Sample size determination
for cost-effectiveness trials
Anne Whitehead
Medical and Pharmaceutical Statistics Research Unit
The University of Reading
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Comparative study
• Parallel group design
• Control treatment (0)
New treatment (1)
• n0 subjects to receive control treatment
n1 subjects to receive new treatment
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Measure of treatment difference
Let  be the measure of the advantage of new over control
 > 0  new better than control
 = 0  no difference
 < 0  new worse than control
Consider frequentist, Bayesian and decision-theoretic
approaches
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1.
Frequentist approach
Focus on hypothesis testing and error rates
- what might happen in repetitions of the trial
e.g.
Test null hypothesis
against alternative
H0 :  = 0
H1+:  > 0
Obtain p-value, estimate and confidence interval
Conclude that new is better than control if the one-sided
p-value is less than or equal to 
Fix
P(conclude new is better than control |  = R) = 1–
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Distribution of ̂
=0
 = R


k
Fail to Reject H0
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Reject H0
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A general parametric approach
Assume ˆ ~ N  , 1 
 w
Reject H0 if ̂ > k
P(ˆ  k |   0)  


1  k w  
k  z
w
where  is the standard normal distribution function and
P(Z > z) =  where Z ~ N(0, 1)
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Require
P(ˆ  k |   R )  1  




1   (k  R ) w  1  
1    R  k  w  
(R  k) w  z
 z   z 
w 


R


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Application to cost-effectiveness trials
Briggs and Tambour (1998)
 = k (E1 – E0) – (C1 – C0)
is the net benefit, where
E1, E0
are mean values for efficacy for
new and control treatments
C1, C0 are mean costs for new and
control treatments
k
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is the amount that can be paid for a
unit improvement in efficacy for a
single patient
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ˆ  k(x E1  x E0 )  (x C1  x C0 )
E1 E 0
var(x E1  x E0 ) 

n1
n0
2
2
C1 C 0
var(x C1  x C0 ) 

n1
n0
2
2
cov  (x E1  x E0 ),(x C1  x C0 )  
 var(x E1  x E0 )var(x C1  x C0 )
Set
 z   z 
1
w 


ˆ

var() 
R

2
and solve for n0 and n1
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2.
Bayesian approach
Treat parameters as random variables
Incorporate prior information
Inference via posterior distribution for parameters
Obtain estimate and credibility interval
Conclude that new is better than control if
P( > 0|data) > 1 – 
Fix P0 (conclude new better than control) = 1 – 
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ˆ ~ N  , 1/w 


Likelihood function f (ˆ | )  exp w(ˆ  ) 2 / 2
N  0 , 1/w 0 
Prior h0() is
Posterior h(|data)
 exp{w(ˆ  )2 / 2}exp{w 0 (  0 ) 2 / 2}


 exp (ˆ w  0 w 0 )  2  w  w 0  / 2
i.e. h(|data) is
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 0 w 0  ˆ w
N
,
 w0  w
1 

w0  w 
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P ( > 0|data) > 1 – 
if
0 w 0  ˆ w

w0  w
z
w0  w
i.e.
0 w 0  ˆ w  z  w 0  w
i.e.
ˆ w  0 w 0  z  w 0  w
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Prior to conducting the study,
ˆ  ~ N  , 1 


 w

1 
 ~ N  0 ,

w
0 

so
ˆ ~   , 1  1 
 0

w
w
0 

2


w
ˆw ~  w, w 
 0

w
0 

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Require
P0 ˆ w  0 w 0  z  w 0  w  1  


  w  z w  w   w 
0
0
  1 
1    0 0 
2


w

w
/ w0


  (w  w)  z w  w 

0

1   0 0
2


w

w
/
w
0


0 (w 0  w)  z  w 0  w  z w  w 2 / w 0
0 (w 0  w)  z  w 0  w  z w  w 2 w 0
Express w in terms of n0 and n1, provide values for 0 and w0
and solve for n0 and n1
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Application to cost-effectiveness trials
O’Hagan and Stevens (2001)
 = k (E1 – E0) – (C1 – C0)
ˆ  k(x E1  x E0 )  (x C1  x C0 )
Use multivariate normal distribution for x E1 , x E0 , x C1 , x C0
- separate correlations between efficacy and cost for
each treatment
Allow different prior distributions for the design stage
(slide13) and the analysis stage (slide 11)
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3.
Decision-theoretic approach
Based on Bayesian paradigm
Appropriate when outcome is a decision
Explicitly model costs and benefits from possible actions
Incorporate prior information
Choose action which maximises expected gain
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Actions
Undertake study and collect w units of information on ,
then one of the following actions is taken:
Action 0 : Abandon new treatment
Action 1 : Use new treatment thereafter
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Table of gains
(relative to continuing with control treatment)

0
>0
Action 1
– cw – b
– cw – b + r1
Action 0
– cw
– cw
c = cost of collecting 1 unit of information (w)
b = further development cost of new treatment
r1 = reward if new treatment is better
G0,w() = – cw
G1,w ()   cw  b  r1I(  0)
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Following collection of w units of information, the expected
gain from action a is
Ga, w(x) = E {Ga,w()| x}
Action will be taken to maximise E {Ga,w()|x},
that is a*, w* where
Ga*, w*(x) = max {Ga, w(x)}
(Note: Action 1 will be taken if P ( > 0|data) > b/r1)
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At design stage consider frequentist expectation:
E (Ga*, w(x))
  G a*,w (x) f (x; , w) dx
x
and use this as the gain function Uw ()
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Expected gain from collecting information w is

U w  E0{U w ()}   U w ()h 0 ()d

So optimal choice of w is w*, where
Uw*  max {Uw }
w
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 

U w*  max E0  E max E (G a,w () | x) 


w
a
 max 
w

 max 
 x
a


G a,w ()h( | x)d f (x; , w)dx h 0 ()d
This is the prior expected utility or pre-posterior gain
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Note:


E max E (G a,w () | x)
a
= E{– cw + max(r1 P ( > 0|data) – b, 0)}
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Application to cost-effectiveness trials
Could apply the general decision-theoretic approach taking q
to be the net benefit
The decision-theoretic approach appears to be ideal for this
setting, but does require the specification of an appropriate
prior and gain function
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Table of gains – ‘Simple Societal’
(relative to continuing with control treatment)

0
>0
Action 1
– cw – b
– cw – b + r1
Action 0
– cw
– cw
c = cost of collecting 1 unit of information (w)
b = further development cost of new treatment
r1 = reward if new treatment is more cost-effective
G0,w() = – cw
G1,w ()   cw  b  r1I(  0)
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Gains – ‘Proportional Societal’
(relative to continuing with control treatment)
c = cost of collecting 1 unit of information (w)
b = further development cost of new treatment
r2 = reward if new treatment is more cost-effective
G0,w() = – cw
G1,w ()   cw  b  r2
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Gains – ‘Pharmaceutical Company’
c = cost of collecting 1 unit of information (w)
b = further development cost of new treatment
r3 = reward if new treatment is more cost-effective
G 0,w (x)   cw
G1,w (x)   cw  b  r3I(x  A)
where A is the set of outcomes which leads to Action 1,
e.g. for which P ( > 0|data) > 1 – 
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References
Briggs, A. and Tambour, M. (1998). The design and analysis
of stochastic cost-effectiveness studies for the evaluation of
health care interventions (Working Paper series in Economics
and Finance No. 234). Stockholm, Sweden: Stockholm
School of Economics.
O’Hagan, A. and Stevens, J. W. (2001). Bayesian assessment
of sample size for clinical trials of cost-effectiveness. Medical
Decision Making, 21, 219-230.
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