Sample size determination for cost-effectiveness trials

1. Sample size determination for cost-effectiveness trials Anne Whitehead Medical and Pharmaceutical Statistics Research Unit The University of Reading CHEBS Workshop - April 2003

2. Comparative study Parallel group design Control treatment (0) New treatment (1) n0 subjects to receive control treatment n1 subjects to receive new treatment CHEBS Workshop - April 2003

3. 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 CHEBS Workshop - April 2003

4. 1. Frequentist approach Focus on hypothesis testing and error rates • what might happen in repetitions of the trial e.g. Test null hypothesis H0 :  = 0 against alternative H1+:  > 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– CHEBS Workshop - April 2003

5.  = 0  = R   k Distribution of Fail to Reject H0 Reject H0 CHEBS Workshop - April 2003

6. A general parametric approach Assume Reject H0 if > k where  is the standard normal distribution function and P(Z > z) =  where Z ~ N(0, 1) CHEBS Workshop - April 2003

7. Require CHEBS Workshop - April 2003

8. 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 k is the amount that can be paid for a unit improvement in efficacy for a single patient CHEBS Workshop - April 2003

9. Set and solve for n0 and n1 CHEBS Workshop - April 2003

10. 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 P0 (conclude new better than control) = 1 –  CHEBS Workshop - April 2003

11. Likelihood function Prior h0() is Posterior h(|data) i.e. h(|data) is CHEBS Workshop - April 2003

12. P ( > 0|data) > 1 –  if i.e. i.e. CHEBS Workshop - April 2003

13. Prior to conducting the study, so CHEBS Workshop - April 2003

14. Require P0 Express w in terms of n0 and n1, provide values for 0 and w0 and solve for n0 and n1 CHEBS Workshop - April 2003

15. 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) CHEBS Workshop - April 2003

16. 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 CHEBS Workshop - April 2003

17. 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 CHEBS Workshop - April 2003

18. Table of gains (relative to continuing with control treatment)  Action 1 Action 0  0 – cw – b – cw > 0 – cw – b + r1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r1 = reward if new treatment is better G0,w() = – cw CHEBS Workshop - April 2003

19. Following collection of w units of information, the expected gain from action a is Ga, w(x) = E {Ga,w()| x} Action will be taken to maximise E {Ga,w()|x}, that is a*, w* where Ga*, w*(x) = max {Ga, w(x)} (Note: Action 1 will be taken if P ( > 0|data) > b/r1) CHEBS Workshop - April 2003

20. At design stage consider frequentist expectation: E (Ga*, w(x)) and use this as the gain function Uw () CHEBS Workshop - April 2003

21. Expected gain from collecting information w is So optimal choice of w is w*, where CHEBS Workshop - April 2003

22. This is the prior expected utility or pre-posterior gain CHEBS Workshop - April 2003

23. Note: = E{– cw + max(r1 P ( > 0|data) – b, 0)} CHEBS Workshop - April 2003

24. 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 CHEBS Workshop - April 2003

25. Table of gains – ‘Simple Societal’(relative to continuing with control treatment)  Action 1 Action 0  0 – cw – b – cw > 0 – cw – b + r1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r1 = reward if new treatment is more cost-effective G0,w() = – cw CHEBS Workshop - April 2003

26. 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 r2 = reward if new treatment is more cost-effective G0,w() = – cw CHEBS Workshop - April 2003

27. Gains – ‘Pharmaceutical Company’ c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r3 = 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 –  CHEBS Workshop - April 2003

28. 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. CHEBS Workshop - April 2003