Missing Data Indicators Research Articles

Multiple Imputation with Factor Scores: A Practical Approach for Handling Simultaneous Missingness Across Items in Longitudinal Designs

Missingness in intensive longitudinal data triggered by latent factors constitute one type of nonignorable missingness that can generate simultaneous missingness across multiple items on each measurement occasion. To address this issue, we propose a multiple imputation (MI) strategy called MI-FS, which incorporates factor scores, lag/lead variables, and missing data indicators into the imputation model. In the context of process factor analysis (PFA), we conducted a Monte Carlo simulation study to compare the performance of MI-FS to listwise deletion (LD), MI with manifest variables (MI-MV, which implements MI on both dependent variables and covariates), and partial MI with MVs (PMI-MV, which implements MI on covariates and handles missing dependent variables via full-information maximum likelihood) under different conditions. Across conditions, we found MI-based methods overall outperformed the LD; the MI-FS approach yielded lower root mean square errors (RMSEs) and higher coverage rates for auto-regression (AR) parameters compared to MI-MV; and the PMI-MV and MI-MV approaches yielded higher coverage rates for most parameters except AR parameters compared to MI-FS. These approaches were also compared using an empirical example investigating the relationships between negative affect and perceived stress over time. Recommendations on when and how to incorporate factor scores into MI processes were discussed.

Prevalence and associated factors for poor mental health among young migrants in Sweden: a cross-sectional study

ABSTRACT Background Young migrants face multiple challenges that can affect their mental, sexual and reproductive health. Objective To assess the prevalence of self-reported poor mental health and its associated demographic, post-migration and sexual risk behaviour factors among young migrants (aged 15–25) in Sweden. Methods Data were drawn from a cross-sectional survey conducted with migrants aged 15–65 years old in Sweden between December 2018 and November 2019 (n = 6449). Among these, 990 participants aged 15–25 were eligible for the study. Mental health was measured using the Refugee Health Screener-13. Missing data indicator analysis and multivariable logistic regression models were conducted to estimate the association between mental health, sexual risk behaviour, demographic and migration-related variables. Results Of the 990 participants, 59% reported poor mental health. Participants reporting poor mental health were more likely to be female (AOR:1.63, 95% CI:1.18–2.25), to have lived in Sweden more than three years (AOR:2.16, 95% CI:1.17–3.97), to engage in any sexual risk behaviour (AOR:1.99, 95% CI:1.25–3.17), and to live alone (AOR:1.95, 95% CI:1.25–3.03) or with friends they already knew (AOR:1.60, 95% CI:1.37–4.91). People arriving from the Americas (AOR:0.54, 95% CI:0.33–0.88), Asia (AOR:0.44, 95% CI:0.22–0.86), Europe (AOR:0.30, 95% CI:0.14–0.61) and Africa (AOR 0.37, 95% CI: 0.23–0.60) had lower odds of poor mental health than those arriving from Syria. Conclusion The prevalence of poor mental health among young migrants in Sweden was high, with specific subgroups (women, asylum seekers, people arriving from Syria, and those residing longer in Sweden) being particularly vulnerable. Our results indicate the interconnectedness between poor mental health and sexual risk behaviour in this population. Thus, policies targeting young migrants should ensure that healthcare services screen for both poor sexual and mental health at the same time.

Estimating Toll Road Travel Times Using Segment-Based Data Imputation

Efficient and sustainable transportation is crucial for addressing the environmental and social challenges associated with urban mobility. Accurate estimation of travel time plays a pivotal role in traffic management and trip planning. This study focused on leveraging machine learning models to enhance travel time estimation accuracy on toll roads under diverse traffic conditions. Two models were developed for travel time estimation under a variety of traffic conditions on the Don Muang Tollway, Bangkok, Thailand: a long short-term memory (LSTM) recurrent neural network model and a support vector regression (SVR) model. Missing data were treated using the proposed segment-based data imputation method. Unlike other studies, the effects of missing input data on the travel time model performance were also analyzed. Traffic parameters, such as speed and flow, along with other relevant parameters (time of day, day of the week, holiday indicators, and a missing data indicator), were fed into each model to estimate travel time on each of the four specific routes. The LSTM and SVR results had similar performance levels based on evaluating the all-day pooled data. However, the mean absolute percentage errors were lower for LSTM during peak periods, while SVR performed slightly better during off-peak periods. Additionally, LSTM coped substantially better than SVR with unusual traffic fluctuations. The sensitivity analysis of the missing input data in this study also revealed that the LSTM model was more robust to the high degree of missing data than the SVR model.

Multiple imputation with missing data indicators.

Multiple imputation is a well-established general technique for analyzing data with missing values. A convenient way to implement multiple imputation is sequential regression multiple imputation, also called chained equations multiple imputation. In this approach, we impute missing values using regression models for each variable, conditional on the other variables in the data. This approach, however, assumes that the missingness mechanism is missing at random, and it is not well-justified under not-at-random missingness without additional modification. In this paper, we describe how we can generalize the sequential regression multiple imputation imputation procedure to handle missingness not at random in the setting where missingness may depend on other variables that are also missing but not on the missing variable itself, conditioning on fully observed variables. We provide algebraic justification for several generalizations of standard sequential regression multiple imputation using Taylor series and other approximations of the target imputation distribution under missingness not at random. Resulting regression model approximations include indicators for missingness, interactions, or other functions of the missingness not at random missingness model and observed data. In a simulation study, we demonstrate that the proposed sequential regression multiple imputation modifications result in reduced bias in the final analysis compared to standard sequential regression multiple imputation, with an approximation strategy involving inclusion of an offset in the imputation model performing the best overall. The method is illustrated in a breast cancer study, where the goal is to estimate the prevalence of a specific genetic pathogenic variant.

The association of working alliance, outcome expectation, adherence and self-efficacy with clinical outcomes for Achilles tendinopathy: A feasibility cohort study (the MAP study).

This study evaluated the feasibility of a large longitudinal cohort study utilizing an online platform to investigate the association and predictive relationship of working alliance, outcome expectations, adherence and self-efficacy with outcome in Achilles tendinopathy. The objectives were: (1) to determine the recruitment and retention rate and (2) to carry out preliminary data analysis of the selected variables and clinical outcomes. A multi-centred, longitudinal feasibility cohort study was used. Eligible participants were directed to a bespoke online platform hosting study information and the outcome measures in the form of an online questionnaire. Responses from the online questionnaire were collected on three occasions: at baseline, at 6 and at 12 weeks following completion of the first questionnaire. Feasibility outcomes (recruitment and retention rates) were described using descriptive statistics. The website recorded a total 55 views. These 55 views resulted in 24 participants consenting to join the study. The questionnaire at baseline was started 63 times and completed on 60 separate occasions resulting in a 95% conversion rate. Retainment for completion of the questionnaire for a second time was 83.3% and for the third time was 66.6%. All questionnaires were completed fully yielding a missing data indicator of 0%. Feasibility studies ask the question 'can this be done'? Based on the data from recruitment and rates and exploratory correlation analysis a future study can be done; this previously untested online platform appears feasible, but changes could be useful before proceeding to a much larger study.

Diagnosing missing always at random in multivariate data

Summary Models for analysing multivariate datasets with missing values require strong, often unassessable, assumptions. The most common of these is that the mechanism that created the missing data is ignorable, which is a two-fold assumption dependent on the mode of inference. The first part, which is the focus here, under the Bayesian and direct-likelihood paradigms requires that the missing data be missing at random; in contrast, the frequentist-likelihood paradigm demands that the missing data mechanism always produce missing at random data, a condition known as missing always at random. Under certain regularity conditions, assuming missing always at random leads to a condition that can be tested using the observed data alone, namely that the missing data indicators depend only on fully observed variables. In this note we propose three different diagnostic tests that not only indicate when this assumption is incorrect but also suggest which variables are the most likely culprits. Although missing always at random is not a necessary condition to ensure validity under the Bayesian and direct-likelihood paradigms, it is sufficient, and evidence of its violation should encourage the careful statistician to conduct targeted sensitivity analyses.

Identifying change-dropout patterns during an Internet-based intervention for depression by applying the Muthen-Roy model

ABSTRACTTo date, only few studies have attempted to investigate non-ignorable dropout during Internet-based interventions by applying an NMAR model, which includes missing data indicators in its equations. Here, the Muthen-Roy model was used to investigate change and dropout patterns in a sample of patients with mild-to-moderate depression symptoms (N = 483) who were randomized to a 12-week Internet-based intervention (deprexis, identifier: NCT01636752). Participants completed the PHQ-9 biweekly during the treatment. We identified four change-dropout patterns: Participants showing high impairment, improvement and low dropout probability (C3, N = 134) had the highest rate of reliable change at 6- and 12-month follow-up. A further pattern was characterized by high impairment, deterioration and high dropout probability (C2, N = 32), another by low impairment, improvement and high dropout probability (C1, N = 198). The last pattern was characterized by high impairment, no change and low dropout probability (C4, N = 119). In addition to deterioration, also rapid improvement may lead to dropout as a result of a perceived “good enough” dosage of treatment. This knowledge may strengthen sensitivity for the mechanisms of dropout and help to consider its meaning in efforts to optimize treatment selection.

Using missing types to improve partial identification with application to a study of HIV prevalence in Malawi

Frequently, empirical studies are plagued with missing data. When the data are missing not at random, the parameter of interest is not identifiable in general. Without additional assumptions, we can derive bounds of the parameters of interest, which, unfortunately, are often too wide to be informative. Therefore, it is of great importance to sharpen these worst-case bounds by exploiting additional information. Traditional missing data analysis uses only the information of the binary missing data indicator, that is, a certain data point is either missing or not. Nevertheless, real data often provide more information than a binary missing data indicator, and they often record different types of missingness. In a motivating HIV status survey, missing data may be due to the units' unwillingness to respond to the survey items or their hospitalization during the visit, and may also be due to the units' temporarily absence or relocation. It is apparent that some missing types are more likely to be missing not at random, but other missing types are more likely to be missing at random. We show that making full use of the missing types results in narrower bounds of the parameters of interest. In a real-life example, we demonstrate substantial improvement of more than 50% reduction in bound widths for estimating the prevalence of HIV in rural Malawi. As we illustrate using the HIV study, our strategy is also useful for conducting sensitivity analysis by gradually increasing or decreasing the set of types that are missing at random. In addition, we propose an easy-to-implement method to construct confidence intervals for partially identified parameters with bounds expressed as the minimums and maximums of finite parameters, which is useful for not only our problem but also many other problems involving bounds.

Bayesian Modeling and Inference for Nonignorably Missing Longitudinal Binary Response Data with Applications to HIV Prevention Trials.

Missing data are frequently encountered in longitudinal clinical trials. To better monitor and understand the progress over time, one must handle the missing data appropriately and examine whether the missing data mechanism is ignorable or nonignorable. In this article, we develop a new probit model for longitudinal binary response data. It resolves a challenging issue for estimating the variance of the random effects, and substantially improves the convergence and mixing of the Gibbs sampling algorithm. We show that when improper uniform priors are specified for the regression coefficients of the joint multinomial model via a sequence of one-dimensional conditional distributions for the missing data indicators under nonignorable missingness, the joint posterior distribution is improper. A variation of Jeffreys prior is thus established as a remedy for the improper posterior distribution. In addition, an efficient Gibbs sampling algorithm is developed using a collapsing technique. Two model assessment criteria, the deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML), are used to guide the choices of prior specifications and to compare the models under different missing data mechanisms. We report on extensive simulations conducted to investigate the empirical performance of the proposed methods. The proposed methodology is further illustrated using data from an HIV prevention clinical trial.

A Bayesian Hierarchical Selection Model for Academic Growth With Missing Data

ABSTRACTUsing a sample of schools testing annually in grades 9–11 with a vertically linked series of assessments, a latent growth curve model is used to model test scores with student intercepts and slopes nested within school. Missed assessments can occur because of student mobility, student dropout, absenteeism, and other reasons. Missing data indicators are modeled using logistic regression, with grade 9 and potentially unobserved growth scores used as covariates. Under a hierarchical selection model, estimates of school effects on academic growth and missingness are obtained. The results from the selection model are compared to a model that ignores the missing data process.

A Correlated Random Effects Model for Nonignorable Missing Data in Value-Added Assessment of Teacher Effects

Value-added models have been widely used to assess the contributions of individual teachers and schools to students’ academic growth based on longitudinal student achievement outcomes. There is concern, however, that ignoring the presence of missing values, which are common in longitudinal studies, can bias teachers’ value-added scores. In this article, a flexible correlated random effects model is developed that jointly models the student responses and the student missing data indicators. Both the student responses and the missing data mechanism depend on latent teacher effects as well as latent student effects, and the correlation between the sets of random effects adjusts teachers’ value-added scores for informative missing data. The methods are illustrated with data from calculus classes at a large public university and with data from an elementary school district.

On analyzing ordinal data when responses and covariates are both missing at random

In many occasions, particularly in biomedical studies, data are unavailable for some responses and covariates. This leads to biased inference in the analysis when a substantial proportion of responses or a covariate or both are missing. Except a few situations, methods for missing data have earlier been considered either for missing response or for missing covariates, but comparatively little attention has been directed to account for both missing responses and missing covariates, which is partly attributable to complexity in modeling and computation. This seems to be important as the precise impact of substantial missing data depends on the association between two missing data processes as well. The real difficulty arises when the responses are ordinal by nature. We develop a joint model to take into account simultaneously the association between the ordinal response variable and covariates and also that between the missing data indicators. Such a complex model has been analyzed here by using the Markov chain Monte Carlo approach and also by the Monte Carlo relative likelihood approach. Their performance on estimating the model parameters in finite samples have been looked into. We illustrate the application of these two methods using data from an orthodontic study. Analysis of such data provides some interesting information on human habit.

Missing observations in paired comparison data

This paper considers the analysis of paired comparison experiments in the presence of missing responses. Various scenarios for how missing data might arise in paired comparisons are considered, and it is suggested that the most common types of missing data mechanism would be either missing completely at random or missing not at random. A new model is then proposed based on the paired comparison set of responses augmented by a set of missing data indicators for each comparison. Taking a sample selection approach, the proposed new method is based on the classical Bradley-Terry model for the response outcomes and a multinomial model for the missing indicators. Different models for the two missing data mechanisms—missing completely at random (MCAR) and missing not at random (MNAR)—are then discussed and a blockwise composite link formulation is used to construct the likelihood. Additionally, an extension to account for dependence between the paired comparison items is introduced. The methodology is illustrated by a survey paired comparison experiment on five distinct teaching qualities of teachers. We show that there is little evidence of a MNAR process in this dataset. A discussion on the sizes of problems that can be fitted using this approach concludes the paper.

한반도 연안 조위자료의 결측 양상

우리나라 연안 조위자료의 결측양상을 분석 제시하였다. 결측구간은 결측정보지시행렬을 이용하여 전체적인 결측양상을 파악할 수 있도록 도시하였으며, 시간적 공간적인 결측비율도 분석하여 제시하였다. 전반적으로 조위의 결측비율은 낮은 수준이나, 결측이 특정 조위관측소에 집중되는 경향을 보이고 있다. 또한 연속적인 결측자료 발생간격에 대한 자기상관함수를 분석한 결과, 조위자료의 결측은 무작위적으로 발생하고 있는 것으로는 파악되었다. The missing data patterns of tidal elevation data in Korean coasts are analysed and provided. The missing interval of the data is displayed for all stations using the missing data indicator matrix in order to identify the overall missing pattern. The spatial and temporal missing rates are also estimated. The total missing rate of tidal elevation data is low. However, most of the missing is mainly derived from just 1 or 2 specific stations. The autocorrelation function of the consecutive missing interval data also shows that the missing interval occurs randomly.

A Pseudo‐Bayesian Shrinkage Approach to Regression with Missing Covariates

We consider the linear regression of outcome Y on regressors W and Z with some values of W missing, when our main interest is the effect of Z on Y, controlling for W. Three common approaches to regression with missing covariates are (i) complete-case analysis (CC), which discards the incomplete cases, and (ii) ignorable likelihood methods, which base inference on the likelihood based on the observed data, assuming the missing data are missing at random (Rubin, 1976b), and (iii) nonignorable modeling, which posits a joint distribution of the variables and missing data indicators. Another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling (DV, for drop variables). DV does not lead to bias when either (i) the regression coefficient of W is zero or (ii) W and Z are uncorrelated. We propose a pseudo-Bayesian approach for regression with missing covariates that compromises between the CC and DV estimates, exploiting information in the incomplete cases when the data support DV assumptions. We illustrate favorable properties of the method by simulation, and apply the proposed method to a liver cancer study. Extension of the method to more than one missing covariate is also discussed.

Block-Conditional Missing at Random Models for Missing Data

Two major ideas in the analysis of missing data are (a) the EM algorithm [Dempster, Laird and Rubin, J. Roy. Statist. Soc. Ser. B 39 (1977) 1–38] for maximum likelihood (ML) estimation, and (b) the formulation of models for the joint distribution of the data Z and missing data indicators M, and associated “missing at random” (MAR) condition under which a model for M is unnecessary [Rubin, Biometrika 63 (1976) 581–592]. Most previous work has treated Z and M as single blocks, yielding selection or pattern-mixture models depending on how their joint distribution is factorized. This paper explores “block-sequential” models that interleave subsets of the variables and their missing data indicators, and then make parameter restrictions based on assumptions in each block. These include models that are not MAR. We examine a subclass of block-sequential models we call block-conditional MAR (BCMAR) models, and an associated block-monotone reduced likelihood strategy that typically yields consistent estimates by selectively discarding some data. Alternatively, full ML estimation can often be achieved via the EM algorithm. We examine in some detail BCMAR models for the case of two multinomially distributed categorical variables, and a two block structure where the first block is categorical and the second block arises from a (possibly multivariate) exponential family distribution.

Weighted Generalized Estimating Functions for Longitudinal Response and Covariate Data That Are Missing at Random

Longitudinal studies often feature incomplete response and covariate data. It is well known that biases can arise from naive analyses of available data, but the precise impact of incomplete data depends on the frequency of missing data and the strength of the association between the response variables and covariates and the missing-data indicators. Various factors may influence the availability of response and covariate data at scheduled assessment times, and at any given assessment time the response may be missing, covariate data may be missing, or both response and covariate data may be missing. Here we show that it is important to take the association between the missing data indicators for these two processes into account through joint models. Inverse probability-weighted generalized estimating equations offer an appealing approach for doing this. Here we develop these equations for a particular model generating intermittently missing-at-random data. Empirical studies demonstrate that the consistent estimators arising from the proposed methods have very small empirical biases in moderate samples. Supplemental materials are available online.

The performance of multiple imputation for missing covariate data within the context of regression relative survival analysis

Relative survival assesses the effects of prognostic factors on disease-specific mortality when the cause of death is uncertain or unavailable. It provides an estimate of patients' survival, allowing for the effects of other independent causes of death. Regression-based relative survival models are commonly used in population-based studies to model the effects of some prognostic factors and to estimate net survival. Most often, studies focus on routinely collected prognostic factors for which the proportion of missing values is usually low (around 5 per cent). However, in some cases, additional factors are collected with a greater proportion of missingness. In the present article, we systematically assess the performance of multiple imputation in regression analysis of relative survival through a series of simulation experiments. According to the assumptions concerning the missingness mechanism (completely at random, at random, and not at random) and the missingness pattern (monotone, non-monotone), several strategies were considered and compared: all cases analysis, complete cases analysis, missing data indicator analysis, and multiple imputation by chained equations (MICE) analysis. We showed that MICE performs well in estimating the hazard ratios and the baseline hazard function when the missing mechanism is missing at random (MAR) conditionally on the vital status. In the situations where the missing mechanism was not MAR conditionally on vital status, complete case behaves consistently. As illustration, we used data of the French Cancer Registries on relative survival of patients with colorectal cancer.

Modeling Nonignorable Missing Data in Speeded Tests

In tests with time limits, items at the end are often not reached. Usually, the pattern of missing responses depends on the ability level of the respondents; therefore, missing data are not ignorable in statistical inference. This study models data using a combination of two item response theory (IRT) models: one for the observed response data and one for the missing data indicator. The missing data indicator is modeled using a sequential model with linear restrictions on the item parameters. The models are connected by the assumption that the respondents' latent proficiency parameters have a joint multivariate normal distribution. Model parameters are estimated by maximum marginal likelihood. Simulations show that treating missing data as ignorable can lead to considerable bias in parameter estimates. Including an IRT model for the missing data indicator removes this bias. The method is illustrated with data from an intelligence test with a time limit.

A copula model for repeated measurements with non-ignorable non-monotone missing outcome

A normal copula-based selection model is proposed for continuous longitudinal data with a non-ignorable non-monotone missing-data process. The normal copula is used to combine the distribution of the outcome of interest and that of the missing-data indicators given the covariates. Parameters in the model are estimated by a pseudo-likelihood method. We first use the GEE with a logistic link to estimate the parameters associated with the marginal distribution of the missing-data indicator given the covariates, assuming that covariates are always observed. Then we estimate other parameters by inserting the estimates from the first step into the full likelihood function. A simulation study is conducted to assess the robustness of the assumed model under different missing-data processes. The proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias.