Gianluca Baio

Bayesian Methods in Health Economics

1Introdution to health eonomi evaluationThis hapter is written by Rahael M. Hunter (Department of Primary Care andPopulation Health, UCL) and Gianlua Baio1.1 IntrodutionIn reent years health eonomis has beome an inreasingly important disi-pline in medial researh, espeially with the transition from the paradigm ofevidene based mediine to that of translational researh (Berwik, 2005; Leanet al., 2008), whih aims at making basi researh appliable in the ontextof real pratie, and under budget onstraints, in order to enhane patients'a

ess to optimal health are.Sine the 1970s, health are servies have undergone dramati hanges:inreasing demand for health are has generated an inrease in the numberof available interventions, whih have sometimes been applied regardless ononsiderations about the atual quality and the osts assoiated.Consequently, deision-makers responsible for the provision of health areare inreasingly faing ritial appraisal proesses of the modality in whihthey manage the available resoures and they need to adjust the managementand the evaluation of the proesses used, with respet to some measures oflinial bene�t. The main reasons for the neessity of ontaining ost assoi-ated with health are are essentially the following: The progressive inrease of the proportion of the �older� (above 65 years)population; the inrease of life expetany and of the inidene of hroni and degen-erative pathologies; the re�nement of diagnosti tehniques; the availability of innovative health tehnologies and therapeuti toolsassoiated with better linial outomes but also with higher osts.In this perspetive, the systemati analysis of organised data provides afundamental ontribution to the identi�ation of eonomially appropriate1

2 Bayesian Methods in Health Eonomisstrategies. This in turns has helped the integration between several linialand quantitative (e.g. statistis and eonomis) disiplines, so muh that itan be reasonably argued that health eonomis is in fat a ombination ofmedial researh, epidemiology, statistis and eonomis. Figure 1.1 shows thisonept graphially and highlights the fat that health eonomis enompassesmore than the mere ost evaluations.Epidemiology Experimentalstudies Cost analysisDeisiontheory HealtheonomisCausalinferene Generalisation and integration ofstatistis, epidemiology, eonometrisand �nanial analysis⇓How muh does it ost totreat a patient withintervention t?⇓ Finanial analysis Budgeting⇓FIGURE 1.1Health eonomi evaluation as the integration of di�erent disiplines. Costanalysis only represents one side of the storyIn this hapter we present some onepts that are relevant to the de�nitionand desription of health eonomi evaluations. In partiular we disuss theharateristis of the two dimensions along whih eonomi evaluations areonduted: osts and linial bene�ts. The latter an be de�ned in severaldi�erent ways, eah of whih gives rise to a spei� method of analysis. Wepresent the main ones in 1.6. Finally, we give a �rst introdution to theproblem of omparative evaluation of two or more health interventions, whihwill be disussed in more tehnial detail in hapters 3 and 5.

1.2 Health eonomi evaluationHealth eonomis an be formally desribed as the appliation of eonomitheory to health (de�ned as �a state of omplete physial, mental, and soialwell-being and not merely the absene of disease or in�rmity�; WHO, 2012)and health are, i.e. the diagnosis, treatment and prevention of disease and

24 Bayesian Methods in Health EonomisTABLE 1.4An example of alulation of QALYs from utility sores. In the two treatmentgroups (t = 0, 1), the measurements onsist of the utility sore utj and theosts ctj , for j = 0, . . . , 4 o

asionsBaseline 6 12 18 24 Totalmonths months months monthsTreatment group (t = 1)Utility sore 0.656 0.744 0.85 0.744 0.744QALYs 0.350 0.399 0.399 0.372 1.519Costs ¿2 300 ¿300 ¿300 ¿300 ¿3 200Control group (t = 0)Utility sore 0.656 0.656 0.656 0.656 0.744QALYs 0.328 0.328 0.328 0.350 1.334Costs ¿300 ¿300 ¿300 ¿300 ¿1 200Di�erenein QALYs (E[∆e])a 0.185in osts (E[∆c])a ¿2 000Cost per QALYa ¿10 811a These quantities are de�ned in 1.7where

δj =time between measurements j and (j − 1), in years1 yearFor example, the QALYs at 6 months (i.e. at time j = 1) for treatment

t = 0 are omputed asq01 =

(0.656 + 0.656

2

)(0.5

1

)= 0.164,sine the time between the two measurements, 6 months, is only half a year.Similarly, for t = 1 the omputation gives

q11 =

(0.656 + 0.744

2

)(0.5

1

)= 0.35.Overall, for eah treatment the QALYs an be omputed by summing the qtjterms aross all the time periods. In the present example, the measurementsare repeated at every 6 month intervals and thus all values are added upover the 2 years. We de�ne the QALYs using the notation et (to indiate the�e�etiveness� of the treatment) as

et =

J∑

j=1

qtj .

Introdution to health eonomi evaluation 25This produes a result of 1.519 extra QALYs for the patient under t = 1 andonly 1.334 extra QALYs for the single patient under t = 0.Notie that, in more realisti ases, instead of a single patient per group, wewould have a

ess to a sample of patients and therefore the relevant measureswould be the population average omputed aross all relevant individuals. �Eah of the three main types of eonomi evaluation desribed above hasstrengths and weaknesses, and although eah has their own spei� hara-teristis most eonomi evaluations generally ombine aspets of eah. Morein depth information an be found in Drummond et al. (2005). NICE's De-ision Support Unit also has extensive guidane to support tehnial ap-praisals http://www.nicedsu.org.uk/. Table 1.5 summarises the main dif-ferenes among them.TABLE 1.5A omparison of the harateristis of the main types of eonomievaluation. Adapted from Meltzer and Teutsh (1998)Type Costs inludeda OutomesDiret IndiretCost-bene�t X X Monetary unitCost-e�etiveness X often Health outomebCost-utility X rarely Utility measureca All future osts and bene�ts should be disounted to the referene year (fr. 1.5)b For example: number of deaths avertedc For example: QALYs (fr. 1.6.4)

1.7 Comparing health interventionsAs disussed earlier, the purpose of eonomi evaluations is to provide infor-mation to deision-makers about the osts and outomes of health are optionsto help with resoure alloation deisions. Generally, eonomi summaries areomputed in the form of �ost-per-outome� ratios.Moreover, in order to ompare the two interventions (t = 0, 1), we an de�nesuitable inremental population summaries, suh as the population averageinrement in bene�ts, suitably measured as utilities (as in a CUA) or bymeans of hard linial outomes (as in a CEA):E[∆e] = e1 − e0 (1.2)

2Introdution to Bayesian inferene2.1 IntrodutionIn the ontext of statistial problems, the frequentist (or empirial) interpre-tation of probability has played a predominant role throughout the twentiethentury, espeially in the medial �eld. In this approah, probability is de�nedas the limiting frequeny of o

urrene in an in�nitely repeated experiment.The underlying assumption is that of a ��xed� onept of probability, whihis unknown but an be theoretially dislosed by means of repeated trials,under the same experimental onditions. Moreover, probability is generallyregarded in lassial statistis as a physial property of the objet of theanalysis.However, although the frequentist approah still plays the role of the stan-dard in various applied areas, there are many other possible oneptualisa-tions of probability haraterising di�erent philosophies behind the problemof statistial inferene. Among these, an inreasingly popular is the Bayesian(sometimes referred to as subjetivist, in its ontemporary form), originatedby the posthumous work of Reverend Thomas Bayes (1763) and the inde-pendent ontributions by Pierre Simone Laplae (1774, 1812) � see Howie(2002), Senn (2003), Fienberg (2006) or Bertsh MGrayne (2011) for a his-torial a

ount of Bayesian statistis.The main feature of this approah is that probability is interpreted as asubjetive degree of belief in the o

urrene of an event, representing theindividual level of unertainty in its atual realisation (fr. de Finetti, 1974,probably the most omprehensive a

ount of subjetive probability). One ofthe main impliations of subjetivism is that there is no requirement thatone should be able to speify, or even oneive of some relevant sequene ofrepetitions of the event in question, as happens in the frequentist framework,with the advantage that �one-o�� type of events an be assessed onsistently(Dawid, 2005).In the Bayesian philosophy, the probability assigned to any event dependson the individual whose unertainty is being expressed and on the state ofbakground information in light of whih this assessment is being made. Asany of these fators hanges, so too might the probability. Consequently, underthe subjetivist view, there is no assumption of a unique, orret (or �true�)29

30 Bayesian Methods in Health Eonomisvalue for the probability of any unertain event. Rather, eah individual isentitled to their own subjetive probability and a

ording to the evidenethat beomes sequentially available, they tend to update their belief.The development of Bayesian applied researh has been limited probablybeause of the ommon pereption among pratitioners that Bayesian meth-ods are �more omplex�. In fat, in our opinion the apparent higher degreeof omplexity is more than ompensated by at least the two following onse-quenes. First, Bayesian methods allow taking into a

ount, through a formaland onsistent model, all the available information, e.g. the results of previousstudies. Moreover, the inferential proess is straightforward, as it is possibleto make probabilisti statements diretly on the quantities of interest (i.e.some unobservable feature of the proess under study, typially � but notneessarily � represented by a set of parameters).In our opinion, Bayesian methods allow the pratitioner to make the mostof the evidene: in just the situation of �repeated trials�, after observing theoutomes (e.g. su

esses and failures) of many past trials (and no other ol-lateral information), all subjetivists will be drawn to an assessment of theprobability of obtaining a su

ess on the next event that is extremely loseto the observed proportion of su

esses so far. However, if past data are notsu�iently extensive, it may reasonably be argued that there should indeedbe sope for interpersonal disagreement as to the impliations of the evidene.Therefore the Bayesian approah provides a more general framework for theproblem of statistial inferene.Justi�ations for the use of Bayesian methods in health are evaluation havebeen detailed by Spiegelhalter et al. (2004), in terms of the formal quantita-tive inlusion of external evidene in all aspets of linial researh, inludingdesign, analysis, interpretation and poliy-making. In partiular, the Bayesianapproah is valuable beause: i) it proves more �exible and apable of adapt-ing to eah unique situation; ii) it represents a more e�ient inferential tool,making use of all available evidene and not restriting formal evaluationsto just the urrent data at hand; iii) it is partiularly e�etive in produingpreditions and inputs for deision-making.Jakman (2009) suggests that performing statistial analysis by means ofBayesian methods also produes advantages from a pragmati point of view:beause of the wide availability of fast and heap omputing power, simulation-based proedures have allowed researhers to exploit more and more omplexstatistial models, espeially under the Bayesian paradigm. Examples inludethe possibility of omputing interval estimations in a straightforward way,without the need to rely on asymptoti arguments. This in turns has thepotential of rendering �hard statistial problems easier�.The a

ount of Bayesian statistis that is presented in this hapter is farfrom exhaustive � more omprehensive referenes are O'Hagan (1994), Berry(1996), Bernardo and Smith (1999), Lindley (2000), Robert (2001), Lee (2004),Spiegelhalter et al. (2004), Gelman et al. (2004), Lindley (2006), Lanaster(2008), Carlin and Louis (2009), Jakman (2009) and Christensen et al. (2011).

Introdution to Bayesian inferene 67

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(d)FIGURE 2.10Gibbs sampling simulation for the semi-onjugated Normal model. The num-bers indiate the simulations sequene. Panels (a)-(d) show the situation after10, 30, 100 and 1000 iterations respetively. In this ase, already 100, or even30 simulations seem to over the relevant portion of the parametri spaegeneri omponent θk an be estimated as:V̂ar(θk | y) = S − 1S

W (θk) +1

SB(θk),where W (θk) and B(θk) are the average within-hain variane and the

68 Bayesian Methods in Health Eonomis

0 100 200 300 400 500

−10

0−

500

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020

0

Iteration

Chain 1Chain 2

Burn−in Sample after convergence

FIGURE 2.11Graphial assessment of onvergene for a Markov hain; in this ase twohains are set up, starting from di�erent initial points. After the burn-in pe-riod, the two hains onverge to the stationary distributionbetween-hains variane, and S is the length of the MCMC sample. Con-vergene is then monitored by assessing the potential sale redutionR̂ =

√ V̂ar(θk | y)W (θk)

, (2.19)whih represents the fator by whih the sale of the urrent estimated poste-rior distribution of θk an be further redued. If R̂ is large, then onsideringa longer MCMC run will potentially improve the inferene about the targetdistribution. As a rule of thumb, values of R̂ ≤ 1.1 are generally a

epted asindiative of su�ient onvergene.2.4.7 MCMC autoorrelationThe seond ritial aspet of MCMC proedures is that the iterations pro-dued by a Markov hain are by de�nition orrelated, sine the urrent obser-vation depends on the previous one. Therefore, intuitively the atual numberof iterations stored to produe the inferene does not give in general the sameinformation provided by a sample of iid observations of the same size. In otherwords, the higher the autoorrelation, the lower the degree of equivalene be-tween the MCMC output and a proper iid sample of the same size.

3Statistial ost-e�etiveness analysis3.1 IntrodutionIn the last ten years, health eonomi evaluations have built on more advanedstatistial deision-theoreti foundations, e�etively beoming a branh of ap-plied statistis (Briggs et al., 2006;Willan and Briggs, 2006), inreasingly oftenunder a Bayesian statistial approah (O'Hagan and Stevens, 2001; O'Haganet al., 2001; Parmigiani, 2002b; Spiegelhalter and Best, 2003; Spiegelhalteret al., 2004).As suggested by Spiegelhalter et al. (2004), this an be asribed to thefat that �the subjetive interpretation of probability is essential, sine theexpressions of unertainty required for a deision analysis an rarely be basedpurely on empirial data�.Even though the proess is, tehnially, a simple appliation of standarddeision-theoreti preepts (desribed for example in Lindley, 1985), healtheonomis is ompliated by issues related to other important fators that playa major role in real pratie medial deision making. Among these are thedi�ulty of applying standard ost-e�etiveness tehniques to the regulatoryproess (Baio and Russo, 2009), and the neessity of properly a

ounting forthe impat of unertainty in the inputs of deision proesses, an issue knownas sensitivity analysis (Parmigiani, 2002b; Saltelli et al., 2004). This latter inpartiular is fundamental and is a required basi omponent of any new drugapproval or reimbursement dossier in settings regulated by deision-makingbodies suh as NICE in the UK (Claxton et al., 2005).In this hapter we �rst brie�y review the main harateristis of deisiontheory. As in hapter 2 we proeed by introduing the more abstrat theory,in order to make the point that rational deision-making is e�eted by max-imising the expetation of a suitably de�ned utility funtion. This is used toquantify the value assoiated with the unertain onsequenes of a possibleintervention.Next we link the general methodology to the spei� problem faed in healtheonomi evaluation. This requires the spei�ation of the problem in termsof a omposite response, a

ounting for both ost and bene�ts. We presenta relatively simple running example and, as in hapter 2, we swith betweenthe development of the theory and its appliation throughout. 75

76 Bayesian Methods in Health EonomisWe then onentrate on the development of sensitivity analysis tehniques,whih as suggested earlier play a fundamental role in health eonomi evalua-tions. Finally, we present some more advaned issues assoiated with the mainassumptions on whih ost-e�etiveness or ost-utility analyses are based: inpartiular, we onsider the problems of risk-aversion and the impat of marketonstraints (e.g. in the ase of regulatory proesses).3.2 Deision theory and expeted utility3.2.1 The problemHealth eonomi evaluations are a typial problem of deision-making underunertainty. The main objetive is to evaluate omparatively the unknownonsequenes of a given health intervention against at least another. A suitableapproah to deal with this kind of problems is based on expeted utility theory,whih we brie�y review in this setion. More substantial referenes are Savage(1954), Rai�a (1968), Lindley (1985), Berger (1985), Smith (1988), Bernardoand Smith (1999), Parmigiani (2002b), Jordaan (2005) and Smith (2011).Formally, a deision problem is haraterised by some fundamental ele-ments: �rst, we onsider the possible deisions (interventions, ations, treat-ments) t ∈ T , representing the alternatives available to the deision-maker.The seletion of eah possible intervention has some onsequenes (outomes)o ∈ O, de�ned in general as funtions of suitable random quantities ω ∈ Ω.Every onsequene an be expressed as o = (ω, t), i.e. as the result of hoosingintervention t and the fat that a series of random quantities ω will obtain inthe future. The set of onsequenes an be then represented as O = Ω× T .In addition to these fundamental quantities, the deision-maker needs tode�ne a sheme of preferenes among the many deisions and onsequenes;this relationship of preferene is generially indiated by the symbol `�'. Thenotation t1 � t2 indiates that the random onsequenes of ation t1 are notpreferred to those of ation t2. If t1 � t2 and simultaneously t2 � t1, the twoations are indi�erent: t1 ∼ t2.The Bayesian deision proess is based on a set of presriptive axioms. Theseare the riteria that should hold in order to make rational deisions. The �rstset of axioms is related to the oherene of the deision making and involves: omparability of the onsequenes. This assumes that the deision-makeris apable of produing some form of ranking of the possible outomes, sothat there exist at least one pair of onsequenes o1 and o2 for whih theformer is preferred to the latter; transitivity of the preferenes. This axiom implies that if the deision-maker has a preferene for ation t2 over ation t1 and for ation t3 over

86 Bayesian Methods in Health Eonomisnint is the number of interventions being ompared∗ (hapter 4 disusses howto run a Bayesian model, store and post-proess its results using R).In this ase, nint = 2 sine we are omparing two interventions and weuse nsim = 500 simulations. The relevant health eonomi quantities an beprodued by running the funtion bcea, by means of the following ommands(fr. 4.7 for a more detailed desription).

treats

Statistial ost-e�etiveness analysis 87possible omparisons. The seond option spei�es the value of the willingness-to-pay to use as referene. In this ase, we have hosen the default value ofk = 25 000, whih is usually reommended by NICE as the referene ost-per-QALY.

Cost effectiveness plane New Chemotherapy vs Old Chemotherapy

Effectiveness differential

Cos

t diff

eren

tial

−100 −50 0 50 100 150 200

−50

0000

050

0000

1000

000

• ICER=6698.11

k = 25000

FIGURE 3.1Cost-e�etiveness plane for the hemotherapy example. The dots representthe simulations from the posterior distribution of (∆e,∆c), while the shadedpart of the graph shows the �sustainability area�, i.e. the portion of the planein whih the points are below the willingness-to-pay threshold, whih is set to25 000 in this aseThe result is depited in Figure 3.1, in whih the dots are the simulationsfrom the posterior distribution of (∆e,∆c). The graph also shows the lineobtained in orrespondene of the set value of k. The shaded area below theline represent the portion of the plane where the simulated values are belowthat threshold and therefore it an be onsidered as a �sustainability area�.The red dot represents the ICER (fr. 1.7). As in this ase it lies in thesustainability area, we an onlude that, at the willingness-to-pay thresholdseleted, the new drug is a ost-e�etive alternative with respet to the statusquo. With respet to Figure 1.6, the urrent analysis also presents a quanti�-ation of the unertainty underlying the point estimation represented by theICER, beause it is based on the entire distribution of (∆e,∆c), rather thanjust on its expetations.A more omprehensive analysis is provided by Figure 3.2, whih is produed

4Bayesian analysis in pratie4.1 IntrodutionAs disussed in hapter 2, if it is possible to sample from the full onditionaldistributions, Gibbs sampling algorithms an be programmed in a relativelyeasy way. However, in most pratial situations, the required onditional dis-tributions are not analytially tratable and therefore it is neessary to ap-proximate them (e.g. by means of algorithms suh as Metropolis-Hastings orslie sampling) before Gibbs sampling an be performed.The most popular software that allows the semi-automatisation of MCMCproedures is BUGS, and partiularly its MS Windows inarnations WinBUGS(Spiegelhalter et al., 2002) and OpenBUGS (Lunn et al., 2009), whose widespreaduse has arguably ontributed to the establishment of applied Bayesian statis-tis in the last twenty years.The aronym BUGS stands for Bayesian analysis Using Gibbs Sampling andthe program essentially onsists of two main parts. The �rst is a parser, whihinspets the set of delarations provided by the user to de�ne the statistialmodel (in terms of data and parameter distributions and, possibly, other de-terministi relationships among the variables in the problem). In partiular,the parser odi�es the statistial model in terms of the orresponding DAG,trying to make use of the onditional independene relationships implied bythe model assumed by the user. These generally simplify the omputationssine the full onditional distribution for any (set of) node(s) only involvesa loal omputation on the graph. Thus, only a small portion of the wholemodel needs to be onsidered at any given time (Lunn et al., 2009).The seond part is an expert system that is used to dedue the form of thefull onditional distributions generated by the problem. When possible, BUGStries to exploit onjugay to speed up the proess; when this is not feasible,suitable omplementary sampling algorithms are applied together with theGibbs sampling to obtain the required MCMC estimation.While both BUGS and WinBUGS an run as stand-alone software, in reentyears several programs have been written to interfae them with standardstatistial software suh as R, Matlab, Stata or SAS, whih makes the proessof data analysis easier (we disuss this aspet later).Despite their wide su

ess, WinBUGS or BUGS are not the only possible al-115

Bayesian analysis in pratie 137baby, knowing that the mother has gone into labour at exatly that estimatedgestational age), or a hypothetial value (e.g. if the dotor wants to plandi�erent are strategies for a mother who has not gone into labour yet).We an then run the model using the following ode.X.star

5Health eonomi evaluation in pratie5.1 IntrodutionIn this �nal hapter we present some examples of health eonomi evaluation.In partiular we fous on three �typial� ases; the �rst onerns the analysisof individual level data, spei�ally from a RCT, in whih a sample of individ-uals is observed in terms of the relevant measures of ost and linial outome.The seond example fousses on the proess of evidene synthesis, a situationpartiularly relevant when individual data are not available. In these situa-tions, the relevant random quantities an be estimated by the ombination ofthe available evidene, e.g. oming from published studies, or expert opinions.Within the Bayesian framework, this is very muh linked to the developmentof hierarhial models, whih we brie�y review before presenting the example.Finally, we onsider the analysis of Markov models, an inreasingly populartool in health eonomi evaluation, whih allow the simulation of a follow upanalysis on a �virtual� ohort of patients.While the problems highlighted in eah of the following setions an beonsidered as typial of the situations onsidered in applied health eonomis,they are far from representing an exhaustive set: in real appliations, thereare ountless subtleties and nuisanes that need to be addressed spei�ally.In partiular in the Bayesian approah, this require a areful spei�ation ofthe model to be used, mainly in terms of the prior distributions, but also interms of the possible orrelation levels among the observed and unobservedrandom variables.Nevertheless, we takle some of the most relevant issues arising from theanalysis of health eonomi data, trying to point out possible solutions andreferenes where more detailed modelling strategies are presented. All theexamples are worked out starting from the desription of the problem, thespei�ation of the Bayesian model and then the ode used to run the MCMCanalysis and the post-proessing neessary to derive the relevant health eo-nomi quantities used to produe the deision-making proess.

153

176 Bayesian Methods in Health Eonomisp1

p0 ρ

µγ σ2γ µδ σ2δ

γh αs δs

βh mh π(0)s n

(0)s π

(1)s n

(1)s

xh r(0)s r

(1)s

h = 1, . . . , H s = 1, . . . , SFIGURE 5.7Graphial representation of the evidene syntheses in the model. In the graph,solid arrows indiate probabilisti links, while dashed arrows indiate logialdependene. H studies are used to investigate the overall population prob-ability of being infeted by in�uenza, p0. A similar struture ombines theinformation for the S studies investigating the e�etiveness of NIs to derivean odds ratio, whih is ombined with the estimation of p0 to provide an esti-mation of p1, the probability of in�uenza in the senario in whih prophylatitreatment with NIs is made available# the "healthy" adults population (t=0)

for(h in 1:H) {

x[h] ~ dbin(beta[h], m[h])

logit(beta[h])

Health eonomi evaluation in pratie 177delta[s] ~ dnorm(mu.delta,tau.delta)

alpha[s] ~ dnorm(0,0.00001)

}

# Prior distributions

mu.delta ~ dnorm(0,0.00001)

mu.gamma ~ dnorm(0,0.00001)

sigma.delta ~ dunif(0,10)

tau.delta

IndexArbuthnot, 49, 51Bayes, 29, 36�40, 43, 49, 50, 54theorem, 36�40, 43, 49, 54Bayesian inferene, 29�31, 33, 34, 36,37, 39, 41, 42, 44, 47�49,51, 57, 59�63, 75�77, 79,80, 84�86, 89�91, 103, 104,113�116, 120, 133, 151, 154,163, 165, 169, 170, 187, 196,201redibility interval, 48, 59, 61,67, 122, 136HPD, 48exhangeability, 41�43, 81, 133,136, 168�170, 173, 200hierarhial model, 69, 129,151, 168�170, 200model average, 106, 148, 150,164DIC, 104�106, 119, 121, 138,148, 150point estimates, 87, 196posterior preditive hek, 166predition, 30, 133, 166BCEA, 85, 86, 92, 94, 99, 102, 108,111ea.plot, 94eplane.plot, 86CEriskav, 108evi.plot, 98mixedAn, 111sim.table, 92linial outomes, 1, 2, 8�10, 17�19,21, 25, 75, 84, 103, 152�154,159, 160, 163, 183, 191�193,205

EQ-5D, 10, 11, 13, 16, 23QALY, 22�28, 83, 87, 95, 153,154, 157, 158, 160�162, 164SF6D, 10, 11, 13, 15, 23, 78, 153valuingstandard gamble, 15, 77, 78time-trade o�, 14�16ost, 1, 2, 4, 6�9, 17�28, 82, 83, 89,95, 98, 105, 152�154, 157,158, 160, 161, 163�166, 171,176, 179, 181�183, 191�193,205�xed, 186loal �nanial information, 8opportunity, 22present value, 17, 18variable, 8ost-bene�t, 18, 21analysis, 18�21ratio, 21ost-e�etiveness, 6, 8, 18, 21, 22, 26,75, 80, 84, 86, 89, 93�95,103, 105, 111, 139, 153, 157,159, 163, 170, 177, 178, 183,184, 193, 197a

eptability urve, 93, 94, 105,106, 109, 159, 178analysis, 18, 21�23, 25, 27, 84,86, 93, 94, 105, 106, 109,111, 139, 157, 159, 163, 177,178, 193, 205plane, 26, 86, 95, 105, 159ost-minimisation analysis, 18ost-utility analysis, 18, 22, 23, 25,75de Finetti, 29, 35, 41, 42 223

224 Indexdeision theory, 75, 77, 78, 80, 84expeted utility, 76, 78, 79, 85,89, 91�93, 97, 101, 102, 111,146, 179inremental bene�t, 84, 91�93,108, 109mixed strategy, 111, 112net bene�t, 84, 85, 89, 92, 94,103, 107, 108risk aversion, 108direted ayli graph, 40, 41, 43,113, 116, 119, 120, 134, 173disount, 17, 25, 26, 183, 192, 193,205present value, 17, 18distributionBernoulli, 165Beta, 50, 53�55, 58, 59, 82, 83,140, 187Binomial, 46, 50, 52�55, 57, 82,116, 169, 172, 187, 197, 198Dirihlet, 187�189, 193Exponential, 55Gamma, 55, 60, 62, 64, 129, 152,160, 162, 164, 166, 190, 199logNormal, 83, 140, 142, 152,157, 160�162, 164, 166, 171,177, 199Multinomial, 187, 188Normal, 7, 51, 54�56, 59, 62, 64,82, 83, 118, 129�132, 134,137, 152, 153, 158, 163, 164,169, 171, 172, 199Poisson, 55, 60�62Uniform, 50, 117, 120, 130, 135,160, 162, 177, 189evidene synthesis, 151, 167, 169,170, 172, 173, 176, 183,197�200inremental ost-e�etiveness ratio,25�28, 84, 85, 87, 88

JAGS language, 73, 114�121, 125�133, 135�137, 139, 142, 143,146, 147, 202interfae with R, 72, 121Laplae, 29, 49, 50, 69, 115, 122, 125,127Markov Chain Monte Carlo, 63, 67�69, 85, 101, 102, 104�106,108, 113�115, 118, 120, 122,123, 126, 130, 144, 146, 147,151, 163, 170, 190, 201onvergene diagnostis, 63, 67,71, 119, 121�123, 130, 135,146, 147autoorrelation, 67�70, 72, 122,138, 177e�etive sample size, 69, 136,137, 143Gelman-Rubin statistis, 68,69, 72, 117, 121, 127, 135,137, 143, 157Gibbs sampling, 63�65, 67, 69�73, 113, 115, 131initial values, 117, 118, 120, 126,130, 135, 137, 143, 146, 147,189methods, 63thinning, 69, 71, 72, 119, 123,130, 135, 146, 147Markov model, 10, 151, 180, 184deision tree, 179transition probability, 10, 180,182�184, 187, 190, 195�202probabilityoherene, 34, 36, 76, 79posterior, 37, 39, 43, 44, 46, 48�51, 54�57, 59�64, 67�73, 80,82, 85�87, 89, 97, 101, 103�105, 118, 121�123, 125, 126,129, 133, 135, 137, 144, 157,158, 163, 164, 166, 181, 183,189, 190, 196, 201

Index 225prior, 18, 37, 39, 43, 45, 46, 48,49, 51�62, 64, 65, 69, 70,80�83, 94, 103, 104, 115�117, 129�132, 139, 140, 151,154, 155, 160, 162, 171�173,187, 189, 193, 198rules of, 34, 36, 51subjetive, 29, 30, 32, 33, 41, 49Duth book, 35Ramsey, 35sensitivity analysis, 16, 17, 75, 96, 98,160probabilisti, 91�93, 96, 98, 103,108, 109, 164, 196value of information, 93, 96�100, 109,111expeted, 97, 99, 101�103, 105,106, 109, 111, 159, 164, 178,193partial, 101�103, 144�147sample, 97opportunity loss, 93, 97�100WinBUGS language, 73, 113�118,123, 128, 130, 136, 148, 160interfae with R, 114