomes, are asymptotically normal and consistent for their target parameters when at least a subset of the nuisance models is correctly specified. However, their behavior in finite samples and the impact of model misspecification on inferences remain unclear. In addition, the estimators' nonregularity may negatively affect the inferences under some specific data generating mechanisms. Our objective was to compare five methods, two asymptotic variance formulas (adjusting or not for the estimation of nuisance parameters) to three bootstrap approaches, to construct confidence intervals for the DWSurv parameters in finite samples. Via simulations, we considered practical scenarios, for example, when some nuisance models are misspecified or when nonregularity is problematic. We also compared the five methods in an application about the treatment of rheumatoid arthritis. We found that the bootstrap approaches performed consistently well at the cost of longer computational times. The asymptotic variance with adjustments generally yielded conservative confidence intervals. The asymptotic variance without adjustments yielded nominal coverages for large sample sizes. We recommend using the asymptotic variance with adjustments in small samples and the bootstrap if computationally feasible. Caution should be taken when nonregularity may be an issue.