Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data

Summary Often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value. In the event that the underlying continuous measurements are from a longitudinal study, one can use the repeated‐measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status. Standard Bayesian multiple imputation techniques ( Rubin, 1987 , in Multiple Imputation for Nonresponse in Surveys) that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within‐ and between‐imputation variances are found to be conservative. The frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of Robins and Wang (2000, Biometrika 87, 113–124) is shown to be more efficient. We propose to apply ( Kenward and Roger, 1997 , Biometrics 53, 983–997) degrees of freedom to account for the uncertainty associated with variance–covariance parameter estimates for the repeated measures model.     

Gaussian mixture clustering and imputation of microarray data

MOTIVATION: In microarray experiments, missing entries arise from blemishes on the chips. In large-scale studies, virtually every chip contains some missing entries and more than 90% of the genes are affected. Many analysis methods require a full set of data. Either those genes with missing entries are excluded, or the missing entries are filled with estimates prior to the analyses. This study compares methods of missing value estimation. RESULTS: Two evaluation metrics of imputation accuracy are employed. First, the root mean squared error measures the difference between the true values and the imputed values. Second, the number of mis-clustered genes measures the difference between clustering with true values and that with imputed values; it examines the bias introduced by imputation to clustering. The Gaussian mixture clustering with model averaging imputation is superior to all other imputation methods, according to both evaluation metrics, on both time-series (correlated) and non-time series (uncorrelated) data sets.     

Extension of Peto's Test by Attribution of Tumor Lethality in the Absence of Cause‐of‐Death Information

A new statistical testing approach is developed for rodent tumorigenicity assays that have a single terminal sacrifice or occasionally interim sacrifices but not cause‐of‐death data. For experiments that lack cause‐of‐death data, statistically imputed numbers of fatal tumors and incidental tumors are used to modify Peto's cause‐of‐death test which is usually implemented using pathologist‐assigned cause‐of‐death information. The numbers of fatal tumors are estimated using a constrained nonparametric maximum likelihood estimation method. A new Newton‐based approach under inequality constraints is proposed for finding the global maximum likelihood estimates. In this study, the proposed method is concentrated on data with a single sacrifice experiment without implementing further assumptions. The new testing approach may be more reliable than Peto's test because of the potential for a misclassification of cause‐of‐death by pathologists. A Monte Carlo simulation study for the proposed test is conducted to assess size and power of the test. Asymptotic normality for the statistic of the proposed test is also investigated. The proposed testing approach is illustrated using a real data set.     

Treatment of nonignorable missing data when modeling unobserved heterogeneity with finite mixture models

Multiple imputation has become a widely accepted technique to deal with the problem of incomplete data. Typically, imputation of missing values and the statistical analysis are performed separately. Therefore, the imputation model has to be consistent with the analysis model. If the data are analyzed with a mixture model, the parameter estimates are usually obtained iteratively. Thus, if the data are missing not at random, parameter estimation and treatment of missingness should be combined. We solve both problems by simultaneously imputing values using the data augmentation method and estimating parameters using the EM algorithm. This iterative procedure ensures that the missing values are properly imputed given the current parameter estimates. Properties of the parameter estimates were investigated in a simulation study. The results are illustrated using data from the National Health and Nutrition Examination Survey.     

Analysis of Longitudinal Clinical Trials with Missing Data Using Multiple Imputation in Conjunction with Robust Regression

Summary In a typical randomized clinical trial, a continuous variable of interest (e.g., bone density) is measured at baseline and fixed postbaseline time points. The resulting longitudinal data, often incomplete due to dropouts and other reasons, are commonly analyzed using parametric likelihood‐based methods that assume multivariate normality of the response vector. If the normality assumption is deemed untenable, then semiparametric methods such as (weighted) generalized estimating equations are considered. We propose an alternate approach in which the missing data problem is tackled using multiple imputation, and each imputed dataset is analyzed using robust regression (M‐estimation; Huber, 1973 , Annals of Statistics 1, 799–821.) to protect against potential non‐normality/outliers in the original or imputed dataset. The robust analysis results from each imputed dataset are combined for overall estimation and inference using either the simple Rubin (1987 , Multiple Imputation for Nonresponse in Surveys, New York: Wiley) method, or the more complex but potentially more accurate Robins and Wang (2000 , Biometrika 87, 113–124.) method. We use simulations to show that our proposed approach performs at least as well as the standard methods under normality, but is notably better under both elliptically symmetric and asymmetric non‐normal distributions. A clinical trial example is used for illustration.     

Integrated survival analysis using an event‐time approach in a Bayesian framework

Event‐time or continuous‐time statistical approaches have been applied throughout the biostatistical literature and have led to numerous scientific advances. However, these techniques have traditionally relied on knowing failure times. This has limited application of these analyses, particularly, within the ecological field where fates of marked animals may be unknown. To address these limitations, we developed an integrated approach within a Bayesian framework to estimate hazard rates in the face of unknown fates. We combine failure/survival times from individuals whose fates are known and times of which are interval‐censored with information from those whose fates are unknown, and model the process of detecting animals with unknown fates. This provides the foundation for our integrated model and permits necessary parameter estimation. We provide the Bayesian model, its derivation, and use simulation techniques to investigate the properties and performance of our approach under several scenarios. Lastly, we apply our estimation technique using a piece‐wise constant hazard function to investigate the effects of year, age, chick size and sex, sex of the tending adult, and nesting habitat on mortality hazard rates of the endangered mountain plover (Charadrius montanus) chicks. Traditional models were inappropriate for this analysis because fates of some individual chicks were unknown due to failed radio transmitters. Simulations revealed biases of posterior mean estimates were minimal (≤ 4.95%), and posterior distributions behaved as expected with RMSE of the estimates decreasing as sample sizes, detection probability, and survival increased. We determined mortality hazard rates for plover chicks were highest at <5 days old and were lower for chicks with larger birth weights and/or whose nest was within agricultural habitats. Based on its performance, our approach greatly expands the range of problems for which event‐time analyses can be used by eliminating the need for having completely known fate data.     

Multiple imputation for missing values through conditional Semiparametric odds ratio models

Multiple imputation is a practically useful approach to handling incompletely observed data in statistical analysis. Parameter estimation and inference based on imputed full data have been made easy by Rubin's rule for result combination. However, creating proper imputation that accommodates flexible models for statistical analysis in practice can be very challenging. We propose an imputation framework that uses conditional semiparametric odds ratio models to impute the missing values. The proposed imputation framework is more flexible and robust than the imputation approach based on the normal model. It is a compatible framework in comparison to the approach based on fully conditionally specified models. The proposed algorithms for multiple imputation through the Markov chain Monte Carlo sampling approach can be straightforwardly carried out. Simulation studies demonstrate that the proposed approach performs better than existing, commonly used imputation approaches. The proposed approach is applied to imputing missing values in bone fracture data.     

Estimations of linkage disequilibrium,effective population size and ROH‐based inbreeding coefficients in Spanish Churra sheep using imputed high‐density SNP genotypes

In this study, the availability of the Ovine HD SNP BeadChip (HD‐chip) and the development of an imputation strategy provided an opportunity to further investigate the extent of linkage disequilibrium (LD) at short distances in the genome of the Spanish Churra dairy sheep breed. A population of 1686 animals, including 16 rams and their half‐sib daughters, previously genotyped for the 50K‐chip, was imputed to the HD‐chip density based on a reference population of 335 individuals. After assessing the imputation accuracy for beagle v4.0 (0.922) and fimpute v2.2 (0.921) using a cross‐validation approach, the imputed HD‐chip genotypes obtained with beagle were used to update the estimates of LD and effective population size for the studied population. The imputed genotypes were also used to assess the degree of homozygosity by calculating runs of homozygosity and to obtain genomic‐based inbreeding coefficients. The updated LD estimations provided evidence that the extent of LD in Churra sheep is even shorter than that reported based on the 50K‐chip and is one of the shortest extents compared with other sheep breeds. Through different comparisons we have also assessed the impact of imputation on LD and effective population size estimates. The inbreeding coefficient, considering the total length of the run of homozygosity, showed an average estimate (0.0404) lower than the critical level. Overall, the improved accuracy of the updated LD estimates suggests that the HD‐chip, combined with an imputation strategy, offers a powerful tool that will increase the opportunities to identify genuine marker‐phenotype associations and to successfully implement genomic selection in Churra sheep.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.     

Rounding Strategies for Multiply Imputed Binary Data

Multiple imputation (MI) has emerged in the last two decades as a frequently used approach in dealing with incomplete data. Gaussian and log‐linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings that include a mix of continuous and discrete variables, the lack of flexible models for the joint distribution of different types of variables can make the specification of the imputation model a daunting task. The widespread availability of software packages that are capable of carrying out MI under the assumption of joint multivariate normality allows applied researchers to address this complication pragmatically by treating the discrete variables as continuous for imputation purposes and subsequently rounding the imputed values to the nearest observed category. In this article, we compare several rounding rules for binary variables based on simulated longitudinal data sets that have been used to illustrate other missing‐data techniques. Using a combination of conditional and marginal data generation mechanisms and imputation models, we study the statistical properties of multiple‐imputation‐based estimates for various population quantities under different rounding rules from bias and coverage standpoints. We conclude that a good rule should be driven by borrowing information from other variables in the system rather than relying on the marginal characteristics and should be relatively insensitive to imputation model specifications that may potentially be incompatible with the observed data. We also urge researchers to consider the applied context and specific nature of the problem, to avoid uncritical and possibly inappropriate use of rounding in imputation models.     

Evaluation of measures of correctness of genotype imputation in the context of genomic prediction: a review of livestock applications

In livestock, many studies have reported the results of imputation to 50k single nucleotide polymorphism (SNP) genotypes for animals that are genotyped with low-density SNP panels. The objective of this paper is to review different measures of correctness of imputation, and to evaluate their utility depending on the purpose of the imputed genotypes. Across studies, imputation accuracy, computed as the correlation between true and imputed genotypes, and imputation error rates, that counts the number of incorrectly imputed alleles, are commonly used measures of imputation correctness. Based on the nature of both measures and results reported in the literature, imputation accuracy appears to be a more useful measure of the correctness of imputation than imputation error rates, because imputation accuracy does not depend on minor allele frequency (MAF), whereas imputation error rate depends on MAF. Therefore imputation accuracy can be better compared across loci with different MAF. Imputation accuracy depends on the ability of identifying the correct haplotype of a SNP, but many other factors have been identified as well, including the number of genotyped immediate ancestors, the number of animals with genotypes at the high-density panel, the SNP density on the low- and high-density panel, the MAF of the imputed SNP and whether imputed SNP are located at the end of a chromosome or not. Some of these factors directly contribute to the linkage disequilibrium between imputed SNP and SNP on the low-density panel. When imputation accuracy is assessed as a predictor for the accuracy of subsequent genomic prediction, we recommend that: (1) individual-specific imputation accuracies should be used that are computed after centring and scaling both true and imputed genotypes; and (2) imputation of gene dosage is preferred over imputation of the most likely genotype, as this increases accuracy and reduces bias of the imputed genotypes and the subsequent genomic predictions.     

Estimating hazard ratios in cohort data with missing disease information due to death

In clinical and epidemiological studies information on the primary outcome of interest, that is, the disease status, is usually collected at a limited number of follow‐up visits. The disease status can often only be retrieved retrospectively in individuals who are alive at follow‐up, but will be missing for those who died before. Right‐censoring the death cases at the last visit (ad‐hoc analysis) yields biased hazard ratio estimates of a potential risk factor, and the bias can be substantial and occur in either direction. In this work, we investigate three different approaches that use the same likelihood contributions derived from an illness‐death multistate model in order to more adequately estimate the hazard ratio by including the death cases into the analysis: a parametric approach, a penalized likelihood approach, and an imputation‐based approach. We investigate to which extent these approaches allow for an unbiased regression analysis by evaluating their performance in simulation studies and on a real data example. In doing so, we use the full cohort with complete illness‐death data as reference and artificially induce missing information due to death by setting discrete follow‐up visits. Compared to an ad‐hoc analysis, all considered approaches provide less biased or even unbiased results, depending on the situation studied. In the real data example, the parametric approach is seen to be too restrictive, whereas the imputation‐based approach could almost reconstruct the original event history information.     

The adequacy of aging techniques in vertebrates for rapid estimation of population mortality rates from age distributions

As a key parameter in population dynamics, mortality rates are frequently estimated using mark–recapture data, which requires extensive, long‐term data sets. As a potential rapid alternative, we can measure variables correlated to age, allowing the compilation of population age distributions, from which mortality rates can be derived. However, most studies employing such techniques have ignored their inherent inaccuracy and have thereby failed to provide reliable mortality estimates. In this study, we present a general statistical model linking birth rate, mortality rate, and population age distributions. We next assessed the reliability and data needs (i.e., sample size) for estimating mortality rate of eight different aging techniques. The results revealed that for half of the aging techniques, correlations with age varied considerably, translating into highly variable accuracies when used to estimate mortality rate from age distributions. Telomere length is generally not sufficiently correlated to age to provide reliable mortality rate estimates. DNA methylation, signal‐joint T‐cell recombination excision circle (sjTREC), and racemization are generally more promising techniques to ultimately estimate mortality rate, if a sufficiently high sample size is available. Otolith ring counts, otolithometry, and age‐length keys in fish, and skeletochronology in reptiles, mammals, and amphibians, outperformed all other aging techniques and generated relatively accurate mortality rate estimation with a sample size that can be feasibly obtained. Provided the method chosen is minimizing and estimating the error in age estimation, it is possible to accurately estimate mortality rates from age distributions. The method therewith has the potential to estimate a critical, population dynamic parameter to inform conservation efforts within a limited time frame as opposed to mark–recapture analyses.     

Validating marker-based incidence estimates in repeatedly screened populations

Disease incidence (new cases of disease per person per year) is usually measured by using longitudinal data. However, several recent proposals for measuring the incidence of human immunodeficiency virus (HIV) rely on cross-sectional data only. These methods assume each person is only sampled once; however, in some instances, it is necessary to consider these cross-sectional methods when individuals are represented more than once in the survey sample. We derive an extension of the cross-sectional incidence estimator that is valid for data from repeatedly screened populations and show under what conditions our new estimator reduces to the old estimator. An example involving estimation of HIV incidence among repeat blood donors is presented.     

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

When responding to infectious disease outbreaks, rapid and accurate estimation of the epidemic trajectory is critical. However, two common data collection problems affect the reliability of the epidemiological data in real time: missing information on the time of first symptoms, and retrospective revision of historical information, including right censoring. Here, we propose an approach to construct epidemic curves in near real time that addresses these two challenges by 1) imputation of dates of symptom onset for reported cases using a dynamically-estimated “backward” reporting delay conditional distribution, and 2) adjustment for right censoring using the NobBS software package to nowcast cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number (R_t) in real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We evaluate how these real-time estimates compare with more complete epidemiological data that became available later. We explore the impact of the different assumptions on the estimates, and compare our estimates with those obtained from commonly used surveillance approaches. Our framework can help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health systems in other locations.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Doubly robust multiple imputation using kernel‐based techniques

We consider the problem of estimating the marginal mean of an incompletely observed variable and develop a multiple imputation approach. Using fully observed predictors, we first establish two working models: one predicts the missing outcome variable, and the other predicts the probability of missingness. The predictive scores from the two models are used to measure the similarity between the incomplete and observed cases. Based on the predictive scores, we construct a set of kernel weights for the observed cases, with higher weights indicating more similarity. Missing data are imputed by sampling from the observed cases with probability proportional to their kernel weights. The proposed approach can produce reasonable estimates for the marginal mean and has a double robustness property, provided that one of the two working models is correctly specified. It also shows some robustness against misspecification of both models. We demonstrate these patterns in a simulation study. In a real‐data example, we analyze the total helicopter response time from injury in the Arizona emergency medical service data.     

Non-iterative, regression-based estimation of haplotype associations with censored survival outcomes

The general availability of reliable and affordable genotyping technology has enabled genetic association studies to move beyond small case-control studies to large prospective studies. For prospective studies, genetic information can be integrated into the analysis via haplotypes, with focus on their association with a censored survival outcome. We develop non-iterative, regression-based methods to estimate associations between common haplotypes and a censored survival outcome in large cohort studies. Our non-iterative methods--weighted estimation and weighted haplotype combination--are both based on the Cox regression model, but differ in how the imputed haplotypes are integrated into the model. Our approaches enable haplotype imputation to be performed once as a simple data-processing step, and thus avoid implementation based on sophisticated algorithms that iterate between haplotype imputation and risk estimation. We show that non-iterative weighted estimation and weighted haplotype combination provide valid tests for genetic associations and reliable estimates of moderate associations between common haplotypes and a censored survival outcome, and are straightforward to implement in standard statistical software. We apply the methods to an analysis of HSPB7-CLCNKA haplotypes and risk of adverse outcomes in a prospective cohort study of outpatients with chronic heart failure.     

Mortality rates in subjects with fetal alcohol spectrum disorders and their siblings

BACKGROUND: Our objective was to estimate the mortality rate in subjects with fetal alcohol spectrum disorders (FASD) and their siblings whose FASD status was unknown. METHODS: We used the state FASD Registry to link subjects with FASD to a North Dakota birth certificate. We were able to link 304 of 486 cases (63%). We used the birth certificates to identify the mother and children born to the mother (siblings). We then searched for death certificates for both the FASD cases and their siblings. We then calculated the annual and age‐adjusted mortality rates for the siblings of the Registry cases and compared them with mortality rates from North Dakota. RESULTS: The FASD case mortality rate was 2.4%, with a 4.5% mortality rate for their sibings, accounting for 14% of all deaths when compared to the North Dakota residents matched by age and year of death. The sibling deaths accounted for 21.5% of all cause mortality matched by age and year of death. The age‐standardized mortality ratios were 4.9 for the FASD cases and 2.6 for their siblings whose FASD status was unknown. CONCLUSIONS: Mortality rates for FASD cases and their siblings were increased and represent a substantial proportion of all cause mortality in North Dakota. Prevention of FASD may be a useful strategy to decrease mortality. Birth Defects Research (Part A), 2008. © 2008 Wiley‐Liss, Inc.     

Latent class regression: Inference and estimation with two‐stage multiple imputation

Latent class regression (LCR) is a popular method for analyzing multiple categorical outcomes. While nonresponse to the manifest items is a common complication, inferences of LCR can be evaluated using maximum likelihood, multiple imputation, and two‐stage multiple imputation. Under similar missing data assumptions, the estimates and variances from all three procedures are quite close. However, multiple imputation and two‐stage multiple imputation can provide additional information: estimates for the rates of missing information. The methodology is illustrated using an example from a study on racial and ethnic disparities in breast cancer severity.