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 In the presence of longitudinal ordinal missing data, where the Gaussian assumption is no longer valid, application of the MNI method is questionable.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.     

Navigating choices when applying multiple imputation in the presence of multi-level categorical interaction effects

Multiple imputation (MI) is an appealing option for handling missing data. When implementing MI, however, users need to make important decisions to obtain estimates with good statistical properties. One such decision involves the choice of imputation model–the joint modeling (JM) versus fully conditional specification (FCS) approach. Another involves the choice of method to handle interactions. These include imputing the interaction term as any other variable (active imputation), or imputing the main effects and then deriving the interaction (passive imputation). Our study investigates the best approach to perform MI in the presence of interaction effects involving two categorical variables. Such effects warrant special attention as they involve multiple correlated parameters that are handled differently under JM and FCS modeling. Through an extensive simulation study, we compared active, passive and an improved passive approach under FCS, as JM precludes passive imputation. We additionally compared JM and FCS techniques using active imputation. Performance between active and passive imputation was comparable. The improved passive approach proved superior to the other two particularly when the number of parameters corresponding to the interaction was large. JM without rounding and FCS using active imputation were also mostly comparable, with JM outperforming FCS when the number of parameters was large. In a direct comparison of JM active and FCS improved passive, the latter was the clear winner. We recommend improved passive imputation under FCS along with sensitivity analyses to handle multi-level interaction terms.     

Testing the proportional odds assumption in multiply imputed ordinal longitudinal data

A popular choice when analyzing ordinal data is to consider the cumulative proportional odds model to relate the marginal probabilities of the ordinal outcome to a set of covariates. However, application of this model relies on the condition of identical cumulative odds ratios across the cut-offs of the ordinal outcome; the well-known proportional odds assumption. This paper focuses on the assessment of this assumption while accounting for repeated and missing data. In this respect, we develop a statistical method built on multiple imputation (MI) based on generalized estimating equations that allows to test the proportionality assumption under the missing at random setting. The performance of the proposed method is evaluated for two MI algorithms for incomplete longitudinal ordinal data. The impact of both MI methods is compared with respect to the type I error rate and the power for situations covering various numbers of categories of the ordinal outcome, sample sizes, rates of missingness, well-balanced and skewed data. The comparison of both MI methods with the complete-case analysis is also provided. We illustrate the use of the proposed methods on a quality of life data from a cancer clinical trial.     

A Simple Method to Ensure Plausible Multiple Imputation for Continuous Multivariate Data

Multiple Imputation (MI) is an established approach for handling missing values. We show that MI for continuous data under the multivariate normal assumption is susceptible to generating implausible values. Our proposed remedy, is to: (1) transform the observed data into quantiles of the standard normal distribution; (2) obtain a functional relationship between the observed data and it's corresponding standard normal quantiles; (3) undertake MI using the quantiles produced in step 1; and finally, (4) use the functional relationship to transform the imputations into their original domain. In conclusion, our approach safeguards MI from imputing implausible values.     

Missing data sensitivity analysis for recurrent event data using controlled imputation

Statistical analyses of recurrent event data have typically been based on the missing at random assumption. One implication of this is that, if data are collected only when patients are on their randomized treatment, the resulting de jure estimator of treatment effect corresponds to the situation in which the patients adhere to this regime throughout the study. For confirmatory analysis of clinical trials, sensitivity analyses are required to investigate alternative de facto estimands that depart from this assumption. Recent publications have described the use of multiple imputation methods based on pattern mixture models for continuous outcomes, where imputation for the missing data for one treatment arm (e.g. the active arm) is based on the statistical behaviour of outcomes in another arm (e.g. the placebo arm). This has been referred to as controlled imputation or reference‐based imputation. In this paper, we use the negative multinomial distribution to apply this approach to analyses of recurrent events and other similar outcomes. The methods are illustrated by a trial in severe asthma where the primary endpoint was rate of exacerbations and the primary analysis was based on the negative binomial model. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Nonparametric methods for the estimation of imputation uncertainty

It is cleared in recent researches that the raising of missing values in datasets is inevitable. Imputation of missing data is one of the several methods which have been introduced to overcome this issue. Imputation techniques are trying to answer the case of missing data by covering missing values with reasonable estimates permanently. There are a lot of benefits for these procedures rather than their drawbacks. The operation of these methods has not been clarified, which means that they provide mistrust among analytical results. One approach to evaluate the outcomes of the imputation process is estimating uncertainty in the imputed data. Nonparametric methods are appropriate to estimating the uncertainty when data are not followed by any particular distribution. This paper deals with a nonparametric method for estimation and testing the significance of the imputation uncertainty, which is based on Wilcoxon test statistic, and which could be employed for estimating the precision of the imputed values created by imputation methods. This proposed procedure could be employed to judge the possibility of the imputation process for datasets, and to evaluate the influence of proper imputation methods when they are utilized to the same dataset. This proposed approach has been compared with other nonparametric resampling methods, including bootstrap and jackknife to estimate uncertainty in the imputed data under the Bayesian bootstrap imputation method. The ideas supporting the proposed method are clarified in detail, and a simulation study, which indicates how the approach has been employed in practical situations, is illustrated.     

Multiple Imputation Under Power Polynomials

Although the normality assumption has been regarded as a mathematical convenience for inferential purposes due to its nice distributional properties, there has been a growing interest regarding generalized classes of distributions that span a much broader spectrum in terms of symmetry and peakedness behavior. In this respect, Fleishman's power polynomial method seems to have been gaining popularity in statistical theory and practice because of its flexibility and ease of execution. In this article, we conduct multiple imputation for univariate continuous data under Fleishman polynomials to explore the extent to which this procedure works properly. We also make comparisons with normal imputation models via widely accepted accuracy and precision measures using simulated data that exhibit different distributional features as characterized by competing specifications of the third and fourth moments. Finally, we discuss generalizations to the multivariate case. Multiple imputation under power polynomials that cover most of the feasible area in the skewness-elongation plane appears to have substantial potential of capturing real missing-data trends.     

GRAPHICAL SENSITIVITY ANALYSIS WITH DIFFERENT METHODS OF IMPUTATION FOR A TRIAL WITH PROBABLE NON‐IGNORABLE MISSING DATA

Graphical sensitivity analyses have recently been recommended for clinical trials with non‐ignorable missing outcome. We demonstrate an adaptation of this methodology for a continuous outcome of a trial of three cognitive‐behavioural therapies for mild depression in primary care, in which one arm had unexpectedly high levels of missing data. Fixed‐value and multiple imputations from a normal distribution (assuming either varying mean and fixed standard deviation, or fixed mean and varying standard deviation) were used to obtain contour plots of the contrast estimates with their P‐values superimposed, their confidence intervals, and the root mean square errors. Imputation was based either on the outcome value alone, or on change from baseline. The plots showed fixed‐value imputation to be more sensitive than imputing from a normal distribution, but the normally distributed imputations were subject to sampling noise. The contours of the sensitivity plots were close to linear in appearance, with the slope approximately equal to the ratio of the proportions of subjects with missing data in each trial arm.     

Imputation for missing values and corresponding variance estimation

Imputation is commonly used to compensate for missing data in surveys. We consider the general case where the responses on either the variable of interest y or the auxiliary variable x or both may be missing. We use ratio imputation for y when the associated x is observed and different imputations when x is not observed. We obtain design-consistent linearization and jackknife variance estimators under uniform response. We also report the results of a simulation study on the efficiencies of imputed estimators, and relative biases and efficiencies of associated variance estimators.     

Comparison of multiple imputation and two-phase logistic regression to analyse two-phase case–control studies with rich phase 1: a simulation study

Two-phase case–control studies cope with the problem of confounding by obtaining required additional information for a subset (phase 2) of all individuals (phase 1). Nowadays, studies with rich phase 1 data are available where only few unmeasured confounders need to be obtained in phase 2. The extended conditional maximum likelihood (ECML) approach in two-phase logistic regression is a novel method to analyse such data. Alternatively, two-phase case–control studies can be analysed by multiple imputation (MI), where phase 2 information for individuals included in phase 1 is treated as missing. We conducted a simulation of two-phase studies, where we compared the performance of ECML and MI in typical scenarios with rich phase 1. Regarding exposure effect, MI was less biased and more precise than ECML. Furthermore, ECML was sensitive against misspecification of the participation model. We therefore recommend MI to analyse two-phase case–control studies in situations with rich phase 1 data.     

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.     

Matched case–control data analyses with missing covariates

We consider methods for analysing matched case–control data when some covariates ( W ) are completely observed but other covariates ( X ) are missing for some subjects. In matched case–control studies, the complete-record analysis discards completely observed subjects if none of their matching cases or controls are completely observed. We investigate an imputation estimate obtained by solving a joint estimating equation for log-odds ratios of disease and parameters in an imputation model. Imputation estimates for coefficients of W are shown to have smaller bias and mean-square error than do estimates from the complete-record analysis.     

Doubly robust point and variance estimation in the presence of imputed survey data

Imputation is often used in surveys to treat item nonresponse. It is well known that treating the imputed values as observed values may lead to substantial underestimation of the variance of the point estimators. To overcome the problem, a number of variance estimation methods have been proposed in the literature, including resampling methods such as the jackknife and the bootstrap. In this paper, we consider the problem of doubly robust inference in the presence of imputed survey data. In the doubly robust literature, point estimation has been the main focus. In this paper, using the reverse framework for variance estimation, we derive doubly robust linearization variance estimators in the case of deterministic and random regression imputation within imputation classes. Also, we study the properties of several jackknife variance estimators under both negligible and nonnegligible sampling fractions. A limited simulation study investigates the performance of various variance estimators in terms of relative bias and relative stability. Finally, the asymptotic normality of imputed estimators is established for stratified multistage designs under both deterministic and random regression imputation. The Canadian Journal of Statistics 40: 259–281; 2012 © 2012 Statistical Society of Canada     

Principal component analysis with interval imputed missing values

In this paper some statistical properties of Interval Imputation are derived in the context of Principal Component Analysis. Interval Imputation is a recent proposal for the treatment of missing values, consisting of replacing blanks with intervals and then analyzing the resulting data matrix using Symbolic Data Analysis techniques. The most noticeable virtue of this method is that it does not require a single-valued imputation, so it allows us to take into account that incomplete observations are affected by a degree of uncertainty. Illustrative examples and simulation studies are carried out in order to illustrate the functioning of the technique.     

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.     

Addressing the problem of missing data in decision tree modeling

Tree-based models (TBMs) can substitute missing data using the surrogate approach (SUR). The aim of this study is to compare the performance of statistical imputation against the performance of SUR in TBMs. Employing empirical data, a TBM was constructed. Thereafter, 10%, 20%, and 40% of variable values appeared as the first split was deleted, and imputed with and without the use of outcome variables in the imputation model (IMP? and IMP+). This was repeated one thousand times. Absolute relative bias above 0.10 was defined as sever (SARB). Subsequently, in a series of simulations, the following parameters were changed: the degree of correlation among variables, the number of variables truly associated with the outcome, and the missing rate. At a 10% missing rate, the proportion of times SARB was observed in either SUR or IMP? was two times higher than in IMP+ (28% versus 13%). When the missing rate was increased to 20%, all these proportions were approximately doubled. Irrespective of the missing rate, IMP+ was about 65% less likely to produce SARB than SUR. Results of IMP? and SUR were comparable up to a 20% missing rate. At a high missing rate, IMP? was 76% more likely to provide SARB estimates. Statistical imputation of missing data and the use of outcome variable in the imputation model is recommended, even in the content of TBM.     

Imputation of MĀori Descent for Electoral Calculations in New Zealand

The New Zealand Government Statistician decided that, for electoral purposes, Statistics New Zealand should impute Māori–descent status for individuals not responding Yes or No to theMāori–descent question in the 1996 Census of Population and Dwellings. Imputation provides a sounder basis for calculating electoral populations than the approach used in 1994, when all who had not answered clearly Yes or No in the 1991 Census were effectively allocated to non–Māori descent. For the purposes of imputation, the key variables related to the Māori–descent variable were identified using a statistical technique called CHAID (Chisquared Automatic Interaction Detector). Subgroups were created by cross–classification across five variables—island, iwi, Māori ethnic group, Māori–descent composition of the rest of the household, and age group. Within each subgroup, the proportion who responded Yes or No for Māori descent was used to allocate the remainder to Yes or No. The imputation increased the proportion allocated to Māori descent from 16.0% to 17.4% of the total population. However, the proportion of the population imputed to Māori descent was smaller than the proportion who specified Māori descent originally.     

Missing-Data Adjustments in Large Surveys

Useful properties of a general-purpose imputation method for numerical data are suggested and discussed in the context of several large government surveys. Imputation based on predictive mean matching is proposed as a useful extension of methods in existing practice, and versions of the method are presented for unit nonresponse and item nonresponse with a general pattern of missingness. Extensions of the method to provide multiple imputations are also considered. Pros and cons of weighting adjustments are discussed, and weighting-based analogs to predictive mean matching are outlined.