Equivalence and Bioequivalence: Frequentist and Bayesian views on sample size

1. 1 Equivalence and Bioequivalence:Frequentist and Bayesian views on sample size Mike Campbell ScHARR CHEBS FOCUS fortnight 1/04/03

2. 2 Equivalence Many trials are not designed to prove differences but equivalences Examples : generic drug vs established drug Video vs psychiatrist NHS Direct vs GP Costs of two treatments Alternatively � non-inferiority (one-sided)

3. 3 Efficacy vs cost For some trials (e.g. of generics) one would like to show similar efficacy at less cost Thus can have an equivalence and a cost difference trial in one study

4. 4 Motivating example AHEAD (Health Economics And Depression) Trial of trycyclics, SSRIs and lofepramine Clinical outcome - depression free months Economic outcome � cost Powered to show equivalence to within 5% with 90% power and 5% significance (estimated effect size 0.3 and SD 1.0)

5. 5 Bio-equivalence (diversion) For bio-equivalence we are trying to show that two therapies have same action Usually compare serum profiles by e.g. AUC Often paired studies FDA: 80:20 rule 80% power to detect 20% difference

6. 6 Frequentist view Impossible to prove null hypothesis All we can do is show that differences are at most ? Choose ? to be a difference within which treatments deemed equivalent General approach � perform two one-sided significance tests of H0: �1-�2> ? and �1-�2< -? If both are significant, then can conclude equivalence

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8. 8 CIs

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11. 11 Problems with equivalence trials Poor trials (e.g. poor compliance and larger measurement errors bias trial towards null) Jones et al (1996) suggest using an ITT approach and �per-protocol� and hope they give similar results!

12. 12 Bayesian sample size (O�Hagan and Stevens 2001) Analysis objective Outcome is positive if the data obtained are such that there is a posterior probability of at least ? that t >0 Design objective We require the sample size (n1,n2) be large enough so there is a probability of at least ? of obtaining a positive result. The probability ? is known as the assurance

13. 13 Bayesian assumptions Let prior expectation of (�1,�2)T be ma according to analysis prior and md according to the design prior Let variances be Va and Vd for analysis and design priors respectively Let be ( )T, the observed data Let S be the sampling variance matrix (note this depends on n1 and n2)

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15. 15 Under design prior

16. 16 Frequentist interpretation

17. 17 Bayesian equivalence (after O�Hagan and Stevens(2001) Analysis objective: Outcome of study is positive if the upper limit of the (1-?)% prediction interval for t is < ? (one sided) or upper and lower limits of prediction interval for t are within � ? (two sided). Design objective: Sample size is such that there is a probability of at least ? of obtaining a positive result.

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19. 19 Parameters for non-inferiority

20. 20 What if md and Vd>0 ?A weak design prior

21. 21 What if Va-1>0?A strong analysis prior

22. 22 Conclusions Bayesian approach more natural for equivalence (Can prove H0) More work on getting pragmatic suggestions for Va and Vd needed