Multiple imputation in the presence of high-dimensional data

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 (MI) has been widely used for handling missing data in biomedical research. In the presence of high-dimensional data, regularized regression has been used as a natural strategy for building imputation models, but limited research has been conducted for handling general missing data patterns where multiple variables have missing values. Using the idea of multiple imputation by chained equations (MICE), we investigate two approaches of using regularized regression to impute missing values of high-dimensional data that can handle general missing data patterns. We compare our MICE methods with several existing imputation methods in simulation studies. Our simulation results demonstrate the superiority of the proposed MICE approach based on an indirect use of regularized regression in terms of bias. We further illustrate the proposed methods using two data examples.

Multiple imputation (MI) has been widely applied to missing value problems in biomedical, social and econometric research, in order to avoid improper inference in the downstream data analysis. In the presence of high-dimensional data, imputation models that include feature selection, especially regularized regression (such as Lasso, adaptive Lasso, and Elastic Net), are common choices to prevent the model from underdetermination. However, conducting MI with feature selection is difficult: existing methods are often computationally inefficient and poor in performance. We propose MISNN, a novel and efficient algorithm that incorporates feature selection for MI. Leveraging the approximation power of neural networks, MISNN is a general and flexible framework, compatible with any feature selection method, any neural network architecture, high/low-dimensional data and general missing patterns. Through empirical experiments, MISNN has demonstrated great advantages over state-of-the-art imputation methods (e.g. Bayesian Lasso and matrix completion), in terms of imputation accuracy, statistical consistency and computation speed.

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.

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.

Propensity score analysis is a popular method to control for confounding in observational studies. A challenge in propensity methods is missing values in confounders. Several strategies for handling missing values exist, but guidance in choosing the best method is needed. In this simulation study, we compared four strategies of handling missing covariate values in propensity matching and propensity weighting. These methods include: complete case analysis, missing indicator method, multiple imputation and combining multiple imputation and missing indicator method. Concurrently, we aimed to provide guidance in choosing the optimal strategy. Simulated scenarios varied regarding missing mechanism, presence of effect modification or unmeasured confounding. Additionally, we demonstrated how missingness graphs help clarifying the missing structure. When no effect modification existed, complete case analysis yielded valid causal treatment effects even when data were missing not at random. In some situations, complete case analysis was also able to partially correct for unmeasured confounding. Multiple imputation worked well if the data were missing (completely) at random, and if the imputation model was correctly specified. In the presence of effect modification, more complex imputation models than default options of commonly used statistical software were required. Multiple imputation may fail when data are missing not at random. Here, combining multiple imputation and the missing indicator method reduced the bias as the missing indicator variable can be a proxy for unobserved confounding. The optimal way to handle missing values in covariates of propensity score models depends on the missing data structure and the presence of effect modification. When effect modification is present, default settings of imputation methods may yield biased results even if data are missing at random.

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.

The occurrence of missing data due to protocol deviations is inevitable in clinical trials. When missing data exist, analyses rely on assumptions about the behaviour of the individuals after dropping out. As a result, sensitivity analysis, which is now advocated by regulatory bodies, should be performed to explore the robustness of the inference to those assumptions. These assumptions should be relevant to the estimand of the study and be practically accessible by all parties. The aim of this document is twofold: to assess the statistical validity of a new method for sensitivity analysis, and apply this method to a published Alzheimer’s study. At the beginning of the thesis, a description of the Alzheimer’s study and issues with missing data encountered therein, take place. This study was mainly set up to investigate the effect rosiglitazone, as an adjunct therapy in Alzheimer’s patients. Two different doses of the drug were compared to placebo. The study suffered from a moderate degree of missing data in each treatment arm. The thesis proceeds with a critique on the per-protocol and intention-to-treat estimands, and revisits their meaning when missing data occur. Two new estimands are introduced, which are particularly amenable to studies with missing data. They are termed de-jure and de-facto. Following that, the main methods for dealing with missing data are introduced, with a particular emphasis on multiple imputation, and how it can easily incorporate missing not at random (MNAR) analyses. A thorough presentation of the new methodology is given. This is built around a set of assumptions, that reflect possible distributional behaviours of the subjects after protocol deviation. The assumptions are Randomised-treatment Missing at Random (MAR), Jump to Reference, Last Mean Carried Forward, Copy Increments in Reference, and Copy Reference. Estimation and inference is achieved via multiple imputation, and it is shown how the predictive distribution of the imputation model can be constructed from parameters borrowed from an MAR model and manipulated in a pattern-mixture model approach, to obtain the five assumptions for the 4 unobserved component of groups of individuals. A number of simulations whose aim is to explore the statistical properties of the new method, are carried out. The simulated datasets, which are based on the parameters from the Alzheimer’s study, focus on the estimator bias, the size and power of the methods, the bias of the variance estimator, and coverage. The results obtained from the simulations show the method has sensible properties; no bias for the estimator was detectable and the sizes and power of the methods agreed closely with their theoretical equivalents. The main result however, pertains to Rubin’s variance estimator, which proves to appropriately reflect the loss of information from missing data. It is therefore argued, it is the right estimator to use in this setting. The results from the application of the proposed method on the Alzheimer’s dataset are presented in tables. Inferences from the sensitivity analysis assumptions were consistent with those from the original MAR analyses. The comparison between high dose rosiglitazone and placebo did not show any evidence in favour of the treatment under any sensitivity assumption. Some evidence of treatment difference existed when the low dose treatment was compared to placebo. This finding though, should be interpreted with caution, as the differences were obtained from analyses not subjected to the rigorous inferential process that was used in the Harringtons study. It was further argued, this finding might have been due to chance, and it was not replicated in a different study.

Comparing the performance of different multiple imputation strategies for missing binary outcomes in cluster randomized trials: a simulation study Jinhui Ma,1–3 Parminder Raina,1,2 Joseph Beyene,1 Lehana Thabane1,3–51Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, ON, Canada; 2McMaster University Evidence-based Practice Center, Hamilton, ON, Canada; 3Biostatistics Unit, St Joseph's Healthcare Hamilton, Hamilton, ON, Canada; 4Centre for Evaluation of Medicines, St Joseph's Healthcare Hamilton, Hamilton, ON, Canada; 5Population Health Research Institute, Hamilton Health Sciences, Hamilton, ON, CanadaIntroduction: Although researchers have proposed various strategies to handle missing outcomes in cluster randomized trials (CRTs), limited attention has been paid to the performance of these strategies. Under the assumption of covariate-dependent missingness, the objective of this simulation study is to compare the performance of various strategies in handling missing binary outcomes in CRTs under different design settings.Methods: There are six missing data strategies investigated in this paper, which include complete case analysis, standard multiple imputation (MI) strategies using either logistic regression or Markov chain Monte Carlo (MCMC) method, within-cluster MI strategies using either logistic regression or MCMC method, and MI using logistic regression with cluster as a fixed effect. The performance of these strategies is evaluated through bias, empirical standard error, root mean squared error, and coverage probability.Results: Under the assumption of covariate-dependent missingness and applying the generalized estimating equations approach for fitting the logistic regression, it was shown that complete case analysis yields valid inferences when the percentage of missing outcomes is not large (<20%) for all the designs of CRTs considered in this paper. Standard MI strategies can be adopted when the design effect is small (variance inflation factor [VIF] ≤ 3); however, they tend to underestimate the standard error of treatment effect when the design effect is large. Within-cluster MI strategy using logistic regression is valid for imputation of missing data from CRTs, especially when the cluster size is large (>50) and the design effect is large (VIF > 3). In contrast, within-cluster MI strategy using MCMC method may yield biased estimates of treatment effect for CRTs with small cluster size (≤50). MI using logistic regression with cluster as a fixed effect may substantially overestimate the standard error of the estimated treatment effect when the intracluster correlation coefficient is small. It may also lead to biased estimated treatment effect.Conclusion: Findings from this simulation study provide researchers with quantitative evidence to guide selection of an appropriate strategy to deal with missing binary outcomes.Keywords: missing data, design effect, variance inflation factor

BackgroundDespite recent advances in causal inference methods, outcome regression remains the most widely used approach for estimating causal effects in epidemiological studies with a single-point exposure and outcome. Missing data are common in these studies, and complete-case analysis (CCA) and multiple imputation (MI) are two frequently used methods for handling them. In randomised controlled trials (RCTs), it has been shown that MI should be conducted separately by treatment group. In observational studies, causal inference is now understood as the task of emulating an RCT, which raises the question of whether MI should be conducted by exposure group in such studies.MethodsWe addressed this question by evaluating the performance of seven methods for handling missing data when estimating causal effects with outcome regression. We conducted an extensive simulation study based on an illustrative case study from the Victorian Adolescent Health Cohort Study, assessing a range of scenarios, including seven outcome generation models with exposure-confounder interactions of differing strength.ResultsThe simulation results showed that MI by exposure group led to the least bias when the size of the smallest exposure group was relatively large, followed by MI approaches that included the exposure-confounder interactions.ConclusionsThe findings from our simulation study, which was designed based on a real case study, suggest that current practice for the conduct of MI in causal inference may need to shift to stratifying by exposure group where feasible, or otherwise including exposure-confounder interactions in the imputation model.

ObjectivesResearchers are concerned whether multiple imputation (MI) or complete case analysis should be used when a large proportion of data are missing. We aimed to provide guidance for drawing conclusions from data with a large proportion of missingness. Study Design and SettingVia simulations, we investigated how the proportion of missing data, the fraction of missing information (FMI), and availability of auxiliary variables affected MI performance. Outcome data were missing completely at random or missing at random (MAR). ResultsProvided sufficient auxiliary information was available; MI was beneficial in terms of bias and never detrimental in terms of efficiency. Models with similar FMI values, but differing proportions of missing data, also had similar precision for effect estimates. In the absence of bias, the FMI was a better guide to the efficiency gains using MI than the proportion of missing data. ConclusionWe provide evidence that for MAR data, valid MI reduces bias even when the proportion of missingness is large. We advise researchers to use FMI to guide choice of auxiliary variables for efficiency gain in imputation analyses, and that sensitivity analyses including different imputation models may be needed if the number of complete cases is small.

Background:The National Trauma Data Bank (NTDB) is plagued by the problem of missing physiological data. The Glasgow Coma Scale score, Respiratory Rate and Systolic Blood Pressure are an essential part of risk adjustment strategies for trauma system evaluation and clinical research. Missing data on these variables may compromise the feasibility and the validity of trauma group comparisons.Aims:To evaluate the validity of Multiple Imputation (MI) for completing missing physiological data in the National Trauma Data Bank (NTDB), by assessing the impact of MI on 1) frequency distributions, 2) associations with mortality, and 3) risk adjustment.Methods:Analyses were based on 170,956 NTDB observations with complete physiological data (observed data set). Missing physiological data were artificially imposed on this data set and then imputed using MI (MI data set). To assess the impact of MI on risk adjustment, 100 pairs of hospitals were randomly selected with replacement and compared using adjusted Odds Ratios (OR) of mortality. OR generated by the observed data set were then compared to those generated by the MI data set.Results:Frequency distributions and associations with mortality were preserved following MI. The median absolute difference between adjusted OR of mortality generated by the observed data set and by the MI data set was 3.6% (inter-quartile range: 2.4%-6.1%).Conclusions:This study suggests that, provided it is implemented with care, MI of missing physiological data in the NTDB leads to valid frequency distributions, preserves associations with mortality, and does not compromise risk adjustment in inter-hospital comparisons of mortality.

Severe asthma in the US population and eligibility for mAb therapy

Latent class based multiple imputation approach for missing categorical data

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.