Equivalence and Bioequivalence:
Frequentist and Bayesian views
on sample size
Mike Campbell
ScHARR
CHEBS FOCUS fortnight 1/04/03
1
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)
2
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
3
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)
4
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
5
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< -Δ
If both are significant, then can conclude equivalence
6
Figure from Jones et al (BMJ 1996) showing relationship
between equivalence and confidence intervals
7
Assume sd  is known.
CIs
Then 100(1-)% CI to compare difference in means is
d  z / 2  , d  z / 2
where
  (n11  n21 )1 / 2
Treatments deemed equivalent if interval falls in
-Δ to +Δ
8
Let τ = μA - μB
H 0 :|  |  vs H 1 : |  | 
Let η = Δ - zα/2 σλ
Pr( d  ( , ))  (
 

 z / 2 )   (
  

 z / 2 )
(1)
Maximal Type I error rate is (1) when τ=Δ
Power is defined when (1) has τ=0
1    2 (


 z / 2 )  1
or
 

1   / 2  
 z / 2 
 

9
If treatment groups have same size, n,
then required sample size is
n2
2
2
(za / 2  z / 2 )2
(2)
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
10
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!
11
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
12
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 x be ( x1 , x2 )T, the observed data
• Let S be the sampling variance matrix (note
this depends on n1 and n2)
13
Let Wa =Va-1 ,Ws=S-1 and V*=(Wa+Ws)-1 and a=(1,-1)T
Under analysis prior
Posterior mean of (μ1, μ2) is Normally distributed with
expectation and variance
a T V * {Wa m a  Ws x)
T
*
a V a
14
Under design prior
Unconditional distribution of x is Normal with mean and variance
E ( x)  m d
,Var (x)  Vd  S
From which can get sample size calculation
(See O Hagan and Stevens)
15
Frequentist interpretation
If
Va1  Wa  0 and
Vd  0
then the Bayesian methods for determining sample
size agree with frequentist
If
If
Va-1 =0 – weak analysis prior –’vague’ prior
Vd=0
- strong design prior
16
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.
17
A modification of O’Hagan and Stevens suggests that
for equivalence trials, a positive outcome occurs when
| a T V * {Wa m a  Ws x)   | 
z1 a T V * a
Two-sided and
  a T V * {Wa m a  Ws x)  z1 a T V * a
One-sided
Sample size also a modification of O’Hagan and Stevens
18
Parameters for non-inferiority
ma - the analysis prior mean could be 0
md - the design prior mean could be 0
19
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 trial
then cannot be 80% sure that δ=0 after the trial
20
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
21
Conclusions
• Bayesian approach more natural for
equivalence (Can prove H0)
• More work on getting pragmatic
suggestions for Va and Vd needed
22