1. MPS Research UnitCHEBS Workshop - April 20031 Anne Whitehead Medical and Pharmaceutical Statistics Research Unit The University of Reading Sample size determination for cost-effectiveness trials

2. MPS Research UnitCHEBS Workshop - April 20032 Comparative study Parallel group design Control treatment (0) New treatment (1) n 0 subjects to receive control treatment n 1 subjects to receive new treatment

3. MPS Research UnitCHEBS Workshop - April 20033 Measure of treatment difference Let  be the measure of the advantage of new over control  > 0  new better than control  = 0  no difference  < 0  new worse than control Consider frequentist, Bayesian and decision-theoretic approaches

4. MPS Research UnitCHEBS Workshop - April 20034 1.Frequentist approach Focus on hypothesis testing and error rates - what might happen in repetitions of the trial e.g.Test null hypothesisH 0 :  = 0 against alternativeH 1 + :  > 0 Obtain p-value, estimate and confidence interval Conclude that new is better than control if the one-sided p-value is less than or equal to  Fix P(conclude new is better than control |  =  R ) = 1– 

5. MPS Research UnitCHEBS Workshop - April 20035 Distribution of  = 0  =  R   k Fail to Reject H 0 Reject H 0

6. MPS Research UnitCHEBS Workshop - April 20036 A general parametric approach Assume Reject H 0 if > k where  is the standard normal distribution function and P(Z > z  ) =  where Z ~ N(0, 1)

7. MPS Research UnitCHEBS Workshop - April 20037 Require

8. MPS Research UnitCHEBS Workshop - April 20038 Application to cost-effectiveness trials Briggs and Tambour (1998)  = k (  E1 –  E0 ) – (  C1 –  C0 ) is the net benefit, where  E1,  E0 are mean values for efficacy for new and control treatments  C1,  C0 are mean costs for new and control treatments kis the amount that can be paid for a unit improvement in efficacy for a single patient

9. MPS Research UnitCHEBS Workshop - April 20039 Set and solve for n 0 and n 1

10. MPS Research UnitCHEBS Workshop - April 200310 2.Bayesian approach Treat parameters as random variables Incorporate prior information Inference via posterior distribution for parameters Obtain estimate and credibility interval Conclude that new is better than control if P (  > 0|data) > 1 –  Fix P 0 (conclude new better than control) = 1 – 

11. MPS Research UnitCHEBS Workshop - April 200311 Likelihood function Prior h 0 (  ) is Posterior h(  |data) i.e. h(  |data) is

12. MPS Research UnitCHEBS Workshop - April 200312 P (  > 0|data) > 1 –  if i.e.

13. MPS Research UnitCHEBS Workshop - April 200313 Prior to conducting the study, so

14. MPS Research UnitCHEBS Workshop - April 200314 Require P 0 Express w in terms of n 0 and n 1, provide values for  0 and w 0 and solve for n 0 and n 1

15. MPS Research UnitCHEBS Workshop - April 200315 Application to cost-effectiveness trials O’Hagan and Stevens (2001)  = k (  E1 –  E0 ) – (  C1 –  C0 ) Use multivariate normal distribution for - separate correlations between efficacy and cost for each treatment Allow different prior distributions for the design stage (slide13) and the analysis stage (slide 11)

16. MPS Research UnitCHEBS Workshop - April 200316 3.Decision-theoretic approach Based on Bayesian paradigm Appropriate when outcome is a decision Explicitly model costs and benefits from possible actions Incorporate prior information Choose action which maximises expected gain

17. MPS Research UnitCHEBS Workshop - April 200317 Actions Undertake study and collect w units of information on , then one of the following actions is taken: Action 0 : Abandon new treatment Action 1 : Use new treatment thereafter

18. MPS Research UnitCHEBS Workshop - April 200318 Table of gains ( relative to continuing with control treatment)  Action 1 Action 0  0 – cw – b – cw > 0 – cw – b + r 1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 1 = reward if new treatment is better G 0,w (  ) = – cw

19. MPS Research UnitCHEBS Workshop - April 200319 Following collection of w units of information, the expected gain from action a is G a, w (x) = E {G a,w (  )| x} Action will be taken to maximise E {G a,w (  )|x}, that is a*, w* where G a*, w* (x) = max { G a, w (x)} (Note: Action 1 will be taken if P (  > 0|data) > b/r 1 )

20. MPS Research UnitCHEBS Workshop - April 200320 At design stage consider frequentist expectation: E ( G a*, w (x)) and use this as the gain function U w (  )

21. MPS Research UnitCHEBS Workshop - April 200321 Expected gain from collecting information w is So optimal choice of w is w*, where

22. MPS Research UnitCHEBS Workshop - April 200322 This is the prior expected utility or pre-posterior gain

23. MPS Research UnitCHEBS Workshop - April 200323 Note: = E{– cw + max(r 1 P (  > 0|data) – b, 0)}

24. MPS Research UnitCHEBS Workshop - April 200324 Application to cost-effectiveness trials Could apply the general decision-theoretic approach taking q to be the net benefit The decision-theoretic approach appears to be ideal for this setting, but does require the specification of an appropriate prior and gain function

25. MPS Research UnitCHEBS Workshop - April 200325 Table of gains – ‘Simple Societal’ ( relative to continuing with control treatment)  Action 1 Action 0  0 – cw – b – cw > 0 – cw – b + r 1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 1 = reward if new treatment is more cost-effective G 0,w (  ) = – cw

26. MPS Research UnitCHEBS Workshop - April 200326 Gains – ‘Proportional Societal’ ( relative to continuing with control treatment) c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 2 = reward if new treatment is more cost-effective G 0,w (  ) = – cw

27. MPS Research UnitCHEBS Workshop - April 200327 Gains – ‘Pharmaceutical Company’ c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 3 = reward if new treatment is more cost-effective where A is the set of outcomes which leads to Action 1, e.g. for which P (  > 0|data) > 1 – 

28. MPS Research UnitCHEBS Workshop - April 200328 References Briggs, A. and Tambour, M. (1998). The design and analysis of stochastic cost-effectiveness studies for the evaluation of health care interventions (Working Paper series in Economics and Finance No. 234). Stockholm, Sweden: Stockholm School of Economics. O’Hagan, A. and Stevens, J. W. (2001). Bayesian assessment of sample size for clinical trials of cost-effectiveness. Medical Decision Making, 21, 219-230.