348 trends in analytical chemistry, uol. I, no. >5, 1982

Analytical chemistry: an economic view? The quality of information obtained in a chemical analysis should be weighed against the cost of obtaining it. The principles of economics may therefore be applied to analytical chemistry in order to evaluate analytical

programmes in terms of costs and benefits.

D. L. Massart Brussels, Belgium

In the course of an excellent lecture’ given to the Analytical Division of the Royal Society of Chemistry, Olav Christie asked the question: ‘Why do we produce data?’ He offered three alternative answers:

- because everyone does. - because my instrument produces them (or as

Wangersky2 put it in a recent article in this journal: ‘a lot of mediocre science is done just to keep big instruments running’).

- because we think that data are carriers of relevant information.

It is, of course, to be hoped that most of us can give the last answer. It is equally to be hoped that we would add to this answer: ‘and because we think the value of the information is worth more than the cost of obtaining it’. This means that one should introduce some element of economics. In this article I want to show that the production of analytical information is subject to some of the laws of economics.

One of the most frequent dilemmas with which an analytical chemist is confronted is to decide whether to improve a particular analytical method by, for instance, increasing its speed, its precision or its limit of detection. This invariably means an investment, at least of time, but often of new instrumentation to develop the new method. One complaint frequently heard from analytical chemists in both industry and in large semi-routine laboratories (e.g. clinical laboratories, government control laboratories) is that they have no idea how to achieve an evaluation of the proposed investment in terms of economics. One of the main difficulties they face in doing this is to assess the value of the analytical product, i.e. information.

There are-certainly situations where such an analysis would be necessary. The development of heavy and sophisticated instrumentation is one such situation. Another is clinical analysis, which happens to be the field where the practitioners have best understood the need for techniques from economics, such as cost-benefit analysis. A few examples concerning screening tests will serve to illustrate this. Sandler3, for instance, found that routine tests such as laboratory tests (ECGs, etc.) are of help in the diagnosis of an 0 165~9936/82/cooooMM/$01 .oo

illness in only 5% of the cases studied and he came to the conclusion that this is not sufficient to justify the high cost of these tests. Durbridge4 investigated the value of routine screening tests carried out at the moment of admission of a patient into a hospital. His conclusion was that not only were the tests responsible for greatly increased costs but, in almost every case, they did not yield medical benefits. While we may not agree with the conclusions of these authors, their work does demonstrate that there is a need for evaluation of the utility of analytical programs or for cost-benefit analysis.

It is difficult for analytical chemists to calculate costs. In fact, I do not know of a single book intended for them in which cost is an important subject. This is a pity and, in the hope of stimulating research and education in this direction, I will attempt to show here that it may be rewarding for an analytical chemist to try and evaluate his product in terms of economics as well as in terms of technical characteristics such as precision, limit of detection, etc.

For this reason I will discuss some principles of economics and apply them to analytical chemistry. Since a complete discussion is not possible in a short article such as this, I will focus attention on one law, which is of great importance in economic theory, the

Utility

C

\

Sacks of grain (or analytical information ! 1

Fig. I. The law of marginal utility. The scale of marginal utility is twice the scale of utility.

0 1982 Ekvier Scientific Publishing Company

trends irz analytical chemists, vol. I, no. 15, 1982

law of diminishing marginal utility, or rather on three related ‘marginal’ laws.

The law of diminishing marginal utility and related laws

The law of diminishing marginal utility was derived by Gosser?, one of the more important economists of the Austrian school, a school which led economic thinking for a long time. Gossen concluded that the marginal utility of a product diminishes when the fulfilment of a need increases. Another economist ofthe same school illustrated this with a simple example: In a tropical forest there lives a colonist who has harvested five sacks of grain. He needs the first sack to remain alive, the second to be sufficiently nourished, the third to keep chickens, the fourth to be able to ma.ke brandy and the fifth to befriend the parrots. The law of diminishing marginal utility is shown in Fig. 1.

It is not difficult to translate this into analytical terms. If one analyses mercury in fish and one is able only to arrive at the conclusion that the concentration is less than 0.5 ppm, this will enable one to decide that the fish may be sold; ifone is able to obtain more analytical information and determine that it is 0.4 ppm, this will satisfy a government agency and obtain one a certificate; still more information in the form of the more precise result of 0.42 ppm will allow one to publish the method, and a result of 0.423 ppm will permit one to boast to colleagues about one’s skill as an analyst.

Other laws in which the word ‘marginal’ is used and which also relate to analytical chemistry are the law of marginal returns and the law of marginal costs. The following example is taken from a handbook on economics’. When a gardener works on his land, the production obtained will grow as he puts more work into it. At the beginning, the additional production per day (the marginal return) may grow, but soon this gain per additional day ofwork decreases. In the same way, the marginal costs per unit of product may initially decrease, but will, after a time, increase. This is illustrated by the example of Table I and Fig. 2.

349

Margi nol Cost

\ e

Days

Fig. 2. Marginal production (kg) and cost (BF) as afunction of the investment (in days) as derivedfrom Table I.

The ‘marginal’ laws in analytical chemistry In the preceding section it was demonstrated

intuitively that an increase in precision obeys the law of diminishing marginal utility. In the same intuitive way it is simple to understand that an increase in detection power (or to put it in more familiar analytical terms, a decrease in detection limit) may obey such a law. One observes that at first the utility increases as the detection limit decreases.

In the series titrimetry + calorimetry + flame atomic absorption + neutron activation analysis, the possibilities of applying the method for trace element analysis increase, but so does the investment.

If one considers as returns the number of determinations submitted to the average laboratory which can be carried out with a specific instrument then a plot such as Fig. 2 would result. The additional investment for the calorimeter enables a much greater number of analyses to be carried out, i.e. an appreciable marginal return, as does the purchase of the AAS. However, for average laboratories the marginal return from the installation of neutron activation may prove small (how many times does one need an analysis of Au in sea water?) and certainly the marginal cost would greatly increase. Clearly, it is not easy to quantify the numbers in order to produce a

TABLE I. Production and cost assuming a futed cost of 1000 BF* and constant variable cost/day (from Ref. 6)

Number of

working days

1

2

3 4

5 6

7

8

9

10

11

12

13

*Belgian francs

Total Marginal Total var-

production production iable cost

kg kg BF

37 - 150

88 51 300

147 59 450 216 69 600

290 74 750 366 76 900

441 75 1050 512 71 1200

567 55 1350 610 43 1500 627 17 1650 636 9 1800 637 1 1950

Total cost BF

1150

1300

1450 1600

1750 1900

2050

2200

2350

2500

2650

2800

2950

Marginal Marginal

cost cost/unit BF BF

_ -

150 2.94

150 2.54 150 2.17

150 2.03 150 1.97

150 2.00 150 2.11

150 2.73 150 3.49 150 8.82 150 16.67 150 150.00

3 5 0 trends in analytical chemistry, vol. 1, no. 15, 1982

graph relating returns or utility (as determined from the detection limit) to investment. Since we, as analytical chemists, are professionally trained to produce data and to believe only what these data prove, this simple example may not be very convincing because of the lack of verifiable data. Moreover we are usually also very practical people and since a law in which one cannot put the necessary numbers is not very helpful, I will now try to give some areas of analytical chemistry where the necessary data can be obtained.

A very simple example can be found in the precision with which the concentration of a sample is determined. By carrying out n replicate measurements, the cost of the analysis increases by a factor n, while the precision with which the real concentration is estimated increases by a factor of X/n (because the standard deviation on the mean decreases with the same factor). To avoid the semantic difficulty of a precision which increases, while its numerical value decreases and to be able to restate the result as a marginal law, we can state that the information obtained is inversely related to the numerical value of the precision (in fact, this can be proved easily by the use of information theory). I f we consider - as we should - information as the product of the analytical chemist, one obtains Fig. 3, which is an illustration of the law of marginal return.

Going one step further in the evaluation process, one may wonder whether indeed the increase in precision which we considered as the marginal return is necessary, i.e. we can question the utility. Indeed, since we are trying to put analytical chemistry into an economic perspective, we should relate the utility to the economic objective. One area of analytical chemistry where a relationship between the quality of an analytical method and its utility can be xnade is clinical chemistry. The less analytical errors are made, the better use the physician will be able to make of the result and arrive at a correct diagnosis. Acland and Lipton 7, posed the question, what is the necessary precision for a clinical test? Let us assume that the

[n f o r m a t i o n

r

n , 1 Fig. 3. Information (1 ~ togz'/s) as a function o f the number n of samples (illustration of the law of marginal returns).

U t i l i t y

100

80

60

&O

20

X X

I 2 3 4 In format ion

Fig. 4. Utility ( ~ specificity) as a function of information ( ~ log2 (SN /S A) ) where s A is the analytical information and s u the biological information (illustration of the law of marginal utility).

utility of a clinical test is directly proportional to the percentage of 'normal ' patients it recognizes as such. This is what clinical chemists call the specificity of a method. To answer the question of Acland and Lipton one must determine the probabili ty of finding a normal person outside the normal range due to insufficient precision. Table II gives this probability and the resulting specificity. Since the product of analytical chemistry is information and information is inversely related to precision, one can now produce the graph shown in Fig. 4, which is a perfect illustration of the law of marginal utility.

Clinical chemists know that they should consider not only the specificity of a test but also its selectivity (its ability to correctly detect abnormal values). Specificity and selectivity are determined by the overlap between the values for abnormal and normal populations. Together they determine the diagnostic power or efficiency of a test. This overlap is partly due to natural overlap of both populations, but it is also due to the

TABLE II. Data from Acland and Lipton 7 transformed to show that the data illustrate the law of marginal utility

SMSN P I U

S A

S N ~ p ~

0.1 0.99 3.32 99 0.2 0.99 2.32 99 0.3 0.98 1.73 98 0.4 0.97 1.32 97 0.5 0.95 1.00 95 0.6 0.94 0.73 94 0.8 0.90 0.32 90 1.0 0.86 0.00 86

analytical standard deviation standard deviation of normal range probability of recognizing a normal patient as such (as given by Acland and Lipton)

I = information; the value given in the table is in fact log2 (s~/sA), which is related to the value given in this column.

U = specificity (in %)

trends in analytical chemistry, vol. 1, no. 15, 1982 351

(a) t Loss to society (b)f Utility

Fig. 5. (a) Loss to society as a function of sAlsN (Rej 8) and (6) utility (= ~/LOSS to society) as a function of information (computed as for Fig. 4).

precision of the analytical determinations. Lindberg’, taking into account an overlap between ‘normal’ and ‘abnormal’ populations (as well as ascribing a different weight to the loss incurred by falsely diagnosing a healthy patient as ill and by diagnosing an ill patient as healthy) determined the ‘loss to society’ as a function of precision (and therefore analytical information). Ifone considers the utility as the inverse of the ‘loss to society’, one obtains Fig. 5 and again the marginal law is the result.

All of the above examples relate to the precision and/or accuracy of analysis. In fact, it can be shown that the ‘laws’ are of more general use in decision making in analytical chemistry and that very recent developments in analytical chemistry, such as the application of information theory and pattern recognition have to do with determining the functions describing the marginal laws. One example from each of these domains will demonstrate this.

Dijkstra and. his colleagues’ formalized the idea that information is the product of analytical chemistry and they applied information theory to quantify the information for qualitative analysis. Information theory permits us, for instance, to state that one HPLC system is to be preferred to another for separations of basic drugs because system 1 yields 3 bits of information while the other yields only 2.

Dijkstra et al. applied this idea to many analytical methods such as IR, MS and GLC. In the case of IR

-I/ ,

20 30 40

no of selected peak positions

Fig. 6. Returns (as percentage unique spectra) as a function of number of selected peaks positions (ReJ 9).

they considered 140 possible peak positions and determined the information present in each of these peak positions from a database consisting of 5100 compounds. They then determined the combinations of 2, 3, 4, etc. peak positions with the best information yield. A graph of the number of peak positions used v. information obtained is shown in Fig. 6. This again clearly resembles the marginal return law and, in fact, the same figure is always obtained when one investigates combinations of qualitative information yielding elements (different GC or TLC systems, MS peaks, etc.).

A final demonstration of the applicability of the marginal laws in analytical chemistry comes from pattern recognition. The main reason for applying pattern recognition in analytical chemistry is to classify a sample into one category or another on the basis of a pattern of analytical results obtained for the sample. This pattern consists of the results of different analytical determinations. For instance, one can try and classily patients into one of three categories (euthyroid, hypothyroid or hyperthyroid) using five analytical results (total serum thyroxine - T4, total serum tri-iodothyronine - T3, T3 resin uptake - RT3U, serum thyroid-stimulating hormone - TSH, and increase in TSH after injection of TSH-releasing hormone - ATSH). One question which can be asked is what is the effect on the quality of the diagnosis when one selects a smaller number of tests? So called feature reduction techniques permit one to determine which combination of m tests out of n possibilities permit one to achieve an optimal classification, i.e. diagnostic efficiency. If we consider that the utility of the combination of tests is directly related to its diagnostic efficiency and plot utility against number of tests, again the law of marginal utility is obtained (Fig. 7). Clearly the ‘marginal laws’ are quite generally applicable in analytical chemistry, and, very often, the analyst’s problem is to determine which cost utility ratio is optimal for his purpose. This, in fact, is a cost-benefit or cost-efficiency analysis, another important tool from

% Efficiency-Utility

/ x/x-x- X

1 2 3 5 n Fig. 7. Utility (= correct diagnosis with 0.999 certaintyl as a function of number of tests for the differentiation between euthyroid and hyperthyroid status (Rej 10).

352 trenas in analytical chemistry, vol. 1, no. 1~ 1982

economic sciences. An interesting area of research could evolve from the recognition of the fact that economics does have a link with analytical chemistry. It might well be that some methods from economics could be used with advantage in analytical chemistry and it is worth noting that an allied science, operations research, has already been used 11-13.

References 1 Christie, O., Lecture at the University of Birmingham, U.K., 10

February 1982 2 Wangersky, P.J. (1982) Trends Anal. Chem. 1, 150 3 Sandler, G. (1979) British Medical Journal, 2, 21 4 Durbridge, T. C., Edwards, F., Edwards, R. G. and Atkinson,

M. (1976) Clin. Chem., 22, 968 5 Gossen, H. H. (1854) Entwicklung der Gesetze des Menschlichen

Verkehrs und die daraus fliessenden Regeln fiir das Menschlichen Handeln

6 Van Meerhaege, M. ( 1979) Handboek van de Economie, 8th Edition, H. E. Stenfert Kroese, Leiden, p. 133

7 Acland, J. D. and Lipton, S. (1971)J. Clin. Pathol. 24, 369 8 Lindberg, D. A. B. and Watson, F. R. (1974)Meth. Inf. Med. 13,

151 9 Heite, F. H., Dupuis, P. F., Van 't Klooster, H. A. and Dijkstra,

A. (1978) Anal. Chim. Acta, 103, 313 10 Coomans, D., Broeckaert, I. and Massart, D. L. (1982) Anal.

Chim. Acta, 134, 139 11 Massart, D. L. and Kaufman, L. (1975) Anal. Chem. 47, 1244 A 12 Vandeginste, B. G. M. (1982) Trends Anal. Chem., 1,210 13 Goulden, R. (1974) Analyst, 99, 929

D. L. Massart is Professor of Analytical Chemistry in the Free University of Brussels, Laarbeeklaan 103, B-1090, Brussels, Belgium. His scientific interests are mainly in statistical and mathematical methods, but include separation science, ion-selective electrodes and trace metal analysis. He is an advisory editor for TrA C.

Luminescent immunoassay in clinical analysis Luminescent immunoassay is a novel method for determin- ing the concentrations of numerous organic compounds in complex body fluids. Its advantages over conventional assays are extreme sensitivity, broad range of linear

response, speed of analysis and stability of reagents.

Hartmut R. Schroeder Elkhart, IN, U.S.A.

The numerous analytical applications of luminescence measurements include the detection of trace metals, microbial detection, phagocytosis and cell stimulation, indicator reactions to measure substrates of oxidases and dehydrogenases and the monitoring of immuno- assays 1-4. The major impact of luminescence for clinical analysis is expected in the last two areas. Highly sensitive measurement of light generated in simple monitoring reactions permit detection of as little as 10 -16 mol ATP, 10 -14 mol NADH, 10 -12 mol H ,O , , 10 -15 mol heme catalyst (e.g. catalase) 4 and 10 -17 mol chemilumigenic compound (e.g. aminonaphthalhy- drazide) 5. In most cases the peak light intensity is attained in less than a second and is related to concentration ofanaly te over five orders of magnitude. Consequently, the above compounds are also attractive as labels for monitoring immunoassays.

The principle of luminescent immunoassay (LIA) The basis of LIA is similar to that of radioim-

munoassay (RIA). A fixed amount of labeled antigen

and varying quantities of unlabeled antigen (sample) are allowed to compete for a limited number of antibody binding sites (Fig. 1 ). However, the radiolabel is replaced by a reactant label which is detected in a light-generating reaction. After an incubation step, the ant ibody-bound labeled antigen is separated from the free form (heterogeneous assay). Common separation techniques include: chromatog- raphic separation, adsorption of the free antigen to charcoal, cellulose or ion-exchange resins and precipitation of the ant igen-ant ibody complex by a second antibody directed against the first. Then luminescence of the bound fraction or the free fraction is measured in the monitoring reaction. The concentration of unlabeled antigen is determined by comparison of the assay result to a standard curve.

Homogeneous assays based on enzyme cofactor labels or chemilumigenic labels are similar to heterogeneous assays, except that the separation step is omitted. Because the light-producing activities of the reactant labeled antigens in the monitoring reactions are modified when bound to antibody, it is possible to measure the free and bound species in the presence of each other.

Labeled Unlabeled Specific Antigen Antigen Antibody

Ag-L + A + Ab ,

Labeled Antigen- Antibody Complex

Unlabeled Antigen- Antibody Complex

' Ab" • "Ag-L + Ab" • "Ag

(Free) (Protein Bound)

Fig. 1. Generalized competitive protein binding reaction of LIA. 0 165-9936/82]0000-00001501.00 © 1982 Elsevier Scientific Publishing Company