Control‐based imputation for sensitivity analyses in informative censoring for recurrent event data

In clinical trials, missing data commonly arise through nonadherence to the randomized treatment or to study procedure. For trials in which recurrent event endpoints are of interests, conventional analyses using the proportional intensity model or the count model assume that the data are missing at random, which cannot be tested using the observed data alone. Thus, sensitivity analyses are recommended. We implement the control‐based multiple imputation as sensitivity analyses for the recurrent event data. We model the recurrent event using a piecewise exponential proportional intensity model with frailty and sample the parameters from the posterior distribution. We impute the number of events after dropped out and correct the variance estimation using a bootstrap procedure. We apply the method to an application of sitagliptin study.     

Multiple imputation methods for recurrent event data with missing event category

Frequently in clinical and epidemiologic studies, the event of interest is recurrent (i.e., can occur more than once per subject). When the events are not of the same type, an analysis which accounts for the fact that events fall into different categories will often be more informative. Often, however, although event times may always be known, information through which events are categorized may potentially be missing. Complete‐case methods (whose application may require, for example, that events be censored when their category cannot be determined) are valid only when event categories are missing completely at random. This assumption is rather restrictive. The authors propose two multiple imputation methods for analyzing multiple‐category recurrent event data under the proportional means/rates model. The use of a proper or improper imputation technique distinguishes the two approaches. Both methods lead to consistent estimation of regression parameters even when the missingness of event categories depends on covariates. The authors derive the asymptotic properties of the estimators and examine their behaviour in finite samples through simulation. They illustrate their approach using data from an international study on dialysis.     

Comparison of alternative imputation methods for ordinal data

In this article, we compare alternative missing imputation methods in the presence of ordinal data, in the framework of CUB (Combination of Uniform and (shifted) Binomial random variable) models. Various imputation methods are considered, as are univariate and multivariate approaches. The first step consists of running a simulation study designed by varying the parameters of the CUB model, to consider and compare CUB models as well as other methods of missing imputation. We use real datasets on which to base the comparison between our approach and some general methods of missing imputation for various missing data mechanisms.     

A comparison of various software tools for dealing with missing data via imputation

In real-life situations, we often encounter data sets containing missing observations. Statistical methods that address missingness have been extensively studied in recent years. One of the more popular approaches involves imputation of the missing values prior to the analysis, thereby rendering the data complete. Imputation broadly encompasses an entire scope of techniques that have been developed to make inferences about incomplete data, ranging from very simple strategies (e.g. mean imputation) to more advanced approaches that require estimation, for instance, of posterior distributions using Markov chain Monte Carlo methods. Additional complexity arises when the number of missingness patterns increases and/or when both categorical and continuous random variables are involved. Implementation of routines, procedures, or packages capable of generating imputations for incomplete data are now widely available. We review some of these in the context of a motivating example, as well as in a simulation study, under two missingness mechanisms (missing at random and missing not at random). Thus far, evaluation of existing implementations have frequently centred on the resulting parameter estimates of the prescribed model of interest after imputing the missing data. In some situations, however, interest may very well be on the quality of the imputed values at the level of the individual – an issue that has received relatively little attention. In this paper, we focus on the latter to provide further insight about the performance of the different routines, procedures, and packages in this respect.     

Mann–Whitney test with empirical likelihood methods for pretest–posttest studies

Pretest–posttest studies are an important and popular method for assessing the effectiveness of a treatment or an intervention in many scientific fields. While the treatment effect, measured as the difference between the two mean responses, is of primary interest, testing the difference of the two distribution functions for the treatment and the control groups is also an important problem. The Mann–Whitney test has been a standard tool for testing the difference of distribution functions with two independent samples. We develop empirical likelihood-based (EL) methods for the Mann–Whitney test to incorporate the two unique features of pretest–posttest studies: (i) the availability of baseline information for both groups; and (ii) the structure of the data with missing by design. Our proposed methods combine the standard Mann–Whitney test with the EL method of Huang, Qin and Follmann [(2008), ‘Empirical Likelihood-Based Estimation of the Treatment Effect in a Pretest–Posttest Study’, Journal of the American Statistical Association, 103(483), 1270–1280], the imputation-based empirical likelihood method of Chen, Wu and Thompson [(2015), ‘An Imputation-Based Empirical Likelihood Approach to Pretest–Posttest Studies’, The Canadian Journal of Statistics accepted for publication], and the jackknife empirical likelihood method of Jing, Yuan and Zhou [(2009), ‘Jackknife Empirical Likelihood’, Journal of the American Statistical Association, 104, 1224–1232]. Theoretical results are presented and finite sample performances of proposed methods are evaluated through simulation studies.     

Some efficient random imputation methods

Imputation methods that assign a selection of respondents’ values for missing i tern nonresponses give rise to an addd,tional source of sampling variation, which we term imputation varLance , We examine the effect of imputation variance on the precision of the mean, and propose four procedures for sampling the rEespondents that reduce this additional variance. Two of the procedures employ improved sample designs through selection of respc,ndents by sampling without replacement and by stratified sampl;lng. The other two increase the sample base by the use of multiple imputations.     

Applying the rescaling bootstrap under imputation: a simulation study

Resampling methods are a common measure to estimate the variance of a statistic of interest when data consist of nonresponse and imputation is used as compensation. Applying resampling methods usually means that subsamples are drawn from the original sample and that variance estimates are computed based on point estimators of several subsamples. However, newer resampling methods such as the rescaling bootstrap of Chipperfield and Preston [Efficient bootstrap for business surveys. Surv Methodol. 2007;33:167–172] include all elements of the original sample in the computation of its point estimator. Thus, procedures to consider imputation in resampling methods cannot be applied in the ordinary way. For such methods, modifications are necessary. This paper presents an approach applying newer resampling methods for imputed data. The Monte Carlo simulation study conducted in the paper shows that the proposed approach leads to reliable variance estimates in contrast to other modifications.     

Multiple imputation for gamma outcome variable using generalized linear model

We used a proper multiple imputation (MI) through Gibbs sampling approach to impute missing values of a gamma distributed outcome variable which were missing at random, using generalized linear model (GLM) with identity link function. The missing values of the outcome variable were multiply imputed using GLM and then the complete data sets obtained after MI were analysed through GLM again for the estimation purpose. We examined the performance of the proposed technique through a simulation study with the data sets having four moderate and large proportions of missing values, 10%, 20%, 30% and 50%. We also applied this technique on a real life data and compared the results with those obtained by applying GLM only on observed cases. The results showed that the proposed technique gave better results for moderate proportions of missing values.     

Using multiple imputation to estimate cumulative distribution functions in longitudinal data analysis with data missing at random

In longitudinal clinical studies, after randomization at baseline, subjects are followed for a period of time for development of symptoms. The interested inference could be the mean change from baseline to a particular visit in some lab values, the proportion of responders to some threshold category at a particular visit post baseline, or the time to some important event. However, in some applications, the interest may be in estimating the cumulative distribution function (CDF) at a fixed time point post baseline. When the data are fully observed, the CDF can be estimated by the empirical CDF. When patients discontinue prematurely during the course of the study, the empirical CDF cannot be directly used. In this paper, we use multiple imputation as a way to estimate the CDF in longitudinal studies when data are missing at random. The validity of the method is assessed on the basis of the bias and the Kolmogorov–Smirnov distance. The results suggest that multiple imputation yields less bias and less variability than the often used last observation carried forward method. Copyright © 2013 John Wiley & Sons, Ltd.     

A new method of imputation in survey sampling

In this paper, an alternative estimator of population mean in the presence of non-response has been suggested which comes in the form of Walsh's estimator. The estimator of mean obtained from the proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean-squared error (MSE) of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optimum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. A suggestion to form ‘warm deck’ method of imputation has been suggested. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. The work has been extended to the case of multi-auxiliary information to be used for imputation. Numerical illustrations are also provided.     

A multiple imputation approach to nonlinear mixed-effects models with covariate measurement errors and missing values

In longitudinal studies, nonlinear mixed-effects models have been widely applied to describe the intra- and the inter-subject variations in data. The inter-subject variation usually receives great attention and it may be partially explained by time-dependent covariates. However, some covariates may be measured with substantial errors and may contain missing values. We proposed a multiple imputation method, implemented by a Markov Chain Monte-Carlo method along with Gibbs sampler, to address the covariate measurement errors and missing data in nonlinear mixed-effects models. The multiple imputation method is illustrated in a real data example. Simulation studies show that the multiple imputation method outperforms the commonly used naive methods.     

Sensitivity to imputation models and assumptions in receiver operating characteristic analysis with incomplete data

Modern statistical methods using incomplete data have been increasingly applied in a wide variety of substantive problems. Similarly, receiver operating characteristic (ROC) analysis, a method used in evaluating diagnostic tests or biomarkers in medical research, has also been increasingly popular problem in both its development and application. While missing-data methods have been applied in ROC analysis, the impact of model mis-specification and/or assumptions (e.g. missing at random) underlying the missing data has not been thoroughly studied. In this work, we study the performance of multiple imputation (MI) inference in ROC analysis. Particularly, we investigate parametric and non-parametric techniques for MI inference under common missingness mechanisms. Depending on the coherency of the imputation model with the underlying data generation mechanism, our results show that MI generally leads to well-calibrated inferences under ignorable missingness mechanisms.     

Multiple imputation for ordinal longitudinal data with monotone missing data patterns

Missing data often complicate the analysis of scientific data. Multiple imputation is a general purpose technique for analysis of datasets with missing values. The approach is applicable to a variety of missing data patterns but often complicated by some restrictions like the type of variables to be imputed and the mechanism underlying the missing data. In this paper, the authors compare the performance of two multiple imputation methods, namely fully conditional specification and multivariate normal imputation in the presence of ordinal outcomes with monotone missing data patterns. Through a simulation study and an empirical example, the authors show that the two methods are indeed comparable meaning any of the two may be used when faced with scenarios, at least, as the ones presented here.     

Impact of the non-distinctness and non-ignorability on the inference by multiple imputation in multivariate multilevel data: a simulation assessment

Multiple imputation (MI) is an increasingly popular method for analysing incomplete multivariate data sets. One of the most crucial assumptions of this method relates to mechanism leading to missing data. Distinctness is typically assumed, which indicates a complete independence of mechanisms underlying missingness and data generation. In addition, missing at random or missing completely at random is assumed, which explicitly states under which conditions missingness is independent of observed data. Despite common use of MI under these assumptions, plausibility and sensitivity to these fundamental assumptions have not been well-investigated. In this work, we investigate the impact of non-distinctness and non-ignorability. In particular, non-ignorability is due to unobservable cluster-specific effects (e.g. random-effects). Through a comprehensive simulation study, we show that MI inferences suggest that nonignoriability due to non-distinctness do not immediately imply dismal performance while non-ignorability due to missing not at random leads to quite subpar performance.     

Sequential imputation for models with latent variables assuming latent ignorability

Models that involve an outcome variable, covariates, and latent variables are frequently the target for estimation and inference. The presence of missing covariate or outcome data presents a challenge, particularly when missingness depends on the latent variables. This missingness mechanism is called latent ignorable or latent missing at random and is a generalisation of missing at random. Several authors have previously proposed approaches for handling latent ignorable missingness, but these methods rely on prior specification of the joint distribution for the complete data. In practice, specifying the joint distribution can be difficult and/or restrictive. We develop a novel sequential imputation procedure for imputing covariate and outcome data for models with latent variables under latent ignorable missingness. The proposed method does not require a joint model; rather, we use results under a joint model to inform imputation with less restrictive modelling assumptions. We discuss identifiability and convergence‐related issues, and simulation results are presented in several modelling settings. The method is motivated and illustrated by a study of head and neck cancer recurrence. Imputing missing data for models with latent variables under latent‐dependent missingness without specifying a full joint model.     

Interval estimation of the joint survival function for successive duration times under left truncation and right censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, we propose two inverse-probability-weighted (IPW) estimators for the estimation of the joint survival function of two successive duration times. One of them is similar to the estimator proposed by Chang and Tzeng [Nonparametric estimation of sojourn time distributions for truncated serial event data – a weight adjusted approach, Lifetime Data Anal. 12 (2006), pp. 53–67]. The other is the extension of the nonparametric estimator proposed by Wang and Wells [Nonparametric estimation of successive duration times under dependent censoring, Biometrika 85 (1998), pp. 561–572]. The weak convergence of both estimators are established. Furthermore, the delete-one jackknife and simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to compare the two IPW approaches.     

Comparisons of imputation methods with application to assess factors associated with self efficacy of physical activity in breast cancer survivors

ABSTRACT

Missing data are commonly encountered in self-reported measurements and questionnaires. It is crucial to treat missing values using appropriate method to avoid bias and reduction of power. Various types of imputation methods exist, but it is not always clear which method is preferred for imputation of data with non-normal variables. In this paper, we compared four imputation methods: mean imputation, quantile imputation, multiple imputation, and quantile regression multiple imputation (QRMI), using both simulated and real data investigating factors affecting self-efficacy in breast cancer survivors. The results displayed an advantage of using multiple imputation, especially QRMI when data are not normal.     

Hybrid pairwise‐likelihood estimation methods for incomplete longitudinal binary data

In longitudinal data, missing observations occur commonly with incomplete responses and covariates. Missing data can have a ‘missing not at random’ mechanism, a non‐monotone missing pattern, and moreover response and covariates can be missing not simultaneously. To avoid complexities in both modelling and computation, a two‐stage estimation method and a pairwise‐likelihood method are proposed. The two‐stage estimation method enjoys simplicities in computation, but incurs more severe efficiency loss. On the other hand, the pairwise approach leads to estimators with better efficiency, but can be cumbersome in computation. In this paper, we develop a compromise method using a hybrid pairwise‐likelihood framework. Our proposed approach has better efficiency than the two‐stage method, but its computational cost is still reasonable compared to the pairwise approach. The performance of the methods is evaluated empirically by means of simulation studies. Our methods are used to analyse longitudinal data obtained from the National Population Health Study.     

Inferences on missing information under multiple imputation and two-stage multiple imputation

In the presence of missing values, researchers may be interested in the rates of missing information. The rates of missing information are (a) important for assessing how the missing information contributes to inferential uncertainty about, Q, the population quantity of interest, (b) are an important component in the decision of the number of imputations, and (c) can be used to test model uncertainty and model fitting. In this article I will derive the asymptotic distribution of the rates of missing information in two scenarios: the conventional multiple imputation (MI), and the two-stage MI. Numerically I will show that the proposed asymptotic distribution agrees with the simulated one. I will also suggest the number of imputations needed to obtain reliable missing information rate estimates for each method, based on the asymptotic distribution.