Bayesian survival analysis in clinical trials: what methods are used in practice? A systematic review of the literature

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1 1 Bayesian survival analysis in clinical trials: what methods are used in practice? A systematic review of the literature C. Brard 1,2, G. Le Teu 1,2, L.V Hampson 3, M-C. Le Deley 1,2 1 Biostatistics and Epidemiology unit - Gustave Roussy, Villejuif, France 2 INSERM U CESP, Team 2, Paris-Sud University, Villejuif, France 3 Department of Mathematics and Statistics, Lancaster University, Lancaster, U.K ISCB Utrecht - August

2 1 Bayesian analysis in clinical trials Augment new trial data with existing information: make sense of conicting datasets maximise power when only small sample sizes are feasible Adaptive design Stochastic curtailment based on predictive probabilities to inform early stopping decisions Adaptive randomisation: allocation rules

3 1 Bayesian analysis in clinical trials Augment new trial data with existing information: make sense of conicting datasets maximise power when only small sample sizes are feasible Adaptive design Stochastic curtailment based on predictive probabilities to inform early stopping decisions Adaptive randomisation: allocation rules

4 2 Bayesian survival analysis in clinical trials Semiparametric modelling using frequentist approach very common Development of methods for Bayesian analysis of survival data Development of user-friendly software (WinBUGS, STAN...) or procedures/packages (in SAS, R...) What about their use in practice in clinical trials?

5 3 Systematic review Search of articles in Pubmed and Web of Science published in English, using MeSH terms in title, abstract and keywords Hand search: full text search on the website of journals where at least one paper was found with the systematic search or journals with a high impact factor JCO, Cancer, JNCI Clinical Trials, PlosOne Medicine, BMC-Medicine, BMJ Note : Lancet, JAMA, BJC do not enable to search in full text

6 4 Systematic review Inclusion criteria: clinical trials using Bayesian survival analysis, including re-analyses (post-hoc analyses) Exclusion criteria: observational, cost-eectiveness, dose-nding studies, meta-analyses, and pharmacokinetics studies Examination of title, abstract, +/- full text by one reader Check of the selected papers by a second reader No divergences

7 5 Data extraction Development of a data extraction form 90 items Aims were to review: Use of Bayesian methods to design, conduct, and analyse clinical trials with a time-to-event (TTE) endpoint Bayesian survival models tted Information used to formulate priors Quality of reporting of Bayesian survival analyses Two readers read each paper independently Third reader if any discrepancy in responses

8 6 Flow chart

9 7 Description and Design General characteristics of the selected trials (N=23) Patient population Medical elds Oncology: 21 (91%) Cardiology: 1 (4%) Obstetrics: 1 (4%) Rare diseases: 11 (48%) Patient age Adults: 19 (83%) Children/teenager to adults: 3 (13%) Preborn: 1 (4%)

10 8 Description and Design General characteristics of the selected trials (N=23) Trial design Phase I & II Phase III No information N=9 (39%) N=11 (48%) N=3 (13%) Open-label N=21 (91%) Double blind 1 1 N=2 (9%) Randomised N=19 (83%) Non randomised 4 N=4 (17%) Single arm 4 N=4 (17%) Parallel group N=16 (70%) Factorial design 2 N=2 (9%) Cross over 1 N=1 (4%) Group sequential N=11 (48%) Adaptive randomisation 1 N=1 (4%) Non adaptive N=11 (48%)

11 9 Design Bayesian design (N=23) Bayesian analysis Primary only Secondary only Monitoring Considered endpoint N=6 (26%) N=7 (30%) N=10 (43%) Time-to-event (TTE) N=21 (91%) Survival at a given time point N=2 (9%) The majority of trials used Bayesian methods to monitor the trial or as a secondary analysis Most trials used the whole survival curve

12 Design Bayesian analysis of the TTE endpoint (N=21) Two dierent approaches: Hybrid approach: Posterior distribution of the ln(hazard Ratio:HR) derived from the partial likelihood estimate (Cox modelling in a frequentist framework) combined with prior distribution; or posterior of the HR without specifying the distribution used Full Bayesian regression modelling Primary Secondary Monitoring (N=5) (N=7) (N=9) Bay. only Bay.+Freq. Bay.+Freq. Bay.+Freq. Hybrid approach (N=10) Bayesian regression modelling (N=11) Bay. : Bayesian approach ; Freq. : Frequentist approach 10

13 11 Analysis Bayesian analysis: Hybrid approach (N=10) Prior distributions: 5 assumed ln(hr) Normal 5 Unspecied In 3 articles, authors used external data to formulate the prior distribution of the relative treatment eect Prior of the treatment eect Frequency Historical data only 1 Historical + Expert + non informative 1 Historical + 'O the shelf' + non informative 1 Non informative only 3 Unspecied 4

14 12 Analysis Bayesian analysis: Regression modelling (N=11) TTE Modelling Parameters Frequency Log-normal Relative treatment eect 4 N=4 + Covariate eect + Dispersion parameter Exponential Experimental + Control arm model parameters 3 N=4 Unspecied 1 Parametric unspecied Relative treatment eect 1 N=1 + Covariate eect Semiparametric Relative treatment eect only 1 N=2 Relative treatment eect 1 + Covariate eect

15 13 Analysis Bayesian analysis: Regression modelling (N=11) Prior on the relative treatment eect (N=7) Noninformative prior, Normal distribution, N=4 Noninformative prior, distribution not specied, N=2 No information on the prior, N=1 Exponential modelling of both treatment groups (N=4) Noninformative prior, unspecied distribution, N=2 Unspecied prior, Gamma distribution, N=1 No information on the prior, N=1 No articles specied the use of external data to formulate prior distribution of the treatment eect

16 14 Reporting Reporting of Bayesian methods Among the 20 articles reporting Bayesian survival analysis in the original trial report (clinical trials): 6 did not mention Bayesian survival analysis in the `Methods' 5 used Bayesian approach for monitoring only 1 used Bayesian approach for secondary analysis

17 15 Reporting How were priors reported? (N=23) Prior Frequency Denition of the distribution + Graphical representation 1 (4%) + Measure of location Graphical representation + Credibility intervals 1 (4%) + Measure of location Denition of the distribution 5 (22%) + Measure of location Not reported 16 (70%) Most trials do not report prior distribution

18 16 Reporting How were posteriors reported? (N=23) Posterior Frequency Denition of the distribution 3 (13%) Graphics 3 (13%) Credibility intervals/highest Posterior Density 8 (35%) Posterior/ Predictive probabilities 21 (91%) Measure of location 10 (43%) Not reported 2 (9%) The majority of trials report posterior or predictive probabilities

19 17 What software? (N=23) Software Frequency Survival at a given time point Unspecied 2 (100%) (N=2) Hybrid approach SAS only 3 (30%) (N=10) Fortran only 1 (10%) Unspecied 6 (60%) Regression modelling WinBUGS/OpenBUGS 5 (45%) (N=11) SAS + R 1 (10%) Unspecied 5 (45%) Most trials did not mention what software was used

20 18 Conclusions Limited number of trials used Bayesian survival analysis (N=23) 6/23 used Bayesian methods for the main analysis, 22/23 trials completed frequentist approach by a Bayesian analysis In few trials, authors reported details about the modelling used for Bayesian analysis External data contributed to the prior distribution in only 4/23 trials 21/23 trials presented results with posterior/predictive probabilities, and 3/23 trials used graphical representations of the priors/posteriors

21 19 Conclusions Reporting of Bayesian analyses could be improved by: Justifying the use of Bayesian analysis Clarifying the endpoint used for Bayesian analysis Clarifying what survival models are tted Dening the prior and posterior distributions What about recommendation for reporting (CONSORT, REMARK,...) clinical trials using Bayesian survival analysis? More training and guidance on Bayesian methods is needed to increase uptake of methods

22 20 Thank you for your attention

23 Hornberger J. Introduction to Bayesian reasoning. Int J technol Assess Health Care 17:9-16, References Abrams K, Ashby D, Errington D. Simple Bayesian analysis in clinical trials: a tutorial. Control Clin Trials 15:349-53, 1994 Berry DA. Bayesian statistics and the eciency and ethics of clinical trials. Stat Sci 19: , 2004 Berry DA. introduction to Bayesian methods III: use and interpretation of Bayesian tools in design and analysis. Clin Trials 2(4): , 2005 Berry DA. Bayesian clinical trials. Nat Rev Drug Discov 5(1):27-36, 2006 Brophy JM, Joseph L. Placing trials in context using Bayesian analysis. JAMA 273:871-5, 1995 Fayers PM, Ashby D, Parmar MKB: Tutorial in biostatistics Bayesian data monitoring in clinical trials. Stat Med 16: , 1997 Gelman A, Carlin JB, Stern HS, et al. Bayesian Data Analysis. New York, NY, Chapman & Hall, 1995 Gelman A. Bayesian Data Analysis (ed 2). Boca Raton, FL, Chapman & Hall/CRC, 2004

24 22 References Ibrahim JG, Chen MH, Sinha D. Bayesian Survival Analysis. New York: Springer ; 2001 Litlle R. Calibrated Bayes: A Bayes/frequentist roadmap. Am Stat 60:1-11, 2006 Little R. Calibrated Bayes, for statistics in general, and missing data in particular. Stat Sci 26: , 2011 Lunn DJ, Thomas A, Best N, Spiegelhalter DJ. WinBUGS - A Bayesian modelling framework: concepts, structure, and extensibility. Stat Comput 2000 ; 10: Simon RM, Dixon DO, Freidlin B. A Bayesian model for evaluating specicity of treatment eects in clinical trials, in Thall PF (ed): Recent advances in clinical trials design and analysis. Boston, MA, Kluwer Academic Publishers, 1995, pp Spiegelhalter DJ, Freedman LS, Parmar MKB. Bayesian approaches to randomized trials. J R Stat Soc 157: , 1994 Thall PF, Sung HG: Some extensions and applications of a Bayesian strategy for monitoring multiple outcomes in clinical trials. Stat Med 17: , 1998 Thall PF, Wooten LH, Tannir NM. Monitoring event times in early phase clinical trials: some practical issues. Clin Trials 2:467-78, 2005

25 23 Inclusion and exclusion criteria Search in the Title, Abstract and Keywords Inclusion criteria bayes* OR `prior distribution*' OR `posterior distribution*' trial* OR randomi?ed OR controlled OR `phase*study' survival OR `time-to' OR `failure-time' Exclusion criteria observational OR epidemio* OR case-control cost-eect* dose-nding

26 24 Where are Bayesian survival trials conducted?

27 25 Where are Bayesian survival trials conducted?

28 26 Where are Bayesian survival trials conducted?

29 27 Decision rules Decision rule Frequency Yes, based on an eect size threshold pre-specied 4 (17%) Yes, based on superiority 1 (4%) No 18 (78%) Decision rule Frequency P(Absolute dierence>threshold D) = y 1 P(Hazard Ratio>threshold D) = y 2 Both 2

30 28 Exponential distribution Assume that observed event time E(µ) µ IG(α, β) Posterior distribution IG(α + F, β + E) F = Number of patients with outcome observed E = Total exposure time Authors assume independent prior distributions for the hazards in each treatment group Each treatment arm are independent posterior distributions for λ C and λ E are independent

31 29 Where are Bayesian survival analyses reported? Where are Bayesian reported Frequency Title+Abstract+Text 2 (9%) Abstract+Text 10 (43%) Text Only 11 (48%)

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