1
Equivalence and Bioequivalence:Frequentist and Bayesian views
on sample size
Mike CampbellScHARR
CHEBS FOCUS fortnight 1/04/03
2
Equivalence
• Many trials are not designed to prove differences but equivalencesExamples : generic drug vs established drug
Video vs psychiatristNHS Direct vs GPCosts 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 H0: μ1-μ2> Δ and μ1-μ2< -ΔIf both are significant, then can conclude equivalence
7
Figure from Jones et al (BMJ 1996) showing relationshipbetween equivalence and confidence intervals
8
CIs
2/2/ , zdzd
Assume sd is known.
Then 100(1-)% CI to compare difference in means is
Treatments deemed equivalent if interval falls in-Δ to +Δ
2/112
11 )( nnwhere
9
Let τ = μA - μB ||:|:| 10 HvsH
Let η = Δ - zα/2 σλ
)()()),(Pr( 2/2/
zzd
Power is defined when (1) has τ=0
(1)
1)(21 2/
z
2/2/1
z
or
Maximal Type I error rate is (1) when τ=Δ
10
If treatment groups have same size, n, then required sample size is
22/2/2
2
)(2 zzn a
This is similar to testing for a difference
Excepti) Usually Δ is smaller than for a difference trialii) 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 (n1,n2) 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 (μ1,μ2)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)
x 21 , xx
14
Let Wa =Va-1 ,Ws=S-1 and V*=(Wa+Ws)-1 and a=(1,-1)T
)xWm{WVa saa*T
Posterior mean of (μ1, μ2) is Normally distributed with expectation and variance
aVa *T
Under analysis prior
15
Under design prior
S V x m xd d ) ( , ) (Var E
Unconditional distribution of is Normal with mean and variancex
From which can get sample size calculation (See O Hagan and Stevens)
16
Frequentist interpretation
If 0V0WV da1
a and
then the Bayesian methods for determining sample size agree with frequentist
Va-1 =0 – weak analysis prior –’vague’ prior
If Vd=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
aVa)xWm{WVa *Tsaa
*T 1|| z
A modification of O’Hagan and Stevens suggests that for equivalence trials, a positive outcome occurs when
aVa)xWm{WVa *Tsaa
*T 1z
Two-sided and
One-sided
Sample size also a modification of O’Hagan and Stevens
19
Parameters for non-inferiority
ma - the analysis prior mean could be 0
md - the design prior mean could be 0
20
What if md and Vd>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 trialthen cannot be 80% sure that δ=0 after the trial
21
What if Va-1>0?
A strong analysis prior
CIs will be shifted towards ma
If ma=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 H0)
• More work on getting pragmatic suggestions for Va and Vd needed