Decision Tree Model

Decision Tree Model Building, Probabilistic Analysis and Validation

A decision tree model is a decision analytic model for economic evaluation whose structure is a decision tree, so each strategy's patient pathways carry probabilities, costs and QALYs taken from the evidence, and the model returns an expected cost and expected QALYs for every option. The decision tree supplies the branching structure and the rollback rule, while the model adds the parameter estimates and their sources, a transparent build, a probabilistic analysis and validation. Petrou and Gray call the decision tree the simplest form of decision analytical modelling in economic evaluation. Tree models suit short, one-off episodes such as an elective procedure or an acute infection. Every pathway is visible, so each input can be traced to its source and the result recomputed by hand.

What turns a drawn tree into a decision tree model

A model starts from the decision problem, not from the drawing. Petrou and Gray put the definition of the question first, naming all relevant options, the recipient population, and the location and setting, and they let the decision maker's requirements set the perspective, the time horizon and the outcome measures. The ISPOR-SMDM task force report on conceptualising a model, by Roberts and colleagues, judges that a decision tree may be appropriate for simple models or for problems with very short time horizons or very few outcomes.

The NICE manual (PMG36, section 4.6.3) asks for the conceptual model development process behind the chosen structure to be transparent and justified, including details of expert involvement. For a tree, the case rests on showing that no event recurs and nothing important falls outside the horizon.

Populating the branches with evidence

Each branch probability, cost and health outcome is a parameter with a source. Petrou and Gray state that identifying and synthesising evidence to populate a decision analytical model should follow the general principles of evidence based medicine. Where randomised evidence is missing, inputs come from epidemiological or observational studies, medical records or, more controversially, expert opinion. Branch probabilities after the first chance node are conditional probabilities, so a population-wide estimate is converted before it goes on a branch only some patients reach.

With pay-offs attached to each pathway, a strategy's result can be written as an expected net monetary benefit, which keeps costs and QALYs in one unit.

$$\overline{NMB}_s = \sum_{j \in s} p_j \left( \lambda Q_j - C_j \right)$$

where $\overline{NMB}_s$ is the expected net monetary benefit of strategy $s$, the sum runs over its pathways $j$, $p_j$ is the pathway probability, $Q_j$ and $C_j$ are the QALYs and cost accumulated along pathway $j$, and $\lambda$ is the value placed on a QALY. Petrou and Gray note that discounting is usually applied when the time horizon is over one year.

Chance nodes with more than two branches

Three or more outcomes at one chance node complicate uncertainty analysis. Briggs, Ades and Price report that modellers have commonly been advised to use only two branches at each chance node, to avoid logical inconsistencies when varied branch probabilities no longer sum to one. They argue instead for the Dirichlet distribution, the multivariate equivalent of the beta distribution, whose draws always sum to one.

$$\pi_k = \frac{g_k}{\sum_{m=1}^{K} g_m}, \qquad g_k \sim \text{Gamma}(n_k, 1)$$

where $\pi_k$ is the sampled probability of branch $k$, $K$ is the number of branches, $n_k$ is the number of patients observed on branch $k$, and $g_k$ is a gamma draw with shape $n_k$ and scale 1. Normalising independent gamma draws in this way gives a Dirichlet draw whose mean for branch $k$ is $n_k$ divided by the total count.

A spreadsheet layout with one row per pathway

A small tree model can be built in a spreadsheet, with one row per pathway and every input entered once on a parameter sheet. Microsoft's documentation describes SUMPRODUCT as returning the sum of the products of corresponding ranges, which is the expected value calculation.

Microsoft documents RAND as returning an evenly distributed number of at least 0 and below 1, recalculated with the worksheet, and GAMMA.INV as giving the standard gamma distribution when its beta argument is 1. A probabilistic run recalculates the sheet many times and records each result. A parameter shared by both strategies, such as the cost of a major complication, is drawn once per simulation and used in both.

Worked example of a three-outcome surgical tree

All numbers are illustrative. A new technique for an elective procedure is compared with the standard technique over one year, so nothing is discounted. Case series of 200 patients each give the counts for one chance node with three outcomes. The procedure costs GBP 3,000 with the standard technique and GBP 3,400 with the new one, and one-year QALYs and added costs are as shown.

1. Run the model at the point estimates. The standard technique has an expected cost of 3,000 + 0.15 × 500 + 0.05 × 12,000 = GBP 3,675 and expected QALYs of 0.80 × 0.80 + 0.15 × 0.78 + 0.05 × 0.40 = 0.7770. The new technique has 3,400 + 0.09 × 500 + 0.03 × 12,000 = GBP 3,805 and 0.88 × 0.80 + 0.09 × 0.78 + 0.03 × 0.40 = 0.7862 QALYs.

$$\text{ICER} = \frac{3805 - 3675}{0.7862 - 0.7770} = \frac{130}{0.0092} \approx 14130$$

where the numerators are expected costs in GBP and the denominators expected QALYs, so the ICER is about GBP 14,130 per QALY gained. PMG36 (section 4.2.16) asks for net health benefits at values of GBP 25,000 and GBP 35,000 per QALY, and at GBP 25,000 the incremental net monetary benefit is 25,000 × 0.0092 − 130 = GBP 100 per patient.

2. Set the distributions. The outcome probabilities become Dirichlet(160, 30, 10) and Dirichlet(176, 18, 6). Costs and QALYs are held fixed for brevity. One illustrative set of gamma draws for the standard technique is 152, 36 and 12, which sum to 200 and give probabilities of 0.76, 0.18 and 0.06, an expected cost of 3,000 + 0.18 × 500 + 0.06 × 12,000 = GBP 3,810 and 0.76 × 0.80 + 0.18 × 0.78 + 0.06 × 0.40 = 0.7724 QALYs.

3. Compare the probabilistic mean with the point estimate. The Dirichlet means are the observed shares and the outputs are linear in them, so the probabilistic means equal the deterministic results. A simulation of 200,000 draws gave GBP 3,673.93 and 0.77704 QALYs for the standard technique and GBP 3,805.24 and 0.78619 for the new one, differences that are Monte Carlo error.

4. Measure the decision uncertainty. At GBP 25,000 per QALY the net value of each outcome, $v_k = 25000\,Q_k - c_k$, is GBP 20,000, GBP 19,000 and minus GBP 2,000. With the procedure cost fixed, the variance of each strategy's net monetary benefit follows from the Dirichlet moments.

$$\operatorname{Var}\left(\sum_{k} \pi_k v_k\right) = \frac{\sum_{k} \bar{\pi}_k v_k^2 - \left(\sum_{k} \bar{\pi}_k v_k\right)^2}{n_0 + 1}$$

where $\bar{\pi}_k$ is the mean probability of branch $k$, $v_k$ is its net value, $c_k$ and $Q_k$ are its added cost and QALYs, and $n_0$ is the total count of 200. For the standard technique the numerator is 374,350,000 minus 351,562,500 (the square of 18,750), or 22,787,500, and dividing by 201 gives about 113,371. For the new technique it is 384,610,000 minus 370,562,500, or 14,047,500, giving about 69,888. The series are independent, so the incremental net benefit has a variance of about 183,259 and a standard deviation of about GBP 428. A normal approximation (100 divided by 428 is about 0.23) puts the probability that the new technique is cost effective at about 0.59, and the simulation gave 0.587 at GBP 25,000 and 0.645 at GBP 35,000.

5. Interpret the result. On expected values the new technique is preferred, but the chance that this choice is wrong is about 41% at GBP 25,000, which a cost-effectiveness acceptability curve would show. About 11% of the draws give a QALY loss, so the simulated ICERs range over large positive and negative values and their average changes sign between runs. PMG36 (section 4.2.16) requires a reported ICER to be the ratio of expected additional total cost to expected additional QALYs. The simulated per-patient expected value of perfect information of about GBP 124 is an upper limit on what further research is worth per patient, and Petrou and Gray scale it to the population and compare it with research costs.

Reading and reporting the results

NICE's PMG36 (section 4.7.12) states that the committee's preferred cost-effectiveness estimate should come from a probabilistic analysis when possible unless the model is linear. A tree in which independent parameters each appear at most once on any pathway is multilinear in them, a sum of products of independent parameters, so its probabilistic and deterministic means agree. The agreement breaks when a probability enters a pathway twice, when a probability is derived from a rate, or when parameters are correlated. For a probability distributed as Beta(40, 160) applied to two successive operations, the expected square is 0.04 + 0.000796 = 0.040796, not 0.04.

Section 4.7.15 of the manual asks for the probability that the treatment is cost effective at maximum acceptable ICERs of GBP 25,000 to GBP 35,000 per QALY gained, alongside the expected mean results. The Briggs task force report on parameter estimation calls beta distributions a natural match for binomial data and warns that arbitrary analyses, such as varying each input by 50% either way, should not be used to represent uncertainty. With several strategies, dominance is checked first.

Validating and documenting a decision tree model

A small tree can still be wrong in ways that leave the arithmetic tidy. Petrou and Gray describe three steps of escalating difficulty, face, internal and external validation, and suggest setting some variables at null or extreme values to see whether outputs move as expected. The Eddy task force report on transparency and validation defines verification as checking whether the model's parts behave as intended and the model has been implemented correctly.

For a tree, giving both strategies identical inputs must return zero increments. PMG36 (section 4.6.4) asks for the methods of quality assurance, the methods and results of model validation, and results in a disaggregated format.

Where a decision tree model stops being the right tool

Recurring events and long-term consequences are better handled by a Markov model or another state-transition model, often fed by a short up-front tree. The broader choice of model type is discussed on the health economic model page.

Common errors in building and running decision tree models

Most errors in tree models sit in the build and rarely show in the results. The pathway layout and its checks catch them.

1. Typed complements. A hand-entered 0.15 on a branch that should be one minus another lets probabilities drift away from one in sensitivity analysis.
2. Independent draws on a multi-branch node. Separate beta draws for three outcomes do not sum to one, which a Dirichlet distribution avoids.
3. Unshared draws. Drawing a common parameter separately for each strategy adds noise to the increments.
4. Averaged ratios. The mean of simulated ICERs is reported instead of the ratio of mean incremental cost to mean incremental QALYs.

Sources

• Briggs AH, Ades AE, Price MJ. Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework. Medical Decision Making. 2003;23(4):341-350. https://doi.org/10.1177/0272989X03255922
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6. Value in Health. 2012;15(6):835-842. https://doi.org/10.1016/j.jval.2012.04.014
• Eddy DM, Hollingworth W, Caro JJ, Tsevat J, McDonald KM, Wong JB. Model transparency and validation: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-7. Value in Health. 2012;15(6):843-850. https://doi.org/10.1016/j.jval.2012.04.012
• Microsoft Support. BETA.INV, GAMMA.INV, RAND and SUMPRODUCT functions. Excel function reference, accessed 31 October 2026. https://support.microsoft.com/en-gb/excel/functions/sumproduct-function
• 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, 4.6.3, 4.6.4, 4.7.12 and 4.7.15. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Petrou S, Gray A. Economic evaluation using decision analytical modelling: design, conduct, analysis, and reporting. BMJ. 2011;342:d1766. https://doi.org/10.1136/bmj.d1766
• Roberts M, Russell LB, Paltiel AD, Chambers M, McEwan P, Krahn M. Conceptualizing a model: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2. Value in Health. 2012;15(6):804-811. https://doi.org/10.1016/j.jval.2012.06.016