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.     

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.     

Predictive performance of missing data methods for logistic regression,classification trees and neural networks

Although the effect of missing data on regression estimates has received considerable attention, their effect on predictive performance has been neglected. We studied the performance of three missing data strategies—omission of records with missing values, replacement with a mean and imputation based on regression—on the predictive performance of logistic regression (LR), classification tree (CT) and neural network (NN) models in the presence of data missing completely at random (MCAR). Models were constructed using datasets of size 500 simulated from a joint distribution of binary and continuous predictors including nonlinearities, collinearity and interactions between variables. Though omission produced models that fit better on the data from which the models were developed, imputation was superior on average to omission for all models when evaluating the receiver operating characteristic (ROC) curve area, mean squared error (MSE), pooled variance across outcome categories and calibration X ² on an independently generated test set. However, in about one-third of simulations, omission performed better. Performance was also more variable with omission including quite a few instances of extremely poor performance. Replacement and imputation generally produced similar results except with neural networks for which replacement, the strategy typically used in neural network algorithms, was inferior to imputation. Missing data affected simpler models much less than they did more complex models such as generalized additive models that focus on local structure For moderate sized datasets, logistic regressions that use simple nonlinear structures such as quadratic terms and piecewise linear splines appear to be at least as robust to randomly missing values as neural networks and classification trees.     

A fresh imputing survey methodology using sensible constraints on study and auxiliary variables: dubious random non-response

In this paper, we introduce a fresh methodology for imputing missing values by making use of sensible constraints on both a study variable and auxiliary variables that are correlated with the variable of interest. The resultant estimator based on these imputed values is shown to lead to the regression type method of imputation in survey sampling. Furthermore, when the data are hybrid of both that missing at random and missing complexly at random, the resultant estimator is shown to be a consistent estimator that has asymptotic mean squared error equal to that of the linear regression method of imputation. A generalization to any type of method of imputation is possible and has been included at the end.     

A new variable importance measure for random forests with missing data

Random forests are widely used in many research fields for prediction and interpretation purposes. Their popularity is rooted in several appealing characteristics, such as their ability to deal with high dimensional data, complex interactions and correlations between variables. Another important feature is that random forests provide variable importance measures that can be used to identify the most important predictor variables. Though there are alternatives like complete case analysis and imputation, existing methods for the computation of such measures cannot be applied straightforward when the data contains missing values. This paper presents a solution to this pitfall by introducing a new variable importance measure that is applicable to any kind of data—whether it does or does not contain missing values. An extensive simulation study shows that the new measure meets sensible requirements and shows good variable ranking properties. An application to two real data sets also indicates that the new approach may provide a more sensible variable ranking than the widespread complete case analysis. It takes the occurrence of missing values into account which makes results also differ from those obtained under multiple imputation.     

A new multivariate imputation method based on Bayesian networks

Dealing with incomplete data is a pervasive problem in statistical surveys. Bayesian networks have been recently used in missing data imputation. In this research, we propose a new methodology for the multivariate imputation of missing data using discrete Bayesian networks and conditional Gaussian Bayesian networks. Results from imputing missing values in coronary artery disease data set and milk composition data set as well as a simulation study from cancer-neapolitan network are presented to demonstrate and compare the performance of three Bayesian network-based imputation methods with those of multivariate imputation by chained equations (MICE) and the classical hot-deck imputation method. To assess the effect of the structure learning algorithm on the performance of the Bayesian network-based methods, two methods called Peter-Clark algorithm and greedy search-and-score have been applied. Bayesian network-based methods are: first, the method introduced by Di Zio et al. [Bayesian networks for imputation, J. R. Stat. Soc. Ser. A 167 (2004), 309–322] in which, each missing item of a variable is imputed using the information given in the parents of that variable; second, the method of Di Zio et al. [Multivariate techniques for imputation based on Bayesian networks, Neural Netw. World 15 (2005), 303–310] which uses the information in the Markov blanket set of the variable to be imputed and finally, our new proposed method which applies the whole available knowledge of all variables of interest, consisting the Markov blanket and so the parent set, to impute a missing item. Results indicate the high quality of our new proposed method especially in the presence of high missingness percentages and more connected networks. Also the new method have shown to be more efficient than the MICE method for small sample sizes with high missing rates.     

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.     

A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials

The objective of this research was to demonstrate a framework for drawing inference from sensitivity analyses of incomplete longitudinal clinical trial data via a re‐analysis of data from a confirmatory clinical trial in depression. A likelihood‐based approach that assumed missing at random (MAR) was the primary analysis. Robustness to departure from MAR was assessed by comparing the primary result to those from a series of analyses that employed varying missing not at random (MNAR) assumptions (selection models, pattern mixture models and shared parameter models) and to MAR methods that used inclusive models. The key sensitivity analysis used multiple imputation assuming that after dropout the trajectory of drug‐treated patients was that of placebo treated patients with a similar outcome history (placebo multiple imputation). This result was used as the worst reasonable case to define the lower limit of plausible values for the treatment contrast. The endpoint contrast from the primary analysis was ? 2.79 (p = .013). In placebo multiple imputation, the result was ? 2.17. Results from the other sensitivity analyses ranged from ? 2.21 to ? 3.87 and were symmetrically distributed around the primary result. Hence, no clear evidence of bias from missing not at random data was found. In the worst reasonable case scenario, the treatment effect was 80% of the magnitude of the primary result. Therefore, it was concluded that a treatment effect existed. The structured sensitivity framework of using a worst reasonable case result based on a controlled imputation approach with transparent and debatable assumptions supplemented a series of plausible alternative models under varying assumptions was useful in this specific situation and holds promise as a generally useful framework. Copyright © 2012 John Wiley & Sons, Ltd.     

Illuminate the unknown: evaluation of imputation procedures based on the SAVE survey

Questions about monetary variables (such as income, wealth or savings) are key components of questionnaires on household finances. However, missing information on such sensitive topics is a well-known phenomenon which can seriously bias any inference based only on complete-case analysis. Many imputation techniques have been developed and implemented in several surveys. Using the German SAVE data, a new estimation technique is necessary to overcome the upward bias of monetary variables caused by the initially implemented imputation procedure. The upward bias is the result of adding random draws to the implausible negative values predicted by OLS regressions until all values are positive. To overcome this problem the logarithm of the dependent variable is taken and the predicted values are retransformed to the original scale by Duan’s smearing estimate. This paper evaluates the two different techniques for the imputation of monetary variables implementing a simulation study, where a random pattern of missingness is imposed on the observed values of the variables of interest. A Monte-Carlo simulation based on the observed data shows the superiority of the newly implemented smearing estimate to construct the missing data structure. All waves are consistently imputed using the new method.     

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.     

Tukey's gh Distribution for Multiple Imputation

Tukey proposed a class of distributions, the g-and-h family (gh family), based on a transformation of a standard normal variable to accommodate different skewness and elongation in the distribution of variables arising in practical applications. It is easy to draw values from this distribution even though it is hard to explicitly state the probability density function. Given this flexibility, the gh family may be extremely useful in creating multiple imputations for missing data. This article demonstrates how this family, as well as its generalizations, can be used in the multiple imputation analysis of incomplete data. The focus of this article is on a scalar variable with missing values. In the absence of any additional information, data are missing completely at random, and hence the correct analysis is the complete-case analysis. Thus, the application of the gh multiple imputation to the scalar cases affords comparison with the correct analysis and with other model-based multiple imputation methods. Comparisons are made using simulated datasets and the data from a survey of adolescents ascertaining driving after drinking alcohol.     

A note on dealing with missing standard errors in meta-analyses of continuous outcome measures in WinBUGS

A meta-analysis of a continuous outcome measure may involve missing standard errors. This is not a problem depending on assumptions made about the population standard deviation. Multiple imputation can be used to impute missing values while allowing for uncertainty in the imputation. Markov chain Monte Carlo simulation is a multiple imputation technique for generating posterior predictive distributions for missing data. We present an example of imputing missing variances using WinBUGS. The example highlights the importance of checking model assumptions, whether for missing or observed data.     

Variable selection models based on multiple imputation with an application for predicting median effective dose and maximum effect

The statistical methods for variable selection and prediction could be challenging when missing covariates exist. Although multiple imputation (MI) is a universally accepted technique for solving missing data problem, how to combine the MI results for variable selection is not quite clear, because different imputations may result in different selections. The widely applied variable selection methods include the sparse partial least-squares (SPLS) method and the penalized least-squares method, e.g. the elastic net (ENet) method. In this paper, we propose an MI-based weighted elastic net (MI-WENet) method that is based on stacked MI data and a weighting scheme for each observation in the stacked data set. In the MI-WENet method, MI accounts for sampling and imputation uncertainty for missing values, and the weight accounts for the observed information. Extensive numerical simulations are carried out to compare the proposed MI-WENet method with the other competing alternatives, such as the SPLS and ENet. In addition, we applied the MI-WENet method to examine the predictor variables for the endothelial function that can be characterized by median effective dose (ED50) and maximum effect (Emax) in an ex-vivo phenylephrine-induced extension and acetylcholine-induced relaxation experiment.     

Performance of standard imputation methods for missing quality of life data as covariate in survival analysis based on simulations from the International Breast Cancer Study Group Trials VI and VII*

Abstract

Imputation methods for missing data on a time-dependent variable within time-dependent Cox models are investigated in a simulation study. Quality of life (QoL) assessments were removed from the complete simulated datasets, which have a positive relationship between QoL and disease-free survival (DFS) and delayed chemotherapy and DFS, by missing at random and missing not at random (MNAR) mechanisms. Standard imputation methods were applied before analysis. Method performance was influenced by missing data mechanism, with one exception for simple imputation. The greatest bias occurred under MNAR and large effect sizes. It is important to carefully investigate the missing data mechanism.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Comparison of several imputation methods for missing baseline data in propensity scores analysis of binary outcome

We performed a simulation study comparing the statistical properties of the estimated log odds ratio from propensity scores analyses of a binary response variable, in which missing baseline data had been imputed using a simple imputation scheme (Treatment Mean Imputation), compared with three ways of performing multiple imputation (MI) and with a Complete Case analysis. MI that included treatment (treated/untreated) and outcome (for our analyses, outcome was adverse event [yes/no]) in the imputer's model had the best statistical properties of the imputation schemes we studied. MI is feasible to use in situations where one has just a few outcomes to analyze. We also found that Treatment Mean Imputation performed quite well and is a reasonable alternative to MI in situations where it is not feasible to use MI. Treatment Mean Imputation performed better than MI methods that did not include both the treatment and outcome in the imputer's model. Copyright © 2009 John Wiley & Sons, Ltd.     

Variable selection for high-dimensional generalized linear model with block-missing data

In modern scientific research, multiblock missing data emerges with synthesizing information across multiple studies. However, existing imputation methods for handling block-wise missing data either focus on the single-block missing pattern or heavily rely on the model structure. In this study, we propose a single regression-based imputation algorithm for multiblock missing data. First, we conduct a sparse precision matrix estimation based on the structure of block-wise missing data. Second, we impute the missing blocks with their means conditional on the observed blocks. Theoretical results about variable selection and estimation consistency are established in the context of a generalized linear model. Moreover, simulation studies show that compared with existing methods, the proposed imputation procedure is robust to various missing mechanisms because of the good properties of regression imputation. An application to Alzheimer's Disease Neuroimaging Initiative data also confirms the superiority of our proposed method.     

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 effects of income imputation on microanalyses: evidence from the European Community Household Panel

Summary.  Social surveys are usually affected by item and unit non-response. Since it is unlikely that a sample of respondents is a random sample, social scientists should take the missing data problem into account in their empirical analyses. Typically, survey methodologists try to simplify the work of data users by 'completing' the data, filling the missing variables through imputation. The aim of the paper is to give data users some guidelines on how to assess the effects of imputation on their microlevel analyses. We focus attention on the potential bias that is caused by imputation in the analysis of income variables, using the European Community Household Panel as an illustration.     

MISSING VALUES, IMPUTATION AND ERROR RATE ESTIMATORS IN DISCRIMINANT ANALYSIS

Error rate is a popular criterion for assessing the performance of an allocation rule in discriminant analysis. Training samples which involve missing values cause problems for those error rate estimators that require all variables to be observed at all data points. This paper explores imputation algorithms, their effects on, and problems of implementing them with, eight commonly used error rate estimators (three parametric and five non-parametric) in linear discriminant analysis. The results indicate that imputation should not be based on the way error rate estimators are calculated, and that imputed values may underestimate error rates.