A distance-based rounding strategy for post-imputation ordinal data

Multiple imputation has emerged as a widely used model-based approach in dealing with incomplete data in many application areas. Gaussian and log-linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings which include a mix of continuous and discrete variables, correct specification of the imputation model could be a daunting task owing to the lack of flexible models for the joint distribution of variables of different nature. This complication, along with accessibility to software packages that are capable of carrying out multiple imputation under the assumption of joint multivariate normality, appears to encourage applied researchers for pragmatically treating the discrete variables as continuous for imputation purposes, and subsequently rounding the imputed values to the nearest observed category. In this article, I introduce a distance-based rounding approach for ordinal variables in the presence of continuous ones. The first step of the proposed rounding process is predicated upon creating indicator variables that correspond to the ordinal levels, followed by jointly imputing all variables under the assumption of multivariate normality. The imputed values are then converted to the ordinal scale based on their Euclidean distances to a set of indicators, with minimal distance corresponding to the closest match. I compare the performance of this technique to crude rounding via commonly accepted accuracy and precision measures with simulated data sets.     

Comparison of methods for incomplete repeated measures data analysis in small samples

This paper presents missing data methods for repeated measures data in small samples. Most methods currently available are for large samples. In particular, no studies have compared the performance of multiple imputation methods to that of non-imputation incomplete analysis methods. We first develop a strategy for multiple imputations for repeated measures data under a cell-means model that is applicable for any multivariate data with small samples. Multiple imputation inference procedures are applied to the resulting multiply imputed complete data sets. Comparisons to other available non-imputation incomplete data methods is made via simulation studies to conclude that there is not much gain in using the computer intensive multiple imputation methods for small sample repeated measures data analysis in terms of the power of testing hypotheses of parameters of interest.     

Multiple imputation for continuous variables using a Bayesian principal component analysis

ABSTRACT

We propose a multiple imputation method based on principal component analysis (PCA) to deal with incomplete continuous data. To reflect the uncertainty of the parameters from one imputation to the next, we use a Bayesian treatment of the PCA model. Using a simulation study and real data sets, the method is compared to two classical approaches: multiple imputation based on joint modelling and on fully conditional modelling. Contrary to the others, the proposed method can be easily used on data sets where the number of individuals is less than the number of variables and when the variables are highly correlated. In addition, it provides unbiased point estimates of quantities of interest, such as an expectation, a regression coefficient or a correlation coefficient, with a smaller mean squared error. Furthermore, the widths of the confidence intervals built for the quantities of interest are often smaller whilst ensuring a valid coverage.     

Latent class based multiple imputation approach for missing categorical data

In this paper we propose a latent class based multiple imputation approach for analyzing missing categorical covariate data in a highly stratified data model. In this approach, we impute the missing data assuming a latent class imputation model and we use likelihood methods to analyze the imputed data. Via extensive simulations, we study its statistical properties and make comparisons with complete case analysis, multiple imputation, saturated log-linear multiple imputation and the Expectation–Maximization approach under seven missing data mechanisms (including missing completely at random, missing at random and not missing at random). These methods are compared with respect to bias, asymptotic standard error, type I error, and 95% coverage probabilities of parameter estimates. Simulations show that, under many missingness scenarios, latent class multiple imputation performs favorably when jointly considering these criteria. A data example from a matched case–control study of the association between multiple myeloma and polymorphisms of the Inter-Leukin 6 genes is considered.     

A comparison of multiple imputation and doubly robust estimation for analyses with missing data

Summary.  Multiple imputation is now a well-established technique for analysing data sets where some units have incomplete observations. Provided that the imputation model is correct, the resulting estimates are consistent. An alternative, weighting by the inverse probability of observing complete data on a unit, is conceptually simple and involves fewer modelling assumptions, but it is known to be both inefficient (relative to a fully parametric approach) and sensitive to the choice of weighting model. Over the last decade, there has been a considerable body of theoretical work to improve the performance of inverse probability weighting, leading to the development of 'doubly robust' or 'doubly protected' estimators. We present an intuitive review of these developments and contrast these estimators with multiple imputation from both a theoretical and a practical viewpoint.     

Modeling dropouts by conditional distribution, a copula-based approach

In this paper the concept of copulas is implemented into the methodology for solving the imputation problem in correlated incomplete data. We use the Gaussian copula as alternative to the joint distribution for modeling the conditional distribution, conditioned by the observed values of measurements. The general formula for imputation and its application for compound symmetry correlation structure are given.     

Joint GEEs for multivariate correlated data with incomplete binary outcomes

This study considers a fully-parametric but uncongenial multiple imputation (MI) inference to jointly analyze incomplete binary response variables observed in a correlated data settings. Multiple imputation model is specified as a fully-parametric model based on a multivariate extension of mixed-effects models. Dichotomized imputed datasets are then analyzed using joint GEE models where covariates are associated with the marginal mean of responses with response-specific regression coefficients and a Kronecker product is accommodated for cluster-specific correlation structure for a given response variable and correlation structure between multiple response variables. The validity of the proposed MI-based JGEE (MI-JGEE) approach is assessed through a Monte Carlo simulation study under different scenarios. The simulation results, which are evaluated in terms of bias, mean-squared error, and coverage rate, show that MI-JGEE has promising inferential properties even when the underlying multiple imputation is misspecified. Finally, Adolescent Alcohol Prevention Trial data are used for illustration.     

Multiple imputation using multivariate gh transformations

Multiple imputation has emerged as a popular approach to handling data sets with missing values. For incomplete continuous variables, imputations are usually produced using multivariate normal models. However, this approach might be problematic for variables with a strong non-normal shape, as it would generate imputations incoherent with actual distributions and thus lead to incorrect inferences. For non-normal data, we consider a multivariate extension of Tukey's gh distribution/transformation [38] to accommodate skewness and/or kurtosis and capture the correlation among the variables. We propose an algorithm to fit the incomplete data with the model and generate imputations. We apply the method to a national data set for hospital performance on several standard quality measures, which are highly skewed to the left and substantially correlated with each other. We use Monte Carlo studies to assess the performance of the proposed approach. We discuss possible generalizations and give some advices to practitioners on how to handle non-normal incomplete data.     

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.     

MIMCA: multiple imputation for categorical variables with multiple correspondence analysis

We propose a multiple imputation method to deal with incomplete categorical data. This method imputes the missing entries using the principal component method dedicated to categorical data: multiple correspondence analysis (MCA). The uncertainty concerning the parameters of the imputation model is reflected using a non-parametric bootstrap. Multiple imputation using MCA (MIMCA) requires estimating a small number of parameters due to the dimensionality reduction property of MCA. It allows the user to impute a large range of data sets. In particular, a high number of categories per variable, a high number of variables or a small number of individuals are not an issue for MIMCA. Through a simulation study based on real data sets, the method is assessed and compared to the reference methods (multiple imputation using the loglinear model, multiple imputation by logistic regressions) as well to the latest works on the topic (multiple imputation by random forests or by the Dirichlet process mixture of products of multinomial distributions model). The proposed method provides a good point estimate of the parameters of the analysis model considered, such as the coefficients of a main effects logistic regression model, and a reliable estimate of the variability of the estimators. In addition, MIMCA has the great advantage that it is substantially less time consuming on data sets of high dimensions than the other multiple imputation methods.     

A multiple imputation approach to deal with the unity measure error

Unity measure errors (UME) in numerical survey data can determine serious bias in the estimates of interest. In this paper, a finite Gaussian mixture model is used to identify observations affected by UME and to robustly estimate the target parameters in presence of this type of error. In the proposed model, the mixture components are associated to the different error patterns across the variables. We follow a multiple imputation approach in a Bayesian setting that allows us to handle missing values in data. In this framework, the assessment of the uncertainty associated with both errors and missingness is based on repeatedly drawing from the predictive distribution of the true non contaminated data given the observed data. The draws are obtained through a suitable version of the data augmentation algorithm. Applications to both simulated and real data are presented.     

Analysis of Bivariate Survival Data using Shared Inverse Gaussian Frailty Model

In this article, we consider shared frailty model with inverse Gaussian distribution as frailty distribution and log-logistic distribution (LLD) as baseline distribution for bivariate survival times. We fit this model to three real-life bivariate survival data sets. The problem of analyzing and estimating parameters of shared inverse Gaussian frailty is the interest of this article and then compare the results with shared gamma frailty model under the same baseline for considered three data sets. Data are analyzed using Bayesian approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same group. The variance component estimation provides the estimated dispersion of the random effects. We carried out a test for frailty (or heterogeneity) using Bayes factor. Model comparison is made using information criteria and Bayes factor. We observed that the shared inverse Gaussian frailty model with LLD as baseline is the better fit for all three bivariate data sets.     

Combining the complete-data and nonresponse models for drawing imputations under MAR

In multiple imputation (MI), the resulting estimates are consistent if the imputation model is correct. To specify the imputation model, it is recommended to combine two sets of variables: those that are related to the incomplete variable and those that are related to the missingness mechanism. Several possibilities exist, but it is not clear how they perform in practice. The method that simply groups all variables together into the imputation model and four other methods that are based on the propensity scores are presented. Two of them are new and have not been used in the context of MI. The performance of the methods is investigated by a simulation study under different missing at random mechanisms for different types of variables. We conclude that all methods, except for one method based on the propensity scores, perform well. It turns out that as long as the relevant variables are taken into the imputation model, the form of the imputation model has only a minor effect in the quality of the imputations.     

The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi‐parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI‐GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI‐GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood‐based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI‐GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE‐based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM. Copyright © 2009 John Wiley & Sons, Ltd.     

Missing data techniques for multilevel data: implications of model misspecification

When modeling multilevel data, it is important to accurately represent the interdependence of observations within clusters. Ignoring data clustering may result in parameter misestimation. However, it is not well established to what degree parameter estimates are affected by model misspecification when applying missing data techniques (MDTs) to incomplete multilevel data. We compare the performance of three MDTs with incomplete hierarchical data. We consider the impact of imputation model misspecification on the quality of parameter estimates by employing multiple imputation under assumptions of a normal model (MI/NM) with two-level cross-sectional data when values are missing at random on the dependent variable at rates of 10%, 30%, and 50%. Five criteria are used to compare estimates from MI/NM to estimates from MI assuming a linear mixed model (MI/LMM) and maximum likelihood estimation to the same incomplete data sets. With 10% missing data (MD), techniques performed similarly for fixed-effects estimates, but variance components were biased with MI/NM. Effects of model misspecification worsened at higher rates of MD, with the hierarchical structure of the data markedly underrepresented by biased variance component estimates. MI/LMM and maximum likelihood provided generally accurate and unbiased parameter estimates but performance was negatively affected by increased rates of MD.     

On the Performance of Sequential Regression Multiple Imputation Methods with Non Normal Error Distributions

Sequential regression multiple imputation has emerged as a popular approach for handling incomplete data with complex features. In this approach, imputations for each missing variable are produced based on a regression model using other variables as predictors in a cyclic manner. Normality assumption is frequently imposed for the error distributions in the conditional regression models for continuous variables, despite that it rarely holds in real scenarios. We use a simulation study to investigate the performance of several sequential regression imputation methods when the error distribution is flat or heavy tailed. The methods evaluated include the sequential normal imputation and its several extensions which adjust for non normal error terms. The results show that all methods perform well for estimating the marginal mean and proportion, as well as the regression coefficient when the error distribution is flat or moderately heavy tailed. When the error distribution is strongly heavy tailed, all methods retain their good performances for the mean and the adjusted methods have robust performances for the proportion; but all methods can have poor performances for the regression coefficient because they cannot accommodate the extreme values well. We caution against the mechanical use of sequential regression imputation without model checking and diagnostics.     

A Simulation Study Comparing Multiple Imputation Methods for Incomplete Longitudinal Ordinal Data

Multiple imputation (MI) is now a reference solution for handling missing data. The default method for MI is the Multivariate Normal Imputation (MNI) algorithm that is based on the multivariate normal distribution. In the presence of longitudinal ordinal missing data, where the Gaussian assumption is no longer valid, application of the MNI method is questionable. This simulation study compares the performance of the MNI and ordinal imputation regression model for incomplete longitudinal ordinal data for situations covering various numbers of categories of the ordinal outcome, time occasions, sample sizes, rates of missingness, well-balanced, and skewed data.     

Data augmentation in multi-way contingency tables with fixed marginal totals

We describe and illustrate approaches to data augmentation in multi-way contingency tables for which partial information, in the form of subsets of marginal totals, is available. In such problems, interest lies in questions of inference about the parameters of models underlying the table together with imputation for the individual cell entries. We discuss questions of structure related to the implications for inference on cell counts arising from assumptions about log-linear model forms, and a class of simple and useful prior distributions on the parameters of log-linear models. We then discuss “local move” and “global move” Metropolis–Hastings simulation methods for exploring the posterior distributions for parameters and cell counts, focusing particularly on higher-dimensional problems. As a by-product, we note potential uses of the “global move” approach for inference about numbers of tables consistent with a prescribed subset of marginal counts. Illustration and comparison of MCMC approaches is given, and we conclude with discussion of areas for further developments and current open issues.     

Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

A Bayesian Approach for Zero-Inflated Count Regression Models by Using the Reversible Jump Markov Chain Monte Carlo Method and an Application

In this study, estimation of the parameters of the zero-inflated count regression models and computations of posterior model probabilities of the log-linear models defined for each zero-inflated count regression models are investigated from the Bayesian point of view. In addition, determinations of the most suitable log-linear and regression models are investigated. It is known that zero-inflated count regression models cover zero-inflated Poisson, zero-inflated negative binomial, and zero-inflated generalized Poisson regression models. The classical approach has some problematic points but the Bayesian approach does not have similar flaws. This work points out the reasons for using the Bayesian approach. It also lists advantages and disadvantages of the classical and Bayesian approaches. As an application, a zoological data set, including structural and sampling zeros, is used in the presence of extra zeros. In this work, it is observed that fitting a zero-inflated negative binomial regression model creates no problems at all, even though it is known that fitting a zero-inflated negative binomial regression model is the most problematic procedure in the classical approach. Additionally, it is found that the best fitting model is the log-linear model under the negative binomial regression model, which does not include three-way interactions of factors.