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 Δ Choose Δ to be a difference within which treatments deemed equivalent General approach – perform two one-sided significance tests of H 0: μ 1 -μ 2 > Δ and μ 1 -μ 2 < -Δ If both are significant, then can conclude equivalence

7. 7 Figure from Jones et al (BMJ 1996) showing relationship between equivalence and confidence intervals

8. 8 CIs Assume sd  is known. Then 100(1-  )% CI to compare difference in means is Treatments deemed equivalent if interval falls in -Δ to +Δ where

9. 9 Let τ = μ A - μ B Let η = Δ - z α/2 σλ Power is defined when (1) has τ=0 (1) or Maximal Type I error rate is (1) when τ=Δ

10. 10 If treatment groups have same size, n, then required sample size is This is similar to testing for a difference Except i)Usually Δ is smaller than for a difference trial ii)We use β/2 rather than β (2)

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 ω that τ >0 Design objective We require the sample size (n 1,n 2 ) be large enough so there is a probability of at least ψ of obtaining a positive result. The probability ψ is known as the assurance

13. 13 Bayesian assumptions Let prior expectation of (μ 1, μ 2 ) T be m a according to analysis prior and m d according to the design prior Let variances be V a and V d for analysis and design priors respectively Let be ( ) T, the observed data Let S be the sampling variance matrix (note this depends on n 1 and n 2 )

14. 14 Let W a =V a -1,W s =S -1 and V * =(W a +W s ) -1 and a=(1,-1) T Posterior mean of (μ 1, μ 2 ) is Normally distributed with expectation and variance Under analysis prior

15. 15 Under design prior Unconditional distribution of is Normal with mean and variance From which can get sample size calculation (See O Hagan and Stevens)

16. 16 Frequentist interpretation If then the Bayesian methods for determining sample size agree with frequentist V a -1 =0 – weak analysis prior –’vague’ prior If V d =0 If - strong design prior

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

18. 18 A modification of O’Hagan and Stevens suggests that for equivalence trials, a positive outcome occurs when Two-sided and One-sided Sample size also a modification of O’Hagan and Stevens

19. 19 Parameters for non-inferiority m a - the analysis prior mean could be 0 m d - the design prior mean could be 0

20. 20 What if m d and V d >0 ? A weak design prior Then we have some information about the possible differences, so ‘proving’ the null hypothesis is difficult E.g. if we were 50% sure that δ>0, before the trial then cannot be 80% sure that δ=0 after the trial

21. 21 What if V a -1 >0? A strong analysis prior CIs will be shifted towards m a If m a =0, then probability of a positive event increased 95% CIs will be narrower than for the frequentist approach

22. 22 Conclusions Bayesian approach more natural for equivalence (Can prove H 0 ) More work on getting pragmatic suggestions for V a and V d needed