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DDecision Analytic Modeling

P Muenning,Columbia University, New York, NY, USA

2008 Elsevier Inc. All rights reserved.

Introduction

Life is full of uncertainties. You could invest in a mutual

fund or a certificate of deposit account. You could buy ahome or rent. You could go to public health school orwrite a novel.

If you knew the risks and benefits of each alternative,the decision would be a lot easier. Decision analysis pro-vides a mathematical framework for making decisionsunder conditions of uncertainty. Lets see how this worksby using an example.

Imagine that you have $100 000 to invest. Youve alwayswanted to study public health, but youve also dreamedabout writing a novel ever since you were in college. Ifyou write a novel, you reason, you can simply invest the

money in a mutual fund and earn interest, making smallwithdrawals as needed. Since you are undecided betweenthe two options, you decide to examine which will be thebetter option financially.

Decision analysis is based on a concept called expectedvalue, in which the value of an uncertain event (such asmaking $10000 in the stock market) is weighed against thechances that the event will occur. For example, if you knowthat the historical average increase in a particular mutualfund is 10% per year, then the expected value of the returnon your $100 000 investment is 0.1 $100 000 $10000over one year.

You set your sights five years down the road and assumethat if you go to public health school, all of the $100 000would be spent on your education after living expenses aretaken into account, but you would be able to earn about$50 000 per year working in public health. After taxes andliving expenses, you estimate that you would save about$10 000 over the 3 years after graduation.

If you decide to write a novel, you will have spent your$100000, along with the interest on the mutual fund, onliving expenses over the 5-year period. But if you publishthe novel, you could earn an additional $30 000. Your

college English professor advises you that there is abouta 1% chance that you will be published. Therefore, theinvest and write option will produce a probabilistically

weighted return of $30 000 0.01 $300.Since $10000 is more than $300, you might go for the

career in public health. However, you might not be satisfiedwith your calculations; there is a chance that the mutualfund could do very well, but there is also a chance that youcould lose money. If it does well, you could end up withleftover money from your mutual fund investment at theend of 5 years. So if you write the book, you might makemoney even if you dont publish (perhaps better thanspending it on tuition). But if the mutual fund losesmoney, you might not be able to finish your novel,which would be quite depressing.

There is also a chance that you will not find a job rightout of public health school, which would also be discoura-ging. Ideally, you would want to have a rough idea notonly of the difference in earnings associated with eachdecision, but also the difference in utility, or happiness.This way, youll not only have a better idea of whichdecision is riskiest, but youll also know the chances ofyour relative satisfaction with each choice.

The various options that one is deciding between arecalled competing alternatives. Decision analysis can thusbe described as the process of making an optimal choiceamong competing alternatives under conditions of uncer-tainty (Gold et al., 1996).

In public health or medicine, decision analysis is mostcommonly employed in cost-effectiveness analyses. Here,the competing alternatives are the different health inter-ventions under study. Therefore, rather than net improve-ment in cash flow and personal utility, cost-effectivenessanalysis provides the user with information on incrementalcosts and health gains.

In the past, researchers performing a cost-effectivenessanalysis had to write a computer program that wouldcalculate the cost and effectiveness of different medical

71

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interventions, or attempt to do so by using a spreadsheetprogram. Today, most cost-effectiveness analyses can beeasily assembled with decision analysis software.

Decision analysis software provides a graphical frame-work for estimating the costs and effectiveness values (usu-ally represented as quality-adjusted life years, or QALYs)associated with a given medical or public health strategy.Lets take a look at a concrete example of how this works inpublic health.

Decision Analysis Models

InFigure 1, we wish to compare two strategies for dealingwith influenza virus during the flu season. The first is toprovide supportive care to people who become ill(Muennig and Khan, 2001). The second is to attempt toprevent infection with an influenza vaccination.

In this figure, the two alternatives are represented in

branches separated by a square called a decision node.The decision node is like a referee holding the competingalternatives apart as they set out to do battle. Followingthis square, youll see a circle. The circle is called a chancenode. A chance node is followed by two or more possibleoutcomes. In Figure 1, the outcomes are remains welland becomes ill.

Notice that each remains well and becomes illbranch in the decision analysis tree is associated with acost and a probability. In the supportive care option, thecost associated with remaining well is $0. But in thevaccinate option, everyone gets a $10 influenza vaccineat the start of influenza season. Thus, whether or not folksreceiving the vaccine would have gotten sick, they mustpay for the vaccination.

However, vaccination greatly reduces the chances ofbecoming ill during influenza season. If, during the aver-age season, there is a 10% chance of developing the fluamong unvaccinated persons, there is only a 3% chanceof developing the flu among vaccinated persons. More-over, among those who get sick despite vaccination, the

infection will be much less severe. Therefore, the costswill be lower. (InFigure 1, those becoming ill incur a costof $83 if unvaccinated and $38 if vaccinated.)

So, how does this work? The model merely provides aframework for calculating probabilistically weighted costs.Thus, we see that there is a 0.97 $10 0.03 $38 $11average cost among those receiving vaccination, and a0.1 $84 0.9 $0 $8 average cost among thosereceiving supportive care alone.

Note thatFigure 1only deals with costs for simplicity.Moreover, it is a bit simplistic in other ways; there is a lotmore uncertainty associated with these two competingalternatives than simply remaining well or becoming ill.

Figure 2presents a more complete representation ofthe decision between supportive care and vaccination.Here, we see that someone who becomes ill has a chanceof seeing a doctor or being hospitalized. As before, the riskof each is much lower among vaccinated persons.

There are a fewthings to note about Figure 2. First, the

costs are represented as running totals. It is not necessaryto set up a decision analysis model this way, but it helps toillustrate a key point; each event is associated with a cost,and that cost is added to the events that preceded it. Thus,the cost at the end of each branch in the tree represents theprobabilistically weighted cost of a given pathway ofevents. Thus, the initial cost of becoming ill along thetop branch is $12 (the cost of over-the-counter medica-tions). Since there is only a 10% chance of incurring thiscost, the average cost will be 0.1 $12 $1.20. But if theperson sees a doctor, the cost will be $110 0.2 $22, andthis gets added to the $1.20, for a running total of $23.20 atthe sees doctor event in the pathway.

The second thing to note about Figure 2 is that allcosts are incurred at one point in time. In other words, wecalculate the average cost of each of the events in thepathway as if they happened in one day. This is fine for adisease like influenza, which tends to make someone ill fora short time. However, it might not work so well forcancer, which, depending on the type of cancer a personhas, can drag on for many years. We return to this problemin the next section.

The above example provides an illustration of howdecision analysis modeling works. As you can see, deci-sion analysis provides probabilistically weighted values of

a series of outcomes of interest. The average value of anycompeting option is called its expected value. Althoughall trees provide a probabilistically weighted expectedvalue, different models arrive at this value in very differ-ent ways.

Types of Decision Analysis Models

There are different types of decision analysis models(Goldet al., 1996). The type of model that analysts choosedepends on the problem under evaluation.

Supportive care

Vaccination

Remains well

Remains well

Cost=$0

Cost=$10

Cost=83

Cost=$38

0.9

0.97

0.03

0.1

Becomes ill

Becomes ill

Cost

Cost

Cost

Cost

Figure 1 Decision analysis tree comparing supportive care tovaccination.

72 Decision Analytic Modeling

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Simple decision trees

The most basic is a simple decision tree, such as the onepresented in the above example. Simple decision trees areusually employed to examine events that will occur in the

near future. They are therefore best suited to evaluateinterventions to prevent or treat illnesses of a short dura-tion, such as acute infectious diseases (Muennig et al.,1999). They may also be used to evaluate chronic diseasesthat may be cured (for example, by surgical intervention).When these trees are used to evaluate diseases that changeover time, they sometimes become too unruly to be useful.

Markov modeling

For chronic or complex diseases, it is best to use a statetransition model, also known as a Markov model

(Sonnenberg and Beck, 1993). This type of model allowsthe researcher to incorporate changes in health states overtime into the analysis. For example, if a person has cancer,there is a chance that the person will recover within a year

and then relapse. There is also a chance that the personwill remain sick for some time or will die soon.

With every passing year, the cost of treatment changes.More importantly, the chances of survival, recovery, ordeterioration change. Markov models allow researchersto track changes in the quality of life, the quantity of life,and the cost of a disease over time when different healthinterventions are applied (Sonnenberg and Beck, 1993).

Most interventions in public health or medicine thattarget diseases have some component of time to them. Tomodel screening mammography, for instance, it is necessary

Remains well

Becomes III

0.9

0.1

Cost=Cost+10

Cost=Cost+110

Cost=Cost+0

Cost

Remains well

Becomes III

0.97

0.03

Cost=Cost+12

Cost=Cost+12

Cost=Cost+0

Cost=Cost+0

Cost

0.2

0.8

No doctor

Sees doctor

Cost=Cost+110

Cost=Cost+0

0.1

0.9

No doctor

Sees doctor

Supportive care

Vaccination

Hospitalized

Not hospitalized

Hospitalized

Not hospitalized

0.01

0.99

0.01

0.99

Cost=Cost+0

Hospitalized

Not hospitalized

Cost=Cost+5000

Cost=Cost+5000

Cost=Cost+0

Cost=Cost+0

Cost=Cost+5000

Cost=Cost+5000

Cost=Cost+0

Hospitalized

Not hospitalized

0.001

0.001

0.999

0.999

Cost

Cost

Cost

Cost

Cost

Cost

Cost

Cost

Figure 2 A more complete decision analysis tree comparing supportive care to vaccination.

Decision Analytic Modeling 73

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to have some way to measure the progression of breastcancer over time. This way, differences in the progressionof disease can be compared when treated and when leftuntreated.

Using a Markov model, we can incorporate a temporalelement into our decision analysis model (Sonnenbergand Beck, 1993). Markov models count the years thataccrue as the model completes cycles and the medicalcosts, living costs, and changes in health-related quality oflife scores over these years. Of course, days, months, ordecades can be modeled, too.

For instance, suppose we are interested in calculatingthe lifelong cost associated with breast cancer in patientswho have been diagnosed via a screening mammogram.We would start the model by using the average age ofonset of breast cancer. The number of survivors would bedetermined by using the probability of death specific towomen who have been diagnosed with breast cancer viascreening mammography. Each patient who is still living

at the end of the year gains 1 year of additional life(1 person-year). But she can also be assigned medicalcosts, home health-care costs, transportation costs, andso forth. Women who die do not accrue such costs, sothese women gain neither a year of life nor any costs.

Thus, if the average annual cost of living with breastcancer were $10 000, a group of 10 women would incur acost of 10 $10000 $100 000 in year 1. Now imaginethat by year 2, one woman died. Thus, year 2 costs wouldbe 9 $10 000 $90000. If we continued this processuntil the last subject died, we would not only have theaverage number of life years lived, but we would also havethe total cost incurred by these women over that period.(SeeTable 1.)

In this example, we are measuring various relevanthealth outcomes over a discrete interval of time. We knowthat the average woman lived 48 years of life/10 women 4.8 years over the 6-year interval. The average cost of thistreatment was $450 000/10 women $45000.

In a Markov model,Table 1is represented graphicallyas a chain of events rather than a table. This makessense given that, for most diseases, subjects do not merely

progress slowly toward death due to the disease understudy. Rather, they can become better, die from othercauses, or develop other diseases.

Thus, to model events that unfold over time realisti-cally, we will want some sort of recursive component inour model. A recursive event is one that repeats over andover (see Figure 3). Thus, at the end of each year of asubjects life, he or she is assigned a cost value and a QALYvalue, and then reenters the next year of life (or dies). Thisprocess is repeated until all subjects are dead or theevaluation period ends.

As in the examples evaluating influenza infection viasimple decision analysis trees discussed previously, eacharrow in Figure 3 is assigned a probability value. Forinstance, the likelihood of remaining well over any givenyear might be 0.98 in the typical cohort of healthy persons.

Markov models can also be used to simulate the lifeexperience of the average individual with a particulardisease before and after treatment, the health effects of a

particular medical intervention, or even the benefits ofhaving health insurance over a lifetime (Mandelblattet al.,2004; Muennig et al., 2004, 2005).

For instance, we might wish to compare the advantagesof folate supplementation to no intervention. Since folateprevents birth defects, one branch of the model mightrecord the health-related quality of life and health costs ofthe average infant with a spinal cord defect as time pro-gressed. In the other branch, we might simulate the health-related quality of life and health costs of the typical babywithout a spinal cord defect as time progressed. As in the

Table 1 Progression of a cohort of 10 women with breastcancer over a 6-year period

Year Women surviving

Years lived in

intervala

Cost of treating

breast cancer

1 10 10 $100 0002 9 9.5 $90 0003 8 8.5 $80 0004 7 7.5 $70 0005 6 6.5 $60 0006 5 5.5 $50 000Total 48 $450 000

Well1 QALY

$0

Dead0 QALY

$0

Sick0.5 QALY

$10000

Figure 3 Conceptual representation of a Markov model.Subjects who are well can remain well, become sick (e.g., frombreast cancer), or can die (e.g., from an accident) over the courseof a year. Likewise, subjects who are sick can remain sick,become well, or die. This process is repeated, accruing a meancost and QALY value. (Each arrow represents a transitionprobability over the course of 1 year.)


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table above, each would account for differences in mortalityrates, and thus calculate the average life expectancy of eachgroup. The model would then tell us the differences inQALYs and costs over the entire lifetime of babies whodid and did not have a spinal cord defect.

Worked examples of Markov models can be found athttp://www.pceo.org/. This site provides free self-instruc-tion manuals and links to government websites that pro-vide data free of charge.

Sensitivity Analysis

The value of decision analysis model inputs can be verydifficult to establish with absolute certainty. For instance,health-related quality of life scores a key ingredient inthe calculation of QALYs can be obtained from variousinstruments, each producing a slightly different number(Gold and Muennig, 2002). Because these differences

arise due to differences in the way the studies aredesigned, they represent a form of nonrandom error.Most model inputs will be derived from a sample, andtherefore also contain random error.

We can usually guess the range of plausible valueswithin which the true value might lie. For instance, look-ing through the literature, we might see that the health-related quality of life score for the disease we are studyingvaries by roughly 20% when using different instruments.We might also know the standard error in datasets orpublished values.

The inputs we are least likely to be certain about arethe assumptions we have made. In the earlier influenzaexample, we might have made an assumption about theamount of time it takes for a nurse to administer theinfluenza vaccine. There are other values that we mightbe fairly confident about, such as the cost of the influenzavaccine itself. (The average wholesale price is usually easyto get.)

The parameters the researcher is least certain aboutshould be tested over the widest range of values (becauseit is plausible that the values are much higher or muchlower than our baseline estimate). Parameters that theresearcher is somewhat more confident about can betested over a narrower range of values. When a particular

strategy remains dominant over the range of plausiblevalues for the inputs that we are uncertain about, themodel is said to be robust.

There are many different ways of testing variables in asensitivity analysis. These include a one-way (univariate)sensitivity analysis, in which a single variable is testedover its range of plausible values while all other variablesare held at a constant value; a two-way (bivariate) sensi-tivity analysis, in which two variables are simultaneouslytested over their range of plausible values while all othersare held constant; a multi-way sensitivity analysis, in

which more than two variables are tested; and a tornadoanalysis (or influence diagram), in which each variable issequentially tested in a one-way sensitivity analysis. Thetornado analysis is used to rank order the different vari-ables in order of their overall influence on the magnitudeof the model outputs.

Figure 4presents a typical one-way sensitivity analy-sis. Returning to the influenza example above, lets assumethat we wish to know how changes in the estimated valueof the vaccine cost will influence the overall expectedvalue of each strategy.

In this instance, well add a treatment arm (there areanti-influenza drugs that can be used to treat the infectionin early stages). Because the cost of the vaccine does notinfluence other arms of the analysis, we see that theirexpected value is not influenced by changes in the costof the influenza vaccine.

Ideally, we would want to generate some estimate ofthe impact of all sources of error in the study on the costand effectiveness values generated by the decision analy-sis model. One way to generate such an estimate is viaMonte Carlo simulation (Halpern et al., 2000). Namedafter the famous gambling enclave, this type of analysisallows for the generation of a single confidence intervalaround multiple variables.

In a Monte Carlo simulation, a hypothetical cohort ofsubjects enters into the decision analysis model. As subjectspass through the model, they encounter a number of differ-ent probabilities such as the chance of developing influ-enza-like illness, the chance of seeing a doctor, the chance ofbeing hospitalized, and so on. Each time a subject encoun-ters one of these variables, the value that the variableassumes is determined by its probability distribution.

The net result is a weighted mean value for eachsampled distribution of each subject randomly enteredinto the model. (Although this isnt exactly the way

$90.00

$80.00

$70.00

$60.00

$50.00

$40.00

$30.00

$20.00

$10.00

$0.006.99 13.96 20.92 27.89 34.86 41.82 48.79

Vaccine cost

Costofstrategy(expectedvalue)

Support Treat Vaccinate

Figure 4 One-way sensitivity analysis examining how the costof providing the influenza vaccine influences a vaccination,supportive care, and treatment intervention.

Decision Analytic Modeling 75

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that the models are always calculated, it is the easiest wayto think about it.) The final standard error associated withall subjects who pass through the model is the overallstandard error of all of the distributions sampled.

The basic theory of cost-effectiveness analysis, decisionanalysis modeling, health-related quality of life scores, anddata collection can be found in a number of books andtextbooks (Drummond et al., 2005; Muennig, 2007). Themethods for standardizing cost-effectiveness analysis arealso available in book form (Goldet al., 1996).

Summary

Decision analysis is a formalized approach to making opti-mal choices under conditions of uncertainty. It allows theuser to enter costs, probabilities, and health-related qualityof life values among other inputs of interest, and thencalculates probabilistically weighted means of these out-

come measures. In public health, these outcome measuresusually include costs and QALYs. Typically, therefore, deci-sion analysis is the heart of cost-effectiveness analyses inpublic health and medicine (Gold et al., 1998).

However, just about any outcome measure can be mod-eled, including vaccine-preventable illnesses averted,deaths avoided, and so forth. Therefore, local health depart-ments, pharmaceutical companies, or other agencies canusedecision analysis for internal decision-making processes.Decision analysis is often used by non-health businessesinterested in deciding whether they should release a prod-uct, perform internal restructuring, and so forth.

One great strength of decision analysis modeling is thatit allows for the calculation of a range of possible valuesaround a given mean. This approach, called sensitivityanalysis, allows the user to better understand the chancesthat he or she will make a bad decision if a given strategyis taken.

Decision analysis, like cost-effectiveness analysis, ishighly dependent on the accuracy and completeness ofmodel inputs, as well as the assumptions that the analystsmake. Drugs can have unforeseen side effects, or inter-ventions can have long-term costs that may not be appar-ent to the analysts. Any of these effects can lead tosuboptimal outcomes.

For instance, the optimal treatment strategy for tuber-culosis in most instances is a low-cost combination ofmedications that can be effectively delivered in developingcountries. By using the most cost-effective medications, itis possible to maximize the number of lives saved within agiven budget. However, as Farmer points out, these med-ications will be wasted if delivered to populations with ahigh percentage of drug-resistant tuberculosis (Farmer,2004). Therefore, decision analysis and cost-effectivenessanalysis must be viewed as an adjunct to optimal decisionmaking rather than the final word in health policy.

See also: Children and Adolescents, Measuring the

Quality of Life of; Comparative Risk Assessment;

International Classification Systems for Health;

Measurement and Modelling of Health-Related Quality

of Life; The Measurement and Valuation of Health for

Economic Evaluation.

Citations

Drummond MF, OBrien BO, Stoddart GL, and Torrance GW (2005)

Methods for the Economic Evaluation of Health Care Programmes,

3rd edn. London: Oxford University Press.

Farmer P (2004) Pathologies of Power: Health, Human Rights, and the

New War on the Poor.Berkeley, CA: University of California Press.

Gold M, Siegel J, Russell L, and Weinstein M (1996) Cost-Effectiveness

in Health and Medicine. New York: Oxford University Press.

Gold MR, Franks P, McCoy KI, and Fryback DG (1998) Toward

consistency in cost-utility analyses: Using national measures to

create condition-specific values.Medical Care 36(6): 778792.

Gold MR and Muennig P (2002) Measure-dependent variation in burden

of disease estimates: Implications for policy. Medical Care 40(3):260266.

Halpern EF, Weinstein MC, Hunink MG, and Gazelle GS (2000)

Representing both first- and second-order uncertainties by Monte

Carlo simulation for groups of patients.Medical Decision Making

20(3): 314322.

Mandelblatt JS, Schechter CB, Yabroff KR, et al.(2004) Benefits and

costs of interventions to improve breast cancer outcomes in African

American women. Journal of Clinical Oncology 22(13): 25542566.

Muennig P (2007)Cost-Effectiveness Analysis in Health, a Practical

Approach.San Francisco, CA: Jossey-Bass.

Muennig P, Pallin D, Sell RL, and Chan MS (1999) The cost

effectiveness of strategies for the treatment of intestinal parasites in

immigrants.New England Journal of Medicine 340(10): 773779.

Muennig P, Pallin D, Challah C, and Khan K (2004) The cost-

effectiveness of ivermectin vs. albendazole in the presumptive

treatment of strongyloidiasis in immigrants to the United States.

Epidemiology and Infection 132(6): 10551063.

Muennig P, Franks P, and Gold M (2005) The cost effectiveness of

health insurance.American Journal of Preventive Medicine28(1):

5964.

Muennig PA and Khan K (2001) Cost-effectiveness of vaccination

versus treatment of influenza in healthy adolescents and adults.

Clinical and Infectious Disease33(11): 18791885.

Sonnenberg FA and Beck JR (1993) Markov models in medical decision

making: A practical guide.Medical Decision Making 13(4): 322338.

Further Reading

Drummond MF, OBrien BO, Stoddart GL, and Torrance GW (2005)

Methods for the Economic Evaluation of Health Care Programmes,

3rd edn. London: Oxford University Press.

Gold MR, Siegel JE, Russell LB and Weinstein MC (eds.) (1996)

Cost-Effectiveness in Health and Medicine.New York: Oxford

University Press.

Hunink M, Glasziou P, Siegel JE,et al.(2001)Decision Making in Health

and Medicine. Cambridge, UK: Cambridge University Press.

Muennig P (2002) Designing and Conducting Cost-Effectiveness

Analysis in Health and Medicine.San Francisco, CA: Jossey-Bass.

Muennig P (2007) Cost-Effectiveness Analysis in Health, a Practical

Approach.San Francisco, CA: Jossey-Bass.

Sonnenberg FA and Beck JR (1993) Markov models in medical decision

making: A practical guide. Medical Decision Making 13(4): 322.

Weinstein MC, Fineberg HV, Elstein AS, et al. (1980)Clinical Decision

Analysis.Philadelphia, PA: W. B. Saunders.


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