Bayesian sensitivity models for missing covariates in the analysis of survival data

ObjectiveTo compare the effects of accounting for different missing data mechanisms in an investigation of the role of lifecourse socio-economic position (SEP) on later-life verbal cognitive ability.DesignTwo UK prospective cohort...

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.

ObjectivesTo describe how systematic reviewers are reporting missing data for dichotomous outcomes, handling them in the analysis and assessing the risk of associated bias.MethodsWe searched MEDLINE and the Cochrane Database...

Missing Data and Imputation: A Practical Illustration in a Prognostic Study on Low Back Pain

Missing data result in less precise and possibly biased effect estimates in single studies. Bias arising from studies with incomplete outcome data is naturally propagated in a meta‐analysis. Conventional analysis using only individuals with available data is adequate when the meta‐analyst can be confident that the data are missing at random (MAR) in every study—that is, that the probability of missing data does not depend on unobserved variables, conditional on observed variables. Usually, such confidence is unjustified as participants may drop out due to lack of improvement or adverse effects. The MAR assumption cannot be tested, and a sensitivity analysis to assess how robust results are to reasonable deviations from the MAR assumption is important. Two methods may be used based on plausible alternative assumptions about the missing data. Firstly, the distribution of reasons for missing data may be used to impute the missing values. Secondly, the analyst may specify the magnitude and uncertainty of possible departures from the missing at random assumption, and these may be used to correct bias and reweight the studies. This is achieved by employing a pattern mixture model and describing how the outcome in the missing participants is related to the outcome in the completers. Ideally, this relationship is informed using expert opinion. The methods are illustrated in two examples with binary and continuous outcomes. We provide recommendations on what trial investigators and systematic reviewers should do to minimize the problem of missing outcome data in meta‐analysis.

Although missing outcome data are an important problem in randomized trials and observational studies, methods to address this issue can be difficult to apply. Using simulated data, the authors compared 3 methods to handle missing outcome data: 1) complete case analysis; 2) single imputation; and 3) multiple imputation (all 3 with and without covariate adjustment). Simulated scenarios focused on continuous or dichotomous missing outcome data from randomized trials or observational studies. When outcomes were missing at random, single and multiple imputations yielded unbiased estimates after covariate adjustment. Estimates obtained by complete case analysis with covariate adjustment were unbiased as well, with coverage close to 95%. When outcome data were missing not at random, all methods gave biased estimates, but handling missing outcome data by means of 1 of the 3 methods reduced bias compared with a complete case analysis without covariate adjustment. Complete case analysis with covariate adjustment and multiple imputation yield similar estimates in the event of missing outcome data, as long as the same predictors of missingness are included. Hence, complete case analysis with covariate adjustment can and should be used as the analysis of choice more often. Multiple imputation, in addition, can accommodate the missing-not-at-random scenario more flexibly, making it especially suited for sensitivity analyses.

Dr. Rodu and colleagues have made several comments regarding our article1 that merit a response.

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

Based on this systematic review and meta-analysis of 31 randomized controlled trials including 8672 patients, moderate certainty evidence suggests that probiotics are effective for preventing CDAD (NNTB = 42 patients, 95% CI 32 to 58). Our post hoc subgroup analyses to explore heterogeneity indicated that probiotics are effective among trials with a CDAD baseline risk >5% (NNTB = 12; moderate certainty evidence), but not among trials with a baseline risk ≤5% (low to moderate certainty evidence). Although adverse effects were reported among 32 included trials, there were more adverse events among patients in the control groups. The short-term use of probiotics appears to be safe and effective when used along with antibiotics in patients who are not immunocompromised or severely debilitated. Despite the need for further research, hospitalized patients, particularly those at high risk of CDAD, should be informed of the potential benefits and harms of probiotics.

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates there has been extensive discussion in the literature with the focus being on the Cox proportional hazards models. However, the impact of measurement error on accelerated failure time (AFT) models has received little attention, though AFT models are very useful in survival data analysis. In this paper, we discuss AFT models with error-prone covariates and study the bias induced by the naive approach of ignoring measurement error in covariates. To adjust for such a bias, we describe a simulation and extrapolation method. This method is appealing because it is simple to implement and it does not require modelling the true but error-prone covariate process that is often not observable. Asymptotic normality for the resulting estimators is established. Simulation studies are carried out to evaluate the performance of the proposed method as well as the impact of ignoring measurement error in covariates. The proposed method is applied to analyse a data set arising from the Busselton Health study (Australian J. Public Health 1994; 18:129-135).

The Cox regression model is the most commonly used regression model for survival data. The Cox regression model has an important assumption that hazard rates are proportional over time. The accelerated failure time model is an alternative method for the analysis of survival data when hazards are not proportional. The accelerated failure time models should lead to more efficient parameter estimates than Cox model under certain circumferences. In this study, The accelerated failure time models are described. The data of stomach cancer patients is used to illustrate these models and the results are discussed

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.

Neglecting missing values in diagnostic models can result in unreliable and suboptimal performance on new data. In this study, the authors imputed missing values for the CA-125 tumor marker in a large data set of ovarian tumors that was used to develop models for predicting malignancy. Four imputation techniques were applied: regression imputation, expectation-maximization, data augmentation, and hotdeck. Models using the imputed data sets were compared with models without CA-125 to investigate the important clinical issue concerning the necessity of CA-125 information for diagnostic models and with models using only complete cases to investigate differences between imputation and complete case strategies for missing values. The models are based on Bayesian generalized linear models (GLMs) and Bayesian least squares support vector machines. indicate that the use of CA-125 resulted in small, clinically nonsignificant increases in the AUC of diagnostic models. Minor differences between imputation methods were observed, and imputing CA-125 resulted in minor differences in the AUC compared with complete case analysis (CCA). However, GLM parameter estimates of predictor variables often differed between CCA and models based on imputation. The authors conclude that CA-125 is not indispensable in diagnostic models for ovarian tumors and that missing value imputation is preferred over CCA.

This study aimed to document the ways by which missing data were handled in clinical pharmacy research to provide an insight into the amount of attention paid to the importance of missing data in this field of research. Our cross-sectional descriptive report evaluated 10 journals affiliated with pharmacy organizations in the United States, Canada, the United Kingdom, and Australia. Randomized controlled trials, cohort studies, case-control studies, and cross-sectional studies published in 2018 were included. The primary outcome measure was the proportion of studies that reported the handling of missing data in their methods or results. A total of 178 studies were included in the analysis. Of these, 19.7% (n = 35) mentioned missing data either in their methods (3.4%, n = 6), results (15.2%, n = 27), or in both sections (1.1%, n = 2). Only 4.5% (n = 8) of the studies mentioned how they handled missing data, the most common method being multiple imputation (n = 3), followed by indicator (n = 2), complete case analysis (n = 2), and simple imputation (n = 1). One study using multiple imputation and both studies using an indicator method also combined other strategies to account for missing data. One study only used complete case analysis for subgroup analysis, and the other study only used this method if a specific baseline variable was missing. Very few studies in clinical pharmacy literature report any handling of missing data. This has the potential to lead to biased results. We advocate that researchers should report how missing data were handled to increase the transparency of findings and minimize bias.

Despite its drawbacks, the complete case analysis is commonly used in regression models with incomplete covariates. Understanding when the complete case analysis will lead to consistent parameter estimation is vital before use. Our aim here is to demonstrate when a complete case analysis is consistent for randomly right-censored covariates and to discuss the implications of its use even when consistent. Across the censored covariate literature, different assumptions are made to ensure a complete case analysis produces a consistent estimator, which leads to confusion in practice. We make several contributions to dispel this confusion. First, we summarize the language surrounding the assumptions that lead to a consistent complete case estimator. Then, we show a unidirectional hierarchical relationship between these assumptions, which leads us to one sufficient assumption to consider before using a complete case analysis. Lastly, we conduct a simulation study to illustrate the performance of a complete case analysis with a right-censored covariate under different censoring mechanism assumptions, and we demonstrate its use with a Huntington disease data example.