sample size determination for cost-effectiveness trials
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Sample size determination for cost-effectiveness trials. Anne Whitehead Medical and Pharmaceutical Statistics Research Unit The University of Reading. Comparative study. Parallel group design Control treatment (0) New treatment (1) n 0 subjects to receive control treatment - PowerPoint PPT PresentationTRANSCRIPT
MPS Research Unit CHEBS Workshop - April 2003 1
Anne Whitehead
Medical and Pharmaceutical Statistics Research Unit
The University of Reading
Sample size determination for cost-effectiveness trials
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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 H0 : = 0against alternative 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 Reject H0
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A general parametric approach
Assume
Reject H0 if > k
where is the standard normal distribution function andP(Z > z) = where Z ~ N(0, 1)
1ˆ ~ N ,w
ˆP( k | 0)
1 k w
k z w
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Require
RˆP( k | ) 1
R1 (k ) w 1
R1 k w
R( k) w z
2
R
z zw
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Application to cost-effectiveness trialsBriggs 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 andcontrol treatments
k is the amount that can be paid for aunit improvement in efficacy for asingle patient
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E1 E0 C1 C0
ˆ k(x x ) (x x ) 22
EE 01E1 E0
1 0
22CC 01
C1 C01 0
var(x x )n n
var(x x )n n
E1 E0 C1 C0
E1 E0 C1 C0
cov (x x ),(x x )
var(x x )var(x x )
2
R
z z1w
ˆvar( )
Set
and solve for n0 and n1
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2. Bayesian approach
Treat parameters as random variablesIncorporate prior informationInference via posterior distribution for parametersObtain 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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Likelihood function
Prior h0() is
Posterior h(|data)
i.e. h(|data) is
2ˆ ˆf ( | ) exp w( ) / 2
0 0N ,1/w
2 20 0
ˆexp{ w( ) / 2}exp{ w ( ) / 2}
20 0 0
ˆexp ( w w ) w w / 2
ˆ ~ N ,1/w
0 0
0 0
ˆw w 1N ,
w w w w
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P ( > 0|data) > 1 –
if
i.e.
i.e.
0 0
0 0
ˆw w z
w w w w
0 0 0ˆw w z w w
0 0 0ˆw w z w w
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Prior to conducting the study,
so
1ˆ ~ N ,w
00
1~ N ,
w
00
1 1ˆ ~ ,w w
2
00
wˆw ~ w, ww
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Require
P0 0 0 0ˆw w z w w 1
0 0 0 0
20
w z w w w1 1
w w / w
0 0 0
20
(w w) z w w1
w w / w
20 0 0 0(w w) z w w z w w / w
20 0 0 0(w w) z w w z w w w
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)
Use multivariate normal distribution for- separate correlations between efficacy and cost for each treatment
Allow different prior distributions for the design stage (slide13) and the analysis stage (slide 11)
E1 E0 C1 C0ˆ k(x x ) (x x )
E1 E0 C1 C0x ,x ,x ,x
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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)
Action 1 Action 0 0 – cw – b – cw
> 0 – cw – b + r1 – 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
1,w 1G ( ) cw b r I( 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))
and use this as the gain function Uw ()
a*,w
x
(x) f (x; ,w) dx G
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Expected gain from collecting information w is
So optimal choice of w is w*, where
w 0 w w 0{U ( )} U ( )h ( )d
U E
w* ww
max { }U U
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This is the prior expected utility or pre-posterior gain
w* 0 a,ww a
max E max (G ( ) | x) U E E
a,w 0xw amax max G ( )h( | x)d f (x; ,w)dx h ( )d
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Note:
= E{– cw + max(r1 P ( > 0|data) – b, 0)}
a,wa
E max (G ( ) | x)E
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Application to cost-effectiveness trials
Could apply the general decision-theoretic approach taking 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)
Action 1 Action 0 0 – cw – b – cw
> 0 – cw – b + r1 – 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
1,w 1G ( ) cw b r I( 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
1,w 2G ( ) cw b r
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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
where A is the set of outcomes which leads to Action 1,
e.g. for which P ( > 0|data) > 1 –
1,w 3G (x) cw b r I(x A)
0,wG (x) cw
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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.