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DEBATE THE USEFULNESS OF AVERAGE COST-EFFECTIVE RATIOS EUGENE M. LASKA 1,2 *, MORRIS MEISNER 2,3 AND CAROLE SIEGEL 2,4 1 Statistical Sciences and Epidemiology Division, The Nathan S. Kline Institute for Psychiatric Research, USA 2 New York University School of Medicine, USA 3 Statistical Sciences Laboratory, The Nathan S. Kline Institute for Psychiatric Research, USA 4 Health Services and Epidemiology Laboratory, The Nathan S. Kline Institute for Psychiatric Research, USA SUMMARY We demonstrate that average cost-effectiveness ratios (CERs) play an important role in the evaluation of the cost-effectiveness of treatments. Criticisms of the usefulness of CERs derive mostly from the context of resource allocation under a constrained budget in which some decisions are based on incremental CERs. However, we show that in many cases, these decision rules are equivalent to decision rules on CERs. This follows for mutually exclusive treatments first, because a treatment is eliminated by extended dominance if and only if there is a mixed treatment with a smaller CER, where the mixing parameter lies in a certain interval. Second, after elimination of treatments by domainance and by extended dominance, resources can be allocated in order of increasing CERs. Moreover, the CER is a parameter that characterizes clinical and economical properties of a treatment independent of its comparators. © 1997 by John Wiley & Sons, Ltd. Health Econ. 6: 497–504 (1997) No. of Figures: 1. No. of Tables: 0. No. of References: 12. KEY WORDS — average cost-effectiveness ratios; extended dominance; statistical inference; clinical trials 1. INTRODUCTION In commenting on Laska, Meisner and Siegel, 1 hereafter LMS, Briggs and Finn, 2 hereafter BF, object to our inference procedures for average cost-effectiveness ratios (CERs) because, they state, ‘decisions over the choice of treatment should be made at the margins’. They conclude that ‘a simple comparison of CERs offers little guidance to the efficient choice of treatment’ and that ‘it is misleading to present average cost per unit effect figures’. While acknowledging the relationship between CERs and the incremental CER set forth in the lemma (on p. 231) in Section 2 of LMS, 1 it appears from their remarks that they failed to appreciate its implications. The bases of BF’s arguments are the ‘decision rules for cost-effectiveness analysis’, whose pur- pose is the maximization of health effects under a constrained budget. These have been described by Johannesson and Weinstein, 3 Weinstein 4 and oth- ers. In this note, for the two treatment case considered in LMS, we show that the results of applying these decision rules, which are based in part on comparison of incremental CERs, can be accomplished equivalently by comparison of CERs. More generally, we show that the use of *Correspondence to: Eugene M. Laska, Statistical Sciences and Epidemiology Division, The Nathan S. Kline Institute for Psychiatric Research, 140 Old Orangeburg Road, Orangeburg, NY 10962, USA. Tel. (914) 365 2000. Fax (914) 359 7029. E-mail [email protected]. HEALTH ECONOMICS , VOL. 6: 497–504 (1997) CCC 1057–9230/97/050497–08 $17.50 © 1997 by John Wiley & Sons, Ltd.

The usefulness of average cost-effectiveness ratios

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DEBATE

THE USEFULNESS OF AVERAGECOST-EFFECTIVE RATIOS

EUGENE M. LASKA1,2*, MORRIS MEISNER2,3 AND CAROLE SIEGEL2,4

1Statistical Sciences and Epidemiology Division, The Nathan S. Kline Institute for Psychiatric Research, USA2New York University School of Medicine, USA

3Statistical Sciences Laboratory, The Nathan S. Kline Institute for Psychiatric Research, USA4Health Services and Epidemiology Laboratory, The Nathan S. Kline Institute for Psychiatric Research, USA

SUMMARY

We demonstrate that average cost-effectiveness ratios (CERs) play an important role in the evaluation of thecost-effectiveness of treatments. Criticisms of the usefulness of CERs derive mostly from the context of resourceallocation under a constrained budget in which some decisions are based on incremental CERs. However, we showthat in many cases, these decision rules are equivalent to decision rules on CERs. This follows for mutuallyexclusive treatments first, because a treatment is eliminated by extended dominance if and only if there is a mixedtreatment with a smaller CER, where the mixing parameter lies in a certain interval. Second, after elimination oftreatments by domainance and by extended dominance, resources can be allocated in order of increasing CERs.Moreover, the CER is a parameter that characterizes clinical and economical properties of a treatmentindependent of its comparators. © 1997 by John Wiley & Sons, Ltd.

Health Econ. 6: 497–504 (1997)

No. of Figures: 1. No. of Tables: 0. No. of References: 12.

KEY WORDS — average cost-effectiveness ratios; extended dominance; statistical inference; clinical trials

1. INTRODUCTION

In commenting on Laska, Meisner and Siegel,1

hereafter LMS, Briggs and Finn,2 hereafter BF,object to our inference procedures for averagecost-effectiveness ratios (CERs) because, theystate, ‘decisions over the choice of treatmentshould be made at the margins’. They concludethat ‘a simple comparison of CERs offers littleguidance to the efficient choice of treatment’ andthat ‘it is misleading to present average cost perunit effect figures’. While acknowledging therelationship between CERs and the incremental

CER set forth in the lemma (on p. 231) in Section2 of LMS,1 it appears from their remarks that theyfailed to appreciate its implications.

The bases of BF’s arguments are the ‘decisionrules for cost-effectiveness analysis’, whose pur-pose is the maximization of health effects under aconstrained budget. These have been described byJohannesson and Weinstein,3 Weinstein4 and oth-ers. In this note, for the two treatment caseconsidered in LMS, we show that the results ofapplying these decision rules, which are based inpart on comparison of incremental CERs, can beaccomplished equivalently by comparison ofCERs. More generally, we show that the use of

*Correspondence to: Eugene M. Laska, Statistical Sciences and Epidemiology Division, The Nathan S. Kline Institute forPsychiatric Research, 140 Old Orangeburg Road, Orangeburg, NY 10962, USA. Tel. (914) 365 2000. Fax (914) 359 7029. [email protected].

HEALTH ECONOMICS, VOL. 6: 497–504 (1997)

CCC 1057–9230/97/050497–08 $17.50© 1997 by John Wiley & Sons, Ltd.

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incremental CERs for elimination of treatmentsby extended dominance and for allocation ofresources are equivalent to criteria based onCERs. Thus, the conclusions of BF are simplyincorrect.

In Section 2, we briefly review the decisionrules for independent treatments and for mutuallyexclusive treatments. The former rules are basedon CERs, while the latter rules involve the use ofincremental CERs. In Section 3, for the fre-quently occurring case of two mutually exclusivetreatments considered in LMS, we describe analternative rule for resource allocation that usesCERs in place of incremental CERs. We show, inSection 4, that the two rules are equivalent. InSection 5, we point out the relevance of hypoth-esis testing in the resource allocation framework.For K mutually exclusive treatments, we show, inSection 6, that a treatment may be eliminated byextended dominance if and only if there is amixed strategy, whose mixing parameter lies in aprescribed range, with smaller CER. We show inthe Appendix that after eliminating such treat-ments, resources may be allocated to the remain-ing treatments in order of increasing CERs untilthe budget is exhausted. Finally, we discuss someimplications of our results in Section 7.

2. THE DECISION RULES FORCOST-EFFECTIVENESS ANALYSIS

The process of resource allocation for K treat-ments begins with the elimination of dominatedtreatments. (The Min test, a statistical test fordetermining whether a treatment is dominated,i.e. is inadmissible, was introduced in the cost-effectiveness context by Siegel et al.,5 and issummarized in LMS. The MIN test contraststreatments in terms of sample costs and sampleeffectiveness and identifies those that are inferioron both measures. Neither CER nor incrementalCERs are involved.)

For independent programmes or treatments, acase ignored by BF, the admissible treatments areallocated resources sequentially in order ofincreasing CERs until the budget is exhausted.6

Of course, sampling variation may require thatinference methods be used to determine if theCER of one treatment is statistically different

from the CER of another. Thus, for resourceallocation among independent programs, the use-fulness of the methods presented in LMS forinference about CERs is evident.

For mutually exclusive treatments, the admis-sible treatments are ranked in order of increasingeffectiveness, and then incremental CERs arecomputed.3 An otherwise admissible treatment iseliminated from further consideration, based onthe concept of extended dominance if its incre-mental CER is greater than the incremental CERof the next ranked more effective treatment.Extended domainance in the important case oftwo treatments is discussed, together with anillustrative example, by Cantor.7 After treatmentsmeeting this condition are eliminated, the incre-mental CERs of the remaining treatmentsincrease in order. (We show in the Appendix,using notation introduced in Section 4, that theCERs of these treatments also increase in order.)Resources are allocated sequentially to treat-ments, possibly using mixed strategies, in order ofincreasing incremental CERs (or equivalentlyCERs), until the budget is exhausted. We will callthis approach the incremental CER-based deci-sion rule. The theorem in the appendix demon-strates that allocation may proceed equivalentlyin order of increasing CERs.

3. A CER-BASED DECISION RULE FORTWO TREATMENTS

We now define a CER-based decision rule (asopposed to an incremental CER rule) forresource allocation for two mutually exclusivetreatments. (We return to the K treatment case inSection 6.) In the first step, a dominated treatmentis eliminated in the same way as in the incre-mental CER based decision rule. If both treat-ments are admissible, they are ranked in order ofincreasing effectiveness and then CERs are com-puted. The less effective treatment is eliminatedfrom further consideration if its CER is greaterthan the CER of the more effective treatment. Weshow in the next section that for two treatments,this is merely an alternative criterion for extendeddominance. In the third step, resources are allo-cated in order of increasing CERs until thebudget is exhausted.

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4. EQUIVALENCE OF THE INCREMENTALCER AND CER DECISION RULES FOR

TWO TREATMENTS

For the CER decision rule to be equivalent to theincremental based rule, the resulting allocationsof resources must be the same. Under bothdecision rules, the admissible treatments are thesame. After checking whether a treatment shouldbe eliminated by extended dominance, resourcesare allocated in order of increasing CERs. Sincethe ranking of treatments is the same, whetherthey are ordered in terms of CER or in terms ofincremental CERs, the two decision rules do notdiffer in this stage. Thus, it remains only to showthat the two approaches produce the same resultunder the two criteria for extended dominance.

In the case of two treatments, as in Cantor’sexample,7 standard care may be considered toincur no cost and to produce no effect. Using thenotation of LMS, denote the less effective treat-ment by T1 and the more effective treatment byT2. Then ε = ε1 – ε2 < 0 and γ = γ1 – γ2 < 0, whereεi is the expected effectiveness and γi the expectedcost for treatment Ti, for i = 1 and 2. The truepopulation CERs, γi/εi, are denoted by ρi for i = 1and 2, and the incremental CER, γ/ε, is denotedby ρ. The CER ρ1 may also be thought of as theincremental CER of T1 compared with standardcare. Since both treatments are admissible, ρ > 0.Therefore, according to the incremental CERcriteria for extended dominance, T1 is eliminatedif ρ < ρ1.

Thus, for the incremental CER decision rule,the three conditions required to show extendeddominance are ρ > 0, ε < 0 and ρ < ρ1. Theseconditions correspond exactly to the conditions ofcase 1(b) of the lemma in Section 2 of LMS.1 Thelemma states that the three conditions are equiva-lent to the condition ρ2 < ρ1. Therefore, elimina-tion of T1 from further consideration because ofextended dominance can also be accomplished bya comparison of the average CERs.

The equivalence of comparison of averageCERs and comparison of incremental CERs todetermine if extended dominance obtains mayalso be seen by examining Fig. 1. According to theincremental CER decision rule, T1 is eliminated ifthe (ε, γ) coordinates of T2 lie in region III(dominance) or in region II (extended dom-inance). If the (ε, γ) coordinates of T2 lie in regionI, then resources may be allocated to either

treatments. The slope of the line from the originthrough T1 is the CER ρ1, and the slope of the linejoining T1 and T2 is the incremental CER ρ. For apoint such as T22 in region II, ρ < ρ1 and ρ2 < ρ1,so that both rules eliminate T1. Similarly, for apoint in region I, such as T21, ρ > ρ1 and ρ2 > ρ1,so that both rules retain T1. Points on theboundary between regions I and II, such as T23,have ρ = ρ1 and ρ2 = ρ1. Formally, T1 may beeliminated because a mixed strategy can achievethe same mean cost and effectiveness. However,the pure strategy may be retained for otherreasons. Thus, we have proved the equivalence ofthe two decision rules for resource allocation,and, therefore, either the CER or the incrementalCER can be used.

5. STATISTICAL INFERENCE ON CERs INTHE CASE OF TWO TREATMENTS

In the presence of uncertainty about the truevalues of the cost and effectiveness parameters, astatistical procedure may be helpful in determin-ing the region in which T2 lies. In LMS, twoprocedures, a Bonferroni-based test and a like-lihood ratio test, are presented for testing the nullhypothesis that the two treatments have the sameCER, i.e. ρ1 = ρ2. If the null hypothesis is notrejected, then T1 can be eliminated by extendeddominance because mixed strategies can be used.If the null hypothesis is rejected, then dependingon the relative magnitude of the CERs, it can beconcluded that T2 lies in region I or in region II.Confidence intervals for the individual CERs mayalso be useful here.

6. EXTENDED DOMINANCE IS ACONDITION ON CERs OF MIXED

STRATEGIES

We turn now to the case of K mutually exclusivetreatments. Here we show that a treatment Ti iseliminated by extended dominance if and only ifthe mixed treatments T*(α) have smaller CERs,where T*(α) = αTi–1 + (1 – α)Ti+1, and

(γi+1 – γi)/(γi+1 – γi–1) ≤ α ≤ (εi+1 – εi)/(εi+1 – εi–1)

Assume that the admissible treatments have been

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ordered in terms of increasing effectiveness. Thecondition to eliminate treatment Ti by extendeddominance, described in Section 2, is that itsincremental CER relative to Ti–1 is greater than inthe incremental CER of Ti+1 relative to Ti. Thatis,

(γi – γi–1) /(εi – εi–1) > (γi+1 – γi)/ (εi+1 – εi)

This condition, after algebraic manipulation andrearrangement of terms, is equivalent to

γi > (αuγi–1 + (1 –αu)γi+1)

where αu = (εi+1 – εi)/ (εi+1 – εi–1). Since algebra-ically εi = αuεi–1 + (1 – αu) εi+1, after division ofthe left side of the previous inequality by εi andthe right side by its equivalent expression αuεi–1+ (1 – αu) εi+1, the condition to eliminate treat-

ment Ti by extended dominance becomes

γi/εi > (αuγi–1 + (1 – αu) γi+1)/ (αuεi–1 + (1 – αu)εi+1)

Note that the left side of the inequality is the CERof treatment Ti. The mean effectiveness and meancost of T*(αu) is

Figure 1. The standard C/E decision rules for two treatments.

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[αuεi–1 + (1 – αu) εi+1, αuγi–1 + (1 – αu)γi+1]

= [εi, αuγi–1 + (1 – αu)γi+1]

so that its CER is

[αuγi–1 + (1 – αu) γi+1]/ [αuεi–1 + (1 – αu) εi+1]

= [αuγi–1 + (1 – αu)γi+1]/εi

Thus, Ti is eliminated under extended dominanceif and only if its CER is greater than the CER ofT*(αu). By a similar calculation, the condition toeliminate treatment Ti by extended dominancebased on incremental CERs is also algebraicallyequivalent to the condition

εi < (α1εi–1 + (1 – α1)εi+1)

where α1 = (γi+1 –γi)/ (γi+1 – γi–1). Note that α1 isexactly Cantor’s7 ‘coefficient of inequity’ k*,which, under the mixed strategy, is the minimumproportion of individuals who can receive the lesseffective treatment. After dividing the left side ofthe algebraic identity γi = α1γi–1 + (1 – α1)γi+1by εi, and dividing the right side by the largerquantity α1εi–1 + (1 – α1)εi+1, the conditionbecomes

γi/εi > [α1γi–1 + (1 – α1) γi+1]/[α1εi–1 + (1 – α1)εi+1]

Thus, Ti is eliminated under extended dom-inance if and only if its CER is greater than theCER of T*(α1) whose coordinates are

[α1εi–1 + (1 – α1) εi+1, α1γi–1 + (1 – α1)γi+1]

= [α1εi–1 + (1–α1)εi+1, γi]

As α varies from α1 to αu, the mean effectivenessof T*(α) decreases from α1εi–1 + (1 – α1) εi+1 to εiand the mean cost decreases from γi to αuγi+1 +(1 – αu)γi–1. Thus, for each α in the interval, T*(α)dominates Ti. For each such α the CER of T*(α) isless than the CER of Ti. This follows since as αincreases in the interval, the CER of T*(α) iseither strictly increasing or strictly decreasingdepending on the sign of the intercept of the linethrough the two points (εi–1, γi–1) and (εi+1, γi+1).As shown above, the CER of T*(α) at both end-points is less than γi/εi.

Thus, we have proved that Ti is eliminated by

extended dominance if and only if its CER isgreater than the CER of T*(α), for all such α inthe interval

(γi+1 – γi)/(γi+1 – γi–1) ≤ α ≤ (εi+1 – εi)/(εi+1 – εi–1)

We remark that the results in Section 4 forK = 2 and i = 1 follow from the above results.Also, we wish to acknowledge that the abovetheorem and the theorem given by Cantor7 areclosely related.

As shown in the Appendix, after elimination oftreatments by dominance and by extended dom-inance, the CERs of the remaining treatments arein increasing order.

The above results suggest an alternative deci-sion rule for allocating resources based entirelyon CERs. First, rank the admissible treatments inincreasing order of effectiveness. Next, for eachtreatment Ti, introduce the mixture treatmentTi*(αu), where αu is given above, for i = 1, …,K – 1. Treatment Ti*(αu) has the same meaneffectiveness as Ti. From this new augmented setof 2K – 1 treatments, eliminate those that aredominated or, equivalently, eliminate either Ti orTi*(αu) depending on which has the larger CER.If Ti is eliminated, the subsequent treatments arerenumbered, a new value of Ti*(αu) computedand the process continued. The remaining treat-ments have increasing CER, and resources maybe allocated sequentially until the budget isexhausted. These treatments and mixtures ofadjacent treatments, in any proportion, may beutilized depending on the size of the budget.

7. DISCUSSION

In the presence of uncertainty, LMS presentsapproaches for drawing inferences about cost andeffectiveness parameters that characterize twotreatments. The focus is on statistical methods forconfidence interval estimation and for testinghypotheses. In a clinical trial comparing twohealthcare interventions, parameter estimates,uncertainty measures and tests of the null hypoth-esis of equality of the means of importantoutcome parameters is not only standard practice,but it is a sine qua non for publication. It seemsobvious that the CER is one such parameter, forit provides a summary economic and clinical

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characterization of a treatment, and it is inde-pendent of the comparator treatments. Themethods in LMS can be used to carry out such astatistical analysis.

In sharp contrast, BF and others believe thatCERs not only convey no useful information butalso can be misleading. They present no argumentbut merely cite Gold et al.,8 who ‘caution againstreporting average cost effectiveness ratios’. Asdemonstrated in Section 4, the equivalence ofresource allocation rules for economic evalua-tions based on CERs and those based on incre-mental CERs, both for independent and formutually exclusive treatments, prove that BF’sconclusion is incorrect.

For the more general case of K mutuallyexclusive treatments, the results presented inSection 6 make the decision rules for resourceallocation in the independence and mutuallyexclusive cases comparable. In the independencecase, all treatments are identified at the outset,and resources are allocated in order of increasingCERs. In the mutually exclusive case, mixedstrategies are not identified at the outset, andincremental CERs are used as a technical meansfor elimination of treatments by extended dom-inance after dominated treatments are eliminated.To parallel the independence case, one canaugment the original list of treatments withmixtures having the same mean effectiveness,eliminate those on the enlarged list that aredominated and allocate resources to the remain-der in order of increasing CERs.

For illustrative purposes, Karlsson and Johan-nesson9 analysed a fictitious example involvingthree independent treatment groups, each ofwhich is comprised of several mutually exclusivetreatments. In their illustration, the total numberof individuals per treatment group was con-strained to be less than 1000, and the problem wasto optimally allocate resources from a fixedbudget. They concluded that ‘average CERs giveno guidance for decision making’ and that ‘even ifincremental CERs are calculated, they do not sayanything about whether a treatment should beimplemented or not [sic], except for treatmentsthat are excluded as dominated’. To accomplishthis, they continued, ‘either a fixed budget or aprice per [additional] unit of effectiveness must beintroduced as the decision rule].

It has not been appreciated that the allocationcan equivalently be made according to the aver-age CERs of ‘compound strategies’. A set of

treatments that may be simultaneously imple-mented is called a compound strategy with costsand effectiveness equal to the sum of the respec-tive costs and effectiveness of its componenttreatments. An example of a compound strategyfrom Karlsson and Johannesson’s9 problem is“treat all 1000 patients in group I with B, all 1000patients in group II with F and all 1000 patients ingroup III with M.” This compound strategy andthe compound strategy “treat all 1000 patients ingroup I with D, all 1000 patients in group II withF and all 1000 patients in group III with M” aremutually exclusive. In fact, the set of all suchcompound strategies is a set of mutually exclusivetreatments; so the algorithm of Section 6 may beapplied. The resulting allocation rank ordering isidentical to the ordering obtained from thedecision rules using incremental CERs given byKarlsson and Johannesson.9 The CER of a com-pound strategy is a natural informative measurefor resource allocation giving the cost per unit ofeffectiveness purchased by implementing its com-ponent treatments.

We remark parenthetically that the Johannes-son and Weinstein3 resource allocation decisionrules have been criticized because the optimalityof the procedure depends upon assumptions thatmay not reflect real-world situations. Among suchassumptions are perfect divisibility, constantreturns to scale and equality of the marginalopportunity costs of resources. (For a discussionof these and other issues, see Birch and Gaf-ney.10,11) Stinnett and Paltiel12describe how lin-ear, integer and mixed integer programmingapproaches can be used to allocate resources thatoptimally maximize health effects without requir-ing such restrictive assumptions. Only undersimplifying assumptions does the optimal solutionobtained by these mathematical programmingmethods agree with simple resouce allocationdecision rules.

In summary, standard decision rules for inde-pendent treatments allocate resources usingCERs. For two treatments, the decision rules formutually exclusive treatments may equivalentlyuse CERs or incremental CERs. For K mutuallyexclusive treatments, for computational conven-ience, incremental CERs may be utilized foreliminating treatments by extended dominance,but this is equivalent to a dominance or CERcomparison using a mixed strategy. Thereafter,resources can be allocated to the remainingtreatments in order of increasing CERs. In addi-

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tion, the CER of a treatment is a useful summaryparameter that characterizes a treatment inde-pendent of its comparators. In the presence ofuncertainty, statistical procedures such as thosegiven in LMS are necessary. Thus, average CERsplay an important role in the evaluation of thecost-effectiveness of treatments.

ACKNOWLEDGEMENTS

The work in this research was supported in part by U.S.National Institute of Mental Health Grants MH 50822, P50MH 51359 and MH 42959.

APPENDIX

Theorem

Let γ0 = ε0 = 0. Assume that all dominated andextended dominated treatments have beenremoved and the remaining mutually exclusivetreatments renumbered sequentially. Then ρi = γi/εi is increasing in i.

Proof

The proof proceeds by mathematical induction.By extended dominance, we have that ∆i = (γi –γi–1)/(εi – εi–1) is strictly increasing in i. Since∆1 < ∆2 after cross-multiplication and cancella-tion of like terms, it follows that ρ1 < ρ2. Nowassume the inductive hypothesis that ρj is increas-ing for j = 1, 2, …, i. It must now be shown thatρi < ρi+1. Note that

ρi–2 < ρi–1 ⇔ γi–2εi–1 < γi–1εi–2

⇔ γi–2(εi–1–εi–2) < εi–2(γi–1 – γi–2)

⇔ ρi–2 < ∆i–1.

Since ∆i are increasing, it follows that ρi–2 < ∆i+1,which may be written as

γi–2(εi+1 – εi) < εi–2 (γi+1 – γi). (1)

Further,

∆i < ∆i+1 ⇔ (γi – γi–1) (εi+1 – ε1)

< (γi+1 – γi) (εi – εi–1), (2)

and

∆i–1 < ∆i+1 ⇔ (γi–1 – γi–2) (εi+1 – εi)

< (γi+1 – γi) (εi–1 – εi–2). (3)

Adding inequalities (2) and (3) and cancelling liketerms yields

γiεi+1 – γi–2 (εi+1 – εi) < γi+1εi – εi–2 (γi+1 – γi). (4)

Adding inequality (1) to inequality (4) yields

γiεi+1 < γi+1εi ⇔ ρi < ρi+1

which completes the proof.

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