Beyond 'statistical significance': A nontechnical primer of Bayesian statistics and Bayes factors for health researchers

RATIONALE

Hypothesis testing is integral to health research and is commonly completed through frequentist statistics focused on computing p values. p Values have been long criticized for offering limited information about the relationship of variables and strength of evidence concerning the plausibility, presence and certainty of associations among variables. Bayesian statistics is a potential alternative for inference-making. Despite emerging discussion on Bayesian statistics across various disciplines, the uptake of Bayesian statistics in health research is still limited.

AIM

To offer a primer on Bayesian statistics and Bayes factors for health researchers to gain preliminary knowledge of its use, application and interpretation in health research.

METHODS

Theoretical and empirical literature on Bayesian statistics and methods were used to develop this methodological primer.

CONCLUSIONS

Using Bayesian statistics in health research without a careful and complete understanding of its underlying philosophy and differences from frequentist testing, estimation and interpretation methods can result in similar ritualistic use as done for p values.

IMPLICATIONS

Health researchers should supplement frequentists statistics with Bayesian statistics when analysing research data. The overreliance on p values for clinical decisions making should be avoided. Bayes factors offer a more intuitive measure of assessing the strength of evidence for null and alternative hypothesis.

Two-stage randomized clinical trials with a right-censored endpoint: Comparison of frequentist and Bayesian adaptive designs

Adaptive randomized clinical trials are of major interest when dealing with a time-to-event outcome in a prolonged observation window. No consensus exists either to define stopping boundaries or to combinep $$ p $$ values or test statistics in the terminal analysis in the case of a frequentist design and sample size adaptation. In a one-sided setting, we compared three frequentist approaches using stopping boundaries relying onα $$ \alpha $$ -spending functions and a Bayesian monitoring setting with boundaries based on the posterior distribution of the log-hazard ratio. All designs comprised a single interim analysis with an efficacy stopping rule and the possibility of sample size adaptation at this interim step. Three frequentist approaches were defined based on the terminal analysis: combination of stagewise statistics (Wassmer) or ofp $$ p $$ values (Desseaux), or on patientwise splitting (Jörgens), and we compared the results with those of the Bayesian monitoring approach (Freedman). These different approaches were evaluated in a simulation study and then illustrated on a real dataset from a randomized clinical trial conducted in elderly patients with chronic lymphocytic leukemia. All approaches controlled for the type I error rate, except for the Bayesian monitoring approach, and yielded satisfactory power. It appears that the frequentist approaches are the best in underpowered trials. The power of all the approaches was affected by the violation of the proportional hazards (PH) assumption. For adaptive designs with a survival endpoint and a one-sided alternative hypothesis, the Wassmer and Jörgens approaches after sample size adaptation should be preferred, unless violation of PH is suspected.

The reporting quality and transparency of orthopaedic studies using Bayesian analysis requires improvement: A systematic review

Background

Bayesian methods are being used more frequently in orthopaedics. To advance the use and transparent reporting of Bayesian studies, reporting guidelines have been recommended. There is currently little known about the use or applications of Bayesian analysis in orthopedics including adherence to recommended reporting guidelines. The objective is to investigate the reporting of Bayesian analysis in orthopedic surgery studies; specifically, to evaluate if these papers adhere to reporting guidelines.

Methods

We searched PUBMED to December 2nd, 2020. Two reviewers independently identified studies and full-text screening. We included studies that focused on one or more orthopaedic surgical interventions and used Bayesian methods.

Results

After full-text review, 100 articles were included. The most frequent study designs were meta-analysis or network meta-analysis (56%, 95% CI 46-65) and cohort studies (25%, 95% CI 18-34). Joint replacement was the most common subspecialty (33%, 95% CI 25-43). We found that studies infrequently reported key concepts in Bayesian analysis including, specifying the prior distribution (37-39%), justifying the prior distribution (18%), the sensitivity to different priors (7-8%), and the statistical model used (22%). In contrast, general methodological items on the checklists were largely well reported.

Conclusions

There is an opportunity to improve reporting quality and transparency of orthopaedic studies using Bayesian analysis by encouraging adherence to reporting guidelines such as ROBUST, JASP, and BayesWatch. There is an opportunity to better report prior distributions, sensitivity analyses, and the statistical models used.

Informed Bayesian survival analysis

BACKGROUND

We provide an overview of Bayesian estimation, hypothesis testing, and model-averaging and illustrate how they benefit parametric survival analysis. We contrast the Bayesian framework to the currently dominant frequentist approach and highlight advantages, such as seamless incorporation of historical data, continuous monitoring of evidence, and incorporating uncertainty about the true data generating process.

METHODS

We illustrate the application of the outlined Bayesian approaches on an example data set, retrospective re-analyzing a colon cancer trial. We assess the performance of Bayesian parametric survival analysis and maximum likelihood survival models with AIC/BIC model selection in fixed-n and sequential designs with a simulation study.

RESULTS

In the retrospective re-analysis of the example data set, the Bayesian framework provided evidence for the absence of a positive treatment effect of adding Cetuximab to FOLFOX6 regimen on disease-free survival in patients with resected stage III colon cancer. Furthermore, the Bayesian sequential analysis would have terminated the trial 10.3 months earlier than the standard frequentist analysis. In a simulation study with sequential designs, the Bayesian framework on average reached a decision in almost half the time required by the frequentist counterparts, while maintaining the same power, and an appropriate false-positive rate. Under model misspecification, the Bayesian framework resulted in higher false-negative rate compared to the frequentist counterparts, which resulted in a higher proportion of undecided trials. In fixed-n designs, the Bayesian framework showed slightly higher power, slightly elevated error rates, and lower bias and RMSE when estimating treatment effects in small samples. We found no noticeable differences for survival predictions. We have made the analytic approach readily available to other researchers in the RoBSA R package.

CONCLUSIONS

The outlined Bayesian framework provides several benefits when applied to parametric survival analyses. It uses data more efficiently, is capable of considerably shortening the length of clinical trials, and provides a richer set of inferences.

Improving clinical trials using Bayesian adaptive designs: a breast cancer example

BACKGROUND

To perform virtual re-executions of a breast cancer clinical trial with a time-to-event outcome to demonstrate what would have happened if the trial had used various Bayesian adaptive designs instead.

METHODS

We aimed to retrospectively "re-execute" a randomised controlled trial that compared two chemotherapy regimens for women with metastatic breast cancer (ANZ 9311) using Bayesian adaptive designs. We used computer simulations to estimate the power and sample sizes of a large number of different candidate designs and shortlisted designs with the either highest power or the lowest average sample size. Using the real-world data, we explored what would have happened had ANZ 9311 been conducted using these shortlisted designs.

RESULTS

We shortlisted ten adaptive designs that had higher power, lower average sample size, and a lower false positive rate, compared to the original trial design. Adaptive designs that prioritised small sample size reduced the average sample size by up to 37% when there was no clinical effect and by up to 17% at the target clinical effect. Adaptive designs that prioritised high power increased power by up to 5.9 percentage points without a corresponding increase in type I error. The performance of the adaptive designs when applied to the real-world ANZ 9311 data was consistent with the simulations.

CONCLUSION

The shortlisted Bayesian adaptive designs improved power or lowered the average sample size substantially. When designing new oncology trials, researchers should consider whether a Bayesian adaptive design may be beneficial.

Methods for Extrapolating Survival Analyses for the Economic Evaluation of Advanced Therapy Medicinal Products

There are two significant challenges for analysts conducting economic evaluations of advanced therapy medicinal products (ATMPs): (i) estimating long-term treatment effects in the absence of mature clinical data, and (ii) capturing potentially complex hazard functions. This review identifies and critiques a variety of methods that can be used to overcome these challenges. The narrative review is informed by a rapid literature review of methods used for the extrapolation of survival analyses in the economic evaluation of ATMPs. There are several methods that are more suitable than traditional parametric survival modelling approaches for capturing complex hazard functions, including, cure-mixture models and restricted cubic spline models. In the absence of mature clinical data, analysts may augment clinical trial data with data from other sources to aid extrapolation, however, the relative merits of employing methods for including data from different sources is not well understood. Given the high and potentially irrecoverable costs of making incorrect decisions concerning the reimbursement or commissioning of ATMPs, it is important that economic evaluations are correctly specified, and that both parameter and structural uncertainty associated with survival extrapolations are considered. Value of information analyses allow for this uncertainty to be expressed explicitly, and in monetary terms.

Developing a reference protocol for structured expert elicitation in health-care decision-making: a mixed-methods study

BACKGROUND

Many decisions in health care aim to maximise health, requiring judgements about interventions that may have higher health effects but potentially incur additional costs (cost-effectiveness framework). The evidence used to establish cost-effectiveness is typically uncertain and it is important that this uncertainty is characterised. In situations in which evidence is uncertain, the experience of experts is essential. The process by which the beliefs of experts can be formally collected in a quantitative manner is structured expert elicitation. There is heterogeneity in the existing methodology used in health-care decision-making. A number of guidelines are available for structured expert elicitation; however, it is not clear if any of these are appropriate for health-care decision-making.

OBJECTIVES

The overall aim was to establish a protocol for structured expert elicitation to inform health-care decision-making. The objectives are to (1) provide clarity on methods for collecting and using experts' judgements, (2) consider when alternative methodology may be required in particular contexts, (3) establish preferred approaches for elicitation on a range of parameters, (4) determine which elicitation methods allow experts to express uncertainty and (5) determine the usefulness of the reference protocol developed.

METHODS

A mixed-methods approach was used: systemic review, targeted searches, experimental work and narrative synthesis. A review of the existing guidelines for structured expert elicitation was conducted. This identified the approaches used in existing guidelines (the 'choices') and determined if dominant approaches exist. Targeted review searches were conducted for selection of experts, level of elicitation, fitting and aggregation, assessing accuracy of judgements and heuristics and biases. To sift through the available choices, a set of principles that underpin the use of structured expert elicitation in health-care decision-making was defined using evidence generated from the targeted searches, quantities to elicit experimental evidence and consideration of constraints in health-care decision-making. These principles, including fitness for purpose and reflecting individual expert uncertainty, were applied to the set of choices to establish a reference protocol. An applied evaluation of the developed reference protocol was also undertaken.

RESULTS

For many elements of structured expert elicitation, there was a lack of consistency across the existing guidelines. In almost all choices, there was a lack of empirical evidence supporting recommendations, and in some circumstances the principles are unable to provide sufficient justification for discounting particular choices. It is possible to define reference methods for health technology assessment. These include a focus on gathering experts with substantive skills, eliciting observable quantities and individual elicitation of beliefs. Additional considerations are required for decision-makers outside health technology assessment, for example at a local level, or for early technologies. Access to experts may be limited and in some circumstances group discussion may be needed to generate a distribution.

LIMITATIONS

The major limitation of the work conducted here lies not in the methods employed in the current work but in the evidence available from the wider literature relating to how appropriate particular methodological choices are.

CONCLUSIONS

The reference protocol is flexible in many choices. This may be a useful characteristic, as it is possible to apply this reference protocol across different settings. Further applied studies, which use the choices specified in this reference protocol, are required.

FUNDING

This project was funded by the NIHR Health Technology Assessment programme and will be published in full in Health Technology Assessment; Vol. 25, No. 37. See the NIHR Journals Library website for further project information. This work was also funded by the Medical Research Council (reference MR/N028511/1).

Predictors of swallowing outcomes in patients with combat-injury related dysphagia

BACKGROUND

Traumatic injuries, such as those from combat-related activities, can lead to complicated clinical presentations that may include dysphagia.

METHODS

This retrospective observational database study captured dysphagia-related information for 215 US military service members admitted to the first stateside military treatment facility after sustaining combat-related or combat-like traumatic injuries. A multidimensional relational database was developed to document the nature, course, and management for dysphagia in this unique population and to explore variables predictive of swallowing recovery using Bayesian statistical modeling and inferential statistical methods.

RESULTS

Bayesian statistical modeling revealed the importance of maxillofacial fractures and soft tissue loss as primary predictors of poor swallowing outcomes. The presence of traumatic brain injury (TBI), though common, did not further complicate dysphagia outcomes. A more detailed examination and rating of videofluoroscopic swallow studies from a subset of 161 participants supported greater impairment for participants with maxillofacial trauma and no apparent relationship between having sustained a TBI and swallow functioning.

CONCLUSION

These analyses revealed that maxillofacial trauma is a stronger indicator than TBI of dysphagia severity and slower or incomplete recovery following combat-related injuries.

LEVEL OF EVIDENCE

Therapeutic/Care Management study, level IV.

The adaptive designs CONSORT extension (ACE) statement: a checklist with explanation and elaboration guideline for reporting randomised trials that use an adaptive design

Adaptive designs (ADs) allow pre-planned changes to an ongoing trial without compromising the validity of conclusions and it is essential to distinguish pre-planned from unplanned changes that may also occur. The reporting of ADs in randomised trials is inconsistent and needs improving. Incompletely reported AD randomised trials are difficult to reproduce and are hard to interpret and synthesise. This consequently hampers their ability to inform practice as well as future research and contributes to research waste. Better transparency and adequate reporting will enable the potential benefits of ADs to be realised.This extension to the Consolidated Standards Of Reporting Trials (CONSORT) 2010 statement was developed to enhance the reporting of randomised AD clinical trials. We developed an Adaptive designs CONSORT Extension (ACE) guideline through a two-stage Delphi process with input from multidisciplinary key stakeholders in clinical trials research in the public and private sectors from 21 countries, followed by a consensus meeting. Members of the CONSORT Group were involved during the development process.The paper presents the ACE checklists for AD randomised trial reports and abstracts, as well as an explanation with examples to aid the application of the guideline. The ACE checklist comprises seven new items, nine modified items, six unchanged items for which additional explanatory text clarifies further considerations for ADs, and 20 unchanged items not requiring further explanatory text. The ACE abstract checklist has one new item, one modified item, one unchanged item with additional explanatory text for ADs, and 15 unchanged items not requiring further explanatory text.The intention is to enhance transparency and improve reporting of AD randomised trials to improve the interpretability of their results and reproducibility of their methods, results and inference. We also hope indirectly to facilitate the much-needed knowledge transfer of innovative trial designs to maximise their potential benefits. In order to encourage its wide dissemination this article is freely accessible on the BMJ and Trials journal websites."To maximise the benefit to society, you need to not just do research but do it well" Douglas G Altman.

The Adaptive designs CONSORT Extension (ACE) statement: a checklist with explanation and elaboration guideline for reporting randomised trials that use an adaptive design

Adaptive designs (ADs) allow pre-planned changes to an ongoing trial without compromising the validity of conclusions and it is essential to distinguish pre-planned from unplanned changes that may also occur. The reporting of ADs in randomised trials is inconsistent and needs improving. Incompletely reported AD randomised trials are difficult to reproduce and are hard to interpret and synthesise. This consequently hampers their ability to inform practice as well as future research and contributes to research waste. Better transparency and adequate reporting will enable the potential benefits of ADs to be realised.This extension to the Consolidated Standards Of Reporting Trials (CONSORT) 2010 statement was developed to enhance the reporting of randomised AD clinical trials. We developed an Adaptive designs CONSORT Extension (ACE) guideline through a two-stage Delphi process with input from multidisciplinary key stakeholders in clinical trials research in the public and private sectors from 21 countries, followed by a consensus meeting. Members of the CONSORT Group were involved during the development process.The paper presents the ACE checklists for AD randomised trial reports and abstracts, as well as an explanation with examples to aid the application of the guideline. The ACE checklist comprises seven new items, nine modified items, six unchanged items for which additional explanatory text clarifies further considerations for ADs, and 20 unchanged items not requiring further explanatory text. The ACE abstract checklist has one new item, one modified item, one unchanged item with additional explanatory text for ADs, and 15 unchanged items not requiring further explanatory text.The intention is to enhance transparency and improve reporting of AD randomised trials to improve the interpretability of their results and reproducibility of their methods, results and inference. We also hope indirectly to facilitate the much-needed knowledge transfer of innovative trial designs to maximise their potential benefits.

Bayesian clinical trials at The University of Texas MD Anderson Cancer Center: An update

BACKGROUND/AIMS

In our 2009 article, we showed that Bayesian methods had established a foothold in developing therapies in our institutional oncology trials. In this article, we will document what has happened since that time. In addition, we will describe barriers to implementing Bayesian clinical trials, as well as our experience overcoming them.

METHODS

We reviewed MD Anderson Cancer Center clinical trials submitted to the institutional protocol office for scientific and ethical review between January 2009 and December 2013, the same length time period as the previous article. We tabulated Bayesian methods implemented for design or analyses for each trial and then compared these to our previous findings.

RESULTS

Overall, we identified 1020 trials and found that 283 (28%) had Bayesian components so we designated them as Bayesian trials. Among MD Anderson-only and multicenter trials, 56% and 14%, respectively, were Bayesian, higher rates than our previous study. Bayesian trials were more common in phase I/II trials (34%) than in phase III/IV (6%) trials. Among Bayesian trials, the most commonly used features were for toxicity monitoring (65%), efficacy monitoring (36%), and dose finding (22%). The majority (86%) of Bayesian trials used non-informative priors. A total of 75 (27%) trials applied Bayesian methods for trial design and primary endpoint analysis. Among this latter group, the most commonly used methods were the Bayesian logistic regression model (N = 22), the continual reassessment method (N = 20), and adaptive randomization (N = 16). Median institutional review board approval time from protocol submission was the same 1.4 months for Bayesian and non-Bayesian trials. Since the previous publication, the Biomarker-Integrated Approaches of Targeted Therapy for Lung Cancer Elimination (BATTLE) trial was the first large-scale decision trial combining multiple treatments in a single trial. Since then, two regimens in breast cancer therapy have been identified and published from the cooperative Investigation of Serial Studies to Predict Your Therapeutic Response with Imaging and Molecular Analysis (I-SPY 2), enhancing cooperation among investigators and drug developers across the nation, as well as advancing information needed for personalized medicine. Many software programs and Shiny applications for Bayesian trial design and calculations are available from our website which has had more than 21,000 downloads worldwide since 2004.

CONCLUSION

Bayesian trials have the increased flexibility in trial design needed for personalized medicine, resulting in more cooperation among researchers working to fight against cancer. Some disadvantages of Bayesian trials remain, but new methods and software are available to improve their function and incorporation into cancer clinical research.

Incorporating individual historical controls and aggregate treatment effect estimates into a Bayesian survival trial: a simulation study

Background

Performing well-powered randomised controlled trials (RCTs) of new treatments for rare diseases is often infeasible. However, with the increasing availability of historical data, incorporating existing information into trials with small sample sizes is appealing in order to increase the power. Bayesian approaches enable one to incorporate historical data into a trial’s analysis through a prior distribution.

Methods

Motivated by a RCT intended to evaluate the impact on event-free survival of mifamurtide in patients with osteosarcoma, we performed a simulation study to evaluate the impact on trial operating characteristics of incorporating historical individual control data and aggregate treatment effect estimates. We used power priors derived from historical individual control data for baseline parameters of Weibull and piecewise exponential models, while we used a mixture prior to summarise aggregate information obtained on the relative treatment effect. The impact of prior-data conflicts, both with respect to the parameters and survival models, was evaluated for a set of pre-specified weights assigned to the historical information in the prior distributions.

Results

The operating characteristics varied according to the weights assigned to each source of historical information, the variance of the informative and vague component of the mixture prior and the level of commensurability between the historical and new data. When historical and new controls follow different survival distributions, we did not observe any advantage of choosing a piecewise exponential model compared to a Weibull model for the new trial analysis. However, we think that it remains appealing given the uncertainty that will often surround the shape of the survival distribution of the new data.

Conclusion

In the setting of Sarcome-13 trial, and other similar studies in rare diseases, the gains in power and accuracy made possible by incorporating different types of historical information commensurate with the new trial data have to be balanced against the risk of biased estimates and a possible loss in power if data are not commensurate. The weights allocated to the historical data have to be carefully chosen based on this trade-off. Further simulation studies investigating methods for incorporating historical data are required to generalise the findings.

Electronic supplementary material

The online version of this article (10.1186/s12874-019-0714-z) contains supplementary material, which is available to authorized users.

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.