L applicazione dei metodi Bayesiani nella Farmacoeconomia

Transcription

1 L applicazione dei metodi Bayesiani nella Farmacoeconomia Gianluca Baio Department of Statistical Science, University College London (UK) Department of Statistics, University of Milano Bicocca (Italy) Razionalizzazione della spesa dei farmaci ad alto costo Torino, 5 Ottobre 2012 Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

2 Outline of presentation 1 What is Bayesian statistics? Relationship with standard statistical procedures Prior distributions Bayesian computation 2 How to implement Bayesian statistics in Health Economics? Probabilistic assumptions Decision-theory Sensitivity analysis 3 Example Modelling Cost-effectiveness analysis Probabilistic sensitivity analysis 4 Conclusions Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

3 Outline of presentation 1 What is Bayesian statistics? Relationship with standard statistical procedures Prior distributions Bayesian computation 2 How to implement Bayesian statistics in Health Economics? Probabilistic assumptions Decision-theory Sensitivity analysis 3 Example Modelling Cost-effectiveness analysis Probabilistic sensitivity analysis 4 Conclusions Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

4 Outline of presentation 1 What is Bayesian statistics? Relationship with standard statistical procedures Prior distributions Bayesian computation 2 How to implement Bayesian statistics in Health Economics? Probabilistic assumptions Decision-theory Sensitivity analysis 3 Example Modelling Cost-effectiveness analysis Probabilistic sensitivity analysis 4 Conclusions Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

5 Outline of presentation 1 What is Bayesian statistics? Relationship with standard statistical procedures Prior distributions Bayesian computation 2 How to implement Bayesian statistics in Health Economics? Probabilistic assumptions Decision-theory Sensitivity analysis 3 Example Modelling Cost-effectiveness analysis Probabilistic sensitivity analysis 4 Conclusions Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

6 Variability and statistical models Size N = 10 Mean µ Standard deviation σ Size n = 5 Mean x Standard deviation s x In reality we observe only one such sample (out of the many possible in fact there are 252 different ways of picking at random 5 units out of the population!) and we want to use the information contained in that sample to infer about the population parameters (e.g. the true mean and standard deviation) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

7 Deductive vs inductive inference Hypothesis 1 Hypothesis 2 Hypothesis 3 = 0% = 5% = 10% c 5% 0% 5% 10% 15% c Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

8 Deductive vs inductive inference Deduction Hypothesis 1 Hypothesis 2 Hypothesis 3 = 0% = 5% = 10% c 5% 0% 5% 10% 15% c Standard (frequentist) methods set the value of the parameters (hypotheses) and by deduction infer about the plausibility of the observed data Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

9 Deductive vs inductive inference Deduction Hypothesis 1 Hypothesis 2 Hypothesis 3 Induction = 0% = 5% = 10% c 5% 0% 5% 10% 15% c Standard (frequentist) methods set the value of the parameters (hypotheses) and by deduction infer about the plausibility of the observed data Conversely, Bayesian statistics conditions on the observed data and by induction makes inference on the unobservable parameters (hypotheses) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

10 Bayesian inference p(θ) Prior (subjective knowledge) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

11 Bayesian inference p(y θ) Data (observed evidence) p(θ) Prior (subjective knowledge) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

12 Bayesian inference p(y θ) Data (observed evidence) p(θ) Prior (subjective knowledge) Bayes theorem p(θ)p(y θ) p(y) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

13 Bayesian inference p(y θ) Data (observed evidence) p(θ) Prior (subjective knowledge) Bayes theorem p(θ)p(y θ) p(y) Posterior (updated knowledge) p(θ y) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

14 Bayesian inference updating knowledge Prior Likelihood Posterior θ Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

15 Choice of the prior distribution Non-informative prior Attempts to include minimal information in the prior to let the data speak for themselves (sometimes known as minimally informative ) Need to be careful in defining the scale in which non-informativeness is selected Sometimes helpful as preliminary approximation often leads to essentially the same inference as using maximum likelihood Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

16 Choice of the prior distribution Non-informative prior Attempts to include minimal information in the prior to let the data speak for themselves (sometimes known as minimally informative ) Need to be careful in defining the scale in which non-informativeness is selected Sometimes helpful as preliminary approximation often leads to essentially the same inference as using maximum likelihood Conjugate prior Convenient mathematical formulation Prior and posterior in the same family E.g. Prior = Normal(m 0,s 0) + Data = Normal(µ,σ 2 ) E.g. Posterior = Normal(m 1,s 1) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

17 Choice of the prior distribution Non-informative prior Attempts to include minimal information in the prior to let the data speak for themselves (sometimes known as minimally informative ) Need to be careful in defining the scale in which non-informativeness is selected Sometimes helpful as preliminary approximation often leads to essentially the same inference as using maximum likelihood Conjugate prior Convenient mathematical formulation Prior and posterior in the same family E.g. Prior = Normal(m 0,s 0) + Data = Normal(µ,σ 2 ) E.g. Posterior = Normal(m 1,s 1) Informative prior Proper Bayesian model: express (subjective) knowledge by means of a suitable probability distribution Can be based on hard evidence NB: Informative priors are not necessarily conjugated! Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

18 Non conjugated models Despite their usefulness in computational terms, non-informative and conjugated models are not always the best Too restrictive, might not encode the actual level of prior information Non-informative priors are generally not invariant to scale transformations When more complex (and realistic!) structures considered for instance multiparametric, or generalised linear models conjugacy rarely hold Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

19 Non conjugated models Despite their usefulness in computational terms, non-informative and conjugated models are not always the best Too restrictive, might not encode the actual level of prior information Non-informative priors are generally not invariant to scale transformations When more complex (and realistic!) structures considered for instance multiparametric, or generalised linear models conjugacy rarely hold Since the 1990 s the development of MCMC methods has allowed the use of simulation techniques for Bayesian computation. Software like BUGS 1 or JAGS 2 can be used to perform analysis on most real-life problems If the model is well specified, the level of accuracy of the approximation provided by the simulation technique is very good Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

20 MCMC methods After 10 iterations σ µ Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

21 MCMC methods After 30 iterations σ µ Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

22 MCMC methods After 1000 iterations µ σ Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

23 MCMC methods Burn in Sample after convergence Chain 1 Chain Iteration Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

24 (Bayesian) Decision-making process Typically, we define a health economic response (e, c), where for each intervention (treatment) t e represents a suitable measure of clinical benefits (e.g. QALYs) c are the costs associated with a given intervention Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

25 (Bayesian) Decision-making process Typically, we define a health economic response (e, c), where for each intervention (treatment) t e represents a suitable measure of clinical benefits (e.g. QALYs) c are the costs associated with a given intervention The variables (e,c) are usually defined as functions of a set of relevant parameters θ t which represent some population-level features of the underlying process Probability of some clinical outcome Duration in treatment Reduction in the rate of occurrence of some event Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

26 (Bayesian) Decision-making process Typically, we define a health economic response (e, c), where for each intervention (treatment) t e represents a suitable measure of clinical benefits (e.g. QALYs) c are the costs associated with a given intervention The variables (e,c) are usually defined as functions of a set of relevant parameters θ t which represent some population-level features of the underlying process Probability of some clinical outcome Duration in treatment Reduction in the rate of occurrence of some event There are (at least) two sources of uncertainty Sampling variability is modelled using an intervention-specific distribution p(e,c θ t ) Parametric uncertainty is modelled using a (possibly subjective) prior distribution p(θ t D), based on some background data D NB: Sometimes, we can (should!) consider also structural uncertainty, i.e. about the modelling assumptions used Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

27 (Bayesian) Decision-making process In addition, we define a utility function to describe the quality of t The function u(e, c; t) describes the value associated with applying intervention t, in terms of the future (uncertain) outcomes Uncertainty is expressed through p(e,c,θ) = p(e,c θ)p(θ D) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

28 (Bayesian) Decision-making process In addition, we define a utility function to describe the quality of t The function u(e, c; t) describes the value associated with applying intervention t, in terms of the future (uncertain) outcomes Uncertainty is expressed through p(e,c,θ) = p(e,c θ)p(θ D) NB: typically, the utility function chosen is the monetary net benefit u(e,c;t) := ke t c t k is the willingness to pay, i.e. the cost per extra unit of effectiveness gained Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

29 (Bayesian) Decision-making process In addition, we define a utility function to describe the quality of t The function u(e, c; t) describes the value associated with applying intervention t, in terms of the future (uncertain) outcomes Uncertainty is expressed through p(e,c,θ) = p(e,c θ)p(θ D) NB: typically, the utility function chosen is the monetary net benefit u(e,c;t) := ke t c t k is the willingness to pay, i.e. the cost per extra unit of effectiveness gained Decision making is based on Computing for each intervention t the expected utility U t = E[u(e,c;t)] (computed with respect to both individual and population uncertainty) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

30 (Bayesian) Decision-making process In addition, we define a utility function to describe the quality of t The function u(e, c; t) describes the value associated with applying intervention t, in terms of the future (uncertain) outcomes Uncertainty is expressed through p(e,c,θ) = p(e,c θ)p(θ D) NB: typically, the utility function chosen is the monetary net benefit u(e,c;t) := ke t c t k is the willingness to pay, i.e. the cost per extra unit of effectiveness gained Decision making is based on Computing for each intervention t the expected utility U t = E[u(e,c;t)] (computed with respect to both individual and population uncertainty) Treating the entire homogeneous (sub)population with the most cost-effective treatment, i.e. that associated with the maximum expected utility Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

31 (Bayesian) Decision-making process In addition, we define a utility function to describe the quality of t The function u(e, c; t) describes the value associated with applying intervention t, in terms of the future (uncertain) outcomes Uncertainty is expressed through p(e,c,θ) = p(e,c θ)p(θ D) NB: typically, the utility function chosen is the monetary net benefit u(e,c;t) := ke t c t k is the willingness to pay, i.e. the cost per extra unit of effectiveness gained Decision making is based on Computing for each intervention t the expected utility U t = E[u(e,c;t)] (computed with respect to both individual and population uncertainty) Treating the entire homogeneous (sub)population with the most cost-effective treatment, i.e. that associated with the maximum expected utility Performing sensitivity analysis (to parameter and/or structural uncertainty) to investigate the impact of underlying uncertainty on the decision process Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

32 Example: Chemotherapy t = 0: Old chemotherapy A 0 Ambulatory care (γ) SE 0 Blood-related side effects (π 0) c drug 0 N Standard treatment H 0 Hospital admission (1 γ) A 0 Ambulatory care (γ) N SE 0 No side effects (1 π 0) H 0 Hospital admission (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

33 Example: Chemotherapy t = 0: Old chemotherapy A 0 Ambulatory care (γ) c amb SE 0 Blood-related side effects (π 0) c drug 0 N Standard treatment H 0 Hospital admission c hosp (1 γ) A 0 Ambulatory care (γ) c amb N SE 0 No side effects (1 π 0) H 0 Hospital admission c hosp (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

34 Example: Chemotherapy t = 1: New chemotherapy A 1 Ambulatory care (γ) c amb SE 1 Blood-related side effects (π 1 = π 0ρ) c drug 1 N New treatment H 1 Hospital admission c hosp (1 γ) A 1 Ambulatory care (γ) c amb N SE 1 No side effects (1 π 1) H 1 Hospital admission c hosp (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

35 Prior information vs prior distributions Often, we can use the prior information and formalise it in order to approximate it with a suitable probability distribution Mean 2.5% Median 97.5% Distribution π Beta(27.12, 85.88) ρ Normal(0.8, 0.2) γ Beta(5.80, 13.80) c amb lognormal(4.77, 0.17) c hosp lognormal(8.60, 0.18) c drug c drug Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

36 Prior information vs prior distributions π 0 Beta(27.12, 85.88) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

37 Prior information vs prior distributions Pr(π 0 < ) = Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

38 Prior information vs prior distributions Pr(π 0 < ) = Pr(π 0 > ) = Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

39 Prior information vs prior distributions Pr( π ) = 0.95 Pr(π 0 < ) = Pr(π 0 > ) = Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

40 Bayesian model specification The assumptions underlying the model can be coded in BUGS/JAGS model { pi[1] ~ dbeta(a.pi,b.pi) # Baseline probability of side effects (t=0) pi[2] <- pi[1]*rho # Decreased probability of side effects (t=1) rho ~ dnorm(m.rho,tau.rho) # Decrement rate in side effects for t=1 gamma ~ dbeta(a.gamma,b.gamma) # Probability of ambulatory care c.amb ~ dlnorm(m.amb,tau.amb) # Unit cost of ambulatory care c.hosp ~ dlnorm(m.hosp,tau.hosp) # Unit cost of hospitalisation for (t in 1:2) { SE[t] ~ dbin(pi[t],n) # Predicted no. patients with side effects A[t] ~ dbin(gamma,se[t]) # Predicted no. patients needing ambulatory care H[t] <- SE[t] - A[t] # Predicted no. patients needing hospitalisation } } and then the MCMC analysis can be performed Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

41 Bayesian model specification The assumptions underlying the model can be coded in BUGS/JAGS model { pi[1] ~ dbeta(a.pi,b.pi) # Baseline probability of side effects (t=0) pi[2] <- pi[1]*rho # Decreased probability of side effects (t=1) rho ~ dnorm(m.rho,tau.rho) # Decrement rate in side effects for t=1 gamma ~ dbeta(a.gamma,b.gamma) # Probability of ambulatory care c.amb ~ dlnorm(m.amb,tau.amb) # Unit cost of ambulatory care c.hosp ~ dlnorm(m.hosp,tau.hosp) # Unit cost of hospitalisation for (t in 1:2) { SE[t] ~ dbin(pi[t],n) # Predicted no. patients with side effects A[t] ~ dbin(gamma,se[t]) # Predicted no. patients needing ambulatory care H[t] <- SE[t] - A[t] # Predicted no. patients needing hospitalisation } } and then the MCMC analysis can be performed The MCMC procedure will generate samples from the posterior distributions of the relevant quantities θ t = (π t,γ,ρ,se t,a t,h t,c amb,c hosp,c drug ) These can be combined to compute the variables of cost and benefit, and perform the economic analysis Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

42 Bayesian model convergence A[1] A[2] H[1] A[1] A[2] H[1] iteration iteration iteration H[2] SE[1] SE[2] H[2] SE[1] SE[2] iteration iteration iteration c.hosp c.amb gamma c.hosp c.amb gamma iteration iteration iteration pi[1] pi[2] rho pi[1] pi[2] rho iteration iteration iteration Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

43 Bayesian model posterior distributions π 0 γ ρ SE 0 SE 1 A A 1 H 0 c hosp Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

44 Measures of cost & benefits The total cost associated with each treatment can be computed by multiplying the unit cost of each clinical resource (drug, ambulatory care and hospital admission) by the number of patients consuming it. Thus: c t := c drug t (N SE t )+(c drug t +c amb )A t +(c drug t +c hosp )H t Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

45 Measures of cost & benefits The total cost associated with each treatment can be computed by multiplying the unit cost of each clinical resource (drug, ambulatory care and hospital admission) by the number of patients consuming it. Thus: c t := c drug t (N SE t )+(c drug t +c amb )A t +(c drug t +c hosp )H t Similarly, the measure of effectiveness can be computed as the total number of patients who do not experience side effects e t := (N SE t ) NB: we can (should) extend this to consider QALYs, instead of the hard effectiveness measure in terms of events averted Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

46 Expected incremental benefit Expected Incremental Benefit EIB EIB = U 1 U 0 k* = Willingness to pay Based on the current evidence, choose old chemotherapy if k < 6500 monetary units and new chemotherapy otherwise Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

47 Probabilistic sensitivity analysis (PSA) The quality of the current evidence is often limited During the pre-market authorisation phase, the regulator should decide whether to grant reimbursement to a new product and in some countries also set the price on the basis of uncertain evidence, regarding both clinical and economic outcomes Although it is possible to answer some unresolved questions after market authorisation, relevant decisions such as that on reimbursement (which determines the overall access to the new treatment) have already been taken Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

48 Probabilistic sensitivity analysis (PSA) The quality of the current evidence is often limited During the pre-market authorisation phase, the regulator should decide whether to grant reimbursement to a new product and in some countries also set the price on the basis of uncertain evidence, regarding both clinical and economic outcomes Although it is possible to answer some unresolved questions after market authorisation, relevant decisions such as that on reimbursement (which determines the overall access to the new treatment) have already been taken This leads to the necessity of performing (probabilistic) sensitivity analysis (PSA) Formal quantification of the impact of uncertainty in the parameters on the results of the economic model Standard requirement in many health systems (e.g. for NICE in the UK), but still not universally applied Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

49 PSA to parameter uncertainty Parameters Model structure Decision analysis Old chemotherapy π 0 SE0 Blood-related side effects (π0) A0 Ambulatory care (γ) c amb Old chemotherapy Benefits Costs H0 Hospital admission (1 γ) c hosp c drug 0 N Standard treatment ρ A0 Ambulatory care (γ) c amb N SE0 No side effects (1 π0) H0 Hospital admission (1 γ) c hosp γ New chemotherapy A1 Ambulatory care (γ) c amb New chemotherapy Benefits Costs SE1 Blood-related side effects (π1 = π0ρ) H1 Hospital admission (1 γ) c hosp c hosp c drug 1 N New treatment N SE1 No side effects (1 π1) A1 Ambulatory care (γ) c amb ICER = QALY H1 Hospital admission (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27 c hosp

50 PSA to parameter uncertainty Parameters Model structure Decision analysis Old chemotherapy x π SE0 Blood-related side effects (π0) A0 Ambulatory care (γ) H0 Hospital admission (1 γ) c amb c hosp Old chemotherapy Benefits Costs c drug 0 N Standard treatment ρ A0 Ambulatory care (γ) c amb N SE0 No side effects (1 π0) x γ New chemotherapy H0 Hospital admission (1 γ) A1 Ambulatory care (γ) c hosp c amb New chemotherapy Benefits Costs x SE1 Blood-related side effects (π1 = π0ρ) H1 Hospital admission (1 γ) c hosp x c hosp c drug 1 N New treatment N SE1 No side effects (1 π1) A1 Ambulatory care (γ) c amb ICER = QALY H1 Hospital admission (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27 c hosp

51 PSA to parameter uncertainty Parameters Model structure Decision analysis Old chemotherapy x π SE0 Blood-related side effects (π0) A0 Ambulatory care (γ) H0 Hospital admission (1 γ) c amb c hosp Old chemotherapy Benefits Costs c drug 0 N Standard treatment ρ A0 Ambulatory care (γ) c amb N SE0 No side effects (1 π0) x γ New chemotherapy H0 Hospital admission (1 γ) A1 Ambulatory care (γ) c hosp c amb New chemotherapy Benefits Costs x SE1 Blood-related side effects (π1 = π0ρ) H1 Hospital admission (1 γ) c hosp x c hosp c drug 1 N New treatment N SE1 No side effects (1 π1) A1 Ambulatory care (γ) c amb ICER = QALY H1 Hospital admission (1 γ) Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27 c hosp

52 PSA to parameter uncertainty Parameters Model structure Decision analysis Old chemotherapy π 0 x ρ c drug 0 N Standard treatment SE0 Blood-related side effects (π0) A0 Ambulatory care (γ) H0 Hospital admission (1 γ) A0 Ambulatory care (γ) c amb c hosp c amb Old chemotherapy Benefits Costs N SE0 No side effects (1 π0) x x γ New chemotherapy SE1 Blood-related side effects (π1 = π0ρ) H0 Hospital admission (1 γ) A1 Ambulatory care (γ) H1 Hospital admission (1 γ) c hosp c amb c hosp New chemotherapy Benefits Costs c hosp x c drug 1 N New treatment N SE1 No side effects (1 π1) A1 Ambulatory care (γ) H1 Hospital admission (1 γ) c amb c hosp ICER = ICER= Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

53 Cost-effectiveness plane Cost effectiveness plane contour plot New Chemotherapy vs Old Chemotherapy Pr( e 0, c > 0) = 0.19 Pr( e > 0, c > 0) = Cost differential Pr( e 0, c 0) = 0 Pr( e > 0, c 0) = Effectiveness differential Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

54 Cost-effectiveness plane The economic analysis depends on the willingness-to-pay, which determines the sustainability area Cost effectiveness plane New Chemotherapy vs Old Chemotherapy ICER= Cost differential k = Effectiveness differential Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

55 Cost-effectiveness plane The economic analysis depends on the willingness-to-pay, which determines the sustainability area Cost effectiveness plane New Chemotherapy vs Old Chemotherapy Cost effectiveness plane New Chemotherapy vs Old Chemotherapy ICER= ICER= Cost differential k = Cost differential k = Effectiveness differential Effectiveness differential Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

56 Summarising PSA For any given value of the willingness-to-pay k, we can analyse the possible futures For example, consider k = monetary units Parameters simulations Expected Incremental t = 0 t = 1 utility benefit Iter/n Benefits Costs Benefits Costs U(θ 0 ) U(θ 1 ) IB(θ) U 0 = U 1 = EIB= One way of summarising PSA is to compute the cost-effectiveness acceptability curve CEAC = Pr(IB(θ) D) > 0 Upon varying k, this is the probability that the optimal decision would not be reversed by reduced uncertainty Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

57 Summarising PSA Cost Effectiveness Acceptability Curve Probability of cost effectiveness Willingness to pay Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

58 Summarising PSA NB: CEACs only quantify the probability of cost-effectiveness, but do not say anything about the payoffs associated with taking the wrong decision Parameters simulations Expected Maximum Opportunity t = 0 t = 1 utility utility loss Iter/n Benefits Costs Benefits Costs U(θ 0 ) U(θ 1 ) U (θ) OL(θ) EVDI= At each iteration, the OL is the difference between the maximum utility and the value associated with the intervention with the maximum utility overall The expected value of information is the average opportunity loss EVDI = E[OL(θ)] and quantifies the value of getting more information to reduce uncertainty Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

59 Summarising PSA Expected Value of Information EVPI Willingness to pay Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

60 Conclusions Bayesian modelling allows the incorporation of external, additional information to the current analysis This can come in the form of Previous studies Elicitation of expert opinions Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

61 Conclusions Bayesian modelling allows the incorporation of external, additional information to the current analysis This can come in the form of Previous studies Elicitation of expert opinions In general, Bayesian models are more flexible and allow the inclusion of complex relationships between variables and parameters This is particularly effective in decision-models, where information from different sources may be combined into a single framework Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

62 Conclusions Bayesian modelling allows the incorporation of external, additional information to the current analysis This can come in the form of Previous studies Elicitation of expert opinions In general, Bayesian models are more flexible and allow the inclusion of complex relationships between variables and parameters This is particularly effective in decision-models, where information from different sources may be combined into a single framework Using MCMC methods, it is possible to produce the results in terms of simulations from the posterior distributions These can be used to build suitable variables of cost and benefit Particularly effective for running probabilistic sensitivity analysis Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

63 More info (and shameless marketing) It is possible to standardise the economic analysis derived from the output of a Bayesian model, for example using the R package BCEA BCEA features heavily in the brilliant, forthcoming book on Bayesian methods in health economics (written by me ) In the book, I describe the entire process of making Bayesian analysis in health economics, including pre-processing of the data and running the model Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

64 More info (and shameless marketing) It is possible to standardise the economic analysis derived from the output of a Bayesian model, for example using the R package BCEA BCEA features heavily in the brilliant, forthcoming book on Bayesian methods in health economics (written by me ) In the book, I describe the entire process of making Bayesian analysis in health economics, including pre-processing of the data and running the model More info is available at the webpages and Also, some discussion (and more to come) in a few posts on gianlubaio.blogspot.co.uk Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27

65 Thank you! Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre / 27