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## EXPERIMENTAL EVIDENCES FAVORING QUANTUM MECHANICS

In 1969, Clauser, Horne, Shimony, and Holt generalized Bell’s work so that realizable experiments could be designed. For the first time Einstein and Bohr’s debate found its way out of the foggy world of epistemology to the verifiable domain of experimental physics. Many experiments have since been conducted and all of them have favored quantum mechanics—their results violate the limits imposed by a “local hidden variable” theory of the type favored by Einstein and his collaborators. In spite of the spectacular success, more work needed to be done as conclusions of these experiments are still the subject of considerable interest and debate. Every experimental test of entanglement appears to have one or more loopholes, raising the slim possibility that “some alternative theory, distinct from quantum mechanics and more aligned with Einstein’s intuitions”, could be in play.

### EARLY EXPERIMENTS

The first experiments that put Einstein’s thought experiment into practice were carried out at the University of California, Berkeley and Harvard University in 1972, followed by Texas A&M University in 1976. Freedman and Clauser (Freedman & Clauser, 1972) were the first to measure the linear polarization correlation of the photons emitted in an atomic cascade of calcium. Earlier, Clauser, Horne, Shimony, and Holt’s generalization of Bell’s inequality showed that the existence of local hidden variables imposes restrictions on this correlation that was in conflict with the predictions of quantum mechanics. Freedman and Clauser’s data agreed with quantum mechanics and violated these restrictions to high statistical accuracy, thus providing strong evidence against local hidden-variable theories. More experiments followed. Barring some initial inconsistencies, these experiments showed an agreement with quantum mechanics and a violation of Bell’s inequalities was reported by as much as 6 standard deviations. However, a few loopholes quickly surfaced, allowing the possibility of a local realist interpretation of these results.

### ALAIN ASPECT

In the early 1980’s, Alain Aspect and his colleagues started designing a series of experiments at Orsay, France to test Bells inequalities. To create the entangled photons, calcium atoms were irradiated by a laser beam causing them to transition to a high-energy state. The excited atoms decayed by emitting two photons in opposite directions with their polarizations8 entangled—meaning they were instantaneously connected over any distance in such a way that the measured property of one depended on the other—like a pair of tossed dice that always roles doubles.

 Figure 6: Einstein-Podolsky-Rosen-Bohm Gedanken experiment with photons. The two photons $\nu_1$ and $\nu_2$, emitted in the state $\psi(1,2)$, are analyzed by linear polarizers in orientations a and b. One can measure the probabilities of single or joint detections in the output channels of the polarizers.

Their first experiment used a single detector with a polarizer9 in front of it, on each side of the atomic beam. Some photons passed through, while others were stopped by the polarizer. Each detector recorded photons that passed through for various combinations of the polarizer settings (polarizers were rotated to register photons of other polarization directions). The polarizations of pairs of photons were measured. Although each individual measurement gave random results, Aspect found these results were in fact correlated (as if the photons “rolled doubles” more than a certain fraction of the time), clearly violating Bell’s inequalities for local hidden variable theories. However, a loophole, called the detector efficiency loophole, was quickly spotted. The detectors sometimes could “miss” a photon and the absence of a count in a particular detector was a significant piece of information. It could be a signal—for example, the polarization was horizontal when the polarizer was vertical. This meant that one could explain their result using a local theory that just happened to “miss” photons in a particularly lucky pattern.

To overcome the detector efficiency loophole, the experiment was repeated with four detectors, two on each side, and only counting data when one photon was detected on each side. Again, the measured correlations exceeded the classical limit, by an absurd 40 times the statistical uncertainty in their measurement.

But the fundamental loophole in these experiments, according to Bell, was the “locality loophole.” The result of a measurement at one polarizer should not depend on the orientation of the other. Bell proposed a way to close this loophole. He argued that if the orientation of each polarizer was chosen while the photons were in flight, then the relativistic causality (no influence can travel faster than light) would prevent a polarizer from “knowing” the orientation of the other during measurement. Thus, an ideal test of Bell’s inequalities would require randomly choosing the orientation of each polarizer.

 Figure 7: Aspect’s timing-experiment with optical switches (C1 and C2).

The switch C1 followed by the two polarizers in orientations a and a’ is equivalent to a single polarizer switched between the orientations a and a’. A switching occurs about once in every 10 ns. A similar setup is implemented on the second side. Source: Aspect, 2000.

Aspect subsequently came up with a design involving a switching device and two polarizers in orientations: a and a’ on side I, b and b’ on side II. The optical switch C1 on side I rapidly redirected the photons to either polarizer a or to a’ thus acting like a polarizer whose orientation could be changed between a and a’. Side II had a similar set up with a variable polarizer capable of switching between b and b’. The distance L between the two switches was 13 m, with L / c = 43 ns. The distance L between the switches was large enough (13 m) so that the time taken by the signal to travel between the switches (43 ns) was significantly larger than the delay between two switching operations (about 10 ns) plus the time delay between the emission of the two successive photons (5 ns average).

Aspect’s clever design allowed the orientations of the polarizers to be changed rapidly while the photons were still in flight. It effectively closed the locality loophole according to which the measurement device itself could somehow transmit information about one photon to the other that would explain any coordination between them. Bell’s inequality was still violated by 6 standard deviations.

In spite of the improved design, the polarizer orientations in Aspect’s experiment couldn’t be made fully random because of certain technical difficulties. Also, the distance between the source and polarizers were such that communications between them at the speed of light couldn’t be excluded.

### ANTON ZEILINGER

The definitive experiment was performed by Anton Zeilinger and his team at the University of Innsbruck in 1997. The experimenters used a much improved source of entangled photons and quantum random-number generators to set the detector orientations. Again, a violation of Bell’s inequality was observed by several tens of standard deviations.

The method that Zeilinger’s team used to generate polarization-entangled photon is called spontaneous parametric down-conversion (SPDC). We won’t burden ourselves with the details of SPDC until the next Section, but for now, suffice it to say when a special kind of crystal is irradiated with a strong laser beam, a photon from the laser beam could be converted into two polarization-entangled daughter photons. These photons were subsequently used in the Innsbruck experiment.

The orientations of the polarizers were varied every nanosecond by placing a device called an electro-optical modulator in front of a polarizer (Figure 8). The effect was to rotate the polarizer at a specific angle. This was done on both the sender and receiver sides independently and randomly using a quantum random-number generator (ORNG).

 Figure 8: Set up of the Innsbruck experiment with two independent observers Alice and Bob (Weihs, Jennewein, Simon, Weinfurter, & Zeilinger, 1998). Two entangled photons are dispatched by way of optical fiber cables to Alice and Bob whose measuring stations are located about 400 m apart. The polarization directions of the in-flight photons at each measurement station is set by two independent ORNGs so that events are registered independently on both sides and coincidences can be identified long after the experiment is finished.

Random number sequences are needed in many theoretical and experimental scenarios and usually complex algorithms are summoned to generate them. But the problem with algorithmic ORNGs is that the sequences they generate are never strictly random, as repetitive number patterns are known to emerge with time. Quantum random numbers, on the other hand, are vastly superior because they never repeat patterns. The Innsbruck team was able to harness the intrinsic randomness of the quantum world to build their QRNG. Here is how.

There are many examples of quantum randomness in nature. Imagine the case when a vertically orientated polarizer meets a photon polarized at 45 degrees. The photon has a 50-50 chance of passing through the polarizer and the process is completely random—nobody can predict if a specific photon would pass through or not. A similar kind of randomness is seen in double-slit experiments where it is impossible to predict through which slit a particular photon would emerge.

The Innsbruck team used half-silvered mirrors or beam splitters to construct their ORNG. When a single photon is incident on a half-silvered mirror, it has a 50 percent chance of passing through (transmission) and a 50 percent chance of getting blocked (reflected). If a photon detector is placed behind the half-silvered mirror to register the transmitted photons and a second detector in the front to detect the reflected photons, both detectors will have equal probability of registering photons. If we associate 1 with the “click” of the detector of transmitted photons and 0 with that of the reflected photons, a random sequence of zeros and ones will emerge with a given stream of photons—a fraction of the huge sequence of random numbers obtained by the Innsbruck team is reproduced here (Zeilinger, 2010):

The photon pairs were generated in a building at the center of the campus, and the two photons were directed to two different measurement stations located about 400 meters from each other. The Innsbruck team shined a beam splitter with a weak light source producing random numbers at a rate faster than the time necessary to communicate with the detector on the other side. In other words, the polarization settings on both sides were changed very rapidly and at the last moment with electro-optic modulators that altered the polarizations at a rate proportional to the voltage applied, with the modulators themselves controlled by the QRNGs. Thus the Innsbruck experiment was able to eliminate the nagging communication loophole because any communication between the detectors on either side had to happen at a speed faster than the speed of light. Yet a violation of Bell’s inequality was observed by several tens of standard deviations (S = 2.73 ±0.02) and quantum entanglement was clearly established with completely independent measuring stations.

### FUTURE EXPERIMENTS

This could be the end of the story—all experimental tests conducted so far had irrefutably favored quantum mechanics. But curiously, a particularly subtle loophole called the “free will” loophole still lingered: The possibility that the settings of the detectors that measure the entangled photons could be correlated with any “hidden” information in their shared causal pasts. In Bells own words: “It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty.”

 Figure 8: Schematic of the “Cosmic Bell” experiment. A source sends entangled particles to two distant detectors. While the particles are in flight, light from distant quasars x and y is used to randomly choose the detector settings a and b on each side, which then measure experimental outcomes A and B. Many runs would be compared to confirm that the results always violate the Bell inequalities and are independent of the choices of which quasars to observe. Source: Gallicchio

In fact, according to a recent theoretical study, only a small correlation between the detector settings and any local hidden variables would be sufficient to mimic the predictions of quantum mechanics. Previous experiments were only able to rule this out less than a millisecond or so before the test began. The recently proposed “Cosmic Bell” experiment aims to eliminate this “over almost the entire history of the universe, all the way back to the Big Bang 14 billion years ago”.

This may sound bizarre, but here is how this experiment will work. The detector settings would be determined not by the experimenters—but by two distant cosmic sources (for example, light from distant quasars). A source will send entangled particles to two distant detectors in the usual way. While the particles are in flight, light from two distant cosmic sources will be used to randomly choose the settings (e.g. polarizing filter angles) for each of the two detectors, which will then measure experimental outcomes A and B, as shown in Figure 9. The quasars will act as special kind of random number generators, based on the random arrival times of light from them.

Selecting sufficiently distant cosmic sources will ensure that they have never been in causal contact with each other. Because of the astronomical distance between them, “conspiring” hidden variables will not have enough time to communicate with both quasars as the lights from these quasars were emitted billions of years ago.

8. Polarization is the property that describes light oscillations in specific planes. A polarization filter will only allow light to pass if it oscillates in a specific plane.
9. A polarizer or a polarization filter determines the direction of polarization of light (or photon).


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Health Policy Analysis| Volume 16, ISSUE 4, P629-638, June 01, 2013

# Illustrating Potential Efficiency Gains from Using Cost-Effectiveness Evidence to Reallocate Medicare Expenditures

Open ArchivePublished:May 13, 2013

## Abstract

### Objectives

The Centers for Medicare & Medicaid Services does not explicitly use cost-effectiveness information in national coverage determinations. The objective of this study was to illustrate potential efficiency gains from reallocating Medicare expenditures by using cost-effectiveness information, and the consequences for health gains among Medicare beneficiaries.

### Methods

We included national coverage determinations from 1999 through 2007. Estimates of cost-effectiveness were identified through a literature review. For coverage decisions with an associated cost-effectiveness estimate, we estimated utilization and size of the “unserved” eligible population by using a Medicare claims database (2007) and diagnostic and reimbursement codes. Technology costs originated from the cost-effectiveness literature or were estimated by using reimbursement codes. We illustrated potential aggregate health gains from increasing utilization of dominant interventions (i.e., cost saving and health increasing) and from reallocating expenditures by decreasing investment in cost-ineffective interventions and increasing investment in relatively cost-effective interventions.

Complete information was available for 36 interventions. Increasing investment in dominant interventions alone led to an increase of 270,000 quality-adjusted life-years (QALYs) and savings of $12.9 billion. Reallocation of a broader array of interventions yielded an additional 1.8 million QALYs, approximately 0.17 QALYs per affected Medicare beneficiary. Compared with the distribution of resources prior to reallocation, following reallocation a greater proportion was directed to oncology, diagnostic imaging/tests, and the most prevalent diseases. A smaller proportion of resources went to cardiology, treatments (including drugs, surgeries, and medical devices, as opposed to nontreatments such as preventive services), and the least prevalent diseases. ### Conclusions Using cost-effectiveness information has the potential to increase the aggregate health of Medicare beneficiaries while maintaining existing spending levels. ## Keywords ## Introduction It is well documented that US health care spending growth is unsustainable [ • Keehan S.P. • Sisko A.M. • Truffer C.J. • et al. National health spending projections through 2020: economic recovery and reform drive faster spending growth. , Organisation for Economic Co-operation and Development. OECD Health Data 2010: How Does the United States Compare? Paris, France: OECD Publishing ]. Compared with other developed countries, return on health care spending in the United States is poor, with a significant proportion of the American population lacking health insurance and the health care system performing poorly across key metrics such as life expectancy and infant mortality [ Davis K, Schoen C, Stremikis K. Mirror, mirror on the wall. How the performance of the US health care system compares internationally. 2010 update. 2010. Available from: http://www.commonwealthfund.org/∼/media/Files/Publications/Fund%20Report/2010/Jun/1400_Davis_Mirror_Mirror_on_the_wall_2010.pdf. [Accessed June 8, 2012]. ]. In many countries, cost-effectiveness analysis is used to prioritize scarce health care resources among competing interventions. Despite the immediate need to increase the value of health care spending, however, decision makers in the United States have resisted this approach [ • Bryan S. • Sofaer S. • Siegelberg T. • Gold M. Has the time come for cost-effectiveness analysis in US health care?. ]. More than 46 million Americans, including those 65 years and older and those with certain disabilities, receive health insurance through Medicare. The Centers for Medicare & Medicaid Services (CMS) does not operate with a fixed budget, and program cost has increased annually at a relatively rapid rate. The program’s current annual cost is estimated at upwards of$600 billion, approximately 3.5% of the gross domestic product, and may reach $1 trillion by 2020 [ Centers for Medicare & Medicaid Services. National health expenditure data. Available from: https://www.cms.gov/NationalHealthExpendData/downloads/tables.pdf. [Accessed June 8, 2012]. ]. Research indicates that approximately 30% of Medicare spending may be inappropriate or unnecessary [ • Bentley T.G. • Effros R.M. • Palar K. • Keeler E.B. Waste in the U.S. health care system: a conceptual framework. , • Fisher E.S. • Wennberg D.E. • Stukel T.A. • et al. The implications of regional variations in Medicare spending, part 1: the content, quality, and accessibility of care. , • Garber A. • Goldman D.P. • Jena A.B. The promise of health care cost containment. , United States General Accounting Office. Report to the Chairman, Subcommittee on Health, Committee on Ways and Means, House of Representatives. Medicare: Divided Authority for Policies on Coverage of Procedures and Devices Results in Inequities. Apr 11, 2003, GAO-03-175. Available from: http://www.gao.gov/assets/240/237841.pdf. [Accessed June 8, 2012]. , • Berwick D.M. • Hackbarth A.D. Eliminating waste in US health care. ]. The CMS issues approximately 10 to 15 national coverage determinations (NCDs) each year for interventions deemed to have a significant impact on the Medicare program [ Medicare Program; Revised Process for Making Medicare National Coverage Determinations; Notice of proposed rulemaking, 68 Federal Register 187 (26 September 2003), pp 55634–41 ]. With respect to cost-effectiveness evidence, CMS states that it “is not a factor CMS considers in making NCDs” [ Centers for Medicare & Medicaid Services. Guidance for the public, industry and CMS staff: factors CMS considers in opening a national coverage determination. Available from: http://www.cms.gov/medicare-coverage-database/details/medicare-coverage-document-details.aspx?MCDId=6&McdName=Factors+CMS+Considers+in+Opening+a+National+Coverage+Determination&mcdtypename=Guidance+Documents&MCDIndexType=1&bc=BAAIAAAAAAAA&. [Accessed June 8, 2012]. ]. While research suggests that coverage decisions made in NCDs are broadly consistent with cost-effectiveness evidence—that is, technologies associated with favorable cost-effectiveness estimates tend to be covered—a number of covered interventions are not cost-effective by traditional standards, with incremental cost-effectiveness ratios (ICERs) greater than$250k per quality-adjusted life-year (QALY) gained [
• Chambers J.D.
• Neumann P.J.
• Buxton M.J.
Does Medicare have an implicit cost-effectiveness threshold?.
]. Thus, efficiency gains are possible through disinvestments in cost-ineffective interventions and investments in relatively cost-effective interventions.
The objective of this research was to estimate potential aggregate health gains from increasing utilization of dominant (i.e., cost-saving and health increasing) interventions, and from a hypothetical reallocation of expenditures among interventions subject to NCDs, through the use of a cost-effectiveness decision rule. We also sought to estimate the impact of reallocation on the distribution of expenditures across diseases and types of intervention. We acknowledge that this is an illustrative exercise, but we believe that it is important as the first of its kind to demonstrate the consequences of using cost-effectiveness information to inform resource allocation. We highlight the research challenges, particularly with regards to data limitations.

## Methods

### National Coverage Determinations

We created a database of NCDs issued by the CMS from 1999 through 2007. We excluded incomplete NCDs or those pertaining to minor coding or language changes, as well as those pertaining to off-label treatments, coverage in clinical trials, coverage with evidence development policies, or treatment facilities. Frequently, NCDs include multiple coverage decisions, often for different interventions or patient populations. Furthermore, on occasion coverage is permitted only for patient subgroups that meet certain conditions and restrictions. An entry was made in the database for each separate coverage decision implied within each NCD. We have previously used this database to evaluate factors that predict positive CMS coverage decisions for interventions [
• Chambers J.D.
• Morris S.
• Neumann P.J.
• Buxton M.J.
Factors predicting Medicare national coverage: an empirical analysis.
].

### Reallocation of Expenditures

To facilitate our analysis, we limited our sample to NCDs in which we could find available estimates of: cost-effectiveness; incremental cost; cost of intervention and comparator in the first year of use; incremental health gain; number of Medicare beneficiaries currently receiving the intervention; and the size of the unserved patient population, that is, Medicare beneficiaries who were eligible for the intervention but did not receive it. Each parameter will be discussed further.

### Cost-Effectiveness

On occasion we were able to identify the cost-effectiveness estimate from CMS’s decision memo, which comprises the agency’s public communication about the NCD, including the evidence featured in its review. In the majority of the cases, we identified cost-effectiveness evidence through a literature search by using the PubMed database, the Tufts Medical Center Cost-Effectiveness Analysis Registry, the Health Economic Evaluations Database, and the National Health Service Economic Evaluation Database [

U.S. National Library of Medicine. PubMed: U.S. National Library of Medicine, National Institutes of Health. Available from: http://www.ncbi.nlm.nih.gov/pubmed/. [Accessed June 8, 2012].

,

CEA Registry. Tufts Medical Center Cost-Effectiveness Analysis (CEA) Registry. Available from: http://www.cearegistry.org. [Accessed June 8, 2012].

,

Health Economic Evaluation Database. Health Economic Evaluation Database. Available from: http://www3.interscience.wiley.com/cgi-bin/mrwhome/114130635/HOME. [Accessed June 8, 2012].

,

Centre for Reviews and Dissemination. National Health Service-Economic Evaluation Database. Available from: http://www.crd.york.ac.uk/crdweb/. [Accessed June 8, 2012].

]. The findings of the literature search have been published elsewhere [
• Chambers J.D.
• Neumann P.J.
• Buxton M.J.
Does Medicare have an implicit cost-effectiveness threshold?.
]. Most frequently, the reported ICER was in the form of a cost per QALY gained. On occasion, the ICER was presented in the form of a cost per life-year (LY) gained, and we adjusted incremental survival gain with a utility weight for Americans aged 65 to 69 years to create an estimate of incremental QALY gained [

Erickson P, Wilson R, Shannon I. Years of healthy life. Healthy People 2000 Stat Notes 1995;7:1–15.

]. This adjustment may underestimate the incremental QALY gain as only the years of life extended by the treatment (incremental LYs gained) are accounted for when adjusting for quality of life, not prior years of treatment during which patient quality of life may have been improved. In sensitivity analyses, we included cost-effectiveness studies that estimated the intervention to be “dominant”—that is, more effective and less costly than the comparator—even if the study reported health outcome using disease-specific units, for example, tumors detected, rather than QALYs or LYs. The majority of cost-effectiveness studies were performed in a US health care system setting (26 of 34, 67%), and of those 63% (15 of 26) incorporated Medicare costs. Occasionally, a US study was unavailable and we included a non-US study. In these instances, we converted the ICER into US dollars by using purchasing power parities, and indexed to the year the coverage decision was made by using the health component of the consumer price index [

Organisation for Economic Co-operation and Development. Purchasing power parity (PPP) statistics. 2010. Available from: http://www.oecd.org/dataoecd/61/56/39653523.xls. [Accessed February 27, 2013].

,

U.S. Bureau of Labor Statistics. Measuring Price Change for Medical Care in the CPI. Available from: http://www.bls.gov/cpi/cpifact4.htm. [Accessed June 8, 2012].

].

### Utilization Rate—Served and Unserved Population

We estimated intervention utilization rates by using a database of Medicare inpatient and outpatient claims [

Centers for Medicare & Medicaid Services . Research, statistics, data and systems: limited data sets. Available from: http://www.cms.gov/LimitedDataSets/01_Overview.asp#TopOfPage. [Accessed June 8, 2012].

]. We used International Classification of Diseases, Ninth Revision (ICD-9) diagnostic codes reported in the database to identify Medicare beneficiaries eligible for an intervention, as defined by the parameters of the NCD. The database also includes Common Procedural Terminology (CPT) codes used for physician reimbursement. We estimated utilization rates by calculating the number of beneficiaries who had matching relevant ICD-9 diagnostic and CPT codes. We estimated the size of the unserved eligible population by calculating the difference between the number of beneficiaries who were a match for both ICD-9 diagnostic and CPT codes and those who were a match solely with ICD-9 diagnostic codes.

### Incremental Cost Data

We extracted incremental cost data, that is, the net present value of future expenditures (the numerator of the ICER), from the included cost-effectiveness study, and when necessary adjusted it to 2007 USDs.

### Cost of Intervention and Comparator in First Year of Use

We included the cost of the intervention and the comparator in the first year of use when it was reported in the cost-effectiveness study (64% of cases). When not reported, we estimated the cost of the intervention and the comparator in the year following first use from Medicare and physician reimbursement codes (36% of cases). Pertinent reimbursement codes were identified from Medicare documentation, the included cost-effectiveness study, or the manufacturer’s website. For interventions subject to noncoverage decisions, we obtained the relevant information from the cost-effectiveness study.

### Categorization of Interventions

To analyze the effect of the reallocation exercise on the distribution of expenditures, we categorized interventions with respect to disease (cardiology, oncology, and other), type of intervention (treatment, diagnostic, and other, i.e., education, preventative care, and mobility assistive equipment), and size of the eligible population (>1 million beneficiaries, 50,000–1 million beneficiaries, and<50,000 beneficiaries).

### Reallocation of Expenditures

In the first analysis, we illustrated the effects of increasing the utilization of dominant interventions, while maintaining the existing utilization of nondominant interventions. That is, for dominant interventions we decreased by 50% the size of the unserved population, that is, Medicare beneficiaries who were eligible for the intervention but did not receive it. We assumed a 50% shift for the reallocation, reasoning that shifting all beneficiaries from one intervention to another would be infeasible in practice. To illustrate the possible range of aggregate health gains, we repeated this analysis by adjusting utilization by 10% and 90%, respectively.
In the second analysis, we reallocated existing resources by using an iterative process. First, we ranked interventions in order of cost-effectiveness. Second, to generate resources for investment in more cost-effective interventions, we “disinvested” in the least cost-effective intervention. Disinvestment was achieved by reducing the existing utilization of the least cost-effective intervention by 50%. Third, with the resources generated from the disinvestment, we increased investment in the most cost-effective intervention. This was achieved by decreasing the size of the unserved eligible population by up to 50%. We continued this process by repeating the second and third steps, that is, disinvesting in the next least cost-effective intervention and investing in the next most cost-effective intervention, until no further reallocation of expenditures was possible and there was no net change in expenditure. We also repeated this analysis by adjusting utilization by 10% and 90%, respectively. Finally, we compared the existing and “ideal” distributions of resources across indications, types of technology, and conditions of different prevalence.

### Assumptions

The reallocation exercise adhered to the assumptions necessary for the league table approach described by Johannesson and Weinstein [
• Johannesson M.
• Weinstein M.C.
On the decision rules of cost-effectiveness analysis.
]: perfect divisibility; that is, a partially implemented health care program will maintain the characteristics of the entire program; and constant returns to scale; that is, costs and effects are proportional to the scale of implementation. Our analysis required additional assumptions. First, all Medicare beneficiaries eligible for care, as defined by the parameters of the NCD, received an intervention. Second, the comparator included in the cost-effectiveness study was the alternative intervention received in all cases. Third, we ordered interventions by using the net present value of their total costs (including downstream costs) and assumed that for interventions with a high upfront cost, for example, surgeries, resources were available to fund that “initial investment.”

## Results

Figure 1 shows the process by which we arrived at the final sample of interventions. Thirty-six of 64 interventions associated with an estimate of cost-effectiveness were included (Table 1). Twenty-six of the 28 excluded interventions were removed because of incomplete data, for which the most common reason was our inability to accurately identify the utilization rate for the intervention in the indicated patient population. For example, for ultrasound stimulation for nonunion fracture healing, no beneficiaries had a combination of the necessary ICD-9 codes, that is, ICD-9 code for nonunion fracture healing (733.82) and for fractures of the relevant bones (tibial [823], scaphoid [814], and radius [813]). We excluded two interventions because they were dominated by another intervention in the sample.
Table 1Interventions included in reallocation of expenditures.
InterventionPopulationCoverage statusUtilization (per annum)Cost-effectivenessCosts in year following first useSpending on intervention
Received Tx for diagnosisEligible but did NOT receive txInc. cost ($)Inc. QALYICER ($)Cost of intervention ($)Cost of comparator ($)Existing spending (million) (3.s.f) ($)Reallocated spending Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%. (million) (3.s.f) ($)
Ventricular assist devicesDestination therapy—patients with chronic end-stage heart failure who meet specified criteriaCovered201,474,400416,5450.42986,630331,87865,1779.634.82
Transmyocardial revascularizationPatients with severe angina (stable or unstable), which has been found refractory to standard medical therapyCovered40143,14019,7770.04489,41718,1234,0861.020.511
Liver transplantationPatients suffering from hepatitis BCovered4014,280150,9670.74204,186117,6248,55810.95.45
Ocular photodynamic therapy with verteporfinMacular degeneration— predominately classic subfoveal CNV lesionsCovered1,20072,20014,5040.03195,5669,570018.79.36
Lung volume reduction surgerySevere upper-lobe emphysemaCovered120109,06060,2430.50120,46087,90528,72716.38.17
Implantable cardioverter defibrillators (ICDs)Patients with documented familial or inherited conditions with a high risk of life-threatening ventricular tachyarrhythmiasCovered28,1801,276,88021,1020.16
Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%.
99,78292,78365,8463,6101,800
Pancreas transplantationPancreas transplants—patients who meet the specified criteria (type 1 diabetes etc.)Covered72067,200198,3512.2090,159227,7884,218267134
Positron emission tomographyEsophageal cancerCovered20080,2005,5980.0781,4854,1921,43812.06.00
ICDsNIDCM, documented prior MI, class II and III heart failureCovered03,24077,1131.0176,24437,4747,09000
Deep brain stimulationParkinson’s diseaseCovered39,860687,94047,1210.7265,97053,8535,98824,30012,100
ICDsDocumented sustained ventricular tachyarrhythmiaCovered28,040931,02034,3750.65
Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%.
39,971101,31073,9124,3602,180
Autologous stem cell transplantation (AuSCT)Patients suffering from multiple myelomaCovered801,52083,1231.69
Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%.
37,2752,3961066.963.48
AcupunctureOsteoarthritisNoncovered0744,8605360.0220,38397000
Lumbar artificial disc replacementBack painNoncovered0140,7007,6250.3918,93925,98616,54700
Laparoscopic adjustable gastric banding (LAGB)—bariatric surgeryTreatment of morbid obesityCovered6,6005,976,9008,1000.4518,0283,36614253.510,300
Cochlear implantationPostlingually hearing impaired patientsCovered1,12031,22041,5203.8011,65326,748046.5695
Hyperbaric oxygen therapyHypoxic wounds and diabetic wounds of the lower extremities—diabetic wounds of the lower extremitiesCovered43,8001,196,8001,7710.276,64952403945,770
Electrical bioimpedance for cardiac output monitoringHypertensionNoncovered01,429,0603140.056,408628515017,000
External counterpulsation (ECP) therapyVarious cardiac conditionsNoncovered05,018,5008200.263,2645,343002,060
Positron emission tomographyHead and neck cancersCovered800575,2001,4250.443,2246,0224,5974.821,740
Screening immunoassay fecal-occult blood test—hemoccult IIScreening for colon cancerCovered56,400476,8004000.13
Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%.
1,3185022.6118
Foot careDiabetic peripheral neuropathy with loss of protective sensationCovered400473,200−3860.05Dominates20702.261,340
Cardiac rehabilitation programsAcute MICovered46,400153,800−4700.60Dominates6908032,130
Cardiac rehabilitation programsPercutaneous transluminal coronary angioplastyCovered152,400479,000−4700.60Dominates6902,5606,580
Positron emission tomography (FDG)Breast cancer—initial staging of axillary lymph nodesNoncovered01,257,240609NADominates901002,680
Positron emission tomographyLung cancer (non–small cell)Covered3, 000835,400−698NADominates2,0382,7366.11857
Positron emission tomography (FDG)Breast cancer—staging and restagingCovered2,4001,948,800−759NADominates953022.29,050
Cryosurgery ablationPrimary treatment for clinically localized prostate cancer (stages T1–T3)Covered5,0001,383,600−2,189NADominates6,0178,20630.14,190
Warm-up wound therapy aka noncontact normothermic wound therapy (NNWT)Stage III and IV ulcersNoncovered01,119,120−14,7060.12Dominates5,7538,43103,810
BP, blood pressure; CNV, choroidal neovascularisation; FDG, fludeoxyglucose (18F); ICER, incremental cost-effectiveness ratio; MI, myocardial infarction; NA, not applicable/available; NIDCM, nonischemic dilated cardiomyopathy; QALY, quality-adjusted life-years; Tx, most effective therapeutic option.
* Reallocation through disinvestment in less cost-effective interventions by 50%, and investment in relatively cost-effective interventions by decreasing the size of the unserved eligible population by up to 50%.
We included both positive and noncoverage decisions included in NCDs made from 1999 through 2007. Of the 36 coverage decisions, 29 (81%) were positive and 7 (19%) were noncoverage decisions. We included noncoverage decisions to maximize the sample size and to evaluate the consequences of reallocation when using cost-effectiveness evidence as the sole criterion for resource allocation. While our previous research has demonstrated that covered technologies tend to be more cost-effective than noncovered ones, this was not necessarily the case for the technologies included here [
• Chambers J.D.
• Neumann P.J.
• Buxton M.J.
Does Medicare have an implicit cost-effectiveness threshold?.
]. Indeed, the least cost-effective technologies in our sample were covered, while noncovered technologies were typically associated with favorable ICERs (e.g., the noncovered intervention acupuncture for osteoarthritis had a reported ICER of $18,383) (Table 1). Prior to reallocation, 470,000 beneficiaries received the most effective of the interventions included in the cost-effectiveness analysis, at a cost of approximately$8 billion. Notably, the most cost-ineffective interventions were already in general used least frequently, with interventions associated with ICERs greater than $100,000 per QALY associated with negligible utilization rates (Table 1). For the first analysis, increasing utilization of dominant interventions had a substantial impact on aggregate health gain and cost-savings. When we considered the net present value of future commitments, and included dominant interventions without an estimate of incremental QALY gain, increasing the utilization of dominant interventions while maintaining the existing utilization of interventions associated with a positive ICER resulted in an additional 5.5 million beneficiaries receiving the most effective intervention, an additional 0.27 million QALYs gained, and approximately$12.9 billion of cost-savings (Table 2).
Table 2Estimated gains in aggregate health and cost-savings.
Net present value of future commitmentsYear following first use of the intervention
ReallocationAdditional beneficiaries receiving most effective intervention (millions) (3.s.f)Cost savings (millions) (3.s.f) ($)QALY gain (millions) (3.s.f)Additional beneficiaries receiving most effective intervention (millions) (3.s.f)Cost savings (millions) (3.s.f) ($)QALY gain (millions) (3.s.f)
All interventions (all health outcome units)
Increased utilization of dominant interventions
Increased utilization of dominant interventions through decreasing the size of the unserved population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by 50% (10%–90%).
5.54 (1.11– 9.96)13,000 (2,590– 23,300)0.269 (0.0538–0.484)5.54 (1.11–9.96)2,540 (507–4,560)0.269 (0.0538–0.484)
Reallocation of expenditures
Reallocation through disinvestment in less cost-effective interventions, achieved by decreasing the utilization of the intervention by 50% (10%–90%) and investment in relatively cost-effective interventions, achieved by decreasing the size of the unserved eligible population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by up to 50% (10%–90%).
11.1 (2.22–20.0)NA1.86 (0.373–3.35)6.73 (1.35–12.1)NA0.580 (0.116–1.04)
Interventions with QALY data
Increased utilization of dominant interventions
Increased utilization of dominant interventions through decreasing the size of the unserved population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by 50% (10%–90%).
1.11 (0.223–2.00) 8,470 (1,690– 15,200)0.269 (0.0538– 0.484)1.11 (0.222–2.00)1,430 (286– 2,570)0.269 (0.0538–0.484)
Reallocation of expenditures
Reallocation through disinvestment in less cost-effective interventions, achieved by decreasing the utilization of the intervention by 50% (10%–90%) and investment in relatively cost-effective interventions, achieved by decreasing the size of the unserved eligible population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by up to 50% (10%–90%).
6.14 (1.23–11.1)NA1.61 (0.323–2.91)2.10 (0.420–3.78)NA0.527 (0.105–0.949)
NA, not applicable/available; QALY, quality-adjusted life-year; 3.s.f, three significant figures.
* Increased utilization of dominant interventions through decreasing the size of the unserved population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by 50% (10%–90%).
Reallocation through disinvestment in less cost-effective interventions, achieved by decreasing the utilization of the intervention by 50% (10%–90%) and investment in relatively cost-effective interventions, achieved by decreasing the size of the unserved eligible population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by up to 50% (10%–90%).
For the second analysis, when we considered the net present value of future commitments and included dominant interventions without an estimate of incremental QALY gain, reallocation of expenditures resulted in an additional 11.1 million beneficiaries receiving the most effective intervention (Table 2). This corresponded to an additional 1.86 million QALYs, approximately 0.17 QALYs per beneficiary affected by the reallocation. The ICER of the marginal technology, that is, the least cost-effective intervention for which utilization was increased, was approximately $18,000 per QALY (bariatric surgery for the treatment of morbid obesity). When we reallocated expenditures in the year following the first use of the intervention, the findings were similar, although the magnitude of changes was smaller. The ICER of the marginal technology was approximately$3,300 per QALY (external counterpulsation therapy).
When we considered only interventions with an available estimate of incremental QALY gain, 25 interventions were included in the reallocation. Findings were broadly consistent with the analysis including interventions without a QALY gain estimate, although the magnitude of gain was smaller (Table 2).
The reallocation had a notable impact on the distribution of expenditures across diseases (Table 3). Following reallocation, a greater proportion of expenditures was directed to beneficiaries receiving an oncology-related intervention (approximately 43%), for example, positron emission tomography for various cancers. In contrast, a decreased proportion was directed to those receiving a cardiology-related intervention (approximately 34%), for example, implantable cardioverter defibrillators. Furthermore, a decreased proportion was directed to interventions categorized as other (approximately 24%), for example, warm-up wound therapy for ulcers. With respect to intervention type, following reallocation the proportion of beneficiaries receiving an intervention categorized as treatment, for example, ocular photodynamic therapy with verteporfin for macular degeneration, or diagnostic, for example, screening immunoassay fecal-occult blood test for colorectal cancer, increased (approximately 50% and 44%, respectively), while the proportion of beneficiaries receiving interventions categorized as other, for example, foot care for diabetic peripheral neuropathy with loss of protective sensation, decreased (approximately 7%). Following reallocation, a much greater proportion of expenditures was directed to beneficiaries receiving interventions with an eligible population of more than 1 million beneficiaries (approximately 78%), for example, external counterpulsation therapy, while a decreased proportion was directed to beneficiaries receiving an intervention with an eligible population of 50,000 to 1 million beneficiaries (approximately 22%), for example, cardiac rehabilitation programs for acute myocardial infarction and percutaneous transluminal coronary angioplasty, and less than 50,000 beneficiaries (0.2%), for example, intravenous immune globulin for bullous pemphigoid, respectively.
Table 3Distribution of resources before and after reallocation
Reallocation through disinvestment in less cost-effective interventions, achieved by decreasing the utilization of the intervention by 50% (10%–90%), and investment in relatively cost-effective interventions, achieved by decreasing the size of the unserved eligible population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by up to 50% (10%–90%).
of expenditures.
Prior to reallocationFollowing reallocation
Beneficiaries receiving most effective therapeutic option (3.s.f)Distribution (%)Beneficiaries receiving most effective therapeutic option (millions) (3.s.f)Distribution (%)
Disease area
Cardiology256,88054.73.8933.6
Oncology118,88025.54.9442.6
Other disease area94,26020.12.7623.8
Type of intervention
Treatment155,22033.05.7549.7
Diagnostic115,60024.65.0843.8
Other type of intervention199,20042.40.752 6.5
Size of untreated patient population
Large135,60028.89.0378.0
Medium332,78070.82.5321.8
Small1,640 0.30.0230 0.2
3.s.f, three significant figures.
* Reallocation through disinvestment in less cost-effective interventions, achieved by decreasing the utilization of the intervention by 50% (10%–90%), and investment in relatively cost-effective interventions, achieved by decreasing the size of the unserved eligible population, i.e., Medicare beneficiaries who were eligible for the intervention but who did not receive it, by up to 50% (10%–90%).

## Discussion

Coverage of cost-ineffective interventions generates relatively little health gain for the expenditure and suggests that existing resources could provide greater benefits if directed toward more cost-effective alternative interventions. As others have shown, using cost-effectiveness evidence can lead to more efficient resource allocation [
• Lord J.
• Laking G.
• Fischer A.
Health care resource allocation: is the threshold rule good enough?.
]. For this research we used an empirical approach to estimate efficiency gains by reallocating expenditures among interventions considered in NCDs by using a cost-effectiveness decision rule. This approach differs from studies that highlight inefficiencies by comparing expenditures across jurisdictions when adjusting for differences in populations [
• Matlock D.D.
• Peterson P.N.
• Heidenreich P.A.
• et al.
Regional variation in the use of implantable cardioverter-defibrillators for primary prevention: results from the National Cardiovascular Data Registry.
,
• Song Y.
• Skinner J.S.
• Bynum J.P.W.
• et al.
Regional variations in diagnostic practices.
]. While these studies highlight opportunities for efficiency gains, they do not suggest an approach to making care more efficient. In contrast, this study illustrates the potential to increase the aggregate health of Medicare beneficiaries while maintaining existing spending levels.
Our findings suggest that substantial efficiency gains are achievable by reallocating expenditures in accordance with cost-effectiveness evidence. Simply increasing the utilization of dominant interventions increases aggregate health gain, while generating addition resources, an approach in which no patients would receive a less effective therapeutic option. Reallocating expenditures in accordance with a broader definition of the cost-effectiveness evidence increases the number of beneficiaries receiving the most effective therapeutic option—though inevitably some patients would then receive a less effective option than their current therapy—and results in sizeable aggregate health gains. As we included dominant interventions without an estimate of incremental QALY gain in the analysis, aggregate population health gain estimates are likely conservative.
Reallocation also affects the distribution of expenditures across diseases. Following reallocation, a greater proportion of resources was directed to beneficiaries receiving oncology-related interventions and a lesser proportion to beneficiaries receiving cardiology-related interventions and diseases categorized as other. Interestingly, many cost-effective, oncology-related interventions in our sample were diagnostic imaging modalities (e.g., positron emission tomography for various cancers) and tests, rather than chemotherapies, which are often associated with high ICERs [
• Greenberg D.
• Earle C.
• Fang C.H.
• et al.
When is cancer care cost-effective? A systematic overview of cost-utility analyses in oncology.
].

### Limitations and Challenges

Many of the challenges of this research pertain to data limitations. Unlike other health technology assessment agencies—for example, the National Institute of Health and Clinical Excellence in the United Kingdom—the CMS does not independently perform cost-effectiveness analyses, nor does it require submission of cost-effectiveness evidence. Consequently, few of the cost-effectiveness estimates we used originated from CMS decision memos, and for the most part we relied on estimates identified in the cost-effectiveness literature. As a result, there may have been a lack of consistency among cost-effectiveness studies with respect to methodology, perspective, costing, country of study, and so on.
While necessary, we used ICD-9 diagnostic codes to identify eligible beneficiaries, though this approach was somewhat crude, as such codes do not sufficiently capture all factors that inform patient management in practice, for example, patient preference. The Medicare claims database provides a “snapshot” of interventions received by Medicare beneficiaries and does not distinguish between incident and prevalent cases. Without this information, identifying eligible beneficiaries is imprecise; for example, identifying a beneficiary with Parkinson’s disease is insufficient to confirm his or her eligibility for deep brain stimulation, as this treatment is indicated only once pharmaceutical management is no longer effective. For other interventions, this is less problematic (e.g., foot care for diabetic patients suffering from diabetic peripheral neuropathy with loss of protective sensation), as the ICD-9 diagnostic codes were sufficient to identify this patient population.
An important assumption was that eligible patients who did not receive the intervention instead received the study comparator. While we attempted to ensure that CMS deemed the comparator relevant by affirming that CMS discussed it in the decision memo, this assumption introduces potential bias. In many cases, despite eligibility, Medicare beneficiaries are likely to receive no therapy or an alternative therapy not included in the cost-effectiveness analysis. Thus, there is potential for the ICER to be either overestimated or underestimated. In future research, we plan to explore and try to validate this assumption, potentially utilizing clinical opinion.
The requirement to integrate data from various sources limits the accuracy of study findings. Indeed, our study highlights the challenges of using currently available data for this type of analysis. While US-based recommendations for the performance cost-effectiveness analyses exist, evidence suggests that they are followed inconsistently [
• Phillips K.A.
• Chen J.L.
Impact of the U.S. panel on cost-effectiveness in health and medicine.
].

### Policy Significance and Next Steps

In the United States, cost-effectiveness is often conflated with “rationing” and a reduction in health care provision. This research shows that rather than using it to reduce spending, it can identify how to increase aggregate population health while maintaining existing spending levels. Despite the limited number of considered interventions, our analysis illustrated that an additional 1.86 million QALYs (0.17 QALYs per affected beneficiary) were achievable, suggesting substantial gains in aggregate health if the policy were implemented on a larger scale.
The least cost-effective intervention to which more expenditure was allocated after reallocation was associated with an ICER of $18,000 per QALY (bariatric surgery for the treatment of morbid obesity), less than the often used benchmarks of value in the United States, for example,$50,000 to \$100,000 per QALY [
• Greenberg D.
• Neumann P.J.
Is it cost-effective? It depends on who you ask.
]. This finding suggests that there is much potential for Medicare to increase spending on interventions associated with ICERs judged highly cost-effective by traditional standards, while reducing spending on interventions deemed comparatively cost-ineffective [
• Weinstein M.C.
• Skinner J.A.
Comparative effectiveness and health care spending--implications for reform.
].
Our research is not offered as a precise accounting exercise, but rather as an illustration of the potential of this approach, which, while incorporating a number of assumptions, underscores the substantial aggregate health gains potentially achievable from using cost-effectiveness evidence to inform resource allocation. Indeed, gains may be even greater if one could better account for patient heterogeneity and if beneficiaries who would benefit most were prioritized for treatment.
For a number of included dominant interventions, positive expenditure was required in their first year of use, with aggregate cost-savings achieved in subsequent years, for example, foot care for diabetic patients with neuropathy. This finding emphasizes that considering interventions over a short time horizon may not adequately account for the potential positive financial impact on the entire health care system. The apparent underutilization of dominant interventions provides an opportunity for policymakers. Research suggests that underutilization may be due to a lack of physician referral, insufficient physician reimbursement, and perceived clinical benefits of the intervention, among other factors [
• Daly J.
• Sindone A.P.
• Thompson D.R.
• et al.
Barriers to participation in and adherence to cardiac rehabilitation programs: a critical literature review.
,
• Parkosewich J.A.
Cardiac rehabilitation barriers and opportunities among women with cardiovascular disease.
,
• Thomas R.J.
Cardiac rehabilitation/secondary prevention programs: a raft for the rapids: why have we missed the boat?.
].
Ideally, data available for the research in this article would be more abundant, and of higher quality and greater consistency. Preferably, available cost-effectiveness studies would include Medicare-specific direct costs and account for all relevant competing interventions. Costs would meet the specifications necessary to facilitate potential legislative action, for example, meet the standards of the Congressional Budget Office and CMS’s Office of the Actuary [

Congressional Budget Office. Our processes. Available from: http://www.cbo.gov/about/our-processes [Accessed June 8, 2012]

,

Centers for Medicare & Medicaid Services. CMS's Office of the Actuary. Available from: http://www.cms.gov/About-CMS/Agency-Information/CMSLeadership/Office_OACT.html [Accessed June 8, 2012].

]. Technologies should be evaluated over a time horizon for which costs and consequences, in terms of QALYs, are likely to differ, and reported on an annual basis. Ideally, studies would account for patient heterogeneity, with data available on the effectiveness of the intervention across beneficiaries in the indicated population.
Using a cost-effectiveness decision rule for resource allocation will impact the distribution of resources across Medicare beneficiaries. Despite the small number of included coverage decisions, the findings underscore the trade-offs inherent in resource allocation decisions, for example, the opportunity cost of prioritizing resources to a particular patient group. Indeed, the objective of maximizing aggregate health subject to a budget constraint is not entirely consistent with CMS’s decision-making criteria. Other factors including the nature, strength, and uncertainty of the available evidence; the availability of alternative interventions; potential impact of the decision on access to health care; statutory mandates to cover certain services; and political considerations are important as well [
• Keenan P.S.
• Neumann P.J.
• Phillips K.A.
Biotechnology and Medicare’s new technology policy: lessons from three case studies.
,
• Griffin S.
• Claxton K.
• Palmer S.
• Sculpher M.J.
Dangerous omissions: the consequences of ignoring decision uncertainty.
].
It is important to highlight that a minority of CMS’s coverage decisions for interventions are made at the national level in NCDs and that we included only coverage decisions associated with cost-effectiveness estimates. Consequently, this research is limited to a relatively small, and potentially unrepresentative, selection of interventions. Included interventions may not be those for which a reallocation of expenditures would yield the greatest efficiency gains. Indeed, targeting interventions with the largest potential health gains may be impractical if they reflect widely used and accepted services, despite poor cost-effectiveness. Rather, targeting interventions for which a change in therapeutic management is more feasible may be a more appropriate approach. We chose arbitrarily to use a 50% (range 10%–90%) change in utilization to simulate resource allocation. Accounting for the relative ease of investing/disinvesting in interventions would be one approach for advancing this research and may produce more realistic efficiency gain estimates.
As an illustrative exercise this research could be furthered by focusing on a select group of interventions for which high-quality data are available, potentially interventions from the same class or for the same indication. Having data of sufficient quality would allow the adoption of alternative approaches, for example, integer programming, or a stochastic process to account for uncertainty in the parameter estimates and for disease incidence.
A larger sample would allow a more comprehensive categorization of interventions and facilitate a more thorough examination of the consequences of using cost-effectiveness evidence to guide resource allocation. For example, as research has shown that society has a preference for the treatment of severe diseases, a variable to capture disease severity would be valuable [
• Coast J.
Is economic evaluation in touch with society’s health values?.
,
• Dolan P.
• Cookson R.
A qualitative study of the extent to which health gain matters when choosing between groups of patients.
,
• Nord E.
• Richardson J.
• Street A.
• et al.
Maximizing health benefits vs egalitarianism: an Australian survey of health issues.
,
• Ubel P.A.
Pricing Life: Why It’s Time for Health Care Rationing.
]. By better understanding Medicare beneficiaries’ resource allocation preferences, the appropriateness of alternative resource use patterns could be evaluated.

## Conclusions

While the US health care system has not embraced the use of cost-effectiveness evidence to inform health care resource allocation, this research illustrates the potential value and consequences of such an approach. While it is apparent that available data present challenges with this methodology, our research illustrates that substantial health gains are achievable from a reallocation of expenditures within existing spending levels.
Source of financial support: This research was funded in part by a grant from the Commonwealth Fund (Grant No. 20090203).

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# 7.5: Special Types of Linear Systems

Created by: CK-12

Solutions to a system can have several forms:

• One intersection
• Two or more solutions
• No solutions
• An infinite amount of solutions

## Inconsistent Systems

This lesson will focus on the last two situations: systems with no solutions or systems with an infinite amount of solutions.

A system with parallel lines will have no solutions.

Remember from chapter 5 that parallel lines have the same slope. When graphed, the lines will have the same steepness with different $y-$intercepts. Therefore, parallel lines will never intersect, thus they have no solution.

Algebraically, a system with no solutions looks like this when solved.

$\begin{cases}4y=5-3x\\6x+8y=7 \end{cases}$

The first equation in this system is “almost” solved for $y$. Substitution would be appropriate to solve this system.

$\begin{cases}4y=5-3x\\6x+8y=7 \end{cases} \rightarrow \quad \begin{cases}y=\frac{5}{4}-\frac{3}{4} x\\6x+8y=7 \end{cases}$

Using the Substitution Property, replace the $y-$variable in the second equation with its algebraic expression in equation #1.

$&&6x+8\left ( \frac{5}{4}-\frac{3}{4} x \right )&=7\\\text{Apply the Distributive Property.} && 6x+10-6x&=7\\\text{Add like terms.} && 10&=7$

You have solved the equation correctly, yet the answer does not make sense.

When solving a system of parallel lines, the final equation will be untrue.

Because $10 \neq 7$ and you have done your math correctly, you can say this system has “no solutions.”

A system with no solutions is called an inconsistent system.

## Consistent Systems

Consistent systems, on the contrary, have at least one solution. This means there is at least one intersection of the lines. There are three cases for consistent systems:

• One intersection, as you have practiced the majority of this chapter
• Two or more intersections, as you will see when a quadratic equation intersects a linear equation
• Infinitely many intersections, as with coincident lines

Coincident lines are lines with the same slope and $y-$intercept. The lines completely overlap.

When solving a consistent system involving coincident lines, the solution has the following result.

$\begin{cases}x+y=3\\3x+3y=9 \end{cases}$

Multiply the first equation by –3: $\begin{cases}-3(x+y=3)\\3x+3y=9\end{cases} \rightarrow \quad \begin{cases}-3x-3y=-9\\3x+3y=9 \end{cases}$

$0=0$

There are no variables left and you KNOW you did the math correctly. However, this is a true statement.

When solving a system of coincident lines, the resulting equation will be without variables and the statement will be true. You can conclude the system has an infinite number of solutions. This is called a consistent-dependent system.

Example 1: Identify the system as consistent, inconsistent, or consistent-dependent.

$3x-2y&=4\\9x-6y&=1$

Solution: Because both equations are in standard form, elimination is the best method to solve this system.

Multiply the first equation by 3.

$3(3x-2y=4)&&9x-6y=12\\& \qquad \qquad \qquad \qquad \Rightarrow & \qquad\\9x-6y=1&&9x-6y=1$

Subtract the two equations.

$& \ \ 9x-6y=12\\& \underline{\;\; 9x-6y=1 \;\;}\\& \qquad \quad \ 0=11 \quad \text{This Statement is not true.}$

This is an untrue statement; therefore, you can conclude:

1. These lines are parallel.
2. The system has no solution.
3. The system is inconsistent.

Example 2: Two movie rental stores are in competition. Movie House charges an annual membership of $30 and charges$3 per movie rental. Flicks for Cheap charges an annual membership of $15 and charges$3 per movie rental. After how many movie rentals would Movie House become the better option?

Solution: It should already be clear to see that Movie House will never become the better option, since its membership is more expensive and it charges the same amount per move as Flicks for Cheap.

The lines that describe each option have different $y-$intercepts, namely 30 for Movie House and 15 for Flicks for Cheap. They have the same slope, three dollars per movie. This means that the lines are parallel and the system is inconsistent.

Let’s see how this works algebraically.

Define the variables: Let $x=$ number of movies rented and $y=$ total rental cost

$\begin{cases}y=30+3x\\y=15+3x \end{cases}$

Because both equations are in slope-intercept form, solve this system by substituting the second equation into the first equation.

$15+3x=30+3x\Rightarrow15=30$

This statement is always false. Therefore, the system is inconsistent with no solutions.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

1. Define an inconsistent system. What is true about these systems?
2. What are the three types of consistent systems?
3. You graph a system and see only one line. What can you conclude?
4. You graph a system and see the lines have an intersection point. What can you conclude?
5. The lines you graphed appear parallel. How can you verify the system will have no solution?
6. You graph a system and obtain the following graph. Is the system consistent or inconsistent? How many solutions does the system have?

In 7 – 24, find the solution of each system of equations using the method of your choice. Please state whether the system is inconsistent, consistent, or consistent-dependent.

1. $3x-4y=13\!\\y=-3x-7$
2. $4x+y=3\!\\12x+3y=9$
3. $10x-3y=3\!\\2x+y=9$
4. $2x-5y=2\!\\4x+y=5$
5. $\frac{3x}{5}+y=3\!\\1.2x+2y=6$
6. $3x-4y=13\!\\y=-3x-7$
7. $3x-3y=3\!\\x-y=1$
8. $0.5x-y=30\!\\0.5x-y=-30$
9. $4x-2y=-2\!\\3x+2y=-12$
10. $3x+2y=4\!\\-2x+2y=24$
11. $5x-2y=3\!\\2x-3y=10$
12. $3x-4y=13\!\\y=-3x-y$
13. $5x-4y=1\!\\-10x+8y=-30$
14. $4x+5y=0\!\\3x=6y+4.5$
15. $-2y+4x=8\!\\y-2x=-4$
16. $x-\frac{y}{2}=\frac{3}{2}\!\\3x+y=6$
17. $0.05x+0.25y=6\!\\x+y=24$
18. $x+\frac{2y}{3}=6\!\\3x+2y=2$
19. Peter buys two apples and three bananas for $4. Nadia buys four apples and six bananas for$8 from the same store. How much does one banana and one apple costs?
20. A movie rental store, CineStar, offers customers two choices. Customers can pay a yearly membership of $45 and then rent each movie for$2, or they can choose not to pay the membership fee and rent each movie for $3.50. How many movies would you have to rent before membership becomes the cheaper option? 21. A movie house charges$4.50 for children and $8.00 for adults. On a certain day, 1200 people enter the movie house and$8,375 is collected. How many children and how many adults attended?
22. Andrew placed two orders with an internet clothing store. The first order was for 13 ties and four pairs of suspenders, and it totaled $487. The second order was for six ties and two pairs of suspenders, and it totaled$232. The bill does not list the per-item price but all ties have the same price and all suspenders have the same price. What is the cost of one tie and of one pair of suspenders?
23. An airplane took four hours to fly 2400 miles in the direction of the jet-stream. The return trip against the jet-stream took five hours. What were the airplane’s speed in still air and the jet-stream's speed?
24. Nadia told Peter that she went to the farmer’s market, that she bought two apples and one banana, and that it cost her $2.50. She thought that Peter might like some fruit so she went back to the seller and bought four more apples and two more bananas. Peter thanked Nadia, but he told her that he did not like bananas, so he would pay her for only four apples. Nadia told him that the second time she paid$6.00 for the fruit. Please help Peter figure out how much to pay Nadia for four apples.

Mixed Review

1. A football stadium sells regular and box seating. There are twelve times as many regular seats as there are box seats. The total capacity of the stadium is 10,413. How many box seats are in the stadium? How many regular seats?
2. Find an equation for the line perpendicular to $y=-\frac{3}{5} x-8.5$ containing the point (2, 7).
3. Rewrite in standard form: $y=\frac{1}{6} x-4$.
4. Find the sum: $7 \frac{2}{3}+\frac{4}{5}$.
5. Divide $\frac{7}{8} \div -\frac{2}{3}$.
6. Is the product of two rational numbers always a rational number? Explain your answer.

8 , 9

Feb 22, 2012

Jul 08, 2014


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# Mark and Ann together were allocated n boxes of cookies to sell for a

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Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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15 Jun 2016, 01:30
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Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15
[Reveal] Spoiler: OA

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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15 Jun 2016, 04:59
if n =11
mark sold 1 box
and Ann sold 9 boxes
total 10 < 11

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Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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15 Jun 2016, 05:23
3
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4
This post was
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Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Mary sold n-10 boxes.
Ann sold n-2 boxes.

They have each sold at least one box.

so n-10 is more than or equal to 1. Also n-2 is more than or equal to 1
n>= 11 [2nd condition only gives us n>=3. We already have this info when we say n>=11]

Now

Total boxes are n only.

so (n-10+n-2) < n

2n-12 < n
n <12

Only value. n = 11

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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10 Oct 2016, 04:49
1
This post was
BOOKMARKED
Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Quote:
Mark and Ann together were allocated n boxes of cookies

M + A = n
Quote:
Mark sold 10 boxes less than n

M = n - 10
Quote:
Ann sold 2 boxes less than n.

A = n - 2
Quote:
Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n

M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11

So, Answer will be (A) 11

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Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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28 Nov 2016, 05:23
Sorry, I ment the part where

"M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11"

I understand how 2n-12<n is received, but n<11? Is it a typo or am I misunderstanding the concept?

As I see it, it should be 2n-12 < n <=> 2n - n < 12 <=> n < 12. Got a little bit confused

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Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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28 Nov 2016, 06:24
1
KUDOS
MrBarksdale wrote:

Sorry, I ment the part where

"M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11"

I understand how 2n-12<n is received, but n<11? Is it a typo or am I misunderstanding the concept?

As I see it, it should be 2n-12 < n <=> 2n - n < 12 <=> n < 12. Got a little bit confused

Its a typo. n < 12 is right.
only option A suffices the condition

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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29 Nov 2016, 15:52
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Expert's post
Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

We are given that Mark sold 10 boxes less than n and Ann sold 2 boxes less than n, and that together they have sold less than n boxes. We can create the following inequality:

n - 10 + n - 2 < n

2n - 12 < n

n < 12

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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05 Apr 2017, 21:49
I got it wrong. I think that the catch in this question is to realize that it's an inequality problem.

If you make an equation out of it, you get 12. Else, if you appropriately decode the question, you set up the inequality and you get that $$n<12$$.

$$n-10+n-2=n\\ n-10+n-2<n$$
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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09 Sep 2017, 10:58
Experts please correct me if I have done this wrongly:

Both have sold at least 1 box but together they have sold less than n boxes.

2<=(n-10)+(n-2)<n
2<=2n-12<n
1<=n-6<n/2
7<=n<(n+12)/2

I thought that answer would come from: n< (n+12)/2
From the first part of the inequality, we know that n<=7

2n<n+12
n<12

Only one option fits in. (11)

My query is can we solve only one part of the inequality like I did above or it will impact the answer? Bunuel it would be great if you could give your view about the solution above.
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18 Oct 2017, 04:59
Oh my god... it seems that only me don't understand why mark sold n-10 and ann sold n-2... the title says that mark sold 10 boxes less than n. what does that mean? any expert can told me...

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a [#permalink]

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18 Oct 2017, 05:22
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Expert's post
HHXX wrote:
Oh my god... it seems that only me don't understand why mark sold n-10 and ann sold n-2... the title says that mark sold 10 boxes less than n. what does that mean? any expert can told me...

Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Step-by-step:

1. Mark sold 10 boxes less than n --> Mark sold n - 10 boxes;

2. Ann sold 2 boxes less than n --> Ann sold n - 2 boxes;

3. Mark sold at least one box of cookies: $$n - 10 \geq 1$$ -->$$n \geq 11$$;

4. Together they have sold less than n boxes: $$(n - 10) + (n - 2) < n$$ -->$$n < 12$$.

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a   [#permalink] 18 Oct 2017, 05:22
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text:
Q

The line y is equal to m x plus 1 is a tangent to the curve y raised to 2 is equal to 4x if the value of m is (A) 1 (B) 2 (C) 3 (D) 1 2

21) The line y is equal to  $mx+1$ is a tangent to the curve $y^2 = 4x$if the value of m is
(A) 1

(B) 2

(C) 3

(D)1/2

Views

Standard equation of the straight line
y = mx + c
Where m is lope and c is constant
By comparing it with equation , y = mx + 1
We find that m is the slope
Now,
we know that the slope of the tangent at a given point on the curve is given by $\frac{dy}{dx}$
Given the equation of the curve is
$y^2 = 4x$
$2y\frac{dy}{dx} = 4\\ \frac{dy}{dx} = \frac{2}{y}$
Put this value of m in the given equation
$y = \frac{2}{y}.\frac{y^2}{4}+1 \ \ \ \ \ \ \ \ \ \ (\because y^2 = 4x \ and \ m =\frac{2}{y})\\ y = \frac{y}{2}+1\\ \frac{y}{2} = 1\\ y = 2$
$m = \frac{2}{y} = \frac{2}{2} = 1$
Hence, value of m is 1


text:
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text:
# Python program to solve matrix-chain multiplication

An assignment at school required me to write a Python program for this task:

In the matrix-chain multiplication problem, we are given a sequence of matrices A(1), A(2), …, A(n). The aim is to compute the product A(1) …, A(n) with the minimum number of scalar multiplications. Thus, we have to find an optimal parenthesization of the matrix product A(1) …, A(n) such that the cost of computing the product is minimized.

Here is my solution to this task:

def matrix_product(p):
"""
Return m and s.

m[i][j] is the minimum number of scalar multiplications needed to compute the
product of matrices A(i), A(i + 1), ..., A(j).

s[i][j] is the index of the matrix after which the product is split in an
optimal parenthesization of the matrix product.

p[0... n] is a list such that matrix A(i) has dimensions p[i - 1] x p[i].
"""
length = len(p) # len(p) = number of matrices + 1

# m[i][j] is the minimum number of multiplications needed to compute the
# product of matrices A(i), A(i+1), ..., A(j)
# s[i][j] is the matrix after which the product is split in the minimum
# number of multiplications needed

m = [[-1]*length for _ in range(length)]
s = [[-1]*length for _ in range(length)]

matrix_product_helper(p, 1, length - 1, m, s)

return m, s

def matrix_product_helper(p, start, end, m, s):

"""
Return the minimum number of scalar multiplications needed to compute the
product of matrices A(start), A(start + 1), ..., A(end).

The minimum number of scalar multiplications needed to compute the
product of matrices A(i), A(i + 1), ..., A(j) is stored in m[i][j].

The index of the matrix after which the above product is split in an optimal
parenthesization is stored in s[i][j].

p[0... n] is a list such that matrix A(i) has dimensions p[i - 1] x p[i].
"""
if m[start][end] >= 0:
return m[start][end]

if start == end:
q = 0
else:
q = float('inf')
for k in range(start, end):
temp = matrix_product_helper(p, start, k, m, s) \
+ matrix_product_helper(p, k + 1, end, m, s) \
+ p[start - 1]*p[k]*p[end]
if q > temp:
q = temp
s[start][end] = k

m[start][end] = q
return q

def print_parenthesization(s, start, end):
"""
Print the optimal parenthesization of the matrix product A(start) x
A(start + 1) x ... x A(end).

s[i][j] is the index of the matrix after which the product is split in an
optimal parenthesization of the matrix product.
"""
if start == end:
print('A[{}]'.format(start), end='')
return

k = s[start][end]

print('(', end='')
print_parenthesization(s, start, k)
print_parenthesization(s, k + 1, end)
print(')', end='')

n = int(input('Enter number of matrices: '))
p = []
for i in range(n):
temp = int(input('Enter number of rows in matrix {}: '.format(i + 1)))
p.append(temp)
temp = int(input('Enter number of columns in matrix {}: '.format(n)))
p.append(temp)

m, s = matrix_product(p)
print('The number of scalar multiplications needed:', m[1][n])
print('Optimal parenthesization: ', end='')
print_parenthesization(s, 1, n)

Here is an example output:

Enter number of matrices: 3
Enter number of rows in matrix 1: 10
Enter number of rows in matrix 2: 100
Enter number of rows in matrix 3: 5
Enter number of columns in matrix 3: 50
The number of scalar multiplications needed: 7500
Optimal parenthesization: ((A[1]A[2])A[3])

Here is the time taken for this output:

0:00:15.332220 seconds.

So, I would like to know whether I could make this program shorter and more efficient.


text:
gluonts.nursery.anomaly_detection.supervised_metrics.utils module¶

gluonts.nursery.anomaly_detection.supervised_metrics.utils.labels_to_ranges(labels: List[bool]) → List[range][source]

Converts the given list of labels to list of anomaly (defined by positive label) ranges.

Parameters

labels – Boolean list of labels.

Returns

Return type

List of ranges.

gluonts.nursery.anomaly_detection.supervised_metrics.utils.labels_to_ranges_numba[source]

Converts the given list of labels to list of anomaly (defined by positive label) ranges where range is represented by a pair of integers (to make numba work).

Parameters

labels – Boolean list of labels.

Returns

Return type

List of ranges.

gluonts.nursery.anomaly_detection.supervised_metrics.utils.range_overlap(left_range: range, right_range: range) → bool[source]

Checks if two ranges have an overlap. Here each range is assumed to be consecutive, i.e., step field of range is always 1.

Parameters
• left_range

• right_range

Returns

Return type

True or False depending on the overlap.

gluonts.nursery.anomaly_detection.supervised_metrics.utils.ranges_to_singletons(ranges: List[range]) → List[range][source]

Convenient function to convert list of consecutive ranges to list of singleton ranges.

Parameters

ranges – List of ranges.

Returns

Return type

List of singleton ranges.