Abstract
Background
Acceptability curves have been proposed for quantifying the probability that a treatment under investigation in a clinical trial is costeffective. Various definitions and estimation methods have been proposed. Loosely speaking, all the definitions, Bayesian or otherwise, relate to the probability that the treatment under consideration is costeffective as a function of the value placed on a unit of effectiveness. These definitions are, in fact, expressions of the certainty with which the current evidence would lead us to believe that the treatment under consideration is costeffective, and are dependent on the amount of evidence (i.e. sample size).
Methods
An alternative for quantifying the probability that the treatment under consideration is costeffective, which is independent of sample size, is proposed.
Results
Nonparametric methods are given for point and interval estimation. In addition, these methods provide a nonparametric estimator and confidence interval for the incremental costeffectiveness ratio. An example is provided.
Conclusions
The proposed parameter for quantifying the probability that a new therapy is costeffective is superior to the acceptability curve because it is not sample size dependent and because it can be interpreted as the proportion of patients who would benefit if given the new therapy. Nonparametric methods are used to estimate the parameter and its variance, providing the appropriate confidence intervals and test of hypothesis.
Introduction
In reporting costeffectiveness analyses alongside clinical trials, authors [14] have used various definitions, estimation methods and interpretations for acceptability curves. Acceptability curves provide an excellent means of quantifying the stochastic uncertainty of the estimated incremental costeffectiveness ratio (ICER) in relation to a particular value ascribed to a unit of effectiveness. It is the certainty, expressed as a probability, that the current evidence would lead us to believe that some new therapy is costeffective, insofar as the ICER is less than the value ascribed to a unit of effectiveness. In addition, acceptability curves provide an estimator for the ICER and its confidence limits. However, acceptability curves are often interpreted and expressed as the probability that the new therapy is costeffective. It is argued below that this is subject to misinterpretation, and an alternative definition for a parameter representing the probability that the new therapy is costeffective is introduced. Data from a clinical trial can be used to make statistical inference about this parameter. Furthermore, the inference provides a nonparametric estimator for the ICER and its confidence interval.
In a twoarm randomized control trial let e_{ji} and c_{ji} be the respective measures of effectiveness and cost for patient i on therapy j, where j = T (treatment), S (standard); i = 1, 2, . . .n_{j} ; and n_{j} is the number of patients randomized to therapy j. Typically, e_{ji} is the patient's survival time (perhaps qualityadjusted) from randomization to death or to the end of the period of interest. Let . Define _{c} similarly. Let E(_{e}) = Δ_{e} and E(_{c}) = Δ_{c}, where E is the expectation function. Thus, the incremental costeffectiveness ratio (ICER) is Δ_{c}/Δ_{e}, and is estimated by _{c} / _{e}. In addition, the incremental net benefit [59] (INB) is Δ_{e}λΔ_{c}, and is estimated by _{e}λ  _{c}, where λ is the value given to a unit of effectiveness. Typically the INB is expressed as a function of λ, allowing readers to provide the value they consider most relevant.
van Hout et al. [1] define the acceptability curve as "the probability that the [ICER] is under a certain acceptable limit", say λ. The acceptability curve, then, is a function of λ. If one assumes that the authors are referring to the true ICER ratio, the definition is Bayesian. In the same paper they define the acceptability curve in algebraic terms as , where f is the joint probability density function for the random vector (_{e},_{c})'. Here the definition is not Bayesian, since it is the probability that, in repeated sampling, the random variable _{e}λ  _{c} (i.e., the observed net benefit) is greater than 0. In an illustration, the authors substitute the sample estimates for the model parameters in f to yield an empirical density function, call it , and refer to the acceptability curve as the probability that the ICER is acceptable. Briggs and Fenn [2] refer to the acceptability curve as "the probability an intervention is costeffective in relation to different values of" λ. For estimation they propose using the integration of , as above, or determining the proportion of bootstrap resamples in which _{e}λ  _{c} is greater than 0.
Briggs[3] provides a purely Bayesian approach by defining the acceptability curve as the probability that Δ_{e}λ  Δ_{c} is greater than 0. In an illustration the author interprets the acceptability curve as "probability of costeffective". Assuming f is the density function for a bivariate normal random vector, and using an uninformative prior, the acceptability curve is given by A(λ)=g(x)dx, where g is the probability density function for a normal random variable with mean _{e}λ  _{c} and variance , where and are estimates of the variance of e_{ji} and c_{ji}, respectively, and is an estimate of the correlation between e_{ji} and c_{ji} . This is exactly the same curve as determined by the integration of , given above, and, due to the symmetry, is equal to 1 minus the pvalue of the test of the hypothesis Δ_{e}λ  Δ_{c} < 0. In reporting the results of a costeffectiveness analysis, Raikou et al. [4] interpret the acceptability curve as the "probability that intervention is cost effective".
By rewriting Pr(Δ_{e}λ  Δ_{c} > 0) as Pr(Δ_{c}/Δ_{e} < λ), assuming Δ_{e} > 0, the acceptability curve can be interpreted as the posterior distribution for the ICER. Defining A(λ_{γ}) = γ, the estimate for the ICER and its (1α/2)100% Baysian limits are given by λ_{0.5}, λ_{α/2} and λ_{1α/2}, respectively.
The interpretation that the acceptability curve is the probability that the intervention is costeffective is not entirely accurate and could easily be misunderstood by policy makers. Consider the situation in which the observed INB for treatment is very small, but due to a very large sample size the acceptability curve at the value of λ of interest is 0.99. Attaching the label "the probability that the intervention is costeffective" to this quantity could mislead policy makers into thinking that treatment is highly beneficial compared to standard. What, in fact, is high is our confidence that the INB, however small, is not zero. A more accurate interpretation of the acceptability curve is that it is a measure of the certainty with which the current evidence would lead us to believe that treatment is costeffective, i.e., Δ_{e}λ  Δ_{c} > 0. For a Bayesian, this is Pr(Δ_{e}λ  Δ_{c}) > 0, and for a frequentist, it is, assuming symmetry, 1 minus the pvalue for the test of the hypothesis Δ_{e}λ  Δ_{c} < 0. This is not just the traditional confusion between statistical and clinical significance. In significance testing as the sample size increases, the variance of the estimator decreases, but the magnitude of the parameter being estimated stays the same. However, as sample size increase the magnitude of the acceptability curve for a given λ increases.
In the next section we provide a more accurate definition for the probability that treatment is costeffective. The definition contains no element of certainty, and is the probability of the "next" patient realizing a larger net benefit if he or she is given treatment rather than standard. Using data from a clinical trial, nonparametric methods can be used to estimate this probability, and uncertainty
Methods
The probability that treatment is costeffective
Let b_{ji}(λ) = e_{ji}λ  c_{ji}. The quantity b_{ji}(λ) is the net benefit, expressed in money, realized by patient i on therapy j. An alternative to the acceptability curve for quantifying the probability that treatment is costeffective is defined as: θ(λ) ≡ the probability, for a given λ, that a patient will receive a larger net benefit with treatment rather than standard. Its definition is not Bayesian because it is a probability statement about random variables, not population parameters. Nonetheless, it relates directly to the notion of the probability of treatment being costeffective. A policy maker can genuinely interpret θ(λ) as the probability of the "next" patient realizing a larger net benefit if he or she is given treatment rather than standard. Such a direct interpretation is not provided by the acceptability curve. The acceptability curve is the probability that the average net benefit of a group of patients of the same size as in the clinical trial will be greater is they receive treatment rather than standard.
To estimate θ(λ) we borrow methodology from receiver operating characteristic curves [10]. An estimate of θ(λ) is given by:
If there are no ties, (λ) is the proportion of all the possible comparisons of a patient on treatment with a patient on standard in which the former was observed to have a larger net benefit. The estimate of the variance of (λ), denoted by , is given by:
The 100(1  α)% confidence interval, defined as (λ)±s_{θ}(λ)z_{1α/2}, can be used to express the uncertainty regarding (θ), where Z_{1α/2} is the 100(1  α/2)^{th} percentile of the standard normal distribution. We have made the assumption that in large samples (λ) will be normally distributed.The value λ_{0.5}, defined at θ (λ_{0.5}) = 0.5, provides a nonparametric estimator of the ICER, since it is that value of λ for which treatment and standard are equally costeffective in the sense that Pr [b_{Tk}(λ_{0.5}) > b_{Si}(λ_{0.5})] = 0.5. The quantities λ_{L} and λ_{U}, defined as (λ_{L})+ s_{θ}(λ_{L})Z_{1α/2} = 0.5 and (λ_{U}) s_{θ}(λ_{U})Z_{1α/2} = 0.5, respectively, are the corresponding lower and upper nonparametric confidence intervals for the ICER.
The hypothesis H_{0}: θ (λ) = 0.5 versus H_{1}: θ (λ) > 0.5 can be tested at the level α by rejecting H_{0} if . Rejecting H_{0} leads to the conclusion that the data provide evidence that treatment is costeffective.
Suppose there is an interest in comparing θ (λ) between patient subgroups, say between males and females. To achieve this, let _{M}(λ) and _{F}(λ) be the estimator of θ (λ) for males and females, respectively, with corresponding estimated variances, and . Then the hypothesis that θ (λ) is the same for males and females can be rejected, at the αlevel, 2sided, if
There is an important distinction to be made between A(λ) and (λ). As sample size increases, A(λ) approaches 1 if Δ_{e} λ  Δ_{c} > 0, reflecting the certainty that treatment is costeffective. If Δ_{e}λ  Δ_{c} ≤ 0, A(λ) approaches 0, reflecting the certainty that treatment is not costeffective. The quantity (λ), being independent of sample size, approaches θ(λ) regardless of Δ_{e}λ  Δ_{c}. The certainty with which θ(λ) is estimated is reflected in s_{θ}(λ) which is a decreasing function of the sample size.
Example
In a trial of symptomatic hormone resistant prostate cancer [11,12], 161 patients were randomized between prednisone alone (S) and prednisone plus mitoxantrone (T). Although there was no statistically significant difference in survival, there was better palliation with T. Cost data, including hospital admissions, outpatient visits, investigations, therapies and palliative care, were collected retrospectively on the 114 patients from the three largest centres. Survival was qualityadjusted using the EORTC quality of life questionnaire QLQC30. All patients were followed until death. The sample means and the sample variancecovariance information can be found in Table 1. Cost is given in Canadian dollars (CAD$) and effectiveness in qualityadjusted lifeweeks (QALW).
Table 1. Sample sizes and parameter estimates for prostate example
A plot of (λ), complete with 90% confidence intervals can be found in Figure 1. In this example all three plots have positive slope over the range of λ shown, although this would not be true for all examples. For example, by definition θ (0) = Pr(c_{Tk}  c_{Sj} < 0) and θ (λ) approaches Pr(e_{Tk}  e_{Sj} > 0) as λ becomes arbitrarily large, for any k and j. Therefore, any example in which Pr(c_{Tk}  c_{Sj}< 0) > Pr(e_{Tk}  e_{Sj} > 0) will have negative slope for some interval of positive λ. In the prostate example, since (λ) crosses the 0.5 horizontal at λ = 50, for any value greater than 50, the estimate of (λ) is greater than 0.5. The value 50 is a nonparametric estimate of the ICER, since for that value of λ, treatment and standard are equally costeffective, in that the probability that a patient on treatment has the same net benefit as a patient on standard is 0.5. This compares to the parametric estimate of 134.
Figure 1. Estimate and confidence limits for θ as a function of λ.
Since the lower bound crosses the 0.5 horizontal at 334, for any value greater than 334 the hypothesis θ(λ) < 0.5 can be rejected at the 5% level of significance. Thus, a nonparametric upper bound of the ICER is 334. This compares to 378 using Fieller's theorem [13,14]. For this example a health policy maker can interpret the results as follows: for any positive λ, the estimated probability that treatment is costeffective is greater than 50%; and for any λ greater than 334 per QALW, the probability that treatment is costeffective is statistically significantly greater than 50%.
Discussion
As an alternative to the acceptability curve, the quantity θ(λ) is proposed as a definition for the probability that treatment is costeffective. One advantage is that it is not sample size dependent, i.e. it is a population parameter. Another is that it has an appropriate interpretation, namely, it is the proportion of patients that realize a larger net benefit if given treatment rather than standard. The acceptability curve does not provide this, although the language often used regarding it, implies that it does. The use of θ(λ) should be helpful to policy makers, since it does not confuse the magnitude of the benefit with the certainty of its estimate.
Analysis regarding the quantity θ(λ) is not proposed as an alternative to traditional costeffectiveness analysis for allocating health care resources. When allocating a fixed amount of resources to one of two new treatments, the proportion of patients receiving an increase in net benefit would be maximized by choosing the treatment with the larger θ(λ) to Δ_{c} ratio. However, this would not maximize net benefit, since the ratio may be larger only because the betweenpatient variability in cost and effectiveness is smaller, resulting in a larger θ(λ).
Nonparametric methods can be used to estimate θ(λ) and its variance. This provides the appropriate confidence intervals and test of hypothesis. In addition, nonparametric estimates of the ICER and its confidence intervals can be determined. This is of particular importance in the presence of highly skewed cost data.
Competing interests
None declared.
Acknowledgements
The author wishes to acknowledge the reviewers whose comments led to a much improved manuscript and to thank Gary Foster for help with the figure.
References

van Hout BA, Al MJ, Gordon GS, Rutten EFH: Cost, effects and C/Eratios alongside a clinical trial.
Health Economics 1994, 3:309319. PubMed Abstract

Brigg A, Fenn P: Confidence intervals or surfaces? Uncertainty on the costeffectiveness plane.
Health Economics 1998, 7:723740. PubMed Abstract  Publisher Full Text

Briggs AH: A Bayesian approach to stochastic costeffectiveness analysis.
Health Economics 1999, 8:257261. PubMed Abstract  Publisher Full Text

Raikou M, Gray , Briggs , Stevens A, Cull C, McGuire A, Fenn P, Stratton I, Holman R, Turner R: Cost effectiveness analysis of improved blood pressure control in hypertensive patients with type 2 diabetes: UKPDS 40.
British Medical Journal 1998, 317:720726. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Phelps CE, Mushlin AI: On the (near) equivalence of costeffectiveness and costbenefit analysis.
International Journal of Technology Assessment in Health Care 1991, 7:1221. PubMed Abstract

Ament A, Baltussen R: The interpretation of results of economic evaluation: explicating the value of health.
Health Economics 1997, 6:625635. PubMed Abstract  Publisher Full Text

Stinnett AA, Mallahy J: Net health benefits: a new framework for the analysis of uncertainty in costeffectiveness analsyis.

Tambour M, Zethraeus , Johannesson M: A note on confidence intervals in costeffectiveness analysis.
International Journal of Technology Assessment 1998, 14:467471.

Willan AR, Lin DY: Incremental net benefit in randomized clinical trials.
Statistics in Medicine 2001, 20:15631574. PubMed Abstract  Publisher Full Text

DeLong ER, DeLong DM, ClakePearson DL: Comparing the area under two or more correlated receiver operating characteristic curves: a nonparametric approach.
Biometrics 1988, 44:837845. PubMed Abstract

Tannock IF, Osoba D, Stockler MR, Ernst DS, Neville AJ, Moore MJ, Armitage GR, Wilson JJ, Venner PM, Coppin CM, Murphy KC: Chemotherapy with mitoxantrone plus prednisone or prednisone alone for symptomatic hormoneresistant prostate cancer: a Canadian trial with palliative endpoints.
Journal of Clinical Oncology 1996, 14:17561764. PubMed Abstract

Bloomfield DJ, Krahn MD, Neogi T, Panzarella T, Warde P, Willan AR, Ernst S, Moore MJ, Neville A, Tannock IF: Economic evaluation of chemotherapy with mitoxantrone plus prednisone for symptomatic hormoneresistant prostate cancer: based on a Canadian trial with palliative endpoints.
Journal of Clinical Oncology 1998, 16:22722279. PubMed Abstract

Willan AR, O'Brien BJ: Confidence intervals for costeffectiveness ratios: An application of Fieller's theorem.
Health Economics 1996, 5:297305. PubMed Abstract  Publisher Full Text

Chaudhary MA, Stearns SC: Confidence intervals for costeffectiveness ratios: an example from a randomized trial.
Statistics in Medicine 1996, 15:14471458. PubMed Abstract  Publisher Full Text
Prepublication history
The prepublication history for this paper can be accessed here: