Expected cost and health outcome at a chance node

Replaces a chance node with the probability-weighted averages of the costs and of the health outcomes at the ends of its k branches. Costs and health outcomes are averaged separately because they are measured in different units; in a cost-utility analysis the health outcome is usually QALYs. When a branch leads to a further chance node, its C_j and E_j are the expected values already calculated at that node.

Signature

C_n = sum_(j=1)^k [p_j * C_j]; E_n = sum_(j=1)^k [p_j * E_j]
Inputs
InputsDefinitionUnit
p_jProbability of branch j, conditional on the history of events on the pathway that leads to the nodeprobability from 0 to 1
C_jCost of the pathway ending at branch j, or the expected cost already calculated at the next chance node on that branchcurrency per person
E_jHealth outcome of the pathway ending at branch j, or the expected outcome already calculated at the next chance nodehealth outcome per person, for example QALYs
Output
C_nProbability-weighted average cost at chance node ncurrency per person, for example pounds
E_nProbability-weighted average health outcome at chance node nhealth outcome per person, for example QALYs
  • k Number of branches leaving the node, equal to the length of the p_j, C_j and E_j lists (count)

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

    Expected cost and QALYs at a chance node in two cells

    With the branch probabilities, branch costs and branch QALYs in named ranges of equal length, the two formulas return the expected cost and the expected QALYs at the node, in cells named NodeCost and NodeQALYs.

    =SUMPRODUCT(BranchProbs,BranchCosts); =SUMPRODUCT(BranchProbs,BranchQALYs)

Assumptions

  • Chance-node branches mutually exclusive and exhaustive

    The branches describe outcomes that cannot occur together and that between them cover every possibility, so the branch probabilities sum to one. Treatment failure and a serious adverse event are not mutually exclusive branches if a patient can have both, and a node that omits a possible outcome misstates the expected cost and outcome.

  • Branch probabilities conditional on the pathway to the node

    Each p_j applies to the people who reach the node, given the events before it. A complication probability placed after a treatment-fails branch is the probability among patients whose treatment failed, not the average across all patients.

  • Chance-node values right to left

    Nodes further from the decision are evaluated first, so that C_j and E_j for a branch leading to another chance node are already expected values. The result is the expected value for the average person reaching the node; no individual receives it.

Worked examples

  • Test-result node among people with the disease

    In the article's illustrative test-and-treat example, 90% of people with the disease test positive (cost £2,300, 8.0 QALYs) and 10% test negative (cost £2,800, 6.0 QALYs). The node's expected cost is £2,350 and its expected QALYs are 7.8.

    p_j = [0.90, 0.10]; C_j = [2300, 2800]; E_j = [8.0, 6.0]; k = 2; C_n = 2350; E_n = 7.8

  • Test-result node among people without the disease

    Among people without the disease, 15% test positive and are treated unnecessarily (cost £2,300, 9.8 QALYs) and 85% test negative (cost £300, 10.0 QALYs). The node's expected cost is £600 and its expected QALYs are 9.97, as in the article.

    p_j = [0.15, 0.85]; C_j = [2300, 300]; E_j = [9.8, 10.0]; k = 2; C_n = 600; E_n = 9.97

  • Disease-status node of the test strategy

    With a prevalence of 0.20, the disease-status node takes the expected values of the two test-result nodes as its branch values. The test strategy has an expected cost of £950 and expected QALYs of 9.536, the article's figures.

    p_j = [0.20, 0.80]; C_j = [2350, 600]; E_j = [7.8, 9.97]; k = 2; C_n = 950; E_n = 9.536

  • Single chance node of the treat-nobody strategy

    Treating nobody leaves untreated disease with £2,500 of later care and 6.0 QALYs, and no cost with 10.0 QALYs for people without the disease. The expected cost is £500 and the expected QALYs are 9.2.

    p_j = [0.20, 0.80]; C_j = [2500, 0]; E_j = [6.0, 10.0]; k = 2; C_n = 500; E_n = 9.2

  • Equal branch costs in the treat-everyone strategy

    Treating everyone costs £2,000 on both branches, so the expected cost equals that cost whatever the prevalence, a limiting case. The expected QALYs are 9.44, as in the article.

    p_j = [0.20, 0.80]; C_j = [2000, 2000]; E_j = [8.0, 9.8]; k = 2; C_n = 2000; E_n = 9.44

Common errors

  • Cost counted on a branch and again at the terminal node

    Attaching a cost to a branch and also including it in the terminal value of the same pathway counts it twice. Computed here for illustration: if the £300 test cost were attached to the test branch while every terminal value already included it, the test strategy's expected cost would rise from £950 to £1,250 and its ICER against treating nobody from about £1,339 to about £2,232 per QALY.

  • Averaging subgroup ICERs at a chance node

    A ratio cannot be rolled back. Computed here for illustration: in the article's example, testing against treating nobody saves £150 and gains 1.8 QALYs among people with the disease, and costs £600 more and loses 0.03 QALYs among people without it. Averaging the two subgroup ratios with weights 0.2 and 0.8 gives about minus £16,017 per QALY, whereas the ICER formed from the expected values, HE-FM-ICER-001, is about £1,339 per QALY. Net monetary benefit (HE-FM-NMB-001) can be averaged at the node because expectation is linear.

Sources

  • Expected values at chance nodes in a decision modelling textbook

    Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006. Chapter 2, Key aspects of decision modelling for economic evaluation (pp. 15-44). Decision trees for economic evaluation, in which expected costs and effects are calculated as probability-weighted averages at chance nodes.

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Canonical Identity

Persistent URI
https://healtheconomics.wiki/formula/expected-cost-and-health-outcome-at-a-chance-node

Stable URI · Machine-readable · Resolvable · CC BY 4.0

Expected cost and health outcome at a chance node | HealthEconomics.wiki