This can be better understood through a simple linear transformation of the numerical scale of the utilities. Here, the value 0 is transformed into 1, 0. The co-ordinates A' and B' in Figure 3 correspond to the same co-ordinates shown in Figure 2 ; the only difference is that the utilities Y' are expressed on a scale running from 1 to 3 instead of from 0 to 1.

This inconsistency highlights the fact that the ratios established between the numerical values QALYs assigned by the multiplicative model to the co-ordinates in the Cartesian plane do not correspond to the relationships that these elements have with each other in reality. The reason for this inconsistency is very simple: multiplication and division with interval scales are permissible only with respect to the intervals and not with respect to the scale values [ 13 ].

Multiplying years of life by utilities makes no sense because the utilities are expressed on an interval scale. The Theorem is of fundamental importance in Euclidean Geometry, where it serves as a basis for the definition of distance between two points. A complex number z is an ordered pair of real numbers a , b that has a representation-point P in a plane with co-ordinates a , b Figure 4. The number a is called the real part of z while b is called the imaginary part of z. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

The magnitude or modulus of a complex number z is denoted z and defined as the distance from 0 to P , that is:. The idea of a QALY is in fact similar to a complex number, as it is made up of a real part Length of Life and an imaginary part Utility , in the sense that utilities are intangible and not susceptible to direct observation. The magnitude of the QALY cannot be calculated by multiplying both components but by determining the modulus of such complex composition. Unlike the multiplicative model, the Complex Number model for the calculation of the QALYs is based on operations implying the intervals separating the values on the utility and time scales b and a , respectively and not the values themselves.

The QALYs thus obtained respect the spatial relationship between the numerical values assigned to the elements in the Cartesian plane. We can see that A equals 1. The Complex Number model assigns numerical values to the elements in question so that these maintain the invariance of their ratios, thus respecting the attribute's bi-dimensionality.

Because the utility value lies between 0 and 1, the resulting distance obtained from the Complex Number approach may be dominated by the variable on the X axis Length of Life if it takes a large value i.

One way around this problem is to similarly bound the length of life variable within 0—1 as well. Likewise, to set the maximum possible QALY value at 1, the c value initially obtained by this procedure must be divided by. To reverse the standardization, the value resulting from this operation must be multiplied by the actual length of life.

The new equation for calculating QALYs is therefore as follows:. Table 2 presents the QALYs corresponding to different values on the utility scale when the number of years remains constant and equal to 1. The most noteworthy aspect of this table is that, unlike in the multiplicative model, whenever utility reaches its minimum value the value of the resulting QALY is greater than 0.

To some extent, it could be said that this new way of calculating the value of QALYs gives greater relative weight to the amount of life, making it impossible for this to be reduced to nullity. When dealing with the gains obtained by health care interventions, this seems appropriate as it adds additional weight to the mere fact of survival. For utilities expressed with a level of precision of 0. For example, 2 years lived in a state of health with a utility of 0.

Figure 7 illustrates with a solid line the function which, based on the Complex Number model, relates utility Y-axis and QALYs X-axis when time is set to 1 year. The function takes the form of a hyperbola in which the vertex is located at the co-ordinate 0. Hyperbolic functions like this arise in many problems of mathematics and mathematical physics in which equations involving appear [ 16 ].

Figure 7 also illustrates with a broken line the function which relates utility and QALYs according to the multiplicative model; in this case, the function is linear and intersects the X-axis at the value 0, indicating that when the utility of a state of health is equivalent to 0, the value of the QALY will also be 0, regardless of the number of years lived.

## [BEST SELLING] Cost-Value Analysis in Health Care: Making Sense Out …

QALYs are often represented as the rectangular area resulting from the product of their constituent factors: utility and time. But that is like comparing apples with oranges: quality utility and quantity time of life are of essentially different "data-types" and cannot be combined by a simple product of their numerical values.

Contrary to what is assumed, the very nature of this arithmetical operation means that the resulting QALY value is not expressed in the same units as the time scale years. The QALYs in the multiplicative model are in fact square units, i. The Complex Number model, while capturing the dimensional nature of the time-utility composite, is expressed in linear units, which are also directly comparable to years of life on the time scale: the value of the complex number is equivalent to the distance between the co-ordinate 0,0 of the Cartesian plane and the representation-point P of this number see Figure 4 ; the standardization of this distance by means of its division by means we are using the same units as the time scale.

As Figure 2 showed, when the multiplicative model was applied, the ratio between co-ordinates B and A was equal to 4; however, when the Complex Number model was applied, the ratio between B and A was equal to 2 Figure 5. Thus the ratio is 2 for the Complex Number Model, and 4 2 2 for the multiplicative model solution. This article has pointed out, in addition, that the Complex Number model respects the bi-dimensional spatial relationships established between different combinations of time and utility when these are represented as points or co-ordinates on the Cartesian plane, thus making it a more suitable geometrical model than the multiplicative model for calculating QALYs.

This new means of calculating QALYs does not mean they will be significantly more difficult to estimate, and they can be used and interpreted as before. Table 3 , for example, shows the incremental QALYs pictured in table 1 with the new method of calculation based on the Complex Number model. The new formulation of the QALYs is noteworthy for its less dramatic adjustment of years of life than that implied by the multiplicative model. The disputed degree of validity and reliability of the measurement of utilities as well as the doubts over the possibility of extending it to all of the individuals in the population [ 10 ] are the greatest obstacles so far to assigning the same relative weight to the quality and quantity of life.

By giving the QALY a real part Length of Life and an imaginary part Utility , the Complex Number model provides an appropriate theoretical framework for combining time and health preferences into a single index, yet limiting the power of the latter to modulate the final value of the indicator. QALYs continue to represent the paradigm of the integration of the bio-medical and the psycho-social models and this new approach to calculating QALYs may represent a further step in the integration of different health dimensions without giving up the use of mathematically sound procedures for this purpose.

In attempting to measure the physical attributes of objects or people, scientists and laypeople take care to measure just one attribute of the target at a time. This is because the measurement problem consists of developing procedures that would permit a number to be assigned to each quantity of the attribute in such a way that certain relations between the numbers reflect analogous relations between the quantities.

This procedure is possible for those attributes whose quantities form a one-dimensional series. In this case, the attribute can be conceived as a one-dimensional continuum, analogous to the straight line in geometry. Assigning numbers to quantities is analogous to specification of the position of points on the straight line.

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## Cost-value Analysis In Health Care: Making Sense Out Of Qalys

This focus on one attribute or dimension is usually referred to as unidimensionality. For those attributes, like the QALY, whose quantities form a multidimensional series the measurement problem is more complex. Here the problem is to develop procedures to assign a set of numbers to each quantity quality and quantity of life so that the numbers, when considered in terms of a specified geometrical system, reflect relations among the various quantities. The number of numbers to be assigned to each quantity corresponds to the dimensionality of the attribute. Thus the analogy is now with a multidimensional geometrical space.

Whereas in unidimensional measurement, the attribute corresponds to the straight line a unidimensional space , and the quantity to a point on the line, in multidimensional measurement or scaling, the attribute corresponds to an n-dimensional space, and the quantity to a point in that space. The geometrical models that might be employed are, of course, of many different varieties.

Besides the metric spaces, such as the Euclidean model, other non-Euclidean models might be used [ 12 ]. Regardless of the particular model used, the nature of the problem remains essentially the same. The meaning of the numbers assigned to the elements in the model is specified by the model. Rules of correspondence are established, relating elements and properties of the model to observable data, thus converting the model into a testable theory.

If the theory is verified, numbers are assigned to the quantities of the multidimensional attribute as specified by the theory.

Once this is done, the multidimensional attribute in question has been measured [ 12 ]. WHO Chronicle , 1: 6— N Eng J Med , — A conceptual Model of Patient Outcomes. JAMA , 59— Pharmacoeconomics , — Macran S, Kind P: "Death" and the valuation of health-related quality of life. Med Care , — Lockwood EH: The Hiperbola.

### Introduction

Download references. Our thanks to Dr. Michael Herdman, for his careful reading of the manuscript and helpful suggestions; an to Dr. Alfonso Casado, for his contribution to an early phase of this work. Correspondence to Luis Prieto.

## Cost-Value Analysis of Health Interventions: Introduction and Update on Methods and Preference Data

LP and JAS actively participated in the discussions that lead to the present manuscript. Both authors participated in writing the manuscript and read and approved the final one. Reprints and Permissions.

Search all BMC articles Search. The integrating role of QALYs In an attempt to integrate the biomedical and psycho-social models, a new approach has been proposed which can be labelled the bio-psycho-social model [ 5 , 6 ]. The application of QALYs in the economic analysis of health-care activities Over the last two decades, QALYs have become increasingly widely used as a measure of health outcomes. This book is a comprehensive account of what it means to try to quantify health in distributing resources for health care. It examines the concept of QALYs Quality Adjusted Life Year which supposedly makes it more accurate to talk about life in terms of both quality and quantity of years lived when referring to health care policy.

It offers an elegant new approach to comparing the costs and benefits of medical interventions. Cost-Utility Analysis CUA is a method designed by economists to aid decision makers distribute scarce resources to areas of health care where they will yield the greatest benefits. Erik Nord questions the feasibility of measuring patients' quality of life meaningfully in numerical terms, as CUA presupposes.

He presents an alternative approach called cost-value analysis in which representative samples of the general public express preferences between different health-care programs. In this approach, subjects are allowed to include concerns for fairness that go beyond concerns for efficiency of conventional health economics.

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