Concept Architecture
Concept
Theoretically, a Chance Node is a structural element within a decision tree that represents an uncertain event whose outcome is determined by probability rather than by the decision-maker. It originates from decision analysis and expected utility theory and is used to model stochastic processes affecting clinical, economic or epidemiological outcomes. In health economics, chance nodes represent uncertain events such as treatment response, adverse events, disease progression or mortality, enabling uncertainty to be incorporated explicitly into economic evaluation.
Mathematically, a chance node is represented by a set of mutually exclusive and collectively exhaustive branches, each assigned a probability. The expected value associated with the node is calculated as the probability-weighted sum of the values associated with each possible outcome. The probabilities assigned to all branches leaving a chance node must sum to one.
In practice, probabilities assigned to chance nodes are estimated from clinical trials, observational studies, meta-analyses, disease registries or expert elicitation. Chance nodes are implemented in decision trees and related decision-analytic models to calculate expected costs, health outcomes and cost-effectiveness by propagating uncertainty through the model.
Purpose
Used to represent uncertain events within decision-analytic models, quantify the effect of uncertainty on healthcare decisions and calculate expected costs and health outcomes for competing interventions.
Mathematical Formulae
Primary Formula
EV = ????� p?V?
where:
- EV = expected value of the chance node
- p? = probability of outcome i
- V? = value (such as cost, utility or QALYs) associated with outcome i
subject to:
????� p? = 1
Supporting Formulae
For expected cost:
E(C) = ????� p?C?
For expected health outcome:
E(H) = ????� p?H?
Related Mathematical Methods
- Decision tree analysis
- Expected value analysis
- Expected utility theory
- Probabilistic decision analysis
- Bayesian decision analysis
Example
Following surgery, a patient faces two possible outcomes:
- Successful recovery: probability = 0.85; QALYs = 9.2
- Major complication: probability = 0.15; QALYs = 5.0
The expected health outcome at the chance node is:
EV = (0.85 ? 9.2) + (0.15 ? 5.0)
EV = 7.82 + 0.75 = 8.57 QALYs
The chance node therefore contributes an expected health outcome of 8.57 QALYs to the decision tree.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B4,C2:C4) | Calculates the expected value at a chance node from probabilities and outcomes. |
| SUM | =SUM(B2:B4) | Verifies that branch probabilities sum to one. |
| IF | =IF(SUM(B2:B4)=1,""Valid"",""Check probabilities"") | Confirms that the probability structure is valid before analysis. |
VBA (Optional)
Automate expected value calculations across all chance nodes within a decision tree and validate that branch probabilities satisfy probability constraints.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Raiffa H. Decision Analysis: Introductory Lectures on Choices Under Uncertainty. Addison-Wesley.
- ISPOR-SMDM Modeling Good Research Practices Task Force reports.
Related Concepts (2)
Library
Publications
1
Conceptualizing a Model: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2 — Roberts, Russell, Paltiel, Chambers, McEwan & Krahn, Task Force Report 2 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on model conceptualisation — defining the decision problem, scoping, and choosing an appropriate model structure before implementation.
Journal ArticleView source →
Frequently Asked Questions (6)
What is a chance node in a decision tree?
A point in a decision tree where the pathway followed is determined by probability rather than choice, branches summing to one.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
What kinds of events are placed at a chance node?
A chance node marks a point where what happens next is outside anyone's control and governed by probability. Events placed there are the uncertain turns a patient's course can take, such as whether a test returns positive, whether a treatment succeeds, whether a side effect occurs, or whether the disease recurs. Each is assigned a probability drawn from evidence, and the branches leaving the node cover the possible results. The node captures uncertainty rather than choice. Hunink and colleagues (2014) describe what chance nodes model.
Source: Hunink et al. 2014
How does a chance node differ from a decision node?
A chance node and a decision node differ in what determines the branch followed. At a decision node, the branch represents a choice made by the decision maker, such as which treatment to give, and is selected. At a chance node, the branch represents the outcome of an uncertain event outside the decision maker's control, and is followed according to its probability. Decision nodes model choices; chance nodes model uncertainty, and a decision tree combines both to represent choices made under uncertainty.
Source: Briggs, Claxton & Sculpher 2006
Why must the probabilities at a chance node sum to one?
The probabilities at a chance node must sum to one because the branches represent all the possible outcomes of the event, which are exhaustive and mutually exclusive, so exactly one must occur. If the probabilities summed to less or more than one, some outcome would be unaccounted for or outcomes would be over-counted, and the expected values computed from the tree would be wrong. Ensuring the probabilities sum to one is therefore basic to the chance node representing the uncertainty correctly.
Source: Briggs, Claxton & Sculpher 2006
How are chance nodes used in calculating expected values?
Chance nodes are used in calculating expected values by weighting the outcomes of their branches by the corresponding probabilities. Working back through the tree, the expected value at a chance node is the sum of the values of its branches, each multiplied by its probability. This averages over the uncertain outcomes, giving the expected value of reaching that node. Repeating this back to the start yields the expected value of each option, so chance nodes are where uncertainty is averaged into the results.
Source: Briggs, Claxton & Sculpher 2006
What do chance nodes represent in a health decision model?
In a health decision model, chance nodes represent the uncertain events that affect a patient's course, such as whether a treatment works, whether an adverse effect occurs, or whether disease progresses, with probabilities drawn from clinical evidence. They capture the fact that the outcome of care is uncertain and not chosen, so the model can weigh the possible outcomes by their likelihood. Chance nodes thus embody the clinical uncertainty that decision analysis must account for in comparing options.
Source: Briggs, Claxton & Sculpher 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 29 Sep 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/chance-node
- Term code
- HE-EM-DM-008
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