Pattern‐mixture models for analyzing normal outcome data with proxy respondents

Studies of older adults often involve interview questions regarding subjective constructs such as perceived disability. In some studies, when subjects are unable (e.g. due to cognitive impairment) or unwilling to respond to these questions, proxies (e.g. relatives or other care givers) are recruited to provide responses in place of the subject. Proxies are usually not approached to respond on behalf of subjects who respond for themselves; thus, for each subject, data from only one of the subject or proxy are available. Typically, proxy responses are simply substituted for missing subject responses, and standard complete‐data analyses are performed. However, this approach may introduce measurement error and produce biased parameter estimates. In this paper, we propose using pattern‐mixture models that relate non‐identifiable parameters to identifiable parameters to analyze data with proxy respondents. We posit three interpretable pattern‐mixture restrictions to be used with proxy data, and we propose estimation procedures using maximum likelihood and multiple imputation. The methods are applied to a cohort of elderly hip‐fracture patients. Copyright © 2010 John Wiley & Sons, Ltd.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

Statistical analysis of longitudinal quality of life data with missing measurements

The statistical analysis of longitudinal quality of life data in the presence of missing data is discussed. In cancer trials missing data are generated due to the fact that patients die, drop out, or are censored. These missing data are problematic in the monitoring of the quality of life during the trial. However, by means of assuming that the cause of the missing data lies in the observed history of the patients and not in their unobserved future, the missing data are ignorable. Consequently, all available data can be used to estimate quality of life change patterns with time. The computations that are required are illustrated with real quality of life data and three commonly used computer packages for statistical analysis.Paper read at meeting of the EORTC Quality of Life Study Group, November 1991, Leicester, UK.This research was supported by a grant from the Dutch Science Foundation (NWO).     

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.     

Meta‐analysis with missing study‐level sample variance data

We consider a study‐level meta‐analysis with a normally distributed outcome variable and possibly unequal study‐level variances, where the object of inference is the difference in means between a treatment and control group. A common complication in such an analysis is missing sample variances for some studies. A frequently used approach is to impute the weighted (by sample size) mean of the observed variances (mean imputation). Another approach is to include only those studies with variances reported (complete case analysis). Both mean imputation and complete case analysis are only valid under the missing‐completely‐at‐random assumption, and even then the inverse variance weights produced are not necessarily optimal. We propose a multiple imputation method employing gamma meta‐regression to impute the missing sample variances. Our method takes advantage of study‐level covariates that may be used to provide information about the missing data. Through simulation studies, we show that multiple imputation, when the imputation model is correctly specified, is superior to competing methods in terms of confidence interval coverage probability and type I error probability when testing a specified group difference. Finally, we describe a similar approach to handling missing variances in cross‐over studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Intention‐to‐treat analysis with treatment discontinuation and missing data in clinical trials

Motivated by a recent National Research Council study, we discuss three aspects of the analysis of clinical trials when participants prematurely discontinue treatments. First, we distinguish treatment discontinuation from missing outcome data. Data collection is often stopped after treatment discontinuation, but outcome data could be recorded on individuals after they discontinue treatment, as the National Research Council study recommends. Conversely, outcome data may be missing for individuals who do not discontinue treatment, as when there is loss to follow up or missed clinic visits. Missing outcome data is a standard missing data problem, but treatment discontinuation is better viewed as a form of noncompliance and treated using ideas from the causal literature on noncompliance. Second, the standard intention to treat estimand, the average effect of randomization to treatment, is compared with three alternative estimands for the intention to treat population: the average effect when individuals continue on the assigned treatment after discontinuation, the average effect when individuals take a control treatment after treatment discontinuation, and a summary measure of the effect of treatment prior to discontinuation. We argue that the latter choice of estimand has advantages and should receive more consideration. Third, we consider when follow‐up measures after discontinuation are needed for valid measures of treatment effects. The answer depends on the choice of primary estimand and the plausibility of assumptions needed to address the missing data. Ideas are motivated and illustrated by a reanalysis of a past study of inhaled insulin treatments for diabetes, sponsored by Eli Lilly. Copyright © 2014 John Wiley & Sons, Ltd.     

EMLasso: logistic lasso with missing data

In clinical settings, missing data in the covariates occur frequently. For example, some markers are expensive or hard to measure. When this sort of data is used for model selection, the missingness is often resolved through a complete case analysis or a form of single imputation. An alternative sometimes comes in the form of leaving the most damaged covariates out. All these strategies jeopardise the goal of model selection. In earlier work, we have applied the logistic Lasso in combination with multiple imputation to obtain results in such settings, but we only provided heuristic arguments to advocate the method. In this paper, we propose an improved method that builds on firm statistical arguments and that is developed along the lines of the stochastic expectation–maximisation algorithm. We show that our method can be used to handle missing data in both categorical and continuous predictors, as well as in a nonpenalised regression. We demonstrate the method by applying it to data of 273 lung cancer patients. The objective is to select a model for the prediction of acute dysphagia, starting from a large set of potential predictors, including clinical and treatment covariates as well as a set of single‐nucleotide polymorphisms. Copyright © 2013 John Wiley & Sons, Ltd.     

Sensitivity analysis for a partially missing binary outcome in a two‐arm randomized clinical trial

Although recent guidelines for dealing with missing data emphasize the need for sensitivity analyses, and such analyses have a long history in statistics, universal recommendations for conducting and displaying these analyses are scarce. We propose graphical displays that help formalize and visualize the results of sensitivity analyses, building upon the idea of ‘tipping‐point’ analysis for randomized experiments with a binary outcome and a dichotomous treatment. The resulting ‘enhanced tipping‐point displays’ are convenient summaries of conclusions obtained from making different modeling assumptions about missingness mechanisms. The primary goal of the displays is to make formal sensitivity analysesmore comprehensible to practitioners, thereby helping them assess the robustness of the experiment's conclusions to plausible missingness mechanisms. We also present a recent example of these enhanced displays in amedical device clinical trial that helped lead to FDA approval. Copyright © 2014 John Wiley & Sons, Ltd.     

A U‐statistics‐based approach for modeling Cronbach coefficient alpha within a longitudinal data setting

Cronbach coefficient alpha (CCA) is a classic measure of item internal consistency of an instrument and is used in a wide range of behavioral, biomedical, psychosocial, and health‐care‐related research. Methods are available for making inference about one CCA or multiple CCAs from correlated outcomes. However, none of the existing approaches effectively address missing data. As longitudinal study designs become increasingly popular and complex in modern‐day clinical studies, missing data have become a serious issue, and the lack of methods to systematically address this problem has hampered the progress of research in the aforementioned fields. In this paper, we develop a novel approach to tackle the complexities involved in addressing missing data (at the instrument level due to subject dropout) within a longitudinal data setting. The approach is illustrated with both clinical and simulated data. Copyright © 2010 John Wiley & Sons, Ltd.     

Joint modeling of missing data due to non‐participation and death in longitudinal aging studies

Specific age‐related hypotheses are tested in population‐based longitudinal studies. At specific time intervals, both the outcomes of interest and the time‐varying covariates are measured. When participants are approached for follow‐up, some participants do not provide data. Investigations may show that many have died before the time of follow‐up whereas others refused to participate. Some of these non‐participants do not provide data at later follow‐ups. Few statistical methods for missing data distinguish between ‘non‐participation’ and ‘death’ among study participants. The augmented inverse probability‐weighted estimators are most commonly used in marginal structure models when data are missing at random. Treating non‐participation and death as the same, however, may lead to biased estimates and invalid inferences. To overcome this limitation, a multiple inverse probability‐weighted approach is presented to account for two types of missing data, non‐participation and death, when using a marginal mean model. Under certain conditions, the multiple weighted estimators are consistent and asymptotically normal. Simulation studies will be used to study the finite sample efficiency of the multiple weighted estimators. The proposed method will be applied to study the risk factors associated with the cognitive decline among the aging adults, using data from the Chicago Health and Aging Project (CHAP). Copyright © 2010 John Wiley & Sons, Ltd.     

Multiple imputation for IPD meta‐analysis: allowing for heterogeneity and studies with missing covariates

Recently, multiple imputation has been proposed as a tool for individual patient data meta‐analysis with sporadically missing observations, and it has been suggested that within‐study imputation is usually preferable. However, such within study imputation cannot handle variables that are completely missing within studies. Further, if some of the contributing studies are relatively small, it may be appropriate to share information across studies when imputing. In this paper, we develop and evaluate a joint modelling approach to multiple imputation of individual patient data in meta‐analysis, with an across‐study probability distribution for the study specific covariance matrices. This retains the flexibility to allow for between‐study heterogeneity when imputing while allowing (i) sharing information on the covariance matrix across studies when this is appropriate, and (ii) imputing variables that are wholly missing from studies. Simulation results show both equivalent performance to the within‐study imputation approach where this is valid, and good results in more general, practically relevant, scenarios with studies of very different sizes, non‐negligible between‐study heterogeneity and wholly missing variables. We illustrate our approach using data from an individual patient data meta‐analysis of hypertension trials. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Reference‐based multiple imputation for missing data sensitivity analyses in trial‐based cost‐effectiveness analysis

Missing data are a common issue in cost‐effectiveness analysis (CEA) alongside randomised trials and are often addressed assuming the data are ‘missing at random’. However, this assumption is often questionable, and sensitivity analyses are required to assess the implications of departures from missing at random. Reference‐based multiple imputation provides an attractive approach for conducting such sensitivity analyses, because missing data assumptions are framed in an intuitive way by making reference to other trial arms. For example, a plausible not at random mechanism in a placebo‐controlled trial would be to assume that participants in the experimental arm who dropped out stop taking their treatment and have similar outcomes to those in the placebo arm. Drawing on the increasing use of this approach in other areas, this paper aims to extend and illustrate the reference‐based multiple imputation approach in CEA. It introduces the principles of reference‐based imputation and proposes an extension to the CEA context. The method is illustrated in the CEA of the CoBalT trial evaluating cognitive behavioural therapy for treatment‐resistant depression. Stata code is provided. We find that reference‐based multiple imputation provides a relevant and accessible framework for assessing the robustness of CEA conclusions to different missing data assumptions.     

Imputation of systematically missing predictors in an individual participant data meta‐analysis: a generalized approach using MICE

Individual participant data meta‐analyses (IPD‐MA) are increasingly used for developing and validating multivariable (diagnostic or prognostic) risk prediction models. Unfortunately, some predictors or even outcomes may not have been measured in each study and are thus systematically missing in some individual studies of the IPD‐MA. As a consequence, it is no longer possible to evaluate between‐study heterogeneity and to estimate study‐specific predictor effects, or to include all individual studies, which severely hampers the development and validation of prediction models. Here, we describe a novel approach for imputing systematically missing data and adopt a generalized linear mixed model to allow for between‐study heterogeneity. This approach can be viewed as an extension of Resche‐Rigon's method (Stat Med 2013), relaxing their assumptions regarding variance components and allowing imputation of linear and nonlinear predictors. We illustrate our approach using a case study with IPD‐MA of 13 studies to develop and validate a diagnostic prediction model for the presence of deep venous thrombosis. We compare the results after applying four methods for dealing with systematically missing predictors in one or more individual studies: complete case analysis where studies with systematically missing predictors are removed, traditional multiple imputation ignoring heterogeneity across studies, stratified multiple imputation accounting for heterogeneity in predictor prevalence, and multilevel multiple imputation (MLMI) fully accounting for between‐study heterogeneity. We conclude that MLMI may substantially improve the estimation of between‐study heterogeneity parameters and allow for imputation of systematically missing predictors in IPD‐MA aimed at the development and validation of prediction models. Copyright © 2015 John Wiley & Sons, Ltd.     

Correlation analysis for longitudinal data: applications to HIV and psychosocial research

Correlation analysis is widely used in biomedical and psychosocial research for assessing rater reliability, precision of diagnosis and accuracy of proxy outcomes. The popularity of longitudinal study designs has propelled the proliferation in recent years of new methods for longitudinal and other multi-level clustered data designs, such as the mixed-effect models and generalized estimating equations. Despite these advances, research and methodological development on addressing missing data for correlation analysis is woefully lacking. In this paper, we consider non-parametric inference for the product-moment correlation within a longitudinal data setting and address missing data under both the missing completely at random and missing at random assumptions. We illustrate the approach with real study data in mental health and HIV prevention research.     

Bias and efficiency of multiple imputation compared with complete‐case analysis for missing covariate values

When missing data occur in one or more covariates in a regression model, multiple imputation (MI) is widely advocated as an improvement over complete‐case analysis (CC). We use theoretical arguments and simulation studies to compare these methods with MI implemented under a missing at random assumption. When data are missing completely at random, both methods have negligible bias, and MI is more efficient than CC across a wide range of scenarios. For other missing data mechanisms, bias arises in one or both methods. In our simulation setting, CC is biased towards the null when data are missing at random. However, when missingness is independent of the outcome given the covariates, CC has negligible bias and MI is biased away from the null. With more general missing data mechanisms, bias tends to be smaller for MI than for CC. Since MI is not always better than CC for missing covariate problems, the choice of method should take into account what is known about the missing data mechanism in a particular substantive application. Importantly, the choice of method should not be based on comparison of standard errors. We propose new ways to understand empirical differences between MI and CC, which may provide insights into the appropriateness of the assumptions underlying each method, and we propose a new index for assessing the likely gain in precision from MI: the fraction of incomplete cases among the observed values of a covariate (FICO). Copyright © 2010 John Wiley & Sons, Ltd.     

Joint modeling of longitudinal and survival data with missing and left‐censored time‐varying covariates

We propose a joint model for longitudinal and survival data with time‐varying covariates subject to detection limits and intermittent missingness at random. The model is motivated by data from the Multicenter AIDS Cohort Study (MACS), in which HIV+ subjects have viral load and CD4 cell count measured at repeated visits along with survival data. We model the longitudinal component using a normal linear mixed model, modeling the trajectory of CD4 cell count by regressing on viral load, and other covariates. The viral load data are subject to both left censoring because of detection limits (17%) and intermittent missingness (27%). The survival component of the joint model is a Cox model with time‐dependent covariates for death because of AIDS. The longitudinal and survival models are linked using the trajectory function of the linear mixed model. A Bayesian analysis is conducted on the MACS data using the proposed joint model. The proposed method is shown to improve the precision of estimates when compared with alternative methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Multivariate longitudinal data analysis with censored and intermittent missing responses

The multivariate linear mixed model (MLMM) has emerged as an important analytical tool for longitudinal data with multiple outcomes. However, the analysis of multivariate longitudinal data could be complicated by the presence of censored measurements because of a detection limit of the assay in combination with unavoidable missing values arising when subjects miss some of their scheduled visits intermittently. This paper presents a generalization of the MLMM approach, called the MLMM‐CM, for a joint analysis of the multivariate longitudinal data with censored and intermittent missing responses. A computationally feasible expectation maximization–based procedure is developed to carry out maximum likelihood estimation within the MLMM‐CM framework. Moreover, the asymptotic standard errors of fixed effects are explicitly obtained via the information‐based method. We illustrate our methodology by using simulated data and a case study from an AIDS clinical trial. Experimental results reveal that the proposed method is able to provide more satisfactory performance as compared with the traditional MLMM approach.     

Multiple imputation for handling systematically missing confounders in meta‐analysis of individual participant data

A variable is ‘systematically missing’ if it is missing for all individuals within particular studies in an individual participant data meta‐analysis. When a systematically missing variable is a potential confounder in observational epidemiology, standard methods either fail to adjust the exposure–disease association for the potential confounder or exclude studies where it is missing. We propose a new approach to adjust for systematically missing confounders based on multiple imputation by chained equations. Systematically missing data are imputed via multilevel regression models that allow for heterogeneity between studies. A simulation study compares various choices of imputation model. An illustration is given using data from eight studies estimating the association between carotid intima media thickness and subsequent risk of cardiovascular events. Results are compared with standard methods and also with an extension of a published method that exploits the relationship between fully adjusted and partially adjusted estimated effects through a multivariate random effects meta‐analysis model. We conclude that multiple imputation provides a practicable approach that can handle arbitrary patterns of systematic missingness. Bias is reduced by including sufficient between‐study random effects in the imputation model. Copyright © 2013 John Wiley & Sons, Ltd.