A Bayesian network framework for reject inference

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.

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.

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

Missing data are frequently encountered in various disciplines and can be divided into three categories: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). Valid statistical approaches to missing data depend crucially on correct identification of the underlying missingness mechanism. Although the problem of testing whether this mechanism is MCAR or MAR has been extensively studied, there has been very little research on testing MAR versus MNAR. A critical challenge that is faced when dealing with this problem is the issue of model identification under MNAR. In this paper, under a logistic model for the missing probability, we develop two score tests for the problem of whether the missingness mechanism is MAR or MNAR under a parametric model and a semiparametric location model on the regression function. The implementation of the score tests circumvents the identification issue as it requires only parameter estimation under the null MAR assumption. Our simulations and analysis of human immunodeficiency virus data show that the score tests have well-controlled type I errors and desirable powers.

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.

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.

Reject inference in credit scoring is a statistical approach to adjust for nonrandom sample bias due to rejected applicants. Function estimation approaches are based on the assumption that rejected applicants are not necessary to be included in the estimation, when the missing data mechanism is missing at random. On the other hand, the density estimation approach by using mixture models indicates that reject inference should include rejected applicants in the model. When mixture models are chosen for reject inference, it is often assumed that data follow a normal distribution. If data include missing values, an application of the normal mixture model to fully observed cases may cause another sample bias due to missing values. We extend reject inference by a multivariate normal mixture model to handle incomplete characteristic variables. A simulation study shows that inclusion of incomplete characteristic variables outperforms the function estimation approaches.

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.

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.

Estimation of prevalence and diagnostic test accuracy in tuberculosis (TB) prevalence surveys suffer from reference standard and verification biases. The former is attributed to the imperfect reference test used to bacteriologically confirm TB disease. The latter occurs when only the participants screening positive for any TB-compatible symptom or chest X-ray abnormality are selected for bacteriological testing (verification). Bayesian latent class analysis (LCA) alleviates the reference standard bias but suffers verification bias in TB prevalence surveys. This work aims to identify best-practice approaches to simultaneously alleviate the reference standard and verification biases in the estimates of pulmonary TB prevalence and diagnostic test performance in TB prevalence surveys. We performed a secondary analysis of 9869 participants aged ≥15 years from a community-based multimorbidity screening study in a rural district of KwaZulu-Natal, South Africa (Vukuzazi study). Participants were eligible for bacteriological testing using Xpert Ultra and culture if they reported any cardinal TB symptom or had an abnormal chest X-ray finding. We conducted Bayesian LCA in five ways to handle the unverified individuals: (i) complete-case analysis, (ii) analysis assuming the unverified individuals would be negative if bacteriologically tested, (iii) analysis of multiply-imputed datasets with imputation of the missing bacteriological test results for the unverified individuals using multivariate imputation via chained equations (MICE), and simultaneous imputation of the missing bacteriological test results in the analysis model assuming the missing bacteriological test results were (iv) missing at random (MAR), and (v) missing not at random (MNAR). We compared the results of (i)-(iii) to the analysis based on a composite reference standard (CRS) of Xpert Ultra and culture. Through simulation with an overall true prevalence of 2.0%, we evaluated the ability of the models to alleviate both biases simultaneously. Based on simulation, Bayesian LCA with simultaneous imputation of the missing bacteriological test results under the assumption that the missing data are MAR and MNAR alleviate the reference standard and verification biases. CRS-based analysis and Bayesian LCA assuming the unverified are negative for TB alleviate the biases only when the true overall prevalence is <3.0%. Complete-case analysis produced biased estimates. In the Vukuzazi study, Bayesian LCA with simultaneous imputation of the missing bacteriological test results under the MAR and MNAR assumptions produced overall PTB prevalence of 0.9% (95% Credible Interval (CrI): 0.6-1.9) and 0.7% (95% CrI: 0.5-1.1) respectively alongside realistic estimates of overall diagnostic test sensitivity and specificity with substantially overlapping 95% CrI. The CRS-based analysis and Bayesian LCA assuming the unverified were negative for TB produced 0.7% (95% CrI: 0.5-0.9) and 0.7% (95% CrI: 0.5-1.2) overall PTB prevalence respectively with realistic estimates of overall diagnostic test sensitivity and specificity. Unlike CRS-based analysis, Bayesian LCA of multiply-imputed data using MICE mitigates both biases. The findings demonstrate the efficacy of these advanced techniques in alleviating the reference standard and verification biases, enhancing the robustness of community-based screening programs. Imputing missing values as negative for bacteriological tests is plausible under realistic assumptions.

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.

Offline evaluation of recommender systems mostly relies on historical data, which is often biased by many confounders. In such data, user-item interactions are Missing Not At Random (MNAR). Measures of recommender system performance on MNAR test data are unlikely to be reliable indicators of real-world performance unless something is done to mitigate the bias. One way that researchers try to obtain less biased offline evaluation is by designing new supposedly unbiased performance estimators for use on MNAR test data. We investigate an alternative solution, a sampling approach. The general idea is to use a sampling strategy on MNAR data to generate an intervened test set with less bias --- one in which interactions are Missing At Random (MAR) or, at least, one that is more MAR-like. An example of this is SKEW, a sampling strategy that aims to adjust for the confounding effect that an item's popularity has on its likelihood of being observed. In this paper, we propose a novel formulation for the sampling approach. We compare our solution to SKEW and to two baselines which perform a random intervention on MNAR data (and hence are equivalent to no intervention in practice). We empirically validate for the first time the effectiveness of SKEW and we show our approach to be a better estimator of the performance one would obtain on (unbiased) MAR test data. Our strategy benefits from high generality properties (e.g. it can also be employed for training a recommender) and low overheads (e.g. it does not require any learning).

Chapter 14 - Methods for handling missing data

Longitudinal studies occupy an important role in scientific researches and clinical trials. When taking the analysis of longitudinal data, investigators are often confronted with missing data which will produce potential biases, even in well-controlled condition. In the literature, missing data could be classified as missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). Generalized estimating equations (GEE), Linear mixed effects model (LME) and Pattern-mixture effect model (PME) are the commonly used analysis methods for longitudinal data. In the current work, we carried out simulations on evaluating the performances of the different methods on analyzing longitudinal data. Based on our simulations, we conclude that when missing is MCAR, all the methods give valid estimation; when missing is MAR, GEE and PME give biased estimating results, while LME provides valid estimation. The choice of the patterns in PME may cause biased results; and when missing is MNAR, none of these models works very well, however, the selection of the patterns in PME may deserve further investigation.

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.