9. Multiple Imputation of Incomplete Categorical Data Using Latent Class Analysis

Missing data are nearly always a problem in research, and missing values represent a serious threat to the validity of inferences drawn from findings. Increasingly, social science researchers are turning to multiple imputation to handle missing data. Multiple imputation, in which missing values are replaced by values repeatedly drawn from conditional probability distributions, is an appropriate method for handling missing data when values are not missing completely at random. However, use of this method requires developing an imputation model from the observed data. This is typically a rigorous and time-consuming process. To encourage wider adoption of multiple imputation in social work research, a simple framework for designing imputation models is presented. The framework and its ability to generate unbiased estimates are demonstrated in a simulation study. KEY WORDS: missing data; multiple imputation; nonresponse ********** Missing data are ubiquitous in social research, and missingness or nonresponse can represent a threat to the validity of inferences because of undue effects on efficiency, power, and parameter bias (Shadish, Cook, & Campbell, 2002). Social work researchers are now addressing missing data in a more rigorous manner. Recently, Saunders et al. (2006) and Choi, Golder, Gillmore, and Morrison (2005) described important data imputation methods and dispelled misunderstandings regarding popular imputation methods, such as mean substitution. Recent advances in analytic methods, such as multiple imputation (MI), are taking hold in social work research. With MI, missing values are replaced with values repeatedly drawn from simulated conditional probability distributions (Schafer, 1997), thus creating multiple versions of the data set. Each version of the data set is analyzed according to the data analysis model, and the multiple results are combined into point estimates (Rubin, 1996). A critical task in MI is to devise an imputation model (Allison, 2002) or missing data model (Graham, Olchowski, & Gilreath, 2007), which involves specifying the measures that are putatively associated with the missing values. Although this process adds additional steps, the specification of an imputation model and the creation of multiple data sets can produce less-biased estimates in the presence of missing data across a wide variety of data analysis techniques (Schafer, 1997). Besides MI, there are many other methods for addressing missing data (Schafer, 1999; Schafer & Graham, 2002). An equally rigorous method known as direct or full information maximum likelihood (FIML) estimation can produce unbiased estimates and correct standard errors in the presence of missing data. When the number of imputations is sufficiently large, identical missing data models will produce the same estimates under MI and FIML (Graham et al., 2007). Unlike MI, FIML is limited to maximum likelihood analytic techniques and the missing data model must be included in the analysis model. Although we focus on MI, the steps we describe for developing an imputation model are equally appropriate for use in the missing data model for FIML. On the basis of the MI literature, this article describes a framework for developing an imputation model for use with any free or commercial software package that performs MI. BRIEF REVIEW OF MISSING DATA CONCEPTS Generally, both MI and a broad range of missing data issues have received ample attention in the applied literature (Graham et al., 2007). We briefly discuss the three types of distributions that describe the randomness of nonresponse given that this property has consequences for the development of an imputation model. For a discussion of general missing data concepts that are not critical to understanding our discussion of imputation model development, we refer readers to Schafer and Graham (2002). Distribution of Nonresponse The probability distribution of nonresponse--more frequently referred to as the missing data or nonresponse mechanism (Rubin, 1976)--is both an important factor in the decision to impute with MI and a context for the development of an imputation model. …

Multiple imputation and maximum likelihood estimation (via the expectation‐maximization algorithm) are two well‐known methods readily used for analyzing data with missing values. While these two methods are often considered as being distinct from one another, multiple imputation (when using improper imputation) is actually equivalent to a stochastic expectation‐maximization approximation to the likelihood. In this article, we exploit this key result to show that familiar likelihood‐based approaches to model selection, such as Akaike's information criterion (AIC) and the Bayesian information criterion (BIC), can be used to choose the imputation model that best fits the observed data. Poor choice of imputation model is known to bias inference, and while sensitivity analysis has often been used to explore the implications of different imputation models, we show that the data can be used to choose an appropriate imputation model via conventional model selection tools. We show that BIC can be consistent for selecting the correct imputation model in the presence of missing data. We verify these results empirically through simulation studies, and demonstrate their practicality on two classical missing data examples. An interesting result we saw in simulations was that not only can parameter estimates be biased by misspecifying the imputation model, but also by overfitting the imputation model. This emphasizes the importance of using model selection not just to choose the appropriate type of imputation model, but also to decide on the appropriate level of imputation model complexity.

Abstract. Latent class analysis has been recently proposed for the multiple imputation (MI) of missing categorical data, using either a standard frequentist approach or a nonparametric Bayesian model called Dirichlet process mixture of multinomial distributions (DPMM). The main advantage of using a latent class model for multiple imputation is that it is very flexible in the sense that it can capture complex relationships in the data given that the number of latent classes is large enough. However, the two existing approaches also have certain disadvantages. The frequentist approach is computationally demanding because it requires estimating many LC models: first models with different number of classes should be estimated to determine the required number of classes and subsequently the selected model is reestimated for multiple bootstrap samples to take into account parameter uncertainty during the imputation stage. Whereas the Bayesian Dirichlet process models perform the model selection and the handling of the parameter uncertainty automatically, the disadvantage of this method is that it tends to use a too small number of clusters during the Gibbs sampling, leading to an underfitting model yielding invalid imputations. In this paper, we propose an alternative approach which combined the strengths of the two existing approaches; that is, we use the Bayesian standard latent class model as an imputation model. We show how model selection can be performed prior to the imputation step using a single run of the Gibbs sampler and, moreover, show how underfitting is prevented by using large values for the hyperparameters of the mixture weights. The results of two simulation studies and one real-data study indicate that with a proper setting of the prior distributions, the Bayesian latent class model yields valid imputations and outperforms competing methods.

Latent class based multiple imputation approach for missing categorical data

Missing data and especially dropouts frequently arise in longitudinal data. Maximum likelihood estimates are consistent when data are missing at random (MAR) but, as this assumption is not checkable, pattern mixture models (PMM) have been developed to deal with informative dropout. More recently, latent class models (LCM) have been proposed as a way to relax PMM assumptions. The aim of this paper is to compare PMM and LCM in order to tackle informative dropout in a longitudinal study of cognitive ageing measured by several psychometric tests. Using a multivariate longitudinal model with a latent process, a sensitivity analysis was performed to compare estimates under the MAR assumption, from a PMM and from two LCM. In the PMM, dropout patterns are included as covariates in the multivariate longitudinal model. In the simple LCM, they are predictors of the class membership probabilities while, in the joint LCM, the dropout time is jointly modeled using a proportional hazard model depending on latent classes. We show that parameter interpretation is different in the two kinds of models and thus can lead to different estimated values. PMM parameters are adjusted on the dropout patterns while LCM parameters are adjusted on the latent classes. This difference is highlighted in our data set because the latent classes exhibit much more heterogeneity than dropout patterns. We suggest several complementary analyses to investigate the characteristics of latent classes in order to understand the meaning of the parameters when using LCM to deal with informative dropout.

BackgroundMultiple imputation is frequently used to deal with missing data in healthcare research. Although it is known that the outcome should be included in the imputation model when imputing missing covariate values, it is not known whether it should be imputed. Similarly no clear recommendations exist on: the utility of incorporating a secondary outcome, if available, in the imputation model; the level of protection offered when data are missing not-at-random; the implications of the dataset size and missingness levels.MethodsWe used realistic assumptions to generate thousands of datasets across a broad spectrum of contexts: three mechanisms of missingness (completely at random; at random; not at random); varying extents of missingness (20–80% missing data); and different sample sizes (1,000 or 10,000 cases). For each context we quantified the performance of a complete case analysis and seven multiple imputation methods which deleted cases with missing outcome before imputation, after imputation or not at all; included or did not include the outcome in the imputation models; and included or did not include a secondary outcome in the imputation models. Methods were compared on mean absolute error, bias, coverage and power over 1,000 datasets for each scenario.ResultsOverall, there was very little to separate multiple imputation methods which included the outcome in the imputation model. Even when missingness was quite extensive, all multiple imputation approaches performed well. Incorporating a secondary outcome, moderately correlated with the outcome of interest, made very little difference. The dataset size and the extent of missingness affected performance, as expected. Multiple imputation methods protected less well against missingness not at random, but did offer some protection.ConclusionsAs long as the outcome is included in the imputation model, there are very small performance differences between the possible multiple imputation approaches: no outcome imputation, imputation or imputation and deletion. All informative covariates, even with very high levels of missingness, should be included in the multiple imputation model. Multiple imputation offers some protection against a simple missing not at random mechanism.

Incomplete data are generally a challenge to the analysis of most large studies. The current gold standard to account for missing data is multiple imputation, and more specifically multiple imputation with chained equations (MICE). Numerous studies have been conducted to illustrate the performance of MICE for missing covariate data. The results show that the method works well in various situations. However, less is known about its performance in more complex models, specifically when the outcome is multivariate as in longitudinal studies. In current practice, the multivariate nature of the longitudinal outcome is often neglected in the imputation procedure, or only the baseline outcome is used to impute missing covariates. In this work, we evaluate the performance of MICE using different strategies to include a longitudinal outcome into the imputation models and compare it with a fully Bayesian approach that jointly imputes missing values and estimates the parameters of the longitudinal model. Results from simulation and a real data example show that MICE requires the analyst to correctly specify which components of the longitudinal process need to be included in the imputation models in order to obtain unbiased results. The full Bayesian approach, on the other hand, does not require the analyst to explicitly specify how the longitudinal outcome enters the imputation models. It performed well under different scenarios. Copyright © 2016 John Wiley & Sons, Ltd.

The usual methods for analyzing case-cohort studies rely on sometimes not fully efficient weighted estimators. Multiple imputation might be a good alternative because it uses all the data available and approximates the maximum partial likelihood estimator. This method is based on the generation of several plausible complete data sets, taking into account uncertainty about missing values. When the imputation model is correctly defined, the multiple imputation estimator is asymptotically unbiased and its variance is correctly estimated. We show that a correct imputation model must be estimated from the fully observed data (cases and controls), using the case status among the explanatory variable. To validate the approach, we analyzed case-cohort studies first with completely simulated data and then with case-cohort data sampled from two real cohorts. The analyses of simulated data showed that, when the imputation model was correct, the multiple imputation estimator was unbiased and efficient. The observed gain in precision ranged from 8 to 37 per cent for phase-1 variables and from 5 to 19 per cent for the phase-2 variable. When the imputation model was misspecified, the multiple imputation estimator was still more efficient than the weighted estimators but it was also slightly biased. The analyses of case-cohort data sampled from complete cohorts showed that even when no strong predictor of the phase-2 variable was available, the multiple imputation was unbiased, as precised as the weighted estimator for the phase-2 variable and slightly more precise than the weighted estimators for the phase-1 variables. However, the multiple imputation estimator was found to be biased when, because of interaction terms, some coefficients of the imputation model had to be estimated from small samples. Multiple imputation is an efficient technique for analyzing case-cohort data. Practically, we suggest building the analysis model using only the case-cohort data and weighted estimators. Multiple imputation can eventually be used to reanalyze the data using the selected model in order to improve the precision of the results.

ObjectivesEpidemiological studies often have missing data, which are commonly handled by multiple imputation (MI). Standard (default) MI procedures use simple linear covariate functions in the imputation model. We examine the bias that may be caused by acceptance of this default option and evaluate methods to identify problematic imputation models, providing practical guidance for researchers. Study Design and SettingUsing simulation and real data analysis, we investigated how imputation model mis-specification affected MI performance, comparing results with complete records analysis (CRA). We considered scenarios in which imputation model mis-specification occurred because (i) the analysis model was mis-specified or (ii) the relationship between exposure and confounder was mis-specified. ResultsMis-specification of the relationship between outcome and exposure, or between exposure and confounder, can cause biased CRA and MI estimates (in addition to any bias in the full-data estimate due to analysis model mis-specification). MI by predictive mean matching can mitigate model mis-specification. Methods for examining model mis-specification were effective in identifying mis-specified relationships. ConclusionWhen using MI methods that assume data are MAR, compatibility between the analysis and imputation models is necessary, but not sufficient to avoid bias. We propose a step-by-step procedure for identifying and correcting mis-specification of imputation models.

Severe asthma in the US population and eligibility for mAb therapy

In situations with missing data, statistical analyses are usually limited to subjects with complete data. However, such estimates may be biased. The method of 'filling in' missing data is called imputation. This article aimed to present a multiple imputation method. From a data set of 470 surgical patients, logistic models were developed for death as the outcome. Two incomplete data sets were generated: one with 5% and another with 20% of missing data in a single variable. Logistic models were fitted for the complete and incomplete data sets and for the data set completed by multiple imputations. Estimates obtained for the data set with missing data were different from those observed in the complete data set, mainly in the situation with 20% of missing data. The multiple imputation used here appeared efficient, producing very similar results to those obtained with the complete data set. However, one coefficient became non-significant. The analysis using multiple imputations was considered superior to using the data sets that excluded incomplete cases from the analysis.

BackgroundMissing data are a common problem in prospective studies with a long follow-up, and the volume, pattern and reasons for missing data may be relevant when estimating the cost of illness. We aimed to evaluate the effects of different methods for dealing with missing longitudinal cost data and for costing caregiver time on total societal costs in Alzheimer’s disease (AD).MethodsGERAS is an 18-month observational study of costs associated with AD. Total societal costs included patient health and social care costs, and caregiver health and informal care costs. Missing data were classified as missing completely at random (MCAR), missing at random (MAR) or missing not at random (MNAR). Simulation datasets were generated from baseline data with 10–40 % missing total cost data for each missing data mechanism. Datasets were also simulated to reflect the missing cost data pattern at 18 months using MAR and MNAR assumptions. Naïve and multiple imputation (MI) methods were applied to each dataset and results compared with complete GERAS 18-month cost data. Opportunity and replacement cost approaches were used for caregiver time, which was costed with and without supervision included and with time for working caregivers only being costed.ResultsTotal costs were available for 99.4 % of 1497 patients at baseline. For MCAR datasets, naïve methods performed as well as MI methods. For MAR, MI methods performed better than naïve methods. All imputation approaches were poor for MNAR data. For all approaches, percentage bias increased with missing data volume. For datasets reflecting 18-month patterns, a combination of imputation methods provided more accurate cost estimates (e.g. bias: −1 % vs −6 % for single MI method), although different approaches to costing caregiver time had a greater impact on estimated costs (29–43 % increase over base case estimate).ConclusionsMethods used to impute missing cost data in AD will impact on accuracy of cost estimates although varying approaches to costing informal caregiver time has the greatest impact on total costs. Tailoring imputation methods to the reason for missing data will further our understanding of the best analytical approach for studies involving cost outcomes.Electronic supplementary materialThe online version of this article (doi:10.1186/s12874-016-0188-1) contains supplementary material, which is available to authorized users.

Multiple imputation (MI) is a method for repairing and analyzing data with missing values. MI replaces missing values with a sample of random values drawn from an imputation model. The most popular form of MI, which we call posterior draw multiple imputation (PDMI), draws the parameters of the imputation model from a Bayesian posterior distribution. An alternative, which we call maximum likelihood multiple imputation (MLMI), estimates the parameters of the imputation model using maximum likelihood (or equivalent). Compared to PDMI, MLMI is less computationally intensive, faster, and yields slightly more efficient point estimates. A past barrier to using MLMI was the difficulty of estimating the standard errors of MLMI point estimates. We derive, implement, and evaluate three consistent standard error formulas: (1) one combines variances within and between the imputed datasets, (2) one uses the score function, and (3) one uses the bootstrap to estimate variance components due to sampling and imputation. Formula (1) modifies for MLMI a formula that has long been used under PDMI, while formulas (2) and (3) can be used without modification under either PDMI or MLMI. We have implemented MLMI and the standard error estimators in the mlmi and bootImpute packages for R.

The authors illustrate how to perform maximum-likelihood estimation in latent class (LC) analysis when there are sampling weights. The methods are natural extensions of the approaches proposed by Clogg and Eliason (1987) and Magidson (1987) for dealing with sampling weights in the log-linear analysis of frequency tables. For the log-linear form of the LC model, the approach corresponds to a special case of Haberman's (1979) log-linear LC model with cell weights. This approach can also be applied to the probability formulation of the LC model with cell weights, which can accommodate many indicators. The authors propose an efficient estimation-maximization algorithm for estimating the parameters for this formulation. A small simulation study shows that the probability estimates obtained by this approach compare favorably to other weighting approaches. Several empirical examples are provided to illustrate various possible weighting methods in LC analysis.

Purpose Our objectives were to (a) identify oral and pharyngeal physiologic swallowing impairment severity classes based on latent class analyses (LCAs) of the Modified Barium Swallow Impairment Profile (MBSImP) swallow task scores and (b) quantify the probability of severity class membership given composite MBSImP oral total (OT) and pharyngeal total (PT) scores. Method MBSImP scores were collected from a patient database of 319 consecutive modified barium swallow studies. Because of missing swallow task scores, LCA was performed using 25 multiply imputed data sets. Results LCA revealed a three-class structure for both oral and pharyngeal models. We identified OT and PT score intervals to assign subjects to oral and pharyngeal impairment latent severity classes, respectively, with high probability (probability of class membership ≥ 0.9 given OT or PT scores within specified ranges) and high confidence (95% credible interval [CI] widths ≤ 0.24 for all total scores within specified ranges). OT scores ranging from 0 to 10 and from 14 to 18 yielded assignments in Oral Latent Classes 1 and 2, respectively, while OT = 22 was assigned to Oral Latent Class 3. PT scores ranging from 0 to 13 and from 18 to 24 yielded assignments in Pharyngeal Latent Classes 1 and 2, respectively, while PT = 26 was assigned to Pharyngeal Latent Class 3. Conclusions LCA of MBSImP task-level data revealed significant underlying oral and pharyngeal ordinal class structures representing increasingly severe gradations of physiologic swallow impairment. Clinically meaningful OT and PT score ranges were derived facilitating latent class assignment. Supplemental Material https://doi.org/10.23641/asha.12315677.