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.     

Inverse Probability Weighting with Missing Predictors of Treatment Assignment or Missingness

Inverse probability weighting (IPW) can deal with confounding in non randomized studies. The inverse weights are probabilities of treatment assignment (propensity scores), estimated by regressing assignment on predictors. Problems arise if predictors can be missing. Solutions previously proposed include assuming assignment depends only on observed predictors and multiple imputation (MI) of missing predictors. For the MI approach, it was recommended that missingness indicators be used with the other predictors. We determine when the two MI approaches, (with/without missingness indicators) yield consistent estimators and compare their efficiencies.We find that, although including indicators can reduce bias when predictors are missing not at random, it can induce bias when they are missing at random. We propose a consistent variance estimator and investigate performance of the simpler Rubin’s Rules variance estimator. In simulations we find both estimators perform well. IPW is also used to correct bias when an analysis model is fitted to incomplete data by restricting to complete cases. Here, weights are inverse probabilities of being a complete case. We explain how the same MI methods can be used in this situation to deal with missing predictors in the weight model, and illustrate this approach using data from the National Child Development Survey.     

Application of sensitivity analysis to incomplete longitudinal CD4 count data

In this paper, we investigate the effect of tuberculosis pericarditis (TBP) treatment on CD4 count changes over time and draw inferences in the presence of missing data. We accounted for missing data and conducted sensitivity analyses to assess whether inferences under missing at random (MAR) assumption are sensitive to not missing at random (NMAR) assumptions using the selection model (SeM) framework. We conducted sensitivity analysis using the local influence approach and stress-testing analysis. Our analyses showed that the inferences from the MAR are robust to the NMAR assumption and influential subjects do not overturn the study conclusions about treatment effects and the dropout mechanism. Therefore, the missing CD4 count measurements are likely to be MAR. The results also revealed that TBP treatment does not interact with HIV/AIDS treatment and that TBP treatment has no significant effect on CD4 count changes over time. Although the methods considered were applied to data in the IMPI trial setting, the methods can also be applied to clinical trials with similar settings.     

Data-driven sensitivity analysis to detect missing data mechanism with applications to structural equation modelling

Missing data are a common problem in almost all areas of empirical research. Ignoring the missing data mechanism, especially when data are missing not at random (MNAR), can result in biased and/or inefficient inference. Because MNAR mechanism is not verifiable based on the observed data, sensitivity analysis is often used to assess it. Current sensitivity analysis methods primarily assume a model for the response mechanism in conjunction with a measurement model and examine sensitivity to missing data mechanism via the parameters of the response model. Recently, Jamshidian and Mata (Post-modelling sensitivity analysis to detect the effect of missing data mechanism, Multivariate Behav. Res. 43 (2008), pp. 432–452) introduced a new method of sensitivity analysis that does not require the difficult task of modelling the missing data mechanism. In this method, a single measurement model is fitted to all of the data and to a sub-sample of the data. Discrepancy in the parameter estimates obtained from the the two data sets is used as a measure of sensitivity to missing data mechanism. Jamshidian and Mata describe their method mainly in the context of detecting data that are missing completely at random (MCAR). They used a bootstrap type method, that relies on heuristic input from the researcher, to test for the discrepancy of the parameter estimates. Instead of using bootstrap, the current article obtains confidence interval for parameter differences on two samples based on an asymptotic approximation. Because it does not use bootstrap, the developed procedure avoids likely convergence problems with the bootstrap methods. It does not require heuristic input from the researcher and can be readily implemented in statistical software. The article also discusses methods of obtaining sub-samples that may be used to test missing at random in addition to MCAR. An application of the developed procedure to a real data set, from the first wave of an ongoing longitudinal study on aging, is presented. Simulation studies are performed as well, using two methods of missing data generation, which show promise for the proposed sensitivity method. One method of missing data generation is also new and interesting in its own right.     

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.     

Semiparametric estimation for weighted average derivatives with responses missing at random

When responses are missing at random, we propose a semiparametric direct estimator for the missing probability and density-weighted average derivatives of a general nonparametric multiple regression function. An estimator for the normalized version of the weighted average derivatives is constructed as well using instrumental variables regression. The proposed estimators are computationally simple and asymptotically normal, and provide a solution to the problem of estimating index coefficients of single-index models with responses missing at random. The developed theory generalizes the method of the density-weighted average derivatives estimation of Powell et al. (1989) for the non-missing data case. Monte Carlo simulation studies are conducted to study the performance of the methods.     

A likelihood-based approach for multivariate one-sided tests with missing data

Inequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Comparing diagnostic tests with missing data

When missing data occur in studies designed to compare the accuracy of diagnostic tests, a common, though naive, practice is to base the comparison of sensitivity, specificity, as well as of positive and negative predictive values on some subset of the data that fits into methods implemented in standard statistical packages. Such methods are usually valid only under the strong missing completely at random (MCAR) assumption and may generate biased and less precise estimates. We review some models that use the dependence structure of the completely observed cases to incorporate the information of the partially categorized observations into the analysis and show how they may be fitted via a two-stage hybrid process involving maximum likelihood in the first stage and weighted least squares in the second. We indicate how computational subroutines written in R may be used to fit the proposed models and illustrate the different analysis strategies with observational data collected to compare the accuracy of three distinct non-invasive diagnostic methods for endometriosis. The results indicate that even when the MCAR assumption is plausible, the naive partial analyses should be avoided.     

A unified empirical likelihood approach for testing MCAR and subsequent estimation

For an estimation with missing data, a crucial step is to determine if the data are missing completely at random (MCAR), in which case a complete‐case analysis would suffice. Most existing tests for MCAR do not provide a method for a subsequent estimation once the MCAR is rejected. In the setting of estimating means, we propose a unified approach for testing MCAR and the subsequent estimation. Upon rejecting MCAR, the same set of weights used for testing can then be used for estimation. The resulting estimators are consistent if the missingness of each response variable depends only on a set of fully observed auxiliary variables and the true outcome regression model is among the user‐specified functions for deriving the weights. The proposed method is based on the calibration idea from survey sampling literature and the empirical likelihood theory.     

Comparison Between Two Partial Likelihood Approaches for the Competing Risks Model with Missing Cause of Failure

In many clinical studies where time to failure is of primary interest, patients may fail or die from one of many causes where failure time can be right censored. In some circumstances, it might also be the case that patients are known to die but the cause of death information is not available for some patients. Under the assumption that cause of death is missing at random, we compare the Goetghebeur and Ryan (1995, Biometrika, 82, 821–833) partial likelihood approach with the Dewanji (1992, Biometrika, 79, 855–857)partial likelihood approach. We show that the estimator for the regression coefficients based on the Dewanji partial likelihood is not only consistent and asymptotically normal, but also semiparametric efficient. While the Goetghebeur and Ryan estimator is more robust than the Dewanji partial likelihood estimator against misspecification of proportional baseline hazards, the Dewanji partial likelihood estimator allows the probability of missing cause of failure to depend on covariate information without the need to model the missingness mechanism. Tests for proportional baseline hazards are also suggested and a robust variance estimator is derived.     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

Logistic regression analysis of randomized response data with missing covariates

Randomized response is an interview technique designed to eliminate response bias when sensitive questions are asked. In this paper, we present a logistic regression model on randomized response data when the covariates on some subjects are missing at random. In particular, we propose Horvitz and Thompson (1952)-type weighted estimators by using different estimates of the selection probabilities. We present large sample theory for the proposed estimators and show that they are more efficient than the estimator using the true selection probabilities. Simulation results support theoretical analysis. We also illustrate the approach using data from a survey of cable TV.     

Joint generalized estimating equations for multivariate longitudinal binary outcomes with missing data: an application to acquired immune deficiency syndrome data

Summary.  In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to women who are infected with the human immunodeficiency virus, instead of a single outcome variable, there are multiple binary outcomes (e.g. abnormal heart rate, abnormal blood pressure and abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated by using generalized estimating equations (GEEs), and consistent estimates can be obtained under the assumption of a missingness completely at random mechanism. When the missing data mechanism is missingness at random, i.e. the probability of missing a particular outcome at a time point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models by using a single modified GEE based on an EM-type algorithm. The method proposed is motivated by the longitudinal study of cardiac abnormalities in children who were born to women infected with the human immunodeficiency virus, and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that, under a missingness at random mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided that the correlation model has been correctly specified, whereas estimates from standard GEEs can lead to substantial bias.     

Choice of estimand and analysis methods in diabetes trials with rescue medication

The analysis of clinical trials aiming to show symptomatic benefits is often complicated by the ethical requirement for rescue medication when the disease state of patients worsens. In type 2 diabetes trials, patients receive glucose‐lowering rescue medications continuously for the remaining trial duration, if one of several markers of glycemic control exceeds pre‐specified thresholds. This may mask differences in glycemic values between treatment groups, because it will occur more frequently in less effective treatment groups. Traditionally, the last pre‐rescue medication value was carried forward and analyzed as the end‐of‐trial value. The deficits of such simplistic single imputation approaches are increasingly recognized by regulatory authorities and trialists. We discuss alternative approaches and evaluate them through a simulation study. When the estimand of interest is the effect attributable to the treatments initially assigned at randomization, then our recommendation for estimation and hypothesis testing is to treat data after meeting rescue criteria as deterministically ‘missing’ at random, because initiation of rescue medication is determined by observed in‐trial values. An appropriate imputation of values after meeting rescue criteria is then possible either directly through multiple imputation or implicitly with a repeated measures model. Crucially, one needs to jointly impute or model all markers of glycemic control that can lead to the initiation of rescue medication. An alternative for hypothesis testing only are rank tests with outcomes from patients ‘requiring rescue medication’ ranked worst, and non‐rescued patients ranked according to final visit values. However, an appropriate ranking of not observed values may be controversial. Copyright © 2015 John Wiley & Sons, Ltd.     

Extending the Mann–Whitney–Wilcoxon rank sum test to longitudinal regression analysis

Outliers are commonly observed in psychosocial research, generally resulting in biased estimates when comparing group differences using popular mean-based models such as the analysis of variance model. Rank-based methods such as the popular Mann–Whitney–Wilcoxon (MWW) rank sum test are more effective to address such outliers. However, available methods for inference are limited to cross-sectional data and cannot be applied to longitudinal studies under missing data. In this paper, we propose a generalized MWW test for comparing multiple groups with covariates within a longitudinal data setting, by utilizing the functional response models. Inference is based on a class of U-statistics-based weighted generalized estimating equations, providing consistent and asymptotically normal estimates not only under complete but missing data as well. The proposed approach is illustrated with both real and simulated study data.     

Empirical likelihood-based inference in nonlinear regression models with missing responses at random

This paper investigates the estimations of regression parameters and response mean in nonlinear regression models in the presence of missing response variables that are missing with missingness probabilities depending on covariates. We propose four empirical likelihood (EL)-based estimators for the regression parameters and the response mean. The resulting estimators are shown to be consistent and asymptotically normal under some general assumptions. To construct the confidence regions for the regression parameters as well as the response mean, we develop four EL ratio statistics, which are proven to have the χ² distribution asymptotically. Simulation studies and an artificial data set are used to illustrate the proposed methodologies. Empirical results show that the EL method behaves better than the normal approximation method and that the coverage probabilities and average lengths depend on the selection probability function.     

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.     

Influence Function Based Variance Estimation and Missing Data Issues in Case-Cohort Studies

Recognizing that the efficiency in relative risk estimation for the Cox proportional hazards model is largely constrained by the total number of cases, Prentice (1986) proposed the case-cohort design in which covariates are measured on all cases and on a random sample of the cohort. Subsequent to Prentice, other methods of estimation and sampling have been proposed for these designs. We formalize an approach to variance estimation suggested by Barlow (1994), and derive a robust variance estimator based on the influence function. We consider the applicability of the variance estimator to all the proposed case-cohort estimators, and derive the influence function when known sampling probabilities in the estimators are replaced by observed sampling fractions. We discuss the modifications required when cases are missing covariate information. The missingness may occur by chance, and be completely at random; or may occur as part of the sampling design, and depend upon other observed covariates. We provide an adaptation of S-plus code that allows estimating influence function variances in the presence of such missing covariates. Using examples from our current case-cohort studies on esophageal and gastric cancer, we illustrate how our results our useful in solving design and analytic issues that arise in practice.