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On the estimation of cost-effectiveness ratios
ELSEVIER
Health Policy 31 (1995) 225-229
Second opinion
On the estimation of cost-effectiveness ratios Magnus Johannesson Cenfre for Health Economics, Stockholm School of Economics, Box 6501, S-i I3 83 Stockholm, Sweden
Received16August1994,accepted19August1994
In a recentpaper Birch and Gafni criticisedthe useof cost-effectiveness ratios in decisions about the allocation of health care resources.To support their claim that the useof costeffectivenessratioswill not leadto the maximizationof health effectsfor a given budgetthey usedan example.In this paper it is pointed out that the exampleusedcontainstwo basic errors. The first error is the failure to excludedominatedprogrammesin the estimationof incrementalcost-effectiveness ratios. The seconderror is the failure to distinguishbetweenindependentand mutually exclusiveprogrammes.It is concludedthat to get a more sober discussionabout the useand interpretation of cost-effectiveness analysisit is important that the techniqueis usedcorrectly. Keywords: Cost-effectiveness analysis;Cost-utility analysis;Economicevaluation; Decision
rules
1. Introduction Presently the most common approach to carry out economic evaluations of health care programmes is cost-effectiveness analysis. In cost-effectiveness analysis the costs are expressed in monetary units and the health effects are expressed in nonmonetary units such as life-years gained or quality-adjusted life-years (QALYs) gained [ 1,2]. The underlying rationale for cost-effectiveness analysis is the maximization of health effects for a given budget [3,4]. l
Corresponding author,Tel: +468 7369443; Fax: +468 302115.
0168-8510/95/sO9.50 0 1995ElsevierScience IrelandLtd. All rightsreserved SSDI 0168-8510(94)0676-6
226
M. Johannesson
/Health
Policy
31 (1995)
225-229
Birch and Gafni (BG) recently criticised the use of cost-effectiveness ratios in decisions about the allocation of health care resources [5]. To support their claim that the use of cost-effectiveness ratios will not lead to the maximization of health effects for a given budget they used an example. In this paper it is pointed out that the example used is erroneous. Firstly BG fail to exclude dominated alternatives in the estimation of incremental cost-effectiveness ratios. They furthermore fail to distinguish between independent and mutually exclusive programmes, when they estimate an incremental cost-effectiveness ratio between two independent programmes. Below we start by briefly describing the decision rules of cost-effectiveness analysis, and then show how BG fail to adhere to these decision rules in their example. The paper ends with some concluding remarks. 2. The estimation of cost-effectiveness
ratios
2. I. The decision rules of cost-effectiveness analysis It can first be useful to briefly review the decision rules of cost-effectiveness analysis [3,4]. Cost-effectiveness analysis is based on the maximization of an effectiveness unit subject to a budget constraint. As a decision rule a fixed budget can be specified and used to maximize the health effects, which will implicitly yield a price per effectiveness unit. Alternatively a price per effectiveness unit can be specified, which will implicitly yield a ‘budget’. To keep this short we focus on the fixed budget as decision rule below, since the estimation of cost-effectiveness ratios does not differ between cases. An often misunderstood distinction is that between incremental (marginal) costeffectiveness ratios and average cost-effectiveness ratios. To understand this distinction it is important to differentiate between choices among independent programmes (e.g., blood pressure control versus ulcer treatment) and choices among mutually exclusive programmes (e.g., different drugs to control high blood pressure). If a number of independent programmes compete for a limited budget, the optimal decision rule is to rank the programmes from the lowest to the highest costeffectiveness ratio (e.g., dollars per QALY gained) and to select programmes from this ranking list until the budget is exhausted [3]. Next, we can consider the decision context in which a number of mutually exclusive programmes are available. For example, there may be three different alternative drugs to treat high cholesterol levels. In this case, the programmes should first be ordered according to effectiveness and then the incremental cost-effectiveness ratio for each successively more effective program should be calculated (e.g., the incremental cost divided by the incremental gain in health effects). If any of these incremental ratios turns out to be less than the previous one in the sequence of increasingly more effective mutually exclusive programmes, then the less effective one is ruled out as dominated, and it should never be implemented irrespective of the size of the budget [4]. The incremental cost-effectiveness ratios should then be recalculated with the dominated alternative excluded, and this process continues until a number of programmes with increasing incremental cost-effectiveness ratios remains.
M. Johannesson
/Health
Policy
31 (1995)
225-229
221
This algorithm results in a sequence of programmes with increasing incremental cost-effectiveness ratios. The optimal decision rule is to move up the list of incremental ratios and successively replace a less effective programme with a more effective programme until the budget is exhausted. Note that the ordering of cost-effectiveness ratios of mutually exclusive programmes cannot be interpreted as a ranking list, since only one programme can be implemented, and the choice of which programme is implemented depends on the size of the budget. Finally, consider the most general decision context in which a number of clusters of mutually exclusive programmes are available. In this case the incremental costeffectiveness ratios should be calculated within each cluster of mutually exclusive programmes. This results in a sequence of programmes with increasing incremental cost-effectiveness ratios. The optimal decision rule is to move up the list of incremental ratios across all clusters of alternatives, and replace mutually exclusive programmes with successively more effective mutually exclusive programmes and add new independent programmes until the budget is exhausted. Note also that in this general case the sequence of incremental cost-effectiveness ratios cannot be interpreted as a ranking list, since only one programme within each cluster can be implemented. The maximization of life-years gained for a given budget is based on an assumption of constant returns to scale, which means that it is assumed that the scale of a programme can be reduced without changing the incremental cost-effectiveness ratio. If this assumption does not hold the decision rules does not necessarily lead to the maximization of the health effects for a given budget, and it will be necessary to use non-linear programming techniques to maximize the health effects for a given budget [6]. 2.2. The example by Birch and Gafni
BG use an example to illustrate that the use of cost-effectiveness ratios will not lead to the maximization of health effects for a given budget. In the example two different clinical programmes A, and B1 (stated to be e.g., paediatrics and geriatrics) that are being considered for implementation are compared with two existing programmes A,, and B,-, for the same patient groups. A0 and AI are two mutually exclusive programmes since they are alternative ways to treat the same patient group. Be and Bi are also two mutually exclusive programmes, since they are also alternative programmes to treat the same patient group. A,, and A, are independent programmes with respect to Bo and B1, however, since A and B are different patient groups. A,, costs $100 000 and yields 1 QALY and A, costs $110 000 and yields 2 QALYs. B0 costs $5000 and yields 1 QALY and B, costs $25 000 and yields 2 QALYs. BG first estimate the incremental cost-effectiveness ratios of Al versus A,, and of B, versus Ba. The incremental cost-effectiveness ratio of A, versus A0 is estimated to be $10 000 and the incremental cost-effectiveness ratio of B1 versus B0 is estimated to be $20 000. BG then concludes that these ratios are misleading, since it would be better to implement B1 than Al. The problem is, however, that BG has failed to exclude dominated alternatives. As stated above if the incremental costeffectiveness ratio turns out to be less than the previous one in the sequence of in-
228
hf. Johannesson
/Health
Policy
31 (1995)
225-229
creasingly more effective mutually exclusive programmes, then the less effective programme should be ruled out as dominated. Since the incremental cost-effectiveness ratio of Ai ($10 000) decreases compared to the incremental cost-effectiveness ratio of As ($100 000), As should be excluded from further consideration. The incremental cost-effectiveness ratio should then be recalculated for A, compared to no treatment, which yield an incremental cost-effectiveness ratio of $55 000 ($110 00012). The incremental cost-effectiveness ratio then becomes higher for A1 than B,. Following the decision rules above, the example used by BG is a case with two clusters of mutually exclusive programmes. Applying the decision rules correctly means that the incremental cost-effectiveness ratios are estimated within each cluster. After excluding dominated alternatives the remaining programmes are ordered in terms of their incremental cost-effectiveness ratios. In the example this will lead to the following order of incremental cost-effectiveness ratios (where As has been excluded as dominated): B, $5000 B, $20 000 A, $55 000 If the budget is known it can then be used to maximize the number of QALYs. For instance with a budget of $5000 only B0 will be implemented, with a budget of $25 000 only Bi will be implemented, and with a budget of $135 000 BI and Ai will both be implemented. The above part of the example has also been used in a previous article by BG [7], and it has been pointed out that it was incorrect [S]. This time BG, however, also introduces a second error. They estimate the incremental cost-effectiveness ratio of A, compared to B,, based on the argument that new programmes should be compared to the current programme with the lowest cost per QALY. In the example the current programme with the lowest cost per QALY is Bs, and the incremental costeffectiveness ratio of A, compared to B, is estimated to $105 000 by BG. Such an estimation is, however, meaningless since an incremental cost-effectiveness ratio should not be estimated between two independent programmes such as Ai and Bs. The reason given by BG for doing this estimation is that they argued that it was claimed in a paper by Johannesson and Weinstein [8] to be the appropriate procedure. No such recommendation was given in the paper by Johannesson and Weinstein [8], nor elsewhere in the literature either as far as we know. The incremental cost-effectiveness ratio between two independent programmes should never be estimated and has no meaningful interpretation. 3. Conchdlng
remarks
BG make two fundamental errors in their paper about cost-effectiveness ratios. They fail to exclude dominated alternatives and they fail to distinguish between choices concerning mutually exclusive and independent programmes. To get a more sober discussion about the use and interpretation of cost-effectiveness analysis it is important that the technique is used correctly.
M. Johannesson
/Health
kdicy
31 (1995)
225-229
229
Acknowledgements This work was financially Pharmacies.
supported by the National
Corporation
of Swedish
References Weinstein, M.C. and Stason, W.B., Hypertension: A Policy Perspective, Harvard University Press, Cambridge+ MA, 1976. VI Boyle, M.H., Torrance, G.W., Sinclair, J.C. and Horwood, S.P., Economic evaluation of neonatal intensive care of very-low-birth-weight infants, New England Journal of Medicine, 308 (1983) 1330-1337. I31 Weinstein, MC. and Zeckhauxr, R., Critical ratios and efficient allocation, Journal of Public Economics, 2 (1973) 147-157. 141 Weinstein, M.C., Principles of cost-effective resource allocation in health care organisations, International Journal of Technology Assessment in Health Care, 6 (1990) 93-103. [I Birch, S. and Gafni, A., Cost-effectiveness ratios: in a league of their own, Health Policy, 28 (1994) 133-141. M Winston, W.L., Operations Research: Applications and Algorithms, Second Edition, PWS-Kent Publishing Company, Boston, 1991. 171 Birch, S. and Gafni, A., Cost-effectiveness/utility analyses: do current decision rules lead us to where we want to be?, Journal of Health Economics, 1I (1992) 279-296. 181 Johannesson, M. and Weinstein, MC., On the decision rules of cost-effectivenessanalysis, Journal of Health Economics, 12 (1993) 459-467.
Ul
Health Policy 31 (1995) 225-229
Second opinion
On the estimation of cost-effectiveness ratios Magnus Johannesson Cenfre for Health Economics, Stockholm School of Economics, Box 6501, S-i I3 83 Stockholm, Sweden
Received16August1994,accepted19August1994
In a recentpaper Birch and Gafni criticisedthe useof cost-effectiveness ratios in decisions about the allocation of health care resources.To support their claim that the useof costeffectivenessratioswill not leadto the maximizationof health effectsfor a given budgetthey usedan example.In this paper it is pointed out that the exampleusedcontainstwo basic errors. The first error is the failure to excludedominatedprogrammesin the estimationof incrementalcost-effectiveness ratios. The seconderror is the failure to distinguishbetweenindependentand mutually exclusiveprogrammes.It is concludedthat to get a more sober discussionabout the useand interpretation of cost-effectiveness analysisit is important that the techniqueis usedcorrectly. Keywords: Cost-effectiveness analysis;Cost-utility analysis;Economicevaluation; Decision
rules
1. Introduction Presently the most common approach to carry out economic evaluations of health care programmes is cost-effectiveness analysis. In cost-effectiveness analysis the costs are expressed in monetary units and the health effects are expressed in nonmonetary units such as life-years gained or quality-adjusted life-years (QALYs) gained [ 1,2]. The underlying rationale for cost-effectiveness analysis is the maximization of health effects for a given budget [3,4]. l
Corresponding author,Tel: +468 7369443; Fax: +468 302115.
0168-8510/95/sO9.50 0 1995ElsevierScience IrelandLtd. All rightsreserved SSDI 0168-8510(94)0676-6
226
M. Johannesson
/Health
Policy
31 (1995)
225-229
Birch and Gafni (BG) recently criticised the use of cost-effectiveness ratios in decisions about the allocation of health care resources [5]. To support their claim that the use of cost-effectiveness ratios will not lead to the maximization of health effects for a given budget they used an example. In this paper it is pointed out that the example used is erroneous. Firstly BG fail to exclude dominated alternatives in the estimation of incremental cost-effectiveness ratios. They furthermore fail to distinguish between independent and mutually exclusive programmes, when they estimate an incremental cost-effectiveness ratio between two independent programmes. Below we start by briefly describing the decision rules of cost-effectiveness analysis, and then show how BG fail to adhere to these decision rules in their example. The paper ends with some concluding remarks. 2. The estimation of cost-effectiveness
ratios
2. I. The decision rules of cost-effectiveness analysis It can first be useful to briefly review the decision rules of cost-effectiveness analysis [3,4]. Cost-effectiveness analysis is based on the maximization of an effectiveness unit subject to a budget constraint. As a decision rule a fixed budget can be specified and used to maximize the health effects, which will implicitly yield a price per effectiveness unit. Alternatively a price per effectiveness unit can be specified, which will implicitly yield a ‘budget’. To keep this short we focus on the fixed budget as decision rule below, since the estimation of cost-effectiveness ratios does not differ between cases. An often misunderstood distinction is that between incremental (marginal) costeffectiveness ratios and average cost-effectiveness ratios. To understand this distinction it is important to differentiate between choices among independent programmes (e.g., blood pressure control versus ulcer treatment) and choices among mutually exclusive programmes (e.g., different drugs to control high blood pressure). If a number of independent programmes compete for a limited budget, the optimal decision rule is to rank the programmes from the lowest to the highest costeffectiveness ratio (e.g., dollars per QALY gained) and to select programmes from this ranking list until the budget is exhausted [3]. Next, we can consider the decision context in which a number of mutually exclusive programmes are available. For example, there may be three different alternative drugs to treat high cholesterol levels. In this case, the programmes should first be ordered according to effectiveness and then the incremental cost-effectiveness ratio for each successively more effective program should be calculated (e.g., the incremental cost divided by the incremental gain in health effects). If any of these incremental ratios turns out to be less than the previous one in the sequence of increasingly more effective mutually exclusive programmes, then the less effective one is ruled out as dominated, and it should never be implemented irrespective of the size of the budget [4]. The incremental cost-effectiveness ratios should then be recalculated with the dominated alternative excluded, and this process continues until a number of programmes with increasing incremental cost-effectiveness ratios remains.
M. Johannesson
/Health
Policy
31 (1995)
225-229
221
This algorithm results in a sequence of programmes with increasing incremental cost-effectiveness ratios. The optimal decision rule is to move up the list of incremental ratios and successively replace a less effective programme with a more effective programme until the budget is exhausted. Note that the ordering of cost-effectiveness ratios of mutually exclusive programmes cannot be interpreted as a ranking list, since only one programme can be implemented, and the choice of which programme is implemented depends on the size of the budget. Finally, consider the most general decision context in which a number of clusters of mutually exclusive programmes are available. In this case the incremental costeffectiveness ratios should be calculated within each cluster of mutually exclusive programmes. This results in a sequence of programmes with increasing incremental cost-effectiveness ratios. The optimal decision rule is to move up the list of incremental ratios across all clusters of alternatives, and replace mutually exclusive programmes with successively more effective mutually exclusive programmes and add new independent programmes until the budget is exhausted. Note also that in this general case the sequence of incremental cost-effectiveness ratios cannot be interpreted as a ranking list, since only one programme within each cluster can be implemented. The maximization of life-years gained for a given budget is based on an assumption of constant returns to scale, which means that it is assumed that the scale of a programme can be reduced without changing the incremental cost-effectiveness ratio. If this assumption does not hold the decision rules does not necessarily lead to the maximization of the health effects for a given budget, and it will be necessary to use non-linear programming techniques to maximize the health effects for a given budget [6]. 2.2. The example by Birch and Gafni
BG use an example to illustrate that the use of cost-effectiveness ratios will not lead to the maximization of health effects for a given budget. In the example two different clinical programmes A, and B1 (stated to be e.g., paediatrics and geriatrics) that are being considered for implementation are compared with two existing programmes A,, and B,-, for the same patient groups. A0 and AI are two mutually exclusive programmes since they are alternative ways to treat the same patient group. Be and Bi are also two mutually exclusive programmes, since they are also alternative programmes to treat the same patient group. A,, and A, are independent programmes with respect to Bo and B1, however, since A and B are different patient groups. A,, costs $100 000 and yields 1 QALY and A, costs $110 000 and yields 2 QALYs. B0 costs $5000 and yields 1 QALY and B, costs $25 000 and yields 2 QALYs. BG first estimate the incremental cost-effectiveness ratios of Al versus A,, and of B, versus Ba. The incremental cost-effectiveness ratio of A, versus A0 is estimated to be $10 000 and the incremental cost-effectiveness ratio of B1 versus B0 is estimated to be $20 000. BG then concludes that these ratios are misleading, since it would be better to implement B1 than Al. The problem is, however, that BG has failed to exclude dominated alternatives. As stated above if the incremental costeffectiveness ratio turns out to be less than the previous one in the sequence of in-
228
hf. Johannesson
/Health
Policy
31 (1995)
225-229
creasingly more effective mutually exclusive programmes, then the less effective programme should be ruled out as dominated. Since the incremental cost-effectiveness ratio of Ai ($10 000) decreases compared to the incremental cost-effectiveness ratio of As ($100 000), As should be excluded from further consideration. The incremental cost-effectiveness ratio should then be recalculated for A, compared to no treatment, which yield an incremental cost-effectiveness ratio of $55 000 ($110 00012). The incremental cost-effectiveness ratio then becomes higher for A1 than B,. Following the decision rules above, the example used by BG is a case with two clusters of mutually exclusive programmes. Applying the decision rules correctly means that the incremental cost-effectiveness ratios are estimated within each cluster. After excluding dominated alternatives the remaining programmes are ordered in terms of their incremental cost-effectiveness ratios. In the example this will lead to the following order of incremental cost-effectiveness ratios (where As has been excluded as dominated): B, $5000 B, $20 000 A, $55 000 If the budget is known it can then be used to maximize the number of QALYs. For instance with a budget of $5000 only B0 will be implemented, with a budget of $25 000 only Bi will be implemented, and with a budget of $135 000 BI and Ai will both be implemented. The above part of the example has also been used in a previous article by BG [7], and it has been pointed out that it was incorrect [S]. This time BG, however, also introduces a second error. They estimate the incremental cost-effectiveness ratio of A, compared to B,, based on the argument that new programmes should be compared to the current programme with the lowest cost per QALY. In the example the current programme with the lowest cost per QALY is Bs, and the incremental costeffectiveness ratio of A, compared to B, is estimated to $105 000 by BG. Such an estimation is, however, meaningless since an incremental cost-effectiveness ratio should not be estimated between two independent programmes such as Ai and Bs. The reason given by BG for doing this estimation is that they argued that it was claimed in a paper by Johannesson and Weinstein [8] to be the appropriate procedure. No such recommendation was given in the paper by Johannesson and Weinstein [8], nor elsewhere in the literature either as far as we know. The incremental cost-effectiveness ratio between two independent programmes should never be estimated and has no meaningful interpretation. 3. Conchdlng
remarks
BG make two fundamental errors in their paper about cost-effectiveness ratios. They fail to exclude dominated alternatives and they fail to distinguish between choices concerning mutually exclusive and independent programmes. To get a more sober discussion about the use and interpretation of cost-effectiveness analysis it is important that the technique is used correctly.
M. Johannesson
/Health
kdicy
31 (1995)
225-229
229
Acknowledgements This work was financially Pharmacies.
supported by the National
Corporation
of Swedish
References Weinstein, M.C. and Stason, W.B., Hypertension: A Policy Perspective, Harvard University Press, Cambridge+ MA, 1976. VI Boyle, M.H., Torrance, G.W., Sinclair, J.C. and Horwood, S.P., Economic evaluation of neonatal intensive care of very-low-birth-weight infants, New England Journal of Medicine, 308 (1983) 1330-1337. I31 Weinstein, MC. and Zeckhauxr, R., Critical ratios and efficient allocation, Journal of Public Economics, 2 (1973) 147-157. 141 Weinstein, M.C., Principles of cost-effective resource allocation in health care organisations, International Journal of Technology Assessment in Health Care, 6 (1990) 93-103. [I Birch, S. and Gafni, A., Cost-effectiveness ratios: in a league of their own, Health Policy, 28 (1994) 133-141. M Winston, W.L., Operations Research: Applications and Algorithms, Second Edition, PWS-Kent Publishing Company, Boston, 1991. 171 Birch, S. and Gafni, A., Cost-effectiveness/utility analyses: do current decision rules lead us to where we want to be?, Journal of Health Economics, 1I (1992) 279-296. 181 Johannesson, M. and Weinstein, MC., On the decision rules of cost-effectivenessanalysis, Journal of Health Economics, 12 (1993) 459-467.
Ul