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18

Statistical methods for healthcareeconomic evaluation

Caterina Conigliani1, Andrea Manca2 and Andrea Tancredi31Department of Economics, University of Roma Tre, Rome, Italy2Centre for Health Economics, The University of York, York, UK3Department of Methods and Models for Economics, Territory and Finance,University of Rome ‘La Sapienza’, Rome, Italy

Synopsis

The increasing burden on the financial budgets of healthcare providers worldwide has em-phasised the need to use healthcare resources as efficiently as possible. Healthcare economicevaluation has gained popularity in the last 20 years as a key tool to generate the informationrequired by policy makers to achieve their goal. This chapter describes the statistical issuesassociated with the conduct of healthcare economic evaluation studies, considering both theanalysis based on a single trial and the use of Bayesian comprehensive decision analyticmodels. Further issues such as the role of probabilistic sensitivity analysis and Bayesianevidence synthesis are also presented.

18.1 Introduction

Healthcare systems worldwide are under increasing financial pressure and, consequently, theyneed to use limited healthcare resources as efficiently as possible. This means (among otherthings) to invest in healthcare treatments that provide value for money. Healthcare economicevaluation (a particular form of which is known as cost-effectiveness analysis) is the vehiclethrough which evidence to inform the above decision can be produced. In fact, many countries

Statistical Methods in Healthcare, First Edition. Edited by Frederick W. Faltin, Ron S. Kenett and Fabrizio Ruggeri.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.


366 STATISTICAL METHODS IN HEALTHCARE

have set up agencies whose role is to make recommendations as to which health technologiesshould be used.

Healthcare economic evaluation has been traditionally carried out using data collectedalongside randomised controlled clinical trials (RCTs). Willan and Briggs (2006) focus onthe role and methods of statistical inference in economic evaluation with data originatingfrom RCTs. However, an increasingly important approach is to use the RCT data to developan economic evaluation model to overcome some of the limitations of standard trial-basedcost-effectiveness analysis (CEA) when this is used to inform funding decisions (Sculpher,Claxton and Drummond, 2006). This view is supported by the notion that single trialsare often not the most appropriate vehicle to inform decision-making. Economic analysesbased on single RCTs may in fact (1) be short in duration, (2) not be relevant to the ju-risdiction of interest, (3) have not compared all the healthcare treatment strategies relevantto the decision-maker context, and (4) ignore the presence of other relevant evidence. Inthese cases it is more appropriate to use economic evaluation models to help address theabove issues.

This chapter describes the statistical issues associated with healthcare economic eval-uation, starting from a single trial-based analysis in Section 18.3, and moving on to theuse of Bayesian comprehensive decision analytic models in Section 18.4. The standardtools for cost-effectiveness analysis are presented in Section 18.2, while the final sectiondeals with further issues such as the role of probabilistic sensitivity analysis and Bayesianevidence synthesis.

18.2 Statistical analysis of cost-effectiveness

Health economic evaluation has been defined as ‘the comparison of two or more alternativecourses of action in terms of both their costs and consequences’ (Drummond et al., 1997).Suppose, in particular, one needs to compare two treatments T1 and T2 in a set of new andexisting technologies. Also suppose that under each treatment Tj the population mean cost isγ j and the population mean efficacy is μ j ( j = 1, 2). Note that μ j and γ j are computed overall patients with a given illness or condition that could be given treatment Tj, that is, over afinite population of patients, and in practical applications they are unknown and need to beestimated; this problem will be considered in detail in Section 18.3 and Section 18.4.

The key quantities in CEA are the mean health effect differential �e = μ2 − μ1 and themean cost differential �c = γ2 − γ1. These outcomes are usually expressed using differentnumeraire, that is, monetary units for costs, and typically health outcomes – such as life-yearsgained or quality-adjusted life years (QALYs) – for the health effects.

18.2.1 Incremental cost-effectiveness plane, incrementalcost-effectiveness ratio and incremental net benefit

A natural setting for presenting and comparing cost differentials and effect differentials is theincremental cost-effectiveness plane (see Black, 1999; O’Hagan, Stevens and Montmartin,2000) shown in Figure 18.1; that is, the plane of possible pairs of values (�e,�c) of theunderlying true mean incremental effects and costs. In quadrant II, treatment T2 is lesseffective and more expensive than treatment T1, so it is unconditionally less acceptable; that


STATISTICAL METHODS FOR HEALTHCARE ECONOMIC EVALUATION 367

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Δ c = λΔ

e

Figure 18.1 The incremental cost-effectiveness plane.

is, T1 dominates T2. Similarly, in quadrant IV, T2 is both more effective and cheaper than T1,and is therefore unconditionally preferred; that is, T2 dominates T1. In contrast, the comparisonof the two treatments in quadrants I and III depends on the decision-maker’s threshold valuefor a unit of effectiveness, which can be represented by a line of slope λ. That is, the quantityλ represents the decision-maker’s maximum willingness to pay to obtain one additional unitof effectiveness.

Notice that once we have a threshold value, the decision rule concerning what can beconsidered cost-effective is straightforward. Now, in quadrant I T2 is both more effective andmore costly than T1, but below the line of slope λ there is a sufficiently high increment inefficacy (relative to the increment in cost) for T2 to be preferred. Analogously, in quadrant IIIT2 is both less effective and cheaper than T1, but below the line of slope λ the reduction incost is sufficiently high (relative to the reduction in efficacy) for T2 to be preferred. It followsthat the region of acceptability of treatment T2 relative to T1 is the portion of the plane belowthe line of slope λ.

Note that the region of acceptability of T2 can be expressed in terms of what it is oftenconsidered to be the primary measure of the cost-effectiveness of a treatment, that is, themean incremental net benefit (INB):

bλ = λ�e − �c

(Stinnett and Mulahy, 1998), or in terms of the incremental cost-effectiveness ratio (ICER):

ρ = �c/�e.

However, while the definition in terms of the INB is quite straightforward, in that theregion of acceptability simply equates to the INB being greater than zero, in terms of the



ICER it is necessary to consider both ρ and the sign of �e (as is obvious from Figure 18.1), sothat T2 is cost-effective relative to T1 if, when �e > 0, ρ < λ (i.e. if what the decision-makeris willing to pay for the achieved additional effectiveness is greater than the actual additionalcost) or if , when �e < 0, ρ > λ (i.e. if the loss in effectiveness is associated with negligiblesavings). Because of this, and due to the challenges associated with the statistical analysis ofa ratio parameter (especially in the frequentist approach to statistical inference), basing theCEA on the INB is generally considered much more straightforward (O’Hagan, Stevens andMontmartin, 2000).

18.2.2 The cost-effectiveness acceptability curve

Notice that, since λ is rarely unambiguously determined in practice, inference about the meanINB is generally presented by means of a cost-effectiveness acceptability curve (CEAC).This was introduced by van Hout, Al and Gordon (1994), and plots the probability that T2

is cost-effective relative to T1, that is, the probability that the net benefit is positive, againsta range of possible values that the coefficient λ can take. Equivalently, as pointed out forinstance in Baio (2010), the CEAC can be seen as representing the area of the right tail of thedistribution of the mean INB for a given value of λ.

It is interesting to note that, although the work of van Hout, Al and Gordon (1994) wasapparently intended for a frequentist analysis of cost-effectiveness, the probability that thenet benefit is positive does not exist nor have any meaning in the frequentist approach tostatistical inference. In fact, as pointed out for instance in O’Hagan, Stevens and Montmartin(2000), in frequentist statistics unknown parameters are not random variables and do nothave probability distributions. It is only in a Bayesian framework that parameters such �e

and �c are random variables, so that a probability such as P(bλ > 0) can be computed andinterpreted.

In this sense, it is particularly interesting to consider the work by O’Hagan, Stevens andMontmartin (2000), where a frequentist interpretation of the CEAC is provided. In particular,it is shown that in some cases the CEAC coincides with 1 minus the classical frequentistp-value to test the null hypothesis that T2 is not acceptable (H0 : λ�e − �c ≤ 0) against thealternative hypothesis that it is acceptable (H1 : λ�e − �c > 0), although this correspondenceapplies only under some very strict conditions. In all other cases the CEAC does not reallyhave a frequentist interpretation, which makes a Bayesian approach to CEA a particularlynatural framework of analysis.

Indeed, the analytical framework outlined above should be placed in the context of thedebate about the formal role of decision theory in health policy making (see, for instance,Claxton and Posnett, 1996; Claxton, 1999). In particular, supporters of its use argue thata maximised expected utility is the only criterion for choosing between two options, sothat measures of significance are all irrelevant to clinical decision-making. When decidingbetween two treatments, assuming that the mean INB is the utility function, then maximisingthe expected utility means choosing T2 over T1 iff the expected mean INB is positive. At agiven λ value, the CEAC would then provide a measure of decision uncertainty, given thedata available. As we will see in the final section, the extent to which the decision uncertaintyassociated with a given decision problem is too high given the cost associated with makingthe wrong decision becomes an empirical question that the policy maker can address usingValue of Information Analysis techniques.



18.3 Inference for cost-effectiveness data from clinical trials

Consider the situation in which both cost and efficacy data are obtained on individual patientsin a comparative RCT; that is, data consist of the effect e js and the cost c js of treatment j onpatient s ( j = 1, 2; s = 1, 2, . . . , nj). In order to assess if treatment T2 is more cost-effectivethan treatment T1, the analyst needs to compare the underlying true mean efficacy in eachtrial arm, μ1 and μ2, as well as the underlying true mean costs γ1 and γ2.

Note that cost-effectiveness analysis of RCT data relies on statistical models whichdescribe the distribution of costs and effects and their interrelation across individuals in thetrial. These, however, are rather difficult to determine, mainly because cost data obtained forindividual patients in health economic studies typically exhibit highly skewed and heavy-tailed distributions, with a few patients incurring high costs because of complications or longtreatments. Moreover, the distributions of cost data are often multimodal, for instance with amass at zero. For a recent review on the statistical methods more widely used to handle costdata, and their ability to address these problems, see Mihaylova et al. (2011).

To illustrate this scenario, we present an example using the eVALuate trial (Sculpheret al., 2004). This was a randomised controlled trial that compared laparoscopic assistedhysterectomy with standard hysterectomy. The latter was carried out via either abdominalor vaginal route, depending on the clinical characteristics of the patient. In the abdominalcomparison, a total of 859 women with gynaecological symptoms were randomised (2 : 1ratio) to either laparoscopic assisted hysterectomy (T2) or standard abdominal hysterectomy(T1). The trial had a median follow up of 52 weeks. The aim of the economic evaluationwhich was carried out alongside the eVALuate trial was to assess whether laparoscopicsurgery offered value for money compared to the status quo at the time (i.e abdominalhysterectomy). During the study follow up, for each patient in the trial, the investigatorscollected healthcare resource utilisation and cost (in UK pounds). Health outcomes weremeasured in terms of QALYs at one-year follow up. For simplicity we use here the QALY atsix weeks from hospital discharge. The data are shown in Figure 18.2, and clearly show thatunder both treatments the cost distribution is highly skewed and heavy tailed.

Note that a CEA for this kind of data can be performed both from a classical and aBayesian point of view, and in both settings parametric and nonparametric techniques arein common use. In particular, a parametric analysis requires the introduction of a bivariatedistribution for the data (c js, e js); one example that is often used as a baseline model isthe bivariate normal distribution. Then representing this joint distribution as the product ofthe marginal distribution for costs and the conditional distribution for effects given costs, datalike that in the eVALuate trial can be modelled as

c js ∼ f (c js | θ j) e js | c js ∼ f (e js | c js, φ j).

so that the mean cost γ j depends on θ j, while the mean effect μ j depends on both θ j and φ j ( j =1, 2; s = 1, 2, . . . , nj). This approach will be considered in detail in the next section from aBayesian perspective. For the moment, remaining in the classical parametric framework, notethat maximum likelihood methods can easily be used to estimate the model parameters θ j andφ j, but do not generally provide satisfactory solutions in terms of evaluating the uncertaintyabout the ICER; see Willan and Briggs (2006) for a discussion on different methods fortackling this problem (such as approximated methods based on Taylor expansions, bootstrapsimulations, Fieller method). Because of this, and given that the natural interpretation of



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Figure 18.2 Data from the eVALuate trial.

the CEAC is only applicable in a Bayesian context, here we do not discuss any further theclassical approach to CEA. Nonparametric methods that might be applied without specifyingthe underlying population distribution will be discussed in Section 18.3.2.

18.3.1 Bayesian parametric modelling

Bayesian parametric modelling has often been used in the context of cost-effectiveness anal-ysis for both costs and efficacy data; in fact, at least in principle, it can lead to efficientinference (see, among others, O’Hagan and Stevens, 2001; O’Hagan and Stevens, 2002; Aland Van Hout, 2000; Fryback, Chinnis and Ulvila, 2001), and it easily allows inclusion ofcovariates in the analysis. To quote a few examples, O’Hagan and Stevens (2003) considera lognormal distribution to model costs in a study where the efficacy outcome was binary;Thompson and Nixon (2005) compare modelling strategies for two commonly used distribu-tions for cost data – the gamma and the lognormal distributions – in an example where theeffects data are apparently adequately represented by a normal distribution. Using the samedata set, Conigliani and Tancredi (2009) explore the behaviour of the generalised Pareto and



the Weibull distributions, whereas Conigliani (2010) investigates also the use of the inverseGaussian distribution.

One point emerges clearly from these parametric analyses: the conclusions in terms ofcost-effectiveness are substantially sensitive to the assumptions made about the distributionsof costs. This conclusion is also supported by simulation results published by Briggs et al.(2005). It might be argued that one should choose the parametric model that fits the databetter, and rely on the corresponding conclusions. However the high skewness and kurtosisusually found in cost data implies that inference on costs will typically be very sensitiveto the shape of the right tail of the fitted distribution beyond the range of the data, that isclearly very difficult to model accurately even when there is a substantial amount of data(Thompson and Nixon, 2005); for instance, Nixon and Thompson (2004) found that in theirexample data sets the different population mean estimates, obtained with different parametricmodels, were due almost entirely to possible costs more than twice the observed maximumcost. One consequence of this is that parametric models that fit the data equally well canproduce very different answers, while in some cases models that fit badly can give similarresults to those that fit well, so that better fit does not necessarily translate into more reliableconclusions.

To illustrate this point, consider again the eVALuate trial, and assume that under bothtreatments we model cost data with a lognormal, with a gamma, with a Weibull, or with anormal distribution; clearly we expect that in both arms of the trial the first three models fitthe data much better than the normal distribution, but because it is hard to judge which of thelognormal, the gamma or the Weibull is better, the fit of each model to the data will be judgedin terms of the deviance (minus twice log likelihood at the maximum likelihood estimate),in that a lower deviance indicates a better fit. Moreover, assume that under both treatmentsthe conditional distribution for the effects is normal, and the potential correlation betweencosts and effects is modelled by assuming that μi is a linear function of γi (i = 1, 2). Weaklyinformative priors lead to the results in Table 18.1, Figures 18.3 and 18.4, and confirm thefindings that the conclusions in terms of cost-effectiveness are substantially sensitive to theassumptions made about the distributions of costs. Moreover, by showing that the gamma andthe normal distribution lead to very similar inferences, although the former has a much betterfit than the latter, these results also confirm the finding that better fit does not necessarily leadto more reliable estimates.

Table 18.1 eVALuate trial: deviances of different parametric cost models and posteriorsummaries of cost and effect differentials.

Model Deviance �̂c PCI0.95

P(�C > 0) �̂e PCI0.95

P(�e > 0) ICER

T1 T2

Lognormal 4402 9071 197 110 287 1.00 0.001 −0.001 0.004 0.83 197000Gamma 4527 9256 196 19 370 0.98 0.001 −0.001 0.004 0.82 196000Weibull 4658 9457 196 49 337 0.99 0.001 −0.001 0.004 0.82 196000Normal 4916 9873 193 19 369 0.98 0.001 −0.001 0.004 0.82 193000

PCI0.95 = 95% posterior credible interval.



−0.010 −0.005 0.000 0.005 0.010

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Figure 18.3 eVALuate trial: draws from the posterior distribution of (�e, �c) under differentcost models.

For these reasons, many authors (see, for instance, Nixon and Thompson, 2004; Thompsonand Nixon, 2005; Mihaylova et al., 2011) recommend that the sensitivity of conclusions tothe choice of the model should always be investigated, so that model uncertainty becomes acrucial aspect of analysing cost-effectiveness data (although it is not obvious over what rangeof models the sensitivity analysis should be performed). Other authors (see, for instance,Nixon and Thompson, 2004; Thompson and Nixon, 2005), argue that since costs for individualpatients must have some finite limits in practice, one could control the behaviour of the righttail by truncating a cost distribution at, say, twice the maximum observed cost (althoughideally such a limit should be based on external prior information – for instance consideringthe maximum cost a provider could theoretically spend on an individual). Others advocatethe use of formal model averaging techniques (Hoeting et al., 1999) over a range of plausiblemodels (see, for instance, Conigliani and Tancredi, 2009; Conigliani, 2010) or of modelsbased on mixtures of parametric distributions (see, for instance, Atienza et al., 2008), in order



0 10000 20000 30000 40000 50000

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Figure 18.4 eVALuate trial: CEAC under different cost models.

to obtain flexible models able to accommodate excess zeros, overdispersion and heavy tails.While both approaches are data based, the problem here is the influence of the tail of theassumed distributions beyond the range of the data. Other authors (see, for instance, Nixon andThompson, 2004; Bartkowiak and Sen, 1992; Willan, 2001) focus their attention on samplesizes, arguing that these must be large enough to enable sufficiently accurate modelling of thetail of the cost distribution. But by far the most common alternatives to parametric modellingin the context of cost-effectiveness analyses are nonparametric methods and transformationof the data, which will be considered in detail in the next sections.

18.3.2 Semiparametric modelling and nonparametricstatistical methods

The difficulty of producing realistic probabilistic models for the underlying population distri-bution of costs has made very attractive (both in the literature and in practice) the possibilityof considering nonparametric or semiparametric methods, which might be applied withoutspecifying such a population distribution.

Two simple nonparametric methods are widely used in the context of cost-effectivenessanalysis (O’Hagan and Stevens, 2002, 2003). The first one is based upon assuming that thesample mean follows a normal distribution. This assumption is justified by the Central Limittheorem (CLT), and the sample size needed for this approximation to be valid depends onthe degree of non-normality in the distribution (although in practice 30 observations areusually enough even with the high degree of skewness encountered in the analysis of costdata). The second nonparametric method that is often used is the bootstrap. This entailsrandomly drawing a large number of resamples of size n from the original sample (of sizen) with replacement, and calculating the value of the parameter of interest for each of the



corresponding samples. This yields an empirical estimate of the sampling distribution of theparameter of interest. Notice that the nonparametric bootstrap methods rest on asymptotictheory as do the normal-theory methods. Thus, the sample distribution tends to the populationdistribution as the sample size approaches the population size, and the bootstrap estimate ofthe sampling distribution of the sample mean approaches the true sampling distribution asthe number of bootstrap replicates approaches infinity.

With regard to the performance of the above nonparametric methods, it is interesting tonote the work of Briggs and Gray (1998) and Thompson and Barber (2000). In both papersthe authors advocate the use of normal-theory methods, but advise the reader to check thevalidity of such methods by using the bootstrap, in that if the bootstrap and the normal-theory methods produce substantially different inferences (possibly because the sample sizeis small or the skewness is extreme), then the bootstrap should be preferred. In fact, what thebootstrap estimate of the sampling distribution shows is the extent to which the sample size issufficiently large to justify the assumption of a normal sampling distribution for the parameterof interest. However, when applying these methods in cost-effectiveness data, the analyst willalways have to pay extra care. In fact, as noted for instance in Conigliani and Tancredi (2005)and Conigliani (2010), the right tail of a distribution of cost data is often so heavy that thevariance of the underlying population distribution may well be infinite. In such cases it isobviously not sensible to appeal to the CLT and to trust the results produced by statisticalmethods resulting from it. Moreover, as pointed out for instance in O’Hagan and Stevens(2003), even when (in sufficiently large samples) these simple nonparametric methods mightstrictly be valid, they may produce poor or misleading inferences. The reason is that, althoughin cost evaluations conceived to have an impact on medical policy the main interest is thetotal healthcare cost, so that it is inference on population mean costs that is informative,the sample mean is not necessarily a good basis for the inference. And this is because thesample mean is very sensitive to the kind of extreme sample values that are often encounteredin cost data; it is unbiased, but it is not necessarily the most efficient estimator. It follows thatnonparametric methods such as the normal-theory and the bootstrap, that are based on thesample mean, may be inappropriate in this setting, which warrants further examination of thenature of the underlying population distribution.

Another proposal by Conigliani and Tancredi (2005), which recognises the inability ofstandard parametric approaches to model the right tail of the cost distribution accurately,as well as the high influence of such tails in estimating the population mean, is to modelthe bulk of the data and the tails separately. In particular, Conigliani and Tancredi (2005)consider a distribution composed of a piecewise constant density up to an unknown endpointand a generalised Pareto distribution for the remaining tail data. This mixture model, whichis extremely flexible and able to fit data sets with very different shapes, allows the reportingof model-based inference for mean costs taking into account tail model uncertainty.

18.3.3 Transformation of the data

Another common approach that can be employed to handle cost data is to transform the datato a scale onto which it is reasonable to assume normality or another standard distributionalform. In particular, approaches to find appropriate transformations have considered mainlythe Box–Cox transformations (to achieve symmetry in error), with special attention devotedto log transformations (Mihaylova et al., 2011).



Note, however, that caution needs to be employed with such transformations, since meanvalues and confidence limits may be difficult to interpret on the transformed scales. Inparticular, the comparison of means on the transformed scale does not usually tell us muchabout the comparison of means on the original scale. Moreover, back-transformation onto theoriginal scale is not always straightforward (see, for instance, Thompson and Barber, 2000;Briggs and Gray, 1998). Consider, for instance, a log transformation of the data. Estimatingthe population mean on the log scale using the sample mean of the log-transformed data,and back-transforming the latter to the original scale by taking its exponential, yields thegeometric mean of the original data, not the arithmetic mean, which is what is needed forresource allocation decisions (Thompson and Barber, 2000).

In this respect, it is important to remember that the approaches for back transformationto the original scale are dictated by the nature of the error term on the transformed scale. Ifwe assume for instance that the error term has a normal distribution with parameters ν andτ 2 (i.e. that the original cost data follow a lognormal distribution), the mean on the originalscale can be written in terms of ν and τ 2 as γ = exp(ν + τ 2

2 ). Then, letting ν̂ and τ̂ 2 bethe sample mean and the sample variance of log-costs, an obvious estimator of γ would beγ̂ = exp(ν̂ + τ̂ 2

2 ). Note that, unlike the sample mean, γ̂ is a biased estimator of γ , but isnaturally less affected by extreme observations, so that one can expect this to perform betterin general. In this sense it is interesting to see the work of O’Hagan and Stevens (2003),where it is shown that γ̂ is both asymptotically unbiased and asymptotically more efficientthan the sample mean, and that in presence of outliers it leads to a much smaller estimate ofγ than the sample mean, so that it also appears to be much more accurate.

However the results in O’Hagan and Stevens (2003) are based upon assuming that cost dataare exactly lognormally distributed, which of course is not in general true: the distribution ofthe error term on the transformed scale is usually unknown, and reliance on the assumption ofnormality or homoscedasticity can lead to inconsistent estimates (see, for instance, Mihaylovaet al., 2011; Conigliani and Tancredi, 2009). In this sense, it is interesting to cite the work byRoyall and Tsou (2003), where, for the problem of the estimation of the mean, it is shownthat while assuming, for instance, a gamma distribution, the object of inference continuesto be the mean of the true generating process also when the model fails, if we assume thelognormal working model then what the likelihood represents evidence about when the modelfails is not E f (c) but the quantity exp(E f (log(c)) + 1

2 var f (log(c))). In such cases it mightbe preferable to employ the sample mean rather than γ̂ , since at least for sufficiently largesample sizes the CLT can be invoked.

18.4 Complex decision analysis models

Parmigiani (2002) states ‘Prediction models used in support of clinical and health policydecision-making often need to consider the course of a disease over an extended period oftime, and draw evidence from a broad knowledge base, including epidemiologic cohort andcase control studies, randomized clinical trials, expert opinions, and more.’ This view isconsistent with the notion that RCT evidence will always need to be integrated with other(possibly multiple) sources of evidence to estimate the parameters of interest in the decisionproblem. For instance, one may have several RCTs which assessed the effectiveness of aparticular medication in a given clinical context. It would be erroneous to single out one ofthese studies and ignore the information provided by the rest of the evidence base. Similarly,



RCT evidence may need to be extrapolated over a longer time period than the study followup. The study may have been conducted in a different jurisdiction (e.g. the USA), whereasthe analyst needs to produce cost-effectiveness estimates to inform adoption decisions in theUK, and address issues like generalisability of the study results from the USA to the UK.

18.4.1 Markov models

The economic analysis of healthcare interventions does often involve some degree of mod-elling. Markov-type models, for instance, are very popular in health economics, given theirability to represent the long-term evolution of chronic diseases. Norris (1997) and Grimmetand Stirzaker (2001) represent two standard references providing complete accounts of theprobabilistic aspects of these models, while Davison (2003) and Lindsay (2004) describe in-ference for them. An introduction to the practical aspects of Markov models from the healtheconomics point of view can be found in Briggs and Sculpher (1998).

The basic idea underlying the use of these models for assessing the cost-effectiveness ofa treatment is that each clinical state of the disease can be associated both with a measureof benefit, such as quality of life, and with a monetary measure of cost (e.g. medications,in-patient stay). Modelling the evolution of the disease allows one to predict the expectedlong-term costs and benefits of a medical treatment. In particular, discrete-time Markovmodels (also called Markov chains), with a discrete number of states, represent a natural andsimple framework for representing a patient’s health status over time. With these models thesubject’s history is divided into equally spaced intervals (days, months or years) with eachinterval representing a cycle in the model. During each cycle an individual can be in one of afinite set of health states. The essential feature of the Markov models is that during each cycle,given the entire past history of the subject, the transition probability from one health state toanother depends only on the present state (i.e. ‘Markovian assumption’). Loosely speaking,this means that in the simplest Markov model one assumes that the future is independent ofthe past given the present. This assumption can be relaxed using – where relevant – time-dependent transition probabilities (Hawkins, Sculpher and Epstein, 2005) or ‘tunnel’ states(Briggs and Sculpher, 1998).

In order to describe more formally the generic structure and the use of these models, weindicate with t = 1, . . . , T the cycle index, and we suppose that within each cycle t a patientassumes one of R states. In particular, during the first cycle each patient is assumed to be inone of the mutually exclusive R states, according to a probability distribution represented bythe row vector π1 = (π11, . . . , π1R). For each patient in state r during cycle t the probabilityof moving to state s in the following cycle will be indicated with pt+1,rs. These transitionprobabilities can be organised into the transition matrix Pt+1 comprising, for each row, theprobability vector (pt+1,r1, . . . , pt+1,rR). Hence, the marginal probability distribution πt forthe patient state during cycle t can be obtained through the recursive relationship

πt = πt−1Pt t = 2, . . . T

Note that we are assuming that the transition probabilities of each state given the previousone may vary over time; that is, they depend on the cycle index t. Under this generalassumption the Markov model is said to be non-homogeneous with respect to time. Matterssimplify considerably, especially for the theoretical results concerning the limit behaviour



of the Markov models, when the transition probabilities are the same in each cycle (i.e.homogeneous Markov model).

Long-term predictions of the parameters of interest (e.g. net benefit) can then be carriedout by attaching costs and benefits to each possible health state in the model. For the moment,suppose that as a measure of health benefit we are interested in the expected number of cyclesspent in a given health state. Let Tr be the random variable indicating the total number ofcycles spent by a patient in state r. Since Tr is equal to

∑Tt=1 Xtr where Xtr = 1 if the patient

is in the state r during the cycle t and Xtr = 0 otherwise, we have that

E(Tr) =T∑

t=1

πtr.

A more general benefit measure can be obtained by considering a row vector b comprisingthe benefits associated with spending one cycle in each state of the model. In fact, supposethat these benefits are discounted at the rate δb per cycle; then the total expected benefit foreach patient is given by

me =T∑

t=1

πtb′

(1 + δb)t−1.

Note that by taking δb = 0, bs = 0 for each s �= r, and br = 1, we have exactly theexpression given above for the quantity E(Tr). Moreover, if one of the states of the modelcorresponds to death, and one of the outputs we are interested in is patients’ life expectancy(in terms of cycles), this can be simply obtained by taking a vector b with all the elementsequal to one but the death state which assumes the value 0. Weighting patients’ survival bytheir health-related quality of life leads to the quality-adjusted life years (QALYs), the mostpopular measure of health benefit in use in health economic evaluation, which can easily beachieved by considering different values for elements of the vector b.

Similarly, suppose that the cost, at current prices, of spending a cycle in state r is Cr, r =1, . . . , R. Let C be the row vector (C1, . . . ,CR) and let C0 be a fixed entry cost. The totalexpected cost for each patient in the population is then given by

mc = C0 +T∑

t=1

πtC′

(1 + δc)t−1,

where δc is a discount rate for future costs.Note that in practical applications it is necessary to fix or to estimate the model parameters

π1 and Pt , and to propagate their uncertainty on the quantities me and mc.Data sets from RCTs, hospital registers, population mortality statistics and observational

studies may be used to inform the model. For example, to investigate the consequencesconcerning the choice of prosthesis in total hip replacement (THR), Fitzpatrick et al. (1998)(see also Spiegelhalter and Best, 2003) distinguish between patients suffering an operativedeath after a revision operation and patients dying for ‘other causes’. For the transitionprobabilities to the operative death, a mortality rate estimated by previous clinical trials wasassumed to increase linearly over time (expressed in years), while the national mortality



rates published by the UK Office for National Statistics were employed to fix the transitionprobabilities for the other causes of death (e.g. competing risks).

In some cases, individual longitudinal data from RCTs may be used directly to estimate thetransition probabilities to be used in the model (for one example see Henriksson et al., 2008).For homogeneous chains, maximum likelihood estimates for the transition probabilities canbe obtained by the row proportions of the one-cycle transition matrix N, whose genericelements nrs represent the number of times a transition from state r to state s occurred. Craigand Sendi (2002) discuss maximum likelihood estimation when the common observationinterval and the desired cycle length do not coincide or when the observation intervals varyacross the study.

In order to briefly illustrate a CEA based on Markov models, we consider the exampleillustrated in Simpson et al. (2008), where a transition model for HIV patients has beenproposed. For each model cycle, the cohort of patients is distributed across 12 ordered healthstates plus death according to their CD4 cell count and viral load. The cost of treating AIDSevents for each health status and a transition matrix, P, which has been estimated usinghospital data for highly treatment-experienced HIV-1 patients, are reported in Simpson et al.(2004) and Simpson et al. (2008). Compared to the analyses presented in these papers, whichconsider a non-homogeneous model involving multiple therapeutic failures, here we assumethat in both treatment groups patients do not change therapy. Patients in the control groupwill transit across health states according to the initial transition matrix P, while the transitionmatrix Q of the cohort of patients receiving the new treatment has elements given by

qij =⎧⎨⎩

(αpij)/ (

α∑i

l=1 pil + β∑

l>i pil

)j ≤ i

(βpij)/ (

α∑i

l=1 pil + β∑

l>i pil

)j > i

where α ≥ 1 and 0 ≤ β ≤ 1. For a cohort of 10 000 simulated patients in the control groupentering the study in the healthiest status, we obtained, without considering discounting rates,a mean lifetime survival of 13.2 years and a mean lifetime cost for the treatment of AIDSevents of US$12 347 253. Repeating the experiment for the new treatment group, lettingα = 1.05 and β = 1 in order to assume slightly higher probabilities of improving the healthstatus than in the control group, we obtained a mean lifetime survival of 14.2 years and a meanlifetime cost of US$12 978 433. The corresponding ICER was US$631 180 per additionallife year.

18.5 Further extensions

From a more general perspective, the problem of combining costs and benefits of a given in-tervention into a rational scheme for allocating resources can be formalised using a Bayesiandecision-theoretic approach, in which rational decision-making is achieved through the com-parison of expected utilities (Baio and Dawid, 2008). This approach has received increasingattention, in particular in the last couple of decades, when economic studies started to beaimed at specific decision-makers, and health systems all over the world have begun to useCEA as a formal input into decisions about treatments, interventions and programmes thatshould be funded from collective resources.

Specifically, let u(y, Tj) be a utility function representing the value of applying treatmentTj and obtaining the health economic outcome y = (e, c); a common form of utility is the



monetary net benefit u(y, Tj) = λe − c. Also define the expected utility Uj = E[u(Y, Tj)] asthe expected value of the utility function obtained averaging out the uncertainty about both thepopulation parameters and the individual variations. Then the most cost-effective treatmentis the one which is associated with the maximum expected utility, U∗ = max j Uj; that is, T2

dominates T1 if the expected incremental benefit EIB = U2 − U1 is positive (or if U2 > U1).Notice that when net benefit is used as the utility function, the expected incremental benefitreduces to EIB = λE(�e) − E(�c), that is, to the expectation over the distribution of thepopulation parameters of the INB bλ defined in Section 18.2.1.

18.5.1 Probabilistic sensitivity analysis and valueof information analysis

Note that since both individual variation and imperfect knowledge of the states of theworld are integrated out, the analysis of the expected incremental benefit outlined aboveprovides a rational scheme for allocating resources given the available evidence. A largepart of the health economics literature suggests that the impact of this uncertainty (and inparticular of the parameter uncertainty) in the final decision should be taken into accountthoroughly, by means of a process known as Probabilistic Sensitivity Analysis (PSA; Parmi-giani, 2002). PSA requires that all the inputs of a model (i.e. parameters for modellingRCT data or estimates plugged into an economic evaluation model) are considered as ran-dom variables, and explicitly analyses the uncertainty on these inputs by means of suitableindicators. In particular, the idea behind PSA is to compare the actual decision process,based on the analysis of EIB, to an ideal one, in which the uncertainty on the parame-ters is resolved, and it is typically conducted using a simulation approach (Doubilet et al.,1985). Recently, as pointed out for instance by Claxton et al. (2005), most of the mainagencies for health economic evaluation have updated their guidance for technology assess-ment, requiring the use of PSA as part of the cost-effectiveness models submitted for theirconsideration.

In order to briefly illustrate a PSA, consider again the Markov model for the HIV treat-ments described in Section 18.4.1, and in particular focus attention on the parameter α, thatcontrols the effectiveness of the new treatment with respect to the standard one. Here, insteadof fixing a value for α, we generate values for it from a uniform distribution in the interval[1, 1.1]. The results are shown in Figure 18.5, and point out that while the new treatment isalmost always cost-effective at the threshold level of US$750 000, with the lower thresholdof US$500 000 the probability of cost-effectiveness is only about 30%.

It is interesting to notice, as pointed out for instance in Claxton et al. (2005), that in healtheconomic evaluations it is common to summarise the results of PSA by means of the CEAC,which has the form introduced in Section 18.2.2 provided the net benefit is used as a utilityfunction. The CEAC, in fact, can be seen as the probability that a more precise knowledgeof the distribution of the population parameters in the decision problem at hand would notchange the optimal decision, and therefore can be used to represent the decision uncertaintysurrounding the cost-effectiveness of a treatment (Baio and Dawid, 2008; Fenwick, Claxtonand Schulpher, 2001).

Despite their wide use, some critical limitations of CEACs for presenting uncertaintyhave recently been brought up. Felli and Hazen (1999), for instance, point out that no explicitreference in the analysis of CEACs is made to the costs associated with a wrong decision;Koerkamp et al. (2007) show that very different distributions for the incremental net benefit



1.51.00.50.0

1 00

0 00

050

0 00

0

0

Δe

Δ clambda =750000

lambda =500000

Figure 18.5 Probabilistic sensitivity analysis for the Markov model for HIV progression.

can produce the same CEAC, so that their interpretation is not always straightforward. Forthese reasons, a purely decision-theoretic approach to PSA based on the analysis of a standardtool in Bayesian decision theory, the Expected Value of Distributional Information (EVDI),is becoming increasingly popular in health economic evaluations (Felli and Hazen, 1999;Claxton et al., 2001; Ades, Lu and Claxton, 2004; Baio and Dawid, 2008; Baio, 2010). Infact, by construction, the EVDI produces an indication of both how much we are likely to loseif we take the wrong decision, and how likely it is that we take it (Baio and Dawid, 2008).Moreover, regardless of the form of the utility function, the EVDI is defined as the expectationof the value of obtaining distributional information on the population parameters, that is, asthe maximum amount that we would be willing to pay to obtain such information. In thissense, it also provides a more general answer to the question of whether it is economicallyefficient to make a decision given the data at hand, or if instead the consequences of makingthe wrong decision are too expensive (e.g. irreversibility, sunk costs), so that collection offurther information is warranted. For further reading on this topic see the work by Claxtonet al. (2001) and Claxton, Cohen and Neumann (2005).

18.5.2 The role of Bayesian evidence synthesis

Another area that is closely related to the analysis of the EVDI, and that is becoming increas-ingly popular in health economic evaluation, is that known as quantitative evidence synthesis(Sutton and Abrams, 2001; Ades et al., 2006; Ades and Sutton, 2006) for decision-making.Since economic evaluations require a synthesis of all available evidence into probability dis-tributions for the input parameters, evidence synthesis (e.g. meta-analysis and its extensions)provide a natural approach to quantify the value (and relevant uncertainty) of such parame-ters. Although the role of evidence synthesis can also be explored in clinical epidemiology, itis in a decision-making context that its fundamental role emerges, since typically decisions



cannot be deferred and must be taken on the basis of all the information that is available.Specific advantages conferred by the Bayesian approach to this problem, which have beenrecognised for instance by the National Institute for Health and Clinical Excellence (NICE)in the UK, include the ability to include all pertinent information and full allowance forparameter uncertainty, as well as its coherent link to decision-making.

Specifically, there are different ways in which several items of information can be com-bined in the Bayesian setting for a CEA. In the simplest case, when each trial is assumed toprovide an estimate of the same parameter, the data can be combined using a fixed effectsmeta-analysis, and the common pooled effect can be estimated with a weighted average ofthe individual study effects, the weights being inversely proportional to the within-studyvariances. If instead the aim is to combine information on items from the same type ofstudy that are similar but not identical, then a random-effects meta-analysis can be employed.This represents a somehow intermediate position between assuming (as in the fixed-effectsmeta-analysis) that each observed effect size provides an estimate of the same underlyingquantity, say θ , and assuming at the other extreme that the true study-specific effects, say θi,are totally unrelated; individual studies may not be estimating the same underlying effect,but in random-effects meta-analysis the unobserved θi are assumed to be similar, so that it isreasonable to consider them to have been drawn from a common underlying distribution. Notethat this approach is particularly common especially in the Bayesian setting, due to the strongrelationship between random effects and hierarchical modelling; in particular, the Bayesiannotion of exchangeability provides a framework that includes most of the methods that havebeen proposed for evidence synthesis. Moreover, it is interesting to note that random-effectsmeta-analysis specifically allows for the existence of between-study heterogeneity (as wellas the within-study variability), although it does not provide an explanation for it. Howevercovariates can easily be included in the analysis, and regression-type models (often calledmeta-regression models) can be effectively employed to explain why study results may sys-tematically differ and to identify associations between patient characteristics and outcomethat might assist in individualising treatment regimes (Sutton and Abrams, 2001).

In recent years these simple forms of meta-analysis have been generalised and extendedto increasingly complex data structures. These include combination of information from dif-ferent study designs taking account of potential bias, mixed comparison, different follow-upperiods, combination of evidence on multiple or surrogate end points, and combination of in-formation on individual parameters with information on complex functions of the parameters.The aim of this section is to review some of the problems that arise in these multiparameterevidence syntheses, as well as some of the methods proposed for dealing with them. The needto check for consistency of evidence when using these methods is emphasised for instance inAdes and Sutton (2006).

Consider first the problem of combining studies that are heterogeneous with respect to thedesign and analysis. Much of initial work in this sense was carried out by Eddy, Hasselbladand Shachter (1992), who developed a general framework called the confidence profile method(CPM) that allows the explicit modelling of biases due to study design. Since this initial work,various methods for combining items of information in specific situations have been proposedin the literature. Muller et al. (1999), for instance, considered a hierarchical model to combinecase-control and prospective studies; Li and Begg (1994) presented a model to estimate anoverall treatment effect when some comparative studies are combined with non-comparativesingle-arm studies; Larose and Dey (1997) proposed a random-effects model to combinedouble-blind (closed) studies with open ones in which the investigator had knowledge of the



treatment regime; Prevost, Abrams and Jones (2000) considered the problem of combiningobservational evidence with results from randomised trials. Although in some of these models(for instance Li and Begg, 1994; Larose and Dey, 1997) different study types estimate thesame underlying mean effect, while in others (for instance Prevost, Abrams and Jones, 2000)different study types estimate different type-specific mean effects, one thing that all thesemethods have in common is that they produce a weighted average of the study-type-specificmean effects, with the weights determined by the model structure and the data, particularlythe heterogeneity within each study type (Ades and Sutton, 2006). An alternative to this is tospecify the weights explicitly; see for instance Spiegelhalter and Best (2003).

Another issue that often arises when combining several items of information for the pur-pose of CEA is that the evidence base is often fragmented, and the only way to considerall the evidence in a unified analytical framework is to conduct a mixed treatment compar-ison (Hasselblad, 1998). Indeed, in healthcare economic evaluation there is often absenceof head-to-head trial evidence on all the options being compared, and past analyses haveoften been based on a series of trials making different pair-wise comparisons among thetreatments of interest. Clearly these analyses cannot help to answer the real question thatis of interest for decision-makers; that is, ‘of all the possible courses of action which oneis the most cost-effective?’; instead, mixed treatment comparison methods address exactlythis problem. The basic idea is that in the absence of direct evidence comparing treatmentA and B, the relative treatment effect θAB can be estimated considering the information pro-vided by studies which compared A and B with a third treatment C; using the terminologyof CPM, θAC and θBC can be regarded as basic parameters, while θAB = θAC − θBC is effec-tively a functional parameter, that is, a function of the basic parameters. Ades and Sutton(2006) show how to generalise simple meta-analysis models to analyse mixed treatmentcomparisons, with the effect of reducing the parameter space to a set of basic parameters,assuming exchangeability of treatment effects holds over the entire ensemble of studies. Fora particularly remarkable mixed treatment comparison analysis see for instance Dominiciet al. (1999).

In Ades and Sutton (2006) a number of other issues related to evidence synthesis, aswell as references to methods proposed for dealing with them, are considered. One pointunderlined by the authors and that we find particularly interesting is that the increasing useof formal decision analysis by bodies such as NICE is, among other things, imposing anew rationale for the use of evidence synthesis methods for decision-making. In fact, as wepointed out in the previous section, the realisation that resources for healthcare are limitedand reliance on formal decision modelling are leading to substantial interest in expectedvalue of information theory. Evidence synthesis is highly relevant in this context, sincethe ability to incorporate additional sources of evidence can have a considerable effect onthe uncertainty surrounding the net benefit estimate and the EVDI; thus ignoring for instanceindirect evidence may result in greater uncertainty. In this sense, the decision context clarifiesthe tasks that evidence synthesis might be required to do. Ades and Sutton (2006) concludethat it is reasonable to expect that more attention will be paid in the future to the assessmentof uncertainty, and this will lead to a reconsideration of the way different types of evidenceare combined for decision-making. In particular, special attention needs to be devoted tothe problem of estimating the consistency of sources of evidence and, as there will often beinsufficient evidence to determine this confidently, to the choice of reasonably informativepriors on the degree of heterogeneity between different types of evidence.



18.6 Summary

Healthcare policy makers are required to use the limited resources as efficiently as possible,given the financial pressure under which many healthcare systems currently operate. Oneway to achieve allocative efficiency of healthcare resources is to identify which investmentstrategies (drugs, medical devices, diagnostics, etc.) provide value for money. Economicevaluation in healthcare (also known as cost-effectiveness analysis) generates the informationrequired to address the above question. This chapter provided an overview of the statisticalmethods used in applied healthcare economic evaluation studies and the challenges faced bythe analysts in dealing with the data, emphasising the emerging role that Bayesian statisticalmethods play in this context.

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