Decision Node

Decision Node in a Health Economic Decision Tree

In a decision tree for economic evaluation, the decision node sets out the strategies being compared, such as a new drug against standard care, and the rule applied there when the tree is evaluated decides which strategy the model recommends. A decision node differs from a chance node in how it is resolved when the tree is folded back, and in cost-effectiveness analysis the choice rests on incremental analysis or net benefit rather than on the highest expected value of a single outcome. Decision nodes can also follow a chance node when decisions are sequential, and an illustrative gene-test example below shows how single-outcome rules, the threshold and a missing comparator change the choice, alongside the errors most often made at decision nodes.

What a decision node represents in an economic evaluation

A decision tree is built from three kinds of node, conventionally drawn as a square, a circle and a triangle. The square is the decision node. Each branch leaving it is a strategy that the decision maker could adopt, and no probabilities are attached to those branches because the choice is made, not left to chance. A chance node (circle) carries the probabilities of uncertain events, and a terminal node (triangle) records the costs and health outcomes of a completed pathway.

The branches of the first decision node are the comparators of the analysis, so choosing them is part of defining the decision problem. The ISPOR-SMDM good practice report on model conceptualisation states that all feasible and practical strategies should be considered and that the results are meaningful only in relation to the interventions considered. It adds that a comparator may be a specific alternative, a distribution that reflects routine practice, or no intervention. When current practice is a mix of treatments, that mix enters the model as a chance node within the current-practice branch, with the shares of patients as its probabilities; it does not turn the decision node into a probabilistic one.

How a decision node is resolved when the tree is folded back

Folding back, or rolling back, evaluates the tree from the terminal nodes towards the root. Chance nodes are replaced by their expected value, the probability-weighted average of the values on their branches. A decision node is treated differently. The analyst takes the best of its branches, not an average of them. For a single objective that is to be maximised, the rule is as follows.

$$V_D = \max_{a \in \mathcal{A}} \bar{V}_a$$

where $V_D$ is the value assigned to the decision node, $\mathcal{A}$ is the set of strategies available at that node, and $\bar{V}_a$ is the expected value of strategy $a$ after all chance nodes on its branch have been folded back. The maximum is written for an objective such as health gain; when the objective is a loss, such as cost with outcomes held equal, the minimum is taken instead.

The ISPOR-SMDM report on parameter estimation and uncertainty describes many models as helping decision makers to maximise a given outcome, with the best choice being the one that maximises the outcome subject to any constraint. It also states that a decision maker who must allocate resources now, with no role in commissioning research and no option to delay or review the decision, should base the decision only on expected values, while noting the argument that nonlinear models may need probabilistic sensitivity analysis to generate appropriate expected values. On current evidence, the choice at a decision node rests on expected values, and the uncertainty around them is examined separately.

Why cost-effectiveness analysis needs more than the highest expected value

A cost-effectiveness model carries two outcomes through every pathway, expected cost and expected health effect, usually in QALYs. Taking the maximum of expected QALYs ignores what the extra health costs, and taking the minimum of expected cost ignores the health given up. Neither respects the opportunity cost of a fixed health budget, so the choice combines both outcomes through a threshold value of health.

One way to do this is to convert each strategy to a single value with net monetary benefit and then take the maximum as above.

$$a^{*} = \arg\max_{a \in \mathcal{A}} \left( \lambda \bar{E}_a - \bar{C}_a \right)$$

where $a^{*}$ is the preferred strategy, $\lambda$ is the cost-effectiveness threshold in money per QALY, $\bar{E}_a$ is the expected health effect of strategy $a$ and $\bar{C}_a$ is its expected cost. Because expectation is linear, net monetary benefit can be folded back through chance nodes in the same way as cost and QALYs.

The other way is fully incremental analysis. Strategies are ranked by expected cost, options subject to dominance or extended dominance are removed, and each remaining strategy is compared with the next less costly one on the cost-effectiveness frontier, using the ratio below.

$$ICER_{j,i} = \frac{\bar{C}_j - \bar{C}_i}{\bar{E}_j - \bar{E}_i}$$

where $ICER_{j,i}$ is the incremental cost-effectiveness ratio of moving from strategy $i$ to the next more costly frontier strategy $j$, and the other terms are as defined above. The most costly strategy whose ratio against its predecessor lies below the threshold is chosen, which gives the same answer as the net monetary benefit rule at that threshold. The NICE manual (PMG36) states that reported ICERs must be the ratio of expected additional total cost to expected additional QALYs, asks for results in a fully incremental analysis with dominated and extendedly dominated technologies removed, and asks for expected net health benefits alongside ICERs, using values of £25,000 and £35,000 per QALY gained.

Sequential decisions and decision nodes after chance nodes

Decision nodes are not limited to the root of the tree. A later choice, such as whether to treat after a test result, appears as a decision node placed after the chance node that reveals the result. When the tree is folded back, that downstream node is resolved first, and the strategy at the root then becomes a contingent policy, such as testing and treating only those with a positive result. A downstream choice may use only information available at that point in the pathway.

In a cost-effectiveness model the downstream node is resolved with the same rule as the root, so its answer depends on the threshold. Listing each contingent policy as a separate branch of the first decision node keeps every policy visible in the incremental analysis. The ISPOR-SMDM report on state-transition modelling takes this approach for longer-horizon models. Its best practice is that sequential decisions should not be modelled within the Markov cycle tree but should be part of the specification of the alternative strategies that precede it. The report notes that Markov decision processes generalise state-transition models by allowing sequential decisions to be embedded, so that decisions can be made in several time periods.

Worked example with a gene test, single-outcome rules and a missing comparator

All figures in this example are illustrative. A new oral therapy for a chronic condition works best in patients who carry a gene variant, found in 25% of patients. Standard care costs £5,000 and gives 8.0 discounted QALYs whatever the variant status. The new therapy costs £13,000 in total and gives 8.6 QALYs to carriers and 8.05 to non-carriers. A test that identifies carriers correctly, a simplifying assumption, costs £400. The analysis uses an illustrative threshold of £20,000 per QALY.

1. Resolve the downstream decision nodes after the test. For a carrier, the net monetary benefit is £159,000 with the new therapy (20,000 × 8.6 − 13,000) and £155,000 with standard care (20,000 × 8.0 − 5,000), so the therapy is chosen. For a non-carrier, it is £148,000 with the therapy (20,000 × 8.05 − 13,000) against £155,000 with standard care, so standard care is chosen.

2. Fold back the chance node for variant status. The resulting policy, "test and treat carriers only", has an expected net monetary benefit of £155,600 (0.25 × 159,000 + 0.75 × 155,000 − 400). Its expected cost is £7,400 (400 + 0.25 × 13,000 + 0.75 × 5,000) and its expected QALYs are 8.15 (0.25 × 8.6 + 0.75 × 8.0).

3. List every policy at the first decision node. The test also allows two other contingent policies, testing and then giving everyone standard care, or testing and then giving everyone the therapy. Each costs £400 more than its untested counterpart for the same QALYs, so both are weakly dominated and removed.

4. Apply the incremental rule to the remaining policies. Moving from standard care to testing and treating carriers gives the following ratio.

$$ICER_{T,S} = \frac{7400 - 5000}{8.15 - 8.00} = \frac{2400}{0.15} = 16000$$

where $ICER_{T,S}$ is the cost in pounds per QALY gained by moving from standard care for all to the test policy. Moving from the test policy to the new therapy for all gives the following.

$$ICER_{N,T} = \frac{13000 - 7400}{8.1875 - 8.15} = \frac{5600}{0.0375} = 149333.33$$

where $ICER_{N,T}$ is the cost in pounds per QALY gained by moving from the test policy to the therapy for all. The ratios rise, so no remaining policy is dominated or extendedly dominated. The test policy is chosen at £20,000 per QALY, and the choice holds across NICE's £25,000 to £35,000 range. Net monetary benefits are £195,000, £196,350 and £191,687.50 at £25,000, and £275,000, £277,850 and £273,562.50 at £35,000, for standard care, the test policy and the therapy for all.

5. Compare single-outcome rules and threshold effects. Choosing the highest expected QALYs would select the therapy for all, and choosing the lowest expected cost would select standard care. At the first decision node, standard care is preferred below £16,000 per QALY, the test policy between £16,000 and about £149,333, and the therapy for all above that. The downstream choice flips too. For carriers the therapy adds £8,000 for 0.6 QALYs, about £13,333 per QALY, so below that value the test no longer changes treatment and the test policy becomes "test, then standard care for all", which is dominated.

6. See what happens when a comparator is left out. If the test policy is omitted, the therapy for all is compared only with standard care, as follows.

$$ICER_{N,S} = \frac{13000 - 5000}{8.1875 - 8.00} = \frac{8000}{0.1875} = 42666.67$$

where $ICER_{N,S}$ is the cost in pounds per QALY gained by moving from standard care to the therapy for all. At an illustrative threshold of £50,000 per QALY, the therapy for all would appear cost-effective. With the test policy included, its ratio against the test policy is about £149,333, and the net monetary benefits at £50,000 are £395,000 for standard care, £400,100 for the test policy and £396,375 for the therapy for all, so testing is preferred. The decision node can only choose among the branches it is given.

Decision nodes, chance nodes and related ideas

Decision and chance nodes are evaluated by different operations and answer different questions. The table sets out the distinctions that matter when a tree is built or checked.

The gap between the best of expected values and the expected value of the best choice is the basis of the expected value of perfect information. Choosing the preferred strategy separately in each simulation of a probabilistic sensitivity analysis and averaging gives the expected value with perfect information; the decision on current evidence uses the highest expected net benefit across all simulations.

Errors made at decision nodes

Several errors at decision nodes give plausible-looking results, usually from treating the node as a chance node or applying a single-outcome rule to a two-outcome model. Checking the rule and the branch set at each decision node catches most of them.

• Choosing the highest expected QALYs or the lowest expected cost. In a cost-effectiveness model this ignores one of the two outcomes; the choice should use net benefit or a fully incremental analysis at a stated threshold.
• Attaching probabilities to decision branches. Shares of patients receiving different treatments in current practice belong at a chance node within the current-practice branch.
• Leaving out feasible strategies. A missing contingent policy can make another strategy appear cost-effective, as step 6 of the example shows.
• Using information not yet available. A downstream choice that depends on the true disease state rather than on an observed test result gives the strategy foresight it cannot have.
• Asymmetric branches. The state-transition modelling report explains that symmetric models represent the disease process consistently across strategies, specifying true disease status even where it is not observed; otherwise sensitivity analyses on that probability give errors.
• Re-optimising a downstream node in each simulation. The draws of a probabilistic analysis are not information the decision maker holds, so the downstream policy is chosen on expected values and held fixed across simulations, unless the aim is a value of information calculation. A scenario in which the decision maker does learn new information is different, and there the downstream node is re-optimised.

Sources

• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://eprints.gla.ac.uk/73438/
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• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford: Oxford University Press; 2015.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 4.2.16 and 4.10.8.
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