Decision analysis models: Opening the black box☆☆☆★

Simple decision trees 

A decision model is best illustrated with a simple example, in this case the decision between surgery and watchful waiting in a patient with asymptomatic carotid stenosis. With carotid endarterectomy, patients experience risks of perioperative stroke and death, as well as subsequent risks of stroke. With watchful waiting, patients avoid the short-term risks associated with surgery but experience higher risks of late stroke.

Figure 1 illustrates this decision as a simple decision tree.

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Fig. 1. Calculating the expected values of watchful waiting and surgery for asymptomatic carotid stenosis (folding back the decision tree).

Once the decision tree has been created, probabilities and values are added to the model. In Fig 1, probabilities for each chance outcome are listed at each branch. (Although similar to data from the clinical trials, these estimates are again intended just for illustration.) Note that probabilities for the outcomes of any given chance node must always sum to 1. Values are then assigned to every terminal node in the tree (ie, “final” outcomes not associated with subsequent chance events). Values can be expressed in a variety of units, such as dollars, life expectancy (life years), or utility (relative worth of state of health on scale of 0 to 1). The most popular metric is the quality-adjusted life year (QALY), obtained by multiplying the life expectancy associated with each outcome by the utility reflecting quality of life. For example, assume patients experiencing a perioperative stroke have an average life expectancy of 15 years and an average utility (quality of life) of 0.5. This outcome would be assigned 7.5 QALYs.

After it has been fully parameterized (all the “inputs” are in), the model can calculate the expected value of each strategy (the “outputs”). This process, known as folding back the tree, is simple arithmetic. For each strategy, expected value is obtained by multiplying the value of each outcome by its probability of occurrence and then summing these products. (The probabilities of outcomes at the end of many branch points in the tree are obtained by multiplying the “path” probabilities.) As illustrated in Fig 1, the watchful waiting arm has an expected value of 14.18 QALYs. Because the expected value of surgery is 14.34 QALYs, the surgery strategy is superior in this baseline analysis.

Simple decision trees have several advantages. They are relatively easy to interpret, and their validity can be judged more readily by experts without expertise in decision analysis. These models apply best to decisions in which the timing of events is not important (eg, they all occur over a short time horizon). Unfortunately, many surgical decisions must account for when outcomes occur. In the example of carotid endarterectomy, the simple decision tree model assumes that a late stroke has the same value as one that occurs perioperatively. This assumption is flawed, because it fails to account for time in good health before the stroke, and because most patients place more value in near-term health than health “down the road.” Also, unless the analyst is careful, a simple tree model can overlook competing risks; many patients will die of other causes before a late stroke can occur.

The Markov model 

A Markov model is a powerful and flexible tool for overcoming these important limitations of simple decision trees. Markov models simulate outcomes in large, hypothetical cohorts of identical patients. Patients start the simulation in one or more defined health states. At regular time intervals (eg, monthly or yearly “cycles”), patients may transition to other health states, according to specified transition probabilities. The model continues to iterate for a time period specified by the analyst (eg, 10 years) or until all patients in the hypothetical cohort are “dead” (the typical approach). In the process, the model keeps track of how much value the starting cohort of patients has accumulated over time and calculates an average life expectancy (with or without quality-adjustment).

Figure 2 is a schematic of a Markov model for the carotid endarterectomy decision.

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Fig. 2. Markov model for the decision tree representing the choice between surgery and watchful waiting for asymptomatic carotid stenosis.

Once the model structure is specified, probabilities are assigned to reflect how patients are initially distributed across health states. The analyst must also specify the probabilities for chance events occurring with each cycle of the model. Some events are reflected by simple probabilities (eg, risk of perioperative death). Others occur gradually over time and thus must be tailored to the cycle duration of the model (eg, risk of dying from other causes during any time interval). This generally involves converting annualized rates (derived from the literature) to cycle-specific probabilities for the model.

Values must also be assigned. In a simple decision tree, a simple “lump sum” value (eg, in QALYs) is assigned to each terminal outcome in the model. In a Markov model, the analyst specifies how much value a patient in any given health state accrues with 1 cycle of the model. In the carotid endarterectomy example, assume a model cycle duration of 1 year. A patient in the health state “Well” accrues 1.0 QALYs per cycle. A patient in “Disabling Stroke” accrues 0.5 QALYs (assuming utility of stroke 0.5). A patient in “Dead” accrues 0 QALYs.

On the basis of a simulation of 100,000 hypothetical patients, Table I summarizes how the Markov model “keeps score” for the surgery strategy in the carotid example. For the surgery arm, the model assumes that carotid endarterectomy is performed at time 0. Thus, 97,000 patients start in the health state Well, 2,000 (2%) in Alive with Stroke, and 1,000 (1%) in Dead. With each yearly cycle, patients gradually transition out of Well to Alive with Stroke and, more commonly, to Dead. Almost 41% of the starting cohort is in the health state Dead by year 10 and 100% by year 50. Table I also lists the total value (in QALYs) accrued by the cohort with each cycle and cumulatively. By year 50, a total of approximately 1,599,600 QALYs have been accumulated. Because there were 100,000 patients in the original cohort, this translates to 16.00 years of quality-adjusted life expectancy per patient. Although not shown, the watchful waiting strategy would be simulated in an analogous fashion.

Table I. Spreadsheet summarizing the Markov model evaluation process for the surgery arm of the decision analysis model to treat asymptomatic carotid stenosis

		Health states
		Well (utility = 1.0)		Alive with stroke (utility = 0.5)		Dead (utility = 0)
	Cycle (y)	No. of patients	Cycle QALYs	No. of patients	Cycle QALYs	No. of patients	Cycle QALYs	Total QALYs	Cumulative QALYs
	0	97,000		2,000		1,000
	1	91,228	91,228	2,822	1,411	5,950	0	92,639	92,639
	2	85,800	85,800	3,548	1,774	10,652	0	87,574	180,213
	3	80,695	80,695	4,786	2,093	15,119	0	82,788	263,001
	4	75,893	75,893	4,744	2,372	19,363	0	78,265	341,266
	5	71,377	71,377	5,228	2,614	23,395	0	73,991	415,257
	6	67,130	67,130	5,645	2,822.5	27,225	0	69,953	485,210
	7	63,136	63,136	6,001	3,000.5	30,863	0	66,137	551,347
	8	59,379	59,379	6,301	3,150.5	34,320	0	62,530	613,877
	9	55,846	55,846	6,550	3,275	37,604	0	59,121	672,998
	10	52,523	52,523	6,754	3,377	40,723	0	55,900	728,898
	50	0	0	0	0	100,000	0	0	1,599,600
	Total:								1,599,600
	Average per patient:								16.00

Learning more 

In this primer we have described how decision analysis models work, focusing primarily on issues related to model structure and the evaluation process for simple decision trees and Markov models. However, we have not addressed other important aspects of decision analysis. These include the fine points of structuring the model, deriving annualized rates and probabilities from the literature, estimating disease-specific life expectancy by using life tables and other sources, methods for utility assessment and quality of life, cost-effectiveness analysis and other types of economic evaluation, and discounting. Those readers interested in learning more about these topics should review the series of primer articles from Medical Decision Making2, 3, 4, 5, 6 or see the major texts in these areas.7, 8, 9