Results

1. CALCULATING EXPECTED VALUE OF SAMPLE INFORMATION FOR SURVIVAL TRIALS: BAYESIAN UPDATING FOR THE WEIBULL DISTRIBUTION. CHEBSAlan Brennan, , Samer Kharroubi. University of Sheffield, England. a.brennan@sheffield.ac.uk . • Purpose:a generalised process for calculating partial EVSI for survival trials. • Background: Brennan et al. 1,2 and Claxton et al 3 have promoted the • EVSI concept as a measure of the societal value of research designs to help • identify optimal sample sizes for primary data collection. • The Weibull Distribution for Survival Data • Survival analysis in cost-effectiveness decision models often utilises Weibull survival curves. • Typical data examines “survival” i.e. ‘time to death’ or ‘time to clinical event’ • N patients, • X suffer an event, • over a follow-up duration D, after which data is censored. • Typical data for the ith patient is a pair of numbers (di, ti) where • di = 1 if died or 0 if alive or withdrew and ti = time of death / last measured date of survival. • The classic text4 determines maximum likelihood estimates (MLE) for shape (γ) and • scale (λ) parameters of the Weibull curve. • These are obtained by solving two equations to define the and , which • maximise the log of the likelihood function, usually using Newton-Raphson approach. • To quantify uncertainty in these parameters we also need J= -l’’(λ,γ) • i.e. the 2nd derivative of the likelihood function. J-1 is the variance covariance matrix. • Some software packages e.g. EXCEL function Weibull(α, β) and SPLUS rweibull(λ, β) • use an alternative formulation for the scale parameter defining β as β = (1/ λ)^(1/ γ). • ^ ^ • Mean survival = , where is the mathematical gamma function. • Cost-effectiveness decision models often have Weibull survival parameters alongside • cost and quality of life parameters. EVSI examines the value of gaining more accurate estimates of these parameters for informing the decision between treatments. • The questions: “How valuable would greater sample size (Nnew) be ?” • “How valuable would longer follow-up (Dnew) be ?” • “How valuable would both greater sample size + longer follow-up be ?” • Part C: Simulating a data collection exercise • Decide on Nnew, the number of new patients to study • Decide on Dnew, the duration of followup in the new study • Sample from prior values for λ (λsample) and γ (γsample i.e. βsample) i.e. partB • Simulate Nnew patients survival times from the Weibull distribution • SPLUS code is ….. newdata = rweibull(Nnew, λsample, βsample) • Part D: Updating the Weibull parameters given simulated data • Combine the prior data with the simulated data • Use Newton Raphson algorithm to quantify revised estimates for and • SPLUS … nlimb() to minimise nonlinear functions with constraints • Part E: Quantifying the Value of the Simulated Additional data • Option 1: 2 level algorithm • Evaluate net benefit for each treatment given new data, make ‘revised decision’ • (Re-run the decision model using probabilistic Monte Carlo simulation ) •  = the parameters for the model (uncertain currently). • d = set of possible decisions or strategies. • NB(d, ) = the net benefit for decision d, and parameters  • Net benefit of ‘revised decision’ | simulated data[i]: = (1) • Re-run the Loop, say 1,000 times including evaluating expression (1) • Calculated the overall expected value of the proposed data collection exercise • (2) • Two Nested Expectations • Option 2: 1st Order Laplace Approximation 5 • Evaluate net benefit for each treatment given new data, make ‘revised decision’ • Net benefit of ‘revised decision’ | simulated data[i]: = (3) • is vector posterior mode of model parameters and • 1st order approximation EVSI = (4) • Only One Expectation • Posterior mode is recalculated for each simulated dataset collected Xi • Results • Doubling the sample size from n=50 to 100, increases the value of a 6 month study by 64% but the value of a 3 years study by only 34%. • Similarly doubling the duration from 6 months to 1 year increases the value of a small n=50 study by 70%, but the value of a larger n=1,000 study by only 10%. Loop Methodology Part A: Prior data to estimate prior Weibull parameters Illustrative data 200 patients 100 days follow-up = 0.007836397 = 0.904329012 VarCovar.= Part B: Estimating the effect of current uncertainty in Weibull parameters To sample: Log (λ), log (γ) ~ MultiVNormal (NormalMean, NormalVarCovar) SPLUS program : thetahat = c(0.007836397,0.904329012) NormalMean=log(thetahat * exp(-0.5* diag(VarCovar))) a=matrix(thetahat,ncol=2) NormalVarCovar = log (VarCovar /c(a%*%t(a))+1) samplelogtheta[i,] = rmvnorm(1, mean=NormalMean, cov=NormalVarCovar) Conclusions 1) A simple illustrative model for two treatments, costs and benefits is used to show how Bayesian updating for the Weibull distribution can quantify the value of different survival study options. 2) The results show diminishing returns in the value of information as sample size increases but at a different rate to the diminishing returns as the follow-up duration is varied. 3) This methodology provides a new and valuable approach for EVSI calculation which might be applied generally in survival studies. 1 Brennan, A.,Chilcott, J. B,Kharroubi, S,O'Hagan, A. Calculating Expected Value of Perfect Information:- Resolution of Conflicting Methods via a Two Level Monte Carlo Approach, presented at the 24th Annual Meeting of SMDM, October 23rd, 2002, Washington. 2002. Submitted - Journal of Medical Decision Making 2 Brennan, A. B., Chilcott, J. B., Kharroubi, S., O'Hagan, A. A Two Level Monte Carlo Approach to Calculation Expected Value of Sample Information: How To Value a Research Design. Presented at the 24th Annual Meeting of SMDM, October 23rd, 2002, Washington. 2002. 3 Claxton, K, Ades, T. Efficient Research Design: An Application of Value of Information Analysis to an Economic Model of Zanamivir. Presented at the 24th Annual Meeting of the Society for Medical Decision Making, October 21st, 2002, Washington. 2002. 4 Collett : Modelling Survival Data in Medical Research, Chapman and Hall, 1994 5 Sweeting, J, Kharroubi, S. Some New Formulae for Posterior Expectations and Bartlett Corrections. Sociedad de Estadistica e Investigacion Operative Test, (Accepted) 2003 Log (γ) Log (λ) Analyses Samples sizes from 0 to 1,000 Follow up from 6 months to 3 years Acknowledgements: thankyou to Professor Tony O’Hagan for encouraging our ongoing work