Score test for missing at random or not under logistic missingness models.

In longitudinal studies, missing data are ubiquitous. This article made a comparison of three model-based techniques for handling different types of missing data (i.e., missing at random (MAR)-based maximum likelihood (ML) approach, missing not at random (MNAR)-based Diggle–Kenward (DK) selection model and MNAR-based pattern mixture (PM) model) in longitudinal studies through a Monte Carlo simulation study. Two influential factors were considered: the dropout rates (5%, 10%, 20%, and 40%) and the sample sizes (100, 300, 500, and 1000) under MAR and MNAR missingness mechanisms respectively. The results indicated that the model selection was a crucial issue when researchers were dealing with missing data in longitudinal studies because under MNAR mechanism, DK method outperformed MAR-based ML approach, but PM method performed worse than MAR-based method did. The differences of the parameter estimation among three methods became more significant as the sample size and the dropout rate increased.

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.

A hybrid return to baseline imputation method to incorporate MAR and MNAR dropout missingness

Count data with extra zeros are common in many biomedical applications and the zero-inflated generalized power series (ZIGPS) distribution may be appropriate, in which the baseline discrete distribution is a generalized power series distribution, which is a natural extension of power series distribution. In this article, a Monte Carlo EM (MCEM) algorithm is proposed to obtain maximum likelihood estimates and standard errors for ZIGPS distribution with missing responses. A Monte Carlo approximation method combining the Gibbs sampler and M-H algorithm is used to implement E-step, whereas the M-step is completed via the method of conditional maximization. As classical model selection procedure such as Akaike's information criterion (AIC) becomes problematic for our considered ZIGPS distribution with incomplete data, some variations on AIC are presented under two different missingness mechanisms, namely, missing at random (MAR) and missing not at random (MNAR), respectively. The most attractive point is that our methods cannot only be used to select distributions and variables, but also can be used to find out whether there exists zero-inflation or not, which is conducted via score test or Bayesian test in previous literature. Finally, a simulation study and a real example are used to illustrate the proposed methodology.

In the analyses of incomplete longitudinal clinical trial data, there has been a shift, away from simple ad hoc methods that are valid only if the data are missing completely at random (MCAR), to more principled (likelihood-based or Bayesian) ignorable analyses, which are valid under the less restrictive missing at random (MAR) assumption. The availability of the necessary standard statistical software allows for such analyses in practice. Although the possibility of data missing not at random (MNAR) cannot be ruled out, it is argued that analyses valid under MNAR are not well suited for the primary analysis in clinical trials. Therefore, rather than either forgetting about or blindly shifting to an MNAR framework, the optimal place for MNAR analyses is within a sensitivity analysis context. Such analyses can be used, for example, to assess how sensitive results from an ignorable analysis are to possible departures from MAR and how much results are affected by influential observations. In this article, we apply the local influence sensitivity tool (Verbeke et al., 2001) to a longitudinal depression trial, thereby applying it to continuous outcomes from clinical trials.

We analytically derive the fixed-effects estimates in unconditional linear growth curve models by typical linear mixed-effects modelling (TLME) and by a pattern-mixture (PM) approach with random-slope-dependent two-missing-pattern missing not at random (MNAR) longitudinal data. Results showed that when the missingness mechanism is random-slope-dependent MNAR, TLME estimates of both the mean intercept and mean slope are biased because of incorrect weights used in the estimation. More specifically, the estimate of the mean slope is biased towards the mean slope for completers, whereas the estimate of the mean intercept is biased towards the opposite direction as compared to the estimate of the mean slope. We also discuss why the PM approach can provide unbiased fixed-effects estimates for random-coefficients-dependent MNAR data but does not work well for missing at random or outcome-dependent MNAR data. A small simulation study was conducted to illustrate the results and to compare results from TLME and PM. Results from an empirical data analysis showed that the conceptual finding can be generalized to other real conditions even when some assumptions for the analytical derivation cannot be met. Implications from the analytical and empirical results were discussed and sensitivity analysis was suggested for longitudinal data analysis with missing data.

Missing data are uncollected data but meaningful for the statistical analysis due to clinical relevancy of the data for properly specified estimands in clinical trials. Meanwhile the efforts to prevent or minimize missing data are commonly applied in clinical trials, in practice, missing data still occurs. Choosing a statistical method for imputation that deals with missing data targeting specified estimands provides the more reliable estimates of treatment effects. We considered longitudinal clinical settings that have different degrees of missing data and treatment effects, and simulated different missing mechanisms using data from randomized, double-blind, placebo-controlled phase 3 confirmatory clinical trials of approved drugs. We compared four commonly used statistical methods to deal with missing data in clinical trials. We find that, when the data are missing not at random (MNAR) with higher missing rates, mixed model for repeated measurements (MMRM) method overestimates treatment difference. Pattern-mixture model estimates were seen to be more conservative in our studies than MMRM given MNAR assumptions, which are more realistic with missing datain clinical trials. We emphasize the importance of prevention of missing data and specifying the estimand based on trial objectives beforehand. The specified proper estimand and the proper statistical method might be key features to value the clinical trial results despite missing data.

e15003 Background: Stella Trial is a phase III, multinational, double-blind and randomized study to confirm clinical similarity between MB02 and EU-bevacizumab in patients with stage IIIB/IV no squamous NSCLC. 627 subjects with newly diagnosed or recurrent stage IIIb/IV NSCLC were randomized 1:1 to receive either MB02 or EU-bevacizumab plus chemotherapy (paclitaxel and carboplatin) every 3-week cycle for six cycles (week 18) followed by MB02/bevacizumab in blinded monotherapy until disease progression, treatment intolerance, death, patient withdrawal or end of study (w52). An approach targeting the MI of the continuous sum of target diameters (mm), and subsequent categorization of Overall Response (OR) was performed as sensitivity analysis for the assessment of the primary endpoint. Multiple Imputation (MI) has emerged as a credible method to assess the effects of missing data (MD), an inescapable problem with a potential ability to undermine research results' strength and validity in a clinical study, providing the user with a valuable toolset to sufficiently account for the varying types of MD and appropriately adjust the assumptions. Methods: The primary endpoint of the study was Objective Response Rate (ORR) at week 18 per an Independent Radiological Committee (IRC) in the Intention to treat set (ITT). The ORR was analysed with a Cochran-Mantel-Haenszel model, including the stratification factors of sex, smoking status, disease diagnosis and disease stage, comparing the stratified estimates risk difference (RD). In addition, these results were analysed implementing a pattern-mixture MI process accounting for missing at random (MAR) and missing not at random (MNAR) data based on the sum of target diameters in subjects without tumour response data or falling into the categories NonCR/NonPD or non-evaluable (NE). Results: The clinical equivalence of MB02 with EU-bevacizumab is demonstrated by the efficacy data provided through the primary endpoint and the MI sensitive analysis applied. The MI (MAR and MNAR) represents an added value that supports the biosimilarity of MB02 and EU-bevacizumab. The results for the primary analysis of the RD in ORR at w18 in the ITT set were entirely contained within the boundaries of predefined margins (±12): -4.02 [95% CI: -11.76 to 3.71]). Under MI, the ORR RD showed similarity at 95% CIs (-1.92; 95% CI: -10.02 to 6.19) and using the multiple imputations for subjects without tumour response data (missing, NonCR/NonPD or NE), the ORR RD was -2.22 with 95% CI of (-10.54 to 6.10) at w18 in the ITT. Conclusions: The clinical equivalence of MB02 with EU-bevacizumab is demonstrated by the efficacy data provided through the primary endpoint and the MI sensitive analysis applied. The MI (MAR and MNAR) represent an added value that supports the biosimilarity of MB02 and EU-bevacizumab. Clinical trial information: NCT03296163.

Randomized clinical trials with outcome measured longitudinally are frequently analyzed using either random effect models or generalized estimating equations. Both approaches assume that the dropout mechanism is missing at random (MAR) or missing completely at random (MCAR). We propose a Bayesian pattern-mixture model to incorporate missingness mechanisms that might be missing not at random (MNAR), where the distribution of the outcome measure at the follow-up time , conditional on the prior history, differs across the patterns of missing data. We then perform sensitivity analysis on estimates of the parameters of interest. The sensitivity parameters relate the distribution of the outcome of interest between subjects from a missing-data pattern at time with that of the observed subjects at time . The large number of the sensitivity parameters is reduced by treating them as random with a prior distribution having some pre-specified mean and variance, which are varied to explore the sensitivity of inferences. The missing at random (MAR) mechanism is a special case of the proposed model, allowing a sensitivity analysis of deviations from MAR. The proposed approach is applied to data from the Trial of Preventing Hypertension.

Chapter 14 - Methods for handling missing data

Competence data from low‐stakes educational large‐scale assessment studies allow for evaluating relationships between competencies and other variables. The impact of item‐level nonresponse has not been investigated with regard to statistics that determine the size of these relationships (e.g., correlations, regression coefficients). Classical approaches such as ignoring missing values or treating them as incorrect are currently applied in many large‐scale studies, while recent model‐based approaches that can account for nonignorable nonresponse have been developed. Estimates of item and person parameters have been demonstrated to be biased for classical approaches when missing data are missing not at random (MNAR). In our study, we focus on parameter estimates of the structural model (i.e., the true regression coefficient when regressing competence on an explanatory variable), simulating data according to various missing data mechanisms. We found that model‐based approaches and ignoring missing values performed well in retrieving regression coefficients even when we induced missing data that were MNAR. Treating missing values as incorrect responses can lead to substantial bias. We demonstrate the validity of our approach empirically and discuss the relevance of our results.

Dropout is a common complication in longitudinal studies, especially since the distinction between missing not at random (MNAR) and missing at random (MAR) dropout is intractable. Consequently, one starts with an analysis that is valid under MAR and then performs a sensitivity analysis by considering MNAR departures from it. To this end, specific classes of joint models, such as pattern-mixture models (PMMs) and selection models (SeMs), have been proposed. On the contrary, shared-parameter models (SPMs) have received less attention, possibly because they do not embody a characterization of MAR. A few approaches to achieve MAR in SPMs exist, but are difficult to implement in existing software. In this article, we focus on SPMs for incomplete longitudinal and time-to-dropout data and propose an alternative characterization of MAR by exploiting the conditional independence assumption, under which outcome and missingness are independent given a set of random effects. By doing so, the censoring distribution can be utilized to cover a wide range of assumptions for the missing data mechanism on the subject-specific level. This approach offers substantial advantages over its counterparts and can be easily implemented in existing software. More specifically, it offers flexibility over the assumption for the missing data generating mechanism that governs dropout by allowing subject-specific perturbations of the censoring distribution, whereas in PMMs and SeMs dropout is considered MNAR strictly.

Incomplete data is a pervasive challenge across a wide range of data types, including tabular, sensor, time-series, image, and textual data. Its presence stems from various real-world factors and gives rise to different missingness mechanisms. While much of the existing research focuses on the Missing Completely At Random (MCAR) assumption, the more complex and realistic mechanisms-Missing At Random (MAR) and Missing Not At Random (MNAR)-remain relatively underexplored despite their prevalence and impact. This PhD project aims to systematically investigate the challenges posed by diverse Incomplete data mechanisms and to develop robust machine learning methods that can perform reliably across MCAR, MAR, and MNAR scenarios. The research spans multiple data modalities and focuses on improving both the theoretical understanding and practical handling of incomplete data. By addressing mechanism-specific imputation challenges and proposing broadly applicable solutions, this work contributes to building more resilient and trustworthy data-driven systems in real-world settings.

Objective To explore the optimal fitting path of missing data of the Scale to make the fitting data close to the real situation of patients' data. Methods Based on the complete data set of the SDS of 507 patients with stroke, the data simulation sets of Missing Completely at Random (MCAR), Missing at Random (MAR), and Missing Not at Random (MNAR) were constructed by R software, respectively, with missing rates of 5%, 10%, 15%, 20%, 25%, 30%, 35%, and 40% under three missing mechanisms. Mean substitution (MS), random forest regression (RFR), and predictive mean matching (PMM) were used to fit the data. Root mean square error (RMSE), the width of 95% confidence intervals (95% CI), and Spearman correlation coefficient (SCC) were used to evaluate the fitting effect and determine the optimal fitting path. Results when dealing with the problem of missing data in scales, the optimal fitting path is ① under the MCAR deletion mechanism, when the deletion proportion is less than 20%, the MS method is the most convenient; when the missing ratio is greater than 20%, RFR algorithm is the best fitting method. ② Under the Mar mechanism, when the deletion ratio is less than 35%, the MS method is the most convenient. When the deletion ratio is greater than 35%, RFR has a better correlation. ③ Under the mechanism of MNAR, RFR is the best data fitting method, especially when the missing proportion is greater than 30%. In reality, when the deletion ratio is small, the complete case deletion method is the most commonly used, but the RFR algorithm can greatly expand the application scope of samples and save the cost of clinical research when the deletion ratio is less than 30%. The best way to deal with data missing should be based on the missing mechanism and proportion of actual data, and choose the best method between the statistical analysis ability of the research team, the effectiveness of the method, and the understanding of readers.

Conventional approaches for handling missingness in substance use disorder trials commonly rely upon a single deterministic "worst value" imputation that posits a perfect relationship between missingness and drug use ("missing value = presumed drug use"); this yields biased estimates of treatment effects and their standard errors. Instead, deterministic imputations should be replaced by probabilistic versions that encode researchers prior beliefs that those with missing data are more likely to be using drugs at those occasions. Motivated by this problem, we present a method for handling non-monotone missing binary data in longitudinal studies. Specifically, we consider a joint model that combines a not missing at random (NMAR) selection model with a generalized linear mixed model for longitudinal binary data. The selection model links the distribution of a missing outcome to the corresponding distribution of the outcome for those observed at that occasion via a fixed and known sensitivity parameter. The mixed model for longitudinal binary data assumes the random effects have bridge distributions; the latter yields regression parameters that have both subject-specific and marginal interpretations. This approach is completely transparent about what is being assumed about missing data and can be used as the basis for sensitivity analysis.