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MPS Research UnitCHEBS Workshop - April Anne Whitehead Medical and Pharmaceutical Statistics Research Unit The University of Reading Sample size determination for cost-effectiveness trials

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MPS Research UnitCHEBS Workshop - April Comparative study Parallel group design Control treatment (0) New treatment (1) n 0 subjects to receive control treatment n 1 subjects to receive new treatment

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MPS Research UnitCHEBS Workshop - April 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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MPS Research UnitCHEBS Workshop - April Frequentist approach Focus on hypothesis testing and error rates - what might happen in repetitions of the trial e.g.Test null hypothesisH 0 : = 0 against alternativeH 1 + : > 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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MPS Research UnitCHEBS Workshop - April Distribution of = 0 = R k Fail to Reject H 0 Reject H 0

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MPS Research UnitCHEBS Workshop - April A general parametric approach Assume Reject H 0 if > k where is the standard normal distribution function and P(Z > z ) = where Z ~ N(0, 1)

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MPS Research UnitCHEBS Workshop - April Require

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MPS Research UnitCHEBS Workshop - April 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 kis the amount that can be paid for a unit improvement in efficacy for a single patient

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MPS Research UnitCHEBS Workshop - April Set and solve for n 0 and n 1

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MPS Research UnitCHEBS Workshop - April 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 P 0 (conclude new better than control) = 1 –

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MPS Research UnitCHEBS Workshop - April Likelihood function Prior h 0 ( ) is Posterior h( |data) i.e. h( |data) is

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MPS Research UnitCHEBS Workshop - April P ( > 0|data) > 1 – if i.e.

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MPS Research UnitCHEBS Workshop - April Prior to conducting the study, so

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MPS Research UnitCHEBS Workshop - April Require P 0 Express w in terms of n 0 and n 1, provide values for 0 and w 0 and solve for n 0 and n 1

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MPS Research UnitCHEBS Workshop - April 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)

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MPS Research UnitCHEBS Workshop - April 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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MPS Research UnitCHEBS Workshop - April 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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MPS Research UnitCHEBS Workshop - April Table of gains ( relative to continuing with control treatment) Action 1 Action 0 0 – cw – b – cw > 0 – cw – b + r 1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 1 = reward if new treatment is better G 0,w ( ) = – cw

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MPS Research UnitCHEBS Workshop - April Following collection of w units of information, the expected gain from action a is G a, w (x) = E {G a,w ( )| x} Action will be taken to maximise E {G a,w ( )|x}, that is a*, w* where G a*, w* (x) = max { G a, w (x)} (Note: Action 1 will be taken if P ( > 0|data) > b/r 1 )

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MPS Research UnitCHEBS Workshop - April At design stage consider frequentist expectation: E ( G a*, w (x)) and use this as the gain function U w ( )

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MPS Research UnitCHEBS Workshop - April Expected gain from collecting information w is So optimal choice of w is w*, where

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MPS Research UnitCHEBS Workshop - April This is the prior expected utility or pre-posterior gain

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MPS Research UnitCHEBS Workshop - April Note: = E{– cw + max(r 1 P ( > 0|data) – b, 0)}

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MPS Research UnitCHEBS Workshop - April 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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MPS Research UnitCHEBS Workshop - April Table of gains – ‘Simple Societal’ ( relative to continuing with control treatment) Action 1 Action 0 0 – cw – b – cw > 0 – cw – b + r 1 – cw c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 1 = reward if new treatment is more cost-effective G 0,w ( ) = – cw

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MPS Research UnitCHEBS Workshop - April 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 r 2 = reward if new treatment is more cost-effective G 0,w ( ) = – cw

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MPS Research UnitCHEBS Workshop - April Gains – ‘Pharmaceutical Company’ c = cost of collecting 1 unit of information (w) b = further development cost of new treatment r 3 = 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 –

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MPS Research UnitCHEBS Workshop - April 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,

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