A Bayesian model for time-to-event data with informative censoring

Informative censoring in externally controlled clinical trials: a potential source of bias

BACKGROUND

Cancer researchers frequently consider the use of single-arm and randomized controlled clinical trial designs that leverage external data. The literature has reported extensively on how the use of external data can introduce bias through a variety of distortion mechanisms. In this article, we focus on a distortion mechanism that is often overlooked: informative censoring. Informative censoring arises when there is statistical dependence between patients' censoring times and survival times.

MATERIALS AND METHODS

We used simulations to investigate how informative censoring of external controls (ECs) can influence the results of cancer clinical trials. Our simulations included the following: (i) model-based replicates of clinical trials and in silico glioblastoma trials obtained by resampling patients from completed phase III trials; (ii) single-arm and randomized controlled cancer clinical trial designs; and (iii) different types of informative censoring, with positive or negative associations between censoring times and survival times.

RESULTS

Our simulations showed that informative censoring of EC data can bias cancer clinical trial results. The direction of the bias depends on the censoring mechanism. Similarly, informative censoring can inflate or reduce type I error and power.

CONCLUSIONS

Selection of EC data and the decision to leverage these data in the analysis of clinical trials should account for the risk of bias due to informative censoring. Analyses to detect informative censoring are recommended when the clinical trial design involves external data.

Rehabilitation outcomes of bird-building collision victims in the Northeastern United States

Building collisions are a leading threat to wild birds; however, only those that are found dead or fatally wounded are included in current mortality estimates, with injured or stunned birds largely assumed to survive long-term. Avian building collision victims are often brought to wildlife rehabilitators for care, with the hopes they can be released and resume their natural lives. We examined the wildlife rehabilitation records of over 3,100 building collisions with 152 different avian species collected across multiple seasons to identify patterns of survival and release among patients. The number of admissions varied by season; fall migration had the highest number of cases and winter had the least number of cases, and summer having the lowest release proportion and winter having the highest. The most common reported injury was head trauma and concussion. Our logistic and Poisson models found that mass had a strong positive effect on release probability, and the season of summer had a strong negative effect on release probability. Mass and winter had a strong positive effect on treatment time, and age and the seasons of fall and winter had a strong negative effect on treatment time in these models. Ultimately, about 60% of patients died in care, either by succumbing to their injuries or by euthanasia. Patients that were released remained in care for longer than patients that died. This study reports different data than carcass studies and views bird-building collisions from the perspective of surviving victims to explore longer-term effects of these collisions on mortality. Increased communication and collaboration between wildlife rehabilitators and conservation researchers is recommended to better understand building collisions and how to respond to this leading threat to wild birds. These findings, along with our estimate of delayed mortality, suggest that overall collision mortality estimates based on carcass collection far exceed one billion birds in the U.S. each year.

Clayton copula for survival data with dependent censoring: An application to a tuberculosis treatment adherence data

Ignoring the presence of dependent censoring in data analysis can lead to biased estimates, for example, not considering the effect of abandonment of the tuberculosis treatment may influence inferences about the cure probability. In order to assess the relationship between cure and abandonment outcomes, we propose a copula Bayesian approach. Therefore, the main objective of this work is to introduce a Bayesian survival regression model, capable of taking into account the dependent censoring in the adjustment. So, this proposed approach is based on Clayton's copula, to provide the relation between survival and dependent censoring times. In addition, the Weibull and the piecewise exponential marginal distributions are considered in order to fit the times. A simulation study is carried out to perform comparisons between different scenarios of dependence, different specifications of prior distributions, and comparisons with the maximum likelihood inference. Finally, we apply the proposed approach to a tuberculosis treatment adherence dataset of an HIV cohort from Alvorada-RS, Brazil. Results show that cure and abandonment outcomes are negatively correlated, that is, as long as the chance of abandoning the treatment increases, the chance of tuberculosis cure decreases.

Bayesian sensitivity analyses for longitudinal data with dropouts that are potentially missing not at random: A high dimensional pattern-mixture model

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 t k , 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 t k with that of the observed subjects at time t k . 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.

A review of the use of controlled multiple imputation in randomised controlled trials with missing outcome data

Background

Missing data are common in randomised controlled trials (RCTs) and can bias results if not handled appropriately. A statistically valid analysis under the primary missing-data assumptions should be conducted, followed by sensitivity analysis under alternative justified assumptions to assess the robustness of results. Controlled Multiple Imputation (MI) procedures, including delta-based and reference-based approaches, have been developed for analysis under missing-not-at-random assumptions. However, it is unclear how often these methods are used, how they are reported, and what their impact is on trial results. This review evaluates the current use and reporting of MI and controlled MI in RCTs.

Methods

A targeted review of phase II-IV RCTs (non-cluster randomised) published in two leading general medical journals (The Lancet and New England Journal of Medicine) between January 2014 and December 2019 using MI. Data was extracted on imputation methods, analysis status, and reporting of results. Results of primary and sensitivity analyses for trials using controlled MI analyses were compared.

Results

A total of 118 RCTs (9% of published RCTs) used some form of MI. MI under missing-at-random was used in 110 trials; this was for primary analysis in 43/118 (36%), and in sensitivity analysis for 70/118 (59%) (3 used in both). Sixteen studies performed controlled MI (1.3% of published RCTs), either with a delta-based (n = 9) or reference-based approach (n = 7). Controlled MI was mostly used in sensitivity analysis (n = 14/16). Two trials used controlled MI for primary analysis, including one reporting no sensitivity analysis whilst the other reported similar results without imputation. Of the 14 trials using controlled MI in sensitivity analysis, 12 yielded comparable results to the primary analysis whereas 2 demonstrated contradicting results. Only 5/110 (5%) trials using missing-at-random MI and 5/16 (31%) trials using controlled MI reported complete details on MI methods.

Conclusions

Controlled MI enabled the impact of accessible contextually relevant missing data assumptions to be examined on trial results. The use of controlled MI is increasing but is still infrequent and poorly reported where used. There is a need for improved reporting on the implementation of MI analyses and choice of controlled MI parameters.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-021-01261-6.

A Bayesian natural cubic B-spline varying coefficient method for non-ignorable dropout

BACKGROUND

Dropout is a common problem in longitudinal clinical trials and cohort studies, and is of particular concern when dropout occurs for reasons that may be related to the outcome of interest. This paper reviews common parametric models to account for dropout and introduces a Bayesian semi-parametric varying coefficient model for exponential family longitudinal data with non-ignorable dropout.

METHODS

To demonstrate these methods, we present results from a simulation study and estimate the impact of drug use on longitudinal CD4 + T cell count and viral load suppression in the Women's Interagency HIV Study. Sensitivity analyses are performed to consider the impact of model assumptions on inference. We compare results between our semi-parametric method and parametric models to account for dropout, including the conditional linear model and a parametric frailty model. We also compare results to analyses that fail to account for dropout.

RESULTS

In simulation studies, we show that semi-parametric methods reduce bias and mean squared error when parametric model assumptions are violated. In analyses of the Women's Interagency HIV Study data, we find important differences in estimates of changes in CD4 + T cell count over time in untreated subjects that report drug use between different models used to account for dropout. We find steeper declines over time using our semi-parametric model, which makes fewer assumptions, compared to parametric models. Failing to account for dropout or to meet parametric assumptions of models to account for dropout could lead to underestimation of the impact of hard drug use on CD4 + cell count decline in untreated subjects. In analyses of subjects that initiated highly active anti-retroviral treatment, we find that the estimated probability of viral load suppression is lower in models that account for dropout.

CONCLUSIONS

Non-ignorable dropout is an important consideration when analyzing data from longitudinal clinical trials and cohort studies. While methods that account for non-ignorable dropout must make some unavoidable assumptions that cannot be verified from the observed data, many methods make additional parametric assumptions. If these assumptions are not met, inferences can be biased, making more flexible methods with minimal assumptions important.

GRADE Guidelines: 29. Rating the certainty in time-to-event outcomes-Study limitations due to censoring of participants with missing data in intervention studies

OBJECTIVES

To provide Grading of Recommendations, Assessment, Development, and Evaluation (GRADE) guidance for the consideration of study limitations (risk of bias) due to missing participant outcome data for time-to-event outcomes in intervention studies.

STUDY DESIGN AND SETTING

We developed this guidance through an iterative process that included membership consultation, feedback, presentation, and iterative discussion at meetings of the GRADE working group.

RESULTS

The GRADE working group has published guidance on how to account for missing participant outcome data in binary and continuous outcomes. When analyzing time-to-event outcomes (e.g., overall survival and time-to-treatment failure) data of participants for whom the outcome of interest (e.g., death and relapse) has not been observed are dealt with through censoring. To do so, standard methods require that censored individuals are representative for those remaining in the study. Two types of censoring can be distinguished, end of study censoring and censoring because of missing data, commonly named loss to follow-up censoring. However, both types are not distinguishable with the usual information on censoring available to review authors. Dealing with individuals for whom data are missing during follow-up in the same way as individuals for whom full follow-up is available at the end of the study increases the risk of bias. Considerable differences in the treatment arms in the distribution of censoring over time (early versus late censoring), the overall degree of missing follow-up data, and the reasons why individuals were lost to follow-up may reduce the certainty in the study results. With often only very limited data available, review and guideline authors are required to make transparent and well-considered judgments when judging risk of bias of individual studies and then come to an overall grading decision for the entire body of evidence.

CONCLUSION

Concern for risk of bias resulting from censoring of participants for whom follow-up data are missing in the underlying studies of a body of evidence can be expressed in the study limitations (risk of bias) domain of the GRADE approach.

Reference-based sensitivity analysis for time-to-event data

The analysis of time‐to‐event data typically makes the censoring at random assumption, ie, that—conditional on covariates in the model—the distribution of event times is the same, whether they are observed or unobserved (ie, right censored). When patients who remain in follow‐up stay on their assigned treatment, then analysis under this assumption broadly addresses the de jure, or “while on treatment strategy” estimand. In such cases, we may well wish to explore the robustness of our inference to more pragmatic, de facto or “treatment policy strategy,” assumptions about the behaviour of patients post‐censoring.

This is particularly the case when censoring occurs because patients change, or revert, to the usual (ie, reference) standard of care. Recent work has shown how such questions can be addressed for trials with continuous outcome data and longitudinal follow‐up, using reference‐based multiple imputation. For example, patients in the active arm may have their missing data imputed assuming they reverted to the control (ie, reference) intervention on withdrawal. Reference‐based imputation has two advantages: (a) it avoids the user specifying numerous parameters describing the distribution of patients' postwithdrawal data and (b) it is, to a good approximation, information anchored, so that the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. In this article, we build on recent work in the survival context, proposing a class of reference‐based assumptions appropriate for time‐to‐event data. We report a simulation study exploring the extent to which the multiple imputation estimator (using Rubin's variance formula) is information anchored in this setting and then illustrate the approach by reanalysing data from a randomized trial, which compared medical therapy with angioplasty for patients presenting with angina.

Bayesian randomized clinical trials: From fixed to adaptive design

Randomized controlled studies are the gold standard for phase III clinical trials. Using α-spending functions to control the overall type I error rate, group sequential methods are well established and have been dominating phase III studies. Bayesian randomized design, on the other hand, can be viewed as a complement instead of competitive approach to the frequentist methods. For the fixed Bayesian design, the hypothesis testing can be cast in the posterior probability or Bayes factor framework, which has a direct link to the frequentist type I error rate. Bayesian group sequential design relies upon Bayesian decision-theoretic approaches based on backward induction, which is often computationally intensive. Compared with the frequentist approaches, Bayesian methods have several advantages. The posterior predictive probability serves as a useful and convenient tool for trial monitoring, and can be updated at any time as the data accrue during the trial. The Bayesian decision-theoretic framework possesses a direct link to the decision making in the practical setting, and can be modeled more realistically to reflect the actual cost-benefit analysis during the drug development process. Other merits include the possibility of hierarchical modeling and the use of informative priors, which would lead to a more comprehensive utilization of information from both historical and longitudinal data. From fixed to adaptive design, we focus on Bayesian randomized controlled clinical trials and make extensive comparisons with frequentist counterparts through numerical studies.

Relaxing the independent censoring assumption in the Cox proportional hazards model using multiple imputation

The Cox proportional hazards model is frequently used in medical statistics. The standard methods for fitting this model rely on the assumption of independent censoring. Although this is sometimes plausible, we often wish to explore how robust our inferences are as this untestable assumption is relaxed. We describe how this can be carried out in a way that makes the assumptions accessible to all those involved in a research project. Estimation proceeds via multiple imputation, where censored failure times are imputed under user-specified departures from independent censoring. A novel aspect of our method is the use of bootstrapping to generate proper imputations from the Cox model. We illustrate our approach using data from an HIV-prevention trial and discuss how it can be readily adapted and applied in other settings. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.