Value of information and pricing new healthcare interventions

Transcription

1 University of Wollongong Research Online Australian Health Services Research Institute Faculty of Business 212 Value of information an pricing new healthcare interventions Anrew R. Willan University of Toronto Simon Eckermann University of Wollongong, Publication Details A. R. Willan & S. Eckermann, "Value of information an pricing new healthcare interventions", PharmacoEconomics 3 6 (212) Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.eu.au

2 Value of information an pricing new healthcare interventions Abstract Previous application of value-of-information methos to optimal clinical trial esign have preominantly taken a societal ecision-making perspective, implicitly assuming that healthcare costs are covere through public expeniture an trial research is fune by government or onation-base philanthropic agencies. In this paper, we consier the interaction between interrelate perspectives of a societal ecision maker (e.g. the National Institute for Health an Clinical Excellence [NICE] in the UK) charge with the responsibility for approving new health interventions for reimbursement an the company that hols the patent for a new intervention. We establish optimal ecision making from societal an company perspectives, allowing for trae-offs between the value an cost of research an the price of the new intervention. Given the current level of evience, there exists a maximum (threshol) price acceptable to the ecision maker. Submission for approval with prices above this threshol will be refuse. Given the current level of evience an the ecision maker's threshol price, there exists a minimum (threshol) price acceptable to the company. If the ecision maker's threshol price excees the company's, then current evience is sufficient since any price between the threshols is acceptable to both. On the other han, if the ecision maker's threshol price is lower than the company's, then no price is acceptable to both an the company's optimal strategy is to commission aitional research. The methos are illustrate using a recent example from the literature. Keywors interventions, healthcare, pricing, information, value Publication Details A. R. Willan & S. Eckermann, "Value of information an pricing new healthcare interventions", PharmacoEconomics 3 6 (212) This journal article is available at Research Online:

3 Value of Information an Pricing New Health Care Interventions Anrew R. Willan SickKis Research Institute an University of Toronto, Toronto, Canaa Simon Eckermann University of Wollongong, Wollongong, Australia Conense Running Title: VOI an Pricing New Health Care Interventions Corresponing to: Anrew R Willan CHES, 555 University Avenue, Toronto, ON M5G 1X8, Canaa Phone: Fax: any@anywillan.com Acknowlegement: A. R. Willan is fune by the Discovery Grant Program of the Natural Sciences an Engineering Research Council of Canaa (grant number ).

4 Abstract Previous application of value of information methos to optimal clinical trial esign have preominantly taken a societal ecision making perspective, implicitly assuming that health care costs are covere through public expeniture an trial research is fune by government or onation-base philanthropic agencies. In this paper, we consier the interaction between interrelate perspectives of a societal ecision maker (e.g. NICE in the UK) charge with the responsibility for approving new health interventions for reimbursement an the company that hols the patent for a new intervention. We establish optimal ecision making from societal an company perspectives, allowing for traeoffs between the value an cost of research an the price of the new intervention. Given the current level of evience, there exists a maximum (threshol) price acceptable to the ecision maker. Submission for approval with prices above this threshol will be refuse. Given the current level of evience an the ecision maker s threshol price, there exists a minimum (threshol) price acceptable to the company. If the ecision maker s threshol price excees the company s then current evience is sufficient since any price between the threshols is acceptable to both. On the other han, if the ecision maker s threshol price is lower than the company s then no price is acceptable to both an the company s optimal strategy is to commission aitional research. The methos are illustrate using a recent example from the literature.

5 1. Introuction Recently, there has been much interest in using value of information methos to etermine optimal sample size for ranomize clinical trials [1-28]. Value of information methos are propose as an alternative to traitional frequentist approaches base on tests of hypotheses an arbitrarily etermine quantities, such as the type I an II error probabilities an the smallest clinically important ifference. Using value of information methos, the sample size that maximizes the expecte net gain can be etermine, where the expecte net gain is the ifference between the expecte value of the (sample) information provie by a trial an the expecte total cost. If the maximum expecte net gain is negative, ecision making can be mae base on current information, aopting the new intervention if, an only if, the expecte incremental net benefit is positive. On the other han, if the maximum expecte net gain is positive then a trial is worthwhile, with the optimal sample size being that which maximizes the expecte net gain. Taking a societal perspective, where health care costs are covere through public expeniture an trial research is fune by government or onation-base philanthropic agencies, Willan an Pinto [2] provie a solution uner restrictive assumptions. Subsequent papers [6-9,23,24] provie solutions with the assumptions relaxe. Inustry perspectives can also been taken. Gittins an Pezeshk [11,12], Kikuchi, Pezeshk an Gittins [16], Pezeshk an Gittins [17] an Pezeshk [18] use a ecision theoretic approach to etermine optimal sample size uner the assumptions that the number of patients receiving the new intervention is a function of the observe size of the treatment effect an the associate statistical significance. Willan [22] provies a solution for optimal sample size from an inustrial

6 perspective, in which the value of the information from a new trial relates to the expecte increase in the probability of regulatory approval an market share. The purpose of this paper is to establish a value of information framework for exploring the interaction between the interrelate perspectives of a societal ecision maker (e.g. NICE in the UK) an a company that submits evience in support of a new intervention for the purposes of supporting the approval of the new intervention for reimbursement. As iscusse by Eckermann an Willan [6,8] an Griffin et al. [29], approving a new intervention base solely on the criterion that the current estimate of incremental net benefit is positive ignores the uncertainty associate with the estimate. From a societal perspective it will be optimal to unertake further research if the expecte value of information from such research excees the expecte opportunity cost. Current evience is sufficient (i.e., aopting now is optimal) only if for any potential research esign the expecte cost of research excees its expecte value. Expecte value of research falls as positive INB becomes more certain, or as the price of the new intervention is reuce. The expecte opportunity cost of research increases as expecte INB increases or as price reuces. Consequently, given the option for the ecision maker to request aitional research, our framework can be use to establish a stricter criterion for current evience of incremental net benefit an price at which aopting is optimal, allowing for the uncertainty associate with current evience.

7 Assuming that the ecision maker an the company are acting optimally an are risk neutral, the framework can also be use to establish the maximum (threshol) price acceptable to the ecision maker an a minimum (threshol) price acceptable to the company. If the ecision maker s threshol price excees the company s then the current evience is sufficient for ecision making since any price between the two threshols is acceptable to both. On the other han, if the company s threshol price excees the ecision maker s then no price is acceptable to both an, as we subsequently emonstrate, the company s optimal strategy is to collect aitional evience prior to submitting for approval. Consier the perspective of a societal ecision maker who is charge with the responsibility of eciing whether or not to a a new intervention to the formulary for reimbursement at a given price. The ecision maker can accept the new intervention, reject it outright or request aitional research. To the ecision maker, the value of aitional research is the expecte reuction in opportunity loss from making ecisions in the face of uncertain incremental net benefit. However, assuming it is infeasible to accept the new intervention while research is unertaken, there is also an expecte opportunity cost to the ecision maker of elaying the ecision, since enying the new intervention to patients until the evience is upate forgoes expecte incremental net benefit of the new intervention. We show that as the price of the new intervention increases, the value of aitional research increases, while the opportunity cost ecreases. Consequently, there exists a threshol price for the societal ecision maker, above which the expecte value of sample information from aitional evience excees its expecte cost, i.e. the expecte net gain from aitional evience is positive.

8 The other perspective to consier is that of the company requesting that the intervention be ae to the formulary for reimbursement. The company incurs a financial cost of conucting further research an an opportunity cost from revenue foregone while the research is conucte. The value of aitional research, from a company perspective, relates to expecte increase in the ecision maker s threshol price associate with a reuction in uncertainty an, as we subsequently show, ecreases as the price increases. We also show that as the price increases, the cost in foregone revenue increases. Hence, as the price of the intervention increases over the range for which expecte net benefit is positive, the expecte net gain of aitional evience from the company s perspective ecreases ue to both increasing cost an falling value. Therefore, for the company, there exists a threshol price below which the value of new evience excees its cost, i.e. the expecte net gain is positive, making aitional research worthwhile. If the company s maximum (with respect to research esign) expecte net gain is positive with the price set at the ecision maker s threshol (or, equivalently, if the company s threshol price excees that of the ecision maker) then a further research is optimal from the company s perspective. That is, where there is positive expecte net gain of further research for the company with the price set low enough to be acceptable to the ecision maker, no common price exists at which both parties woul prefer to a the intervention to the formulary. Conversely, if the company s maximum expecte net gain is negative with the price set to the ecision maker s threshol price then it will be optimal to submit a proposal for approval at the ecision maker s threshol price rather than commission further research.

9 Methos for establishing the societal ecision maker s an the company s threshol prices, given current evience an expecte actions an allowing for their interaction, are provie in Section 2 an illustrate in Section 3 with an example taken from a recent publication. Extensions to the moel to account for partial revenue per patient, iscounting an cost of aopting the new intervention are establishe in Section 4. Section 5 iscusses implications of the finings for pricing an reimbursement in processes of health technology assessment within a jurisiction. Further research on optimal solutions across jurisictions an the importance of appropriate threshol value for health outcomes in the etermination of incremental net benefit are also iscusse.

10 2. Methos 2.1. Incremental net benefit an expecte value of information Consier the cost-effectiveness assessment of a new health care intervention, referre to as Treatment (T), versus the appropriate comparator, referre to as Stanar (S). Let e ji, j = T, S be the (clinical) effectiveness for patient i receiving intervention j an let c ji, j = T, S be the total health care cost for patient i receiving intervention j. The cost c Ti inclues the price of the new intervention for patients receiving Treatment. Let ej = E( eji ), cj = E( cji ), e = et es an = c c, where E( ) is the expecte value function. If λ is the ecision maker s threshol c T S value for a unit of effectiveness, then b eλ c is the incremental net benefit. Now, if we separate out the price of the new intervention from other costs in the notation we can explore the consequences of allowing it to vary. If the per-patient price of the new intervention (i.e. revenue per patient to the company) equals R, then c Ti cti R is the health care cost for patient i receiving Treatment, excluing the price of the new intervention, where price is assume to be the same for all patients. Further, let c T = ct R, c = c R an = eλ c. We assume that the ecision maker s threshol value is known to the company. b Suppose that a societal ecision maker is charge with the task of eciing whether or not to approve a submission from a company to have the new intervention ae to the formulary for reimbursement at a price of R. The current evience in support of the new intervention, relative to the appropriate comparator, is expresse as a normal probability istribution function for the incremental net benefit, with mean b an variance v. That is, b = eλ c an 2 = eλ + c λ ec v v v 2 c, where, base on current evience, e an c are the means an v e

11 an v c the variance of e an c, respectively, an c ec is the covariance between e an c. Let b = b + R. The assumption of normality is applie to incremental net benefit an not to the iniviual patient observations, as illustrate in the Section 3 example, where binomial an gamma moels are assume for effectiveness an cost, respectively. If b, it is optimal for the ecision maker to refuse approval or request a price reuction. If b >, potentially optimal ecisions are to approve reimbursement, request a price reuction prior to approval, or request aitional research. Assuming that the aitional evience is from a ranomize clinical trial in which the cost an effectiveness are observe on n patients per arm (Treatment an Stanar), the expecte value of sample information (EVSI ) of the trial to the societal ecision maker is given by Willan an Pinto [2] an Eckermann an Willan [7] as where { } EVSI ( n) = Nn ( ) D F ( n), Nn ( ) is the number of patients to whom the ecision applies; ( ) 2 D = v (2 π) exp b (2 v) b Φ ( b v) I b ; ( ) 32 2 ( ) exp( 2 ) ( 2 ) 2 2 ( ) ( ) F ( n) = v (2 π) σ exp b 2 v ( nv) b Φ b v + v b v v π + bφ b v v v exp bv(2 v) 2 πv; 2 + σ = V( e λ c ) + V( e λ c ) is the sum over treatment groups of the between-patient Ti Ti Si Si variance of net benefit; 2 + v= v +σ n;

12 Φ () is the cumulative istribution function for the stanar normal ranom variable; an I() is the inicator function. The terms D an F ( n) are the pre- an post-trial per-patient expecte opportunity loss, respectively. Their ifference is the amount by which the per-patient expecte opportunity loss is reuce by the trial evience an, when multiplie by the number of patients who can benefit, yiels the expecte value of sample information. Where b >, the ifference D F ( n), which is the per-patient expecte value of sample information (EVSI pp (n)), simplifies to 2 2 ( ) ( ) EVSI ( n) = v exp bv(2 v) 2 πv bφ b v v. pp If h, expresse in years, is the time horizon for the new intervention, k the annual incience of the health conition in question, a the annual patient accrual rate an τ, expresse in years, the uration from when the last patient is recruite until the evience is upate, then, as given in Eckermann an Willan [7], the number of patients to whom the ecision applies is given by { } Nn ( ) = h ( τ+ 2 na) k. If the trial is unertaken by the company, the only cost to the ecision maker is the expecte opportunity cost (EOC ) incurre by those patients who are enie the intervention while the trial is performe an the evience is upate, given by Eckermann an Willan [7] as { } EOC ( n) = ( τ+ 2 nak ) nb. Therefore, the expecte net gain (ENG ) to the ecision maker of another trial of n patients per arm, efine as EVSI EOC, is given by

13 { } { } { } ENG ( n) = h ( τ+ 2 na) k D F ( n) ( τ+ 2 nak ) nb. (1) Let ENG ( n ) be maximize at n R. If elay approval an request another trial with R ENG ( n ) is positive then the optimal ecision is to n R patients per arm. On the other han, if R ENG ( n ) is negative then, if b is positive, the optimal ecision is to approve the intervention for reimbursement at a price of R. The subscript R in the notation for optimal sample size is a reminer that the optimal sample size epens on the submitte price. Griffin et al. [29] provie a criterion similar to Equation 1 for choosing between aoption an rejection which allows for uncertainty as to whether or not aitional research will be conucte. However, they use the current expecte value of perfect information (EVPI), rather than the expecte value of sample information, as the value of aitional research. EVPI oes not allow for optimal ecision making, since it overestimates value of research an has no efine relationship to EVSI, let alone ENG which is require f optimal ecision making. Hence, Eckermann, Karnon an Willan (21) show that use of EVPI in prioritizing research can easily lea to support for research with negative ENG, while also failing to support research with high research return espite small EVPI Decision maker s threshol price By substituting b R for b, where b >, the expecte net gain can be seen as a function of n an R, given as

14 ENG ( nr, ) = EVSI ( nr, ) EOC ( nr, ) { } { } pp = h ( τ+ 2 na) kevsi ( nr, ) ( τ+ 2 nak ) n( b R), (2) where ( 2 2 ) ( ) EVSI ( n, R) = v exp ( b R) v (2 v ) 2 πv ( b R) Φ ( b R) v v. pp Since, if all other variables are hel constant, the EVSI is an increasing function of R an EOC is a ecreasing function of R, there exists a ecision maker s threshol price, enote R, such that if R< R, R ENG ( n ) is negative, while if R> R, R ENG ( n ) is positive. Therefore, if R R, the expecte net gain for another trial is negative, regarless of its size, an the optimal ecision for the ecision maker is to approve the intervention for reimbursement at a price of R. On the other han, if R> R, the optimal ecision is to request evience from another trial, with R n per arm, or to request a reuction in the price to no more than R. Since R is the maximum price acceptable to the ecision maker then b = b R is the minimum acceptable incremental net benefit, referre to as the threshol incremental net benefit. Therefore, because of the uncertainty, the criterion for aoption shoul be b b rather than b >, where b is the estimate of incremental net benefit base on some price R, ie.. b = b R. Note that b > b is equivalent to R < R. > 2.3. Company s threshol Price

15 The maximum price the company can receive following a trial of m patients per arm is R m, the post-trial threshol price for the ecision maker. Therefore, for a company facing a price of R, the expecte value of the sample information is the increase in the post-trial revenue per patient, given by { } { m } c EVSI ( mr, ) = h ( τ+ 2 ma) k E( R ) R, which is simply the post-trial time horizon multiplie by the incience an the expecte increase in price. All other variables constant, EVSI c ( mr, ) is a ecreasing function of R. The financial cost to the company of performing a trial with m patients per arm is given by C f + 2mC, where C f is the fixe cost an C v the per-patient variable cost of performing the v trial. The expecte opportunity cost of foregone revenue experience by the company, facing a price of R, is given by ( τ+ 2 m a) kr, which is simply the uration of the trial multiplie by the incience an the price. Therefore, the expecte total cost for the company (ETC c ) is given by c ETC ( m, R) = C + 2 mc + ( τ+ 2 m a) kr. f All other variables hel constant, ETC c ( mr, ) is an increasing function of R. The expecte net gain to the company (ENG c ) of a trial with m patients per arm is given by c c c ENG ( mr, ) = EVSI ( mr, ) ETC ( mr, ) v { h ( 2 ma) } k{ E( R m) R} { Cf 2 mcv ( 2 makr ) } = τ+ + + τ+ { } = h ( τ+ 2 m a) k E( R ) hkr ( C + 2 mc ). m f v

16 Let ENG c ( mr, ) be maximize at m R. Since c R EVSI ( m, R ) is a ecreasing function of R an c R ETC ( m, R ) is a increasing function of R, there exists a company threshol price, enote R, c such that if R< R, c c R ENG ( m, R ) is positive, while if R> R, c c R ENG ( m, R ) is negative. The threshol price can be etermine by setting c R ENG ( m, R ) = an solving for R, yieling { } m R f R v h ( τ+ 2 m a) ke( R ) ( C + 2 mc) c R R =. hk c The threshol price R epens on R, the price the company faces, an, substituting the maximum pre-trial price the company faces, i.e. R, the company threshol price is { } R m f R h ( τ+ 2 m a) ke( R ) ( C + 2 m Cv) R c R =. (3) hk c If the ecision maker s threshol price is greater than the company s, i.e. R > R, the maximum expecte net gain for another trial is negative an the optimal ecision for the company is to submit for approval at an expecte price of R c. On the other han, if R < R, the maximum expecte net gain for another trial is positive an the optimal ecision for the company is to perform another trial with a sample size of mr an submit for approval at a price of R= R m R when the evience is upate.

17 3. Example The Caet-Hp Trial The CADET-Hp Trial was a ouble-blin, placebo-controlle, parallel-group, multi-centre, ranomize controlle trial performe in 36 family practitioner centres across Canaa. The results are publishe in Chiba et al. [3,31] an Willan [32]. Patients 18 years an over with uninvestigate yspepsia of at least moerate severity presenting to their family physicians were eligible for ranomization, provie they i not have any alarm symptoms an were eligible for empiric rug therapy. Patients were ranomize between T: Omeprazole 2 mg, metroniazole 5 mg an clarithromycin 25 mg; an S: Omeprazole 2 mg, placebo metroniazole an placebo clarithromycin. A total of 288 patients were ranomize, 142 (= n T ) to Treatment an 146 (= n S ) to Stanar. The new intervention (i.e. Treatment) is the regimen of metroniazole 5 mg an clarithromycin 25 mg. Both regimens were given twice aily for seven ays. The binary measure of effectiveness was treatment success, efine as the presence of no or minimal yspepsia symptoms at one year. Costs were etermine from the societal perspective an are given in Canaian ollars. A summary of the trial results are given in Table I. Treatment was observe to increase the probability of treatment success by percentage points an reuce total cost by $75.3 per patient, excluing the price of metroniazole an clarithromycin. If we assume a normal flat prior for incremental net benefit, an assume that the estimator of incremental net benefit from this trial is normally istribute then the current evience in favour of Treatment will be base solely on the ata from this trial, an will be characterize by a normal istribution for incremental net benefit with mean

18 ˆ ( ˆ b = λ + R) =.1371 λ ( R) =.1371λ R e c an variance v = Vˆ( ˆ ) λ + Vˆ( ˆ ) 2 λcˆ ( ˆ, ˆ ) =.3356λ λ(.687), 2 2 e c e c where λ is the threshol value for the willingness-to-pay for a treatment success. Assume a time horizon (h) of 1 years, an annual incience (k) of 8,, an annual accrual rate (a) of 8 an a uration of 1.5 years for follow-up an ata analysis (τ). A plot of the ecision maker s threshol price ( R ) as a function of the threshol value of a treatment success (λ) is given in Figure 1. The quantity R is the maximum price at which the ecision maker woul approve now in preference to requesting another trial, an increases with the threshol value for a treatment success. Also given in Figure 1 is the plot of the threshol incremental net benefit, i.e. =. b b R For λ = 5, the threshol ecision maker s price is $16.53, an the threshol incremental net benefit is $ Thus the ecision maker woul approve for reimbursement if the submitte price is less than $16.53 or, equivalently, if the mean incremental net benefit is greater than $ A plot of the ecision maker s optimal sample size n R ( = ) as a function of price (R) is given in Figure 2 for λ = 5. At a price less than or equal to $16.53 ( = R ), Treatment woul be approve for reimbursement, see Figure 1. At the other en of the scale, if the price excees $143.85, approval woul be refuse since mean incremental net benefit (b ) woul be negative. For a price between $16.53 an $ the ecision maker woul request another trial, with the size of the trial increasing with R over this range, as the incremental net benefit falls towars zero at R = $ For example, at a submitte price of $14.67, the ecision maker woul

19 request a trial of 387 patients per arm. Given the societal ecision maker s threshol price with current evience, the company s optimal behaviour is to submit a request with the price set to $16.53 ( = R ), unless there exists a sample size such that their expecte net gain (ENG c ) is positive. For λ = 5 an fixe (C f ) an variable (C v ) cost of $8, an $2 respectively, Table II contains, from the company s perspective, the expecte value of sample information (EVSI c ), the total cost (TC c ) an the expecte net gain (ENG c ) for various sample sizes. Also given in Table II is the post-trial expecte threshol price for the ecision maker ( E( R m) ), which was etermine by numerical integration, see the Appenix. The optimal sample size lies between 1 an 2 patients per arm. A more exhaustive search reveals that the optimal sample size is 137 patients per arm, corresponing to a pre-trial threshol price to the company ( R ) of $113.6 an an expecte net gain to the company of $6,451,162. The expecte threshol price for the ecision maker following a trial of 137 patients per arm ( E( 137) ) R is $ By contrast, a pre-trial submission by the company at a price of $14.67 woul precipitate a request from the ecision maker for a trial with 387 patients per arm, see Figure 2, which is associate with an expecte net gain to the company of only $1,17,179, see Table II. c 4. Extensions 4.1. Partial revenue per patient In Sections 2 an 3, it was assume that the revenue per patient receive by the company is equal to the price. It is more realistic to assume that the revenue per patient to the company is, instea,

20 a fraction, U, of the price, in which case the expecte net gain an the threshol price to the company become: { } ENG c ( m) = h ( 2 m a) k E( R ) m hkr τ+ U ( C f + 2 mcv) { } R m f R h ( τ+ 2 m a) ke( R ) U ( C + 2 m Cv) R c R =. hk 4.2. Discounting In Sections 2, 3 an 4.1 above, a iscount rate of zero is assume. A iscount rate of r > requires the following ajustments to the formulations for the expecte net gain for the ecision maker an company (ENG an ENG c c respectively) an threshol price to the company ( R ), as given below. h 1 L U t i ENG ( nr, ) = ( t t)(1 + r) + (1 + r) k D( R) F ( nr, ) U i= t { } L t 1 L L t i ( t t )(1 + r) + (1 + r) kb ( R) i= L t 1 L L ta + ( ta ta)(1 + r) + (1 + r) a 2 ( b R) i= a i ( ), where t= 2na+τ is the trial uration; L t is the integer part of t; t = t + 1; t = 2nais the U L a uration of patient accrual; an, L t a is the integer part of t a. h 1 h 1 L c U t i i ENG ( n) = ( t t)(1 + r) + (1 + r) ke( R m) U (1 + r) kr U U i= t i=

21 L ta 1 L L ta i C f + ( ta ta)(1 + r) + (1 + r) acv. i= R L h 1 t 1 a L L t a ( )(1 + ) + (1 + ) E( ) ( )(1 ) (1 ) m U R, U t i L i t t r r k R U C f ta ta r r acv i= t i= c = h 1 i i= (1 + r) ku where R t = 2m a+τ is the optimal trial uration; t L is the integer part of t; U L t = t + 1; t = 2m a is the uration of patient accrual; an t L a is the integer part of t a. a R 4.3 Positive cost of aoption In Sections 2, 3, 4.1 an 4.2 above, the cost of aopting the new intervention is assume to be zero. Let C A be the cost of aopting Treatment. It is reasonable to assume that the aoption of a new health care intervention will incur some up-front costs, such as those associate with conveying public health messages, training an learning by oing as well as capital equipment. For a positive C A, it can be shown that the formulations for D ( R), F ( nr, ) an EOC ( n ) become: 2 { A } { } { } D( R) = v (2 π) exp b R C ( hk) (2 v ) ( ) ( ) b R CA ( hk) Φ b R CA ( hk) v I b R + CA ( hk) ;

22 2 2 + ( { A } ) { A ( )} { A ( )} 32 2 v exp ( { b R CA Nn ( )} 2v) ( v 2 ) { A ( )} { A ( )} 2 2 vexp ( { b R CA Nn ( )} v(2 v) ) 2 v; F ( n, R) = v (2 π) σ exp b R C N( n) 2 v ( nv) ( ) b R C Nn Φ b R C Nn v + π ( ) + b R C Nn Φ b R C Nn v v π an { }{ A } EOC ( n) = ( τ+ 2 n a) k n b R C ( hk). 5. Discussion Previous application of value of information methos to optimal trial esign have preominantly taken a societal ecision making perspective, implicitly assuming that society commissions prospective trials an ecies whether or not to aopt new health interventions. Eckermann an Willan [6-9] emonstrate that optimal societal ecision making an trial esign requires joint consieration of whether to commission another trial or aopt the new intervention, given that the value, cost an feasibility of performing another trial are etermine by whether or not the new intervention is aopte. Optimal ecision making is shown to require a comparison of expecte net gain for elaying the ecision regaring aoption an performing another trial versus aopting immeiately with no trial within jurisiction, with the aitional consieration of expecte net gain for aopting an trialing, where feasible, across jurisictions. Griffin, Claxton an Sculpher [29] suggest that, where societal ecision making is restricte to aopting or rejecting, the ecision coul influence manufacturers through a traeoff between the

23 price of, an level of evience for, a new intervention. The traeoff they suggest is between expecte value of perfect information an incremental net benefit, where expecte value of perfect information is suggeste as the opportunity cost of aopting an incremental net benefit the opportunity cost of elaying. However, the populations to which the value of information an the opportunity costs apply are ifferent. Value of information (the option value of elay) arises for all patients beyon the point that evience is upate, while an opportunity cost of incremental net benefit arises for all patients except those on the treatment arm of the trial, until evience is upate [6-8]. Consequently, a traeoff between value of information an opportunity cost nees to consier time an population ifferences. Further, value of information from elaying shoul be the expecte value of sample information of an optimal trial, rather than expecte value of perfect information, given evience. The expecte opportunity loss of aoption is the expecte value of sample information provie by an optimal trial, not the expecte value of perfect information. Griffin et al. [29] exten their methos to account for changing populations an consier the role of aitional research. However, they still quantify the value of aitional research as the expecte value of perfect information, rather than the expecte value of sample information as require by optimal ecision making, which we have aresse as part of this paper. In this paper we have establishe an illustrate the appropriate traeoff between pricing an the level of evience relevant to the societal ecision of whether to approve health care interventions for reimbursement when companies have sole remit to commission trials. For a given level of evience, it has been illustrate that there exists a maximum threshol price acceptable to the

24 societal ecision maker. For prices above this threshol, the expecte net gain for the ecision maker from another trial is positive an requesting another trial is their optimal strategy. Further, we have shown that the optimal response of manufacturers to the societal threshol price of whether to unertake further research or lower their price epens on their expecte value an cost of research an current evience. Given current evience, there exists a minimum threshol price acceptable to the company, meaning that for prices below the threshol, the expecte net gain for the company from another trial is positive an performing another trial is their optimal strategy. The company s threshol price excees that of the ecision maker if, an only if, there exists a sample size for which the company s expecte net gain is positive. The optimal strategy for a company is to submit for approval at the ecision maker s threshol price when the company s expecte net gain is negative for all sample sizes at this price, or to perform another trial when the maximum expecte net gain for the company is positive. From the company perspective, the optimal sample size of the trial will be that which maximizes their expecte net gain, given the value an cost of trials an revenue foregone. In general, it is suboptimal for the company to submit for approval at a price greater than the ecision maker s threshol, since, at best, it will precipitate a request for another trial with, from their perspective, sub-optimal sample size. Thus, the incentives implicit in the framework presente here iscourage the company from submitting for approval until there is sufficient evience to support the submitte price. This reuces aministrative an analytic buren on ecision makers an companies alike, in turn

25 reucing the associate transaction costs of the approval process. Other consierations, such as the value of being the first to market, the competing uses of research funing, or uncertainty in relation to the threshol value of outcomes in a jurisiction may also influence the expecte revenue an cost of research trae-off face by companies in unertaking ecision making. Hence, the framework presente here coul be generalize to account for these aitional factors where appropriate. Nevertheless, in general, the framework enables optimal traeoffs between the value an cost of further research from both societal an company perspectives an establishes how these traeoffs interact an play out in practice, where companies have control of prospective research an society has control of reimbursement within a jurisiction. The analysis presente has been strictly within jurisiction. Moving beyon a strictly within jurisiction analysis, options arise in relation to aopting an trialing, with the associate avantages in avoiing opportunity cost of elay, an the potential for improving risk sharing arrangements between companies an societal ecision makers [9,1]. Hence, further research is suggeste to exten the within jurisiction framework presente here an explore optimal mechanisms for researching an pricing across jurisictions, given interactions between ecision makers an manufacturers an the potential to aopt an trial. This coul, for example, consier incorporating contractual agreements to ajust pricing in jurisictions where such aoption is optimal while aitional evience is collecte in other jurisictions in which elaying an trialing is optimal. To apply a framework for optimal ecision making an interaction between societal ecision makers an companies, within or across jurisictions, it is critical to establish economically an

26 meaningful societal threshol values for health outcomes. Threshol values are require to etermine the prior istribution of incremental net benefit, the expecte value of sample information an opportunity cost, as well as the consequent threshol prices an optimal research ecisions. There is wie agreement that the threshol value for health outcomes in societal ecision making shoul reflect the opportunity cost of funing new interventions within a fixe buget an the current use of existing interventions. Recently, it has been suggeste that, if the societal objective is restricte to health maximisation, the threshol value for outcomes can be estimate as the shaow price of the least cost-effective (worst performing) interventions to be isplace [33,34-36]. However, even if the objective is restricte to health maximization, the shaow price of contracting or isplacing the least cost-effective interventions will only coincie with that from the best expansion of current interventions (represente by the opportunity cost from financing new interventions) when there is complete allocative efficiency across all activities an interventions [37,38]. Hence, with allocative inefficiency in the current health care system, the opportunity cost an threshol price of, e.g., incremental ollars per QALY gaine will be lower than that of isplacing the least cost effective services. Consequently, evience of the most cost effective expansion of existing technology is require to estimate the true opportunity cost an threshol values for incremental net benefit so that value of information methos can be appropriately applie. Throughout the paper we have assume that the parameters h, k, a an τ are fixe, mostly to focus the attention on the uncertainty regaring incremental net benefit. However, the uncertainty of such parameters coul be ae to the moel. The parameters h, a an τ woul be amenable to sensitivity analyses, since they are somewhat in the control of the investigators. On

27 the other han, the uncertainty regaring k might be best incorporate by using a Bayesian approach since its estimate woul be typically base on empirical evience. We have assume that the prior- an post-stuy istributions for incremental net benefit are erive from ranomize clinical trials ata. However, it is often the case, as in ecision-analytic moels, for example, that incremental net benefit is a complex function of many parameters, the information for which may come from a variety of stuy types, see Aes, Lu an Claxton [1]. This is illustrate in Welton et al. [19] who examine the evience in support of interventions for improving the uptake of breast cancer screening, an by Brennan an Kharroubi [39] who explore methos for EVSI etermination for moels with Weibull survival parameters. Consequently value is suggeste to extening the methos presente in this paper for ranomize clinical trials to other research esigns. Nonetheless, the principle of applying value of information methos for the pricing of new health interventions illustrate in this paper is the same, regarless of the erivation of incremental net benefit. The case for assuming normality for mean incremental net benefit base on iniviual patient ata has been mae by numerous authors, an has been generally accepte. The parametric assumption of bivariate normality for mean cost an effectiveness (an hence, mean incremental net benefit) has been shown to perform well [4-43]. Alternative istributional assumptions for incremental net benefit o not, in general, lea to close form solutions for the expecte value of sample information, requiring the use of numerical integration or Markov Chain-Monte Carlo methos. Consequently, the computer intensiveness of methos require with alternative assumptions may prove to be particularly challenging [1].

28 We have assume that the company is risk-neutral, implying that if the company s threshol price excees the ecision maker s then it is optimal for the company to o aitional research. However, if the company is somewhat risk-averse then they shoul be more willing at the margin to accept the ecision maker s threshol price base on current evience. Hence, while expecte revenue associate with an expecte increase in the ecision maker s threshol price with aitional evience may be greater than the companies irect an opportunity costs, the riskaverse company may not be willing to risk that actual net revenue coul be reuce ue to a potential price reuction with aitional evience.

29 Appenix R m is the ecision maker s threshol price following a trial of m patients per arm. That is, R m is that value of R, such that max {ENG ( nr, )} =, where ENG ( nr, ) is the expecte net gain n m of performing a trial of n patients per arm, once the evience is upate with ata from the trial of m patients per arm. Numerical integration with respect to the istribution f is use to m etermine the expecte value of R, where f is the pf for the observe incremental net benefit from the trial of m patients per arm, which, uner the assumptions we have mae, is normal with mean b an variance v= v +σ + m. 2 m

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35 Table I. Parameter estimates for the CADET-Hp Trial Treatment Stanar Sample size ( = n j ) Proportion of successes ( = ˆ ) ifference =.1371 ( = ˆ e) e j Estimate of mean cost minus cost of metroniazole an clarithromycin (using gamma moel) Estimate variance of proportion of successes ( = eˆ (1 eˆ ) n ) j j j Estimate variance of average cost (using gamma moel) Estimate covariance between proportion of successes an average cost (using gamma moel) ifference = ( ˆ = c ) sum =.3356 ( = Vˆ( ˆ e)) 1,825 2,495 sum = 4,32 ( = Vˆ( ˆ c )) sum = ( ˆ( ˆ, ˆ = C e c)) note: ˆθ is an estimate of θ

36 Table II. From the company s perspective, the expecte value of sample information (EVSI c ), total cost (TC c ), expecte net gain (ENG c ) an the ecision maker s expecte threshol price ( E( R )) as a function of sample size, for the CADET-Hp Trial m Sample Size Per Arm (m) EVSI c TC c ENG c E( R ) m 5 18,252,845 14,65, 3,62, ,539,382 15,9, 4,639, ,276,162 16,825, 6,451, ,53,291 17,15, 5,38, ,796,479 18,4, 6,396, ,679,76 19,65, 4,29, ,283,713 2,9, 3,383, ,325,27 22,15, 1,175, ,245,179 23,75, 1,17, ,126,392 23,4, 726, ,85,97 24,65, -1,564, R = m = m 387 = n c = n R

37 , λ Figure 1. The ecision maker s threshol price ( R ) an threshol mean incremental net benefit b = ( b R ) as a function of the threshol value for treatment success (λ), for the CADET-Hp Trial. At a threshol value for treatment success of $5, the ecision maker s threshol price an threshol mean incremental net benefit are $16.53 an $37.32, respectively.

38 n R R Figure 2. Optimal sample size ( n R ) as a function of price (R) for a threshol value for treatment success (λ) of 5. The ecision maker approves for R < 16.53; refuses approval for R > ; an, requests another trial for R