Sequential Decision-Making Process Under Conditions of Uncertainty in Drug Analysis Ik-Whan Kwon' St. Louis University, St. Louis, MO 63103 Joe H. Kim Trenton State College, Trenton, NJ 08625
In testing newly developed drugs,2 one of the common problems that the researchers and chemists in pharmaceutical firms or in the NCDA (National Center for Drug Analysis) face is to decide whether to develop an automated analytic method or to apply a manual method. The crux of this prohlem of selection is that the choice has to he made under conditions of uncertainty. Decision makers in business and research have attempted to discover and improve methods to . tackle uncertaintiin decision making. Decision theory, also called Bayesian decision theorv. provides strategito attack the problem. Bayesian decision theorv is a tool decision makers can rely on in situations where sever2 choices of action are available but no exact information exists about the determinants of the outcomes of those alternatives. In decision theory, choice is described or prescribed by the interaction of groups of related constructs (I). Modern decision theory is different from classical decision theory in ' t h a t the former requires a systematic way to handle data and algorithms to selectthe bestcoune of act& whereas the latter simply rejects or accepts a certain hypothesis without looking into alternative courses of action. Modern decision theorv is. therefore, particularly useful in cases where the results of one decision may depend upon the outcome of others. Most decision making deals with more than one choice. Evidence of wide applicability of modern decision theory is apparent in such areas as business and economics (3,4,7,10, I I ) , patient management ( l , 9 ) , meteorology (8,12,14), and health care (2,17,18). Taking a systematic approach based on modern decision, theory involves the following major steps: A
.
1. List all possible actions. 2. List all possible outcomes.
3. Assess the probability of each outcome from each action. 4. Choose the best action based on likelihood and utility of the outcome.
For the problem of choosing between two alternative test methods in drug analysis, decision theory can be a n efficient and beneficial tool for the analyst. When considering whether to develop an automated method or manual method, normally the decision is made in a state of uncertainty. If the quantity of drugs to be analyzed is large and the probability of successfully developing an automated method is reasonably high, the automated method can he considered seriouslv . .(19). . However, if the quantity is small and the drug categories numerous, the cost of developing an automated method may outweigh the possible advantages. Furthermore, the research effort to develop an automated method may not succeed. Frequently, researchers and chemists must spend a considerable amount of valuable time before they can successfully develop an accurate and precise automated m e t h o d . V o measure the value of the alternative methods and to do an adequate comparison becomes further complicated if or when a manual method has been either developed already or successfully tested with satisfying degrees of accuracy and precision in similar cases. If a manual method has been developed, 462
Journal of Chemical Education
i t must be tested prior to its deployment to let the analysts be more familiar and experienced with the method. Such practice may he needed also to determine the method's degree of accuracy and precision before it can be finally accepted. From preliminary testing, an estimate of time needed to analyze a given number of samples can be made. If the propwed manual method demands too much time, the analysis cannot be I~*~neficiall) idrried $ r u t due to the excessive manp qk.
a, + p a = a, + p,q 4 k =-
Probability of Developing Successfully Automated Method, p(s) Requested Sample Size Estimated Amount of Time To Develop an Automatic Method Number of Days Required To Do the Analysis. Given Automatic Method Is Developed Estimated Amount of Time for Analyzing Drugs by Manual Method Estimated Amount of Time to Test the Manual Method
Example Problem
In 1977, a regional center for drug analysis began the study of a very important category of drug products. The number of products that were to he covered was so large that an agreement was reached to perform the analysis on the products of one firm a t a time until all had been covered. At this preliminary stage, the center received a reauest from their main office-for arecommendation on the hestway to complete the study on the remaining firms. Althoueh analvsis had not yet begun on these produ&, a method had been developed by the center's research branch for testing similar products. The research branch had gathered preliminary information on the cost of developing the automated method, the mannal method, and other pertinent cost variahles as shown in the table. It was tentatively decided that a sample size of 600 tablets would be appropriate for the entire study. Optimum Decision Under Certainty
From the table, the following coefficients for each parameter can he determined:E
+ Y,')
= P,,(Yn)+ (1 - PdY,')
(6) Since there is always a chance that the development effort may fail, the aggregate cost (Yt) contains the expected cost of an alternative choice (Y,). Once the effort fails, there should he an opportunity loss (L) for not having made an optimum choice a t the beginning. The opportunity loss is usually expressed as a fraction of the original cost, or L = Y,,, X p where o < p < 1.0. Accordingly, the total cost for Y, when the .. effort fails is higher than the original cost, or Y,' > Y.,
The aggregate expected cost (Pt),however, cannot exceed the expected cost of using the manual method (Y,). Otherwise, it k n o t even worth investigating the feasibility of developing the automated method. Accordingly,
~ ( p t=) E(Y,) PAY.) + (1 - PAY,') = Y, (7) Solving eqn. (7) for the break-even prohahility (Pk), Y, - Y,"' Pk=Y. - Y,' The break-even probability (Pk) indicates the minimum probability of success that is required to initiate the development of the automated method. Accordingly, if the expected probability (Po)of successfully developing the automated method is higher than the break-even prohability (Pk), the cost of research and use of the automated method is
Accordingly, the cost functions become
+
Y, = 9,241.60 V for the automated method, and Y , = 1,638+ 15?j for the manual method. By eqn. (41, the break-even quantity of drugs to be analyzed is
Since the requested amount of samples is 600, use of the automated method is feasible and should be further explored. Optimum Decision Under Uncertainty
As stated, once the break-even quantity is determined and i t is decided that the automated method is preferred, the likelihood of successfully developing the automated method should he examined. The table shows that the likelihood is 0.4. If a, > 8, and 8, < Pa, then Y, < Y, always regardless of p For further discussion, see Kwon (10). Ch. 7. "he cost of analysis for the automated machine is 10 man days and for the manual method, 150 man days. Therefore, the ratio is 1:15. Volume 58
Number 6
June 1981
463
Is 0.4 high enough to proceed to developing the automated machine given the cost information? Assuming that p = 0.1, or the loss factor is 10% of the alternative cost (Y,,,), then the total cost for the manual method a t 600 samples including the opportunity loss (L) would he
Using eqn. (9),the posterior prohahility of success given the test result known is
= 0.875.
Ym'=Ym+L = U r n + (Y, X P ) =Y,(l+p)
= (1,638
+ 15 X 600)(1+0.1)
= 11,701.8
By eqn. (81, the hreak-even probahility (Pr) is Px=-
Y, - Y,' Y, - Ym'
Since the minimum probahility of success that is needed to use the automated method (0.572) is higher than the expected prohahility (0.4), the optimum decision is not to develop the automated machine, although ii > qk." Bayesian Approach
When the result of the decision appears to he sensitive to one of the decision parameters, a further analysis of these parameters is desirable (5h7Among parameters to be studied, effort should be directed toward those which contain uncertain elements. In our example, unit costs and the alternative course of action are fairly deterministic whereas the assessment of probability is stochastic. Therefore, it is logical to direct our efforts toward further analvsis of ~robahilitv. Bayes' theorem assures us that the sampling information is oro~erlv . . -and ohiectivelv incoroorated into the orior information. Suppose that recent research on similar cases shows that in 2 out of 3 incidents the automated machine has been developed successfully. Then the revised (or posterior) probahility based on this new information may be computed via Bayes' theorem (11):
The posterior probahility of failure is 1- 0.575 = 0.425. Accordingly, the optimum decision after sample should be to develop the automated machine, since the prohahility of success (0.575) is greater than the required minimum prohability of success (0.572). Conclusion In analyzing drugs, a decision has to be made whether an automatic analytical method or a manual method should he used. Two sequential models to solve the problem were suggested in this paper: a deterministic model and a stochastic model. It was shown that the most important variable in the deterministic model is the quantity of drugs to he analyzed with respect to the break-even quantity, whereas the prohability of successfully developing the automatic machine is the most critical variable in the stochastic model. However, this paper shows that even if the quantity of drugs is large enough to warrant a use of an automatic machine, such a decision should be delayed until it has been proven that the prohahility of success is a t least greater than the reauired minimum probability of success'(hreak-even probahility). When the results of the decision are sensitive to the decision Darameters. Bayesian analysis is appropriate to ascertain the sensitivity zone.
Literature Cited (11 Alhen, u. A,, "Decill0"Theo.y
in Medicine:A Reviewand critique." Millh,"!, Memvriol Fund (luorlarly.56.1978.pp. 162401. 121 Bamque, N. E., and Ihrry, G. A.,"AutomatingJudgmental Decision-Makingfop a Serious Medical Problem: Monagemenl Science,1971, pp. 421-434.
(31 n,,"". D.w., -syntheaia O ~ P I P ~ ~ C ~ ~ Vin~ ~ ~ ~nereareh? k ~J O Ut ~ ~ iO,,iI ~ Mnrkc4nrRe.seamh. May 1979,pp. 280-283. (41 Calabru. W. '"Bayesian Thaw in Evaluation or New Praduct Line." Journal of AcadvmlofMwkelin8Scirnre.Spring1972,pp. 12-24. (5) Evsnr. R. A., "Rayesian Analypis of Risk: Quality Prugrew. 79-12 (February
~
,,,O.$
Am,.,.
u ,,......,, ,a,z .."."""",--., .".".
161 Foreman, J. K., and Sloetwell,P. 8.."AutumatieChamicai Analysis/ London: Ellis A 7.2
where
Pl(SIIT) = Posterior probability of success given test result is known = prior probability of success (0.4) P(So) = prior probahility of failure (0.6) P(Fo) P(T1So) = sampling result, given the outcome is a success (0.67) P(T(Fo) = samplingresult, given the outcome is a failure (0.33).
If the automated machine is developed under P. expected cost at q = 600 would he E ( ? ~ )= 0.4(9,841.16)
< Pk,the total
+ (1- 0.4)(11,101.8)= 10,957.72
whereas the expected cmt for the manual method would be Y , = 10,638. If a change in the size of the coefficient on decision parameters results in an alteration of the optimum decision, the parameter(s) is said to be sensitive to change in value. For further discussion, see Raiffa (15).
(71 Green. P. E. and P. J. Harriron,"Fauhion Farecastingfnra Mail Order Company Usin. a Bayesian A~prcaeh." Oprmlionr Reseorch Qunnrrly.June 1973,pp.193-M5. 181 dulian, P. H., and Maryhy,A. H.. "ProhahilityandStatistics in Meimmlopy:A Review of Some Reoent Developments,"Bullstin oiAmericnn M~i~orolrlyieal Seiiriy,10. 1972, pp. 957-965. w E+*.J. P.,"Cli"ieslApplicationorDpcis m Nudoor Modicinr.Octabor 1978.pp. 324-355. (LO1 Kwo". I. W.,"Stafirtieal DocisionTheory with Burines~and Economic Applications: A Bayesian Apprusch."NewYolk: D. Van Nosfrand, 1978. (11) Kt""". I. W..and wou. we,. "Mearurinzthe Value of Additional Infurmath within Bayesian krsmowork: &them ~ ~ (Fall 19781& ,pp. lb28. ~ (121 Kwon.1. W..and Doty.G. L.,"BayasianDecisionAnalysir ApproaehfoMew~eurol~y: H i ~ and v Unsolved Problems: Fifth Conferenceof Probobilily and Storistics in Atmo$pherirruScience,November 1977. (131 "NstionslFormulary: 14thEd..Eanton.PA:Mach PrintingCo..1975. (I41 Olssn A. R., "Bayeaisn and Ciamiss1 Statistical Methods Applied to Randomized Wealher Modification Experiment? Journal of Applied M~lrornlopy.8.1875,pp. ~~~~
Journal of Chemical Education
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1151 Raiffa. H.,"DedPionAnalysir: I n V d u M r y Lecturer onchoice UnderUncertainty." Rcsdine, MA: Addison-Wesley Publishing Co.. 1968. (161 Khudel. C. R., and Hone, R. Z.,"AutamatedAnalysisofDrugsand Other Drugssnd Other Substances cllPharmaceutiral lnterort: London:Battarworth. 1976. (171 Scheanhsum,S.C., McNei1.B.J..and Kavet. J.,"TheSwine-lnflnnnnanecision."Neui Enylond Jvurnol of Medicine. 295.1976, pp. 759-765. (181 S1immn.D.H.,"UtilityM~mt in I'ublicHedLh h i s i o n Makiny,('Monogemml Science,1969,pp. 17-30. (19)Snyder, L.. Leuine. J.. Stoy, R.,and Cunetta,A . "AutomatedChemieal Anelysls: Update on Centinuuw Flow Approach." Anolytiial Chemidiy, 48:942&956A (October l WCi
(20) '"United S l a m Phsrmacopeia,"19th Ed.. Eastan, PA: Mack Printing Cu, 1915.
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