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

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

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

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