Signature
pi_t = p_1 * p_2
| Inputs | Definition | Unit |
|---|---|---|
p_1 | Probability of the branch taken at the first chance node on the pathway | probability from 0 to 1 |
p_2 | Probability of the branch taken at the second chance node, conditional on the branch taken at the first, written p_(2|1) in the article | probability from 0 to 1 |
pi_t | Probability that a person in the strategy reaches terminal node t | probability from 0 to 1 |
|---|
Function
Chance node evaluation in a decision tree
Maps the branches leaving a chance node, each with a probability conditional on the pathway that leads to the node and a cost and health outcome at its end, to the expected cost and expected health outcome at the node. It also maps the chain of conditional branch probabilities along a pathway to the probability of reaching the terminal node at its end. The branch probabilities at every chance node sum to one. Strategies are compared only after the tree has been evaluated, with the ICER formula HE-FM-ICER-001 or net monetary benefit HE-FM-NMB-001, which are not restated here. Rolling back a whole tree from the terminal nodes to the first decision is covered on the Folding Back page; the formulae below work at the level of one node or one pathway.
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Implementations
Excel
Pathway probability through two chance nodes
With the first-node probability in Prob1 and the conditional second-node probability in Prob2, the cell, named PathwayProb, returns the pathway probability. For a pathway through more nodes, PRODUCT over a range of the conditional probabilities gives the same result.
=Prob1*Prob2
Assumptions
Second-node probability conditional on the first branch
p_2 is the probability among people who took the first branch. Test sensitivity and specificity are conditional on true disease status, so they can be used directly at a test-result node placed after a disease-status node.
Either node order gives the same pathway probability
The tree can place the test result first and disease status second, using predictive values derived through Bayes theorem from prevalence, sensitivity and specificity. When the conditional probabilities are consistent, both orders give the same pathway probabilities and expected values.
Worked examples
True positive pathway in the test strategy
With a prevalence of 0.20 and a sensitivity of 0.90, the probability of having the disease and testing positive is 0.18.
p_1 = 0.20; p_2 = 0.90; pi_t = 0.18
False negative pathway in the test strategy
People with the disease who test negative make up 0.02 of the cohort, the pathway that carries the cost of later care.
p_1 = 0.20; p_2 = 0.10; pi_t = 0.02
False positive pathway in the test strategy
With a specificity of 0.85, 15% of the 80% without the disease test positive, a pathway probability of 0.12.
p_1 = 0.80; p_2 = 0.15; pi_t = 0.12
True negative pathway in the test strategy
The remaining pathway has probability 0.68, so the four pathway probabilities of the strategy sum to 1.00.
p_1 = 0.80; p_2 = 0.85; pi_t = 0.68
Common errors
Adding the probabilities of successive events
Adding the prevalence of 0.20 to the sensitivity of 0.90 gives 1.10, which is not a probability. Successive events along a pathway are multiplied, giving 0.18.
Unconditional probability at the second chance node
Using the overall proportion testing positive, 0.30, as the probability of a positive test after the disease branch gives a true positive pathway probability of 0.06 instead of 0.18, computed here for illustration, and the four pathways no longer sum to one.
Sources
Path probabilities as products of branch probabilities
Grinstead CM, Snell JL. Grinstead and Snell's Introduction to Probability. The CHANCE Project version of 4 July 2006, based on the 2nd edition published by the American Mathematical Society. Section 4.1, Discrete Conditional Probability (pp. 135-136): branch weights in a tree diagram are conditional probabilities, path probabilities are their products, and the reverse tree gives the Bayes probabilities.
Branch probabilities summing to one at multi-branch chance nodes
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. Abstract, which describes the logical inconsistencies that arise in sensitivity analysis when the branch probabilities at a chance node do not sum to 1, and proposes the Dirichlet distribution for multi-branch nodes.
Canonical Identity
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