Chance Node

Chance Node Probabilities, Expected Values and Rollback in Health Economic Models

In a decision tree or Markov cycle tree for economic evaluation, an event that the decision maker cannot control, such as a response to treatment, a test result, an adverse event or a death, typically enters the model at a chance node. The node holds the probability of each possible outcome and is where expected costs and expected quality-adjusted life years are formed, so an error in its branches passes straight into the incremental cost-effectiveness ratio.

Chance, decision and terminal nodes in an economic evaluation

A decision tree for economic evaluation is built from three kinds of node, conventionally drawn as a square, a circle and a triangle. A decision node (square) marks a choice between strategies, such as testing or not testing, and carries no probabilities because the decision maker chooses between its branches. A chance node (circle) marks an uncertain event, and each branch leaving it carries a probability. A terminal node (triangle) closes a pathway, and the costs and health outcomes of that pathway are recorded there.

When the tree is evaluated, the three nodes do different jobs. Terminal nodes supply values, chance nodes average them, and the decision node compares the averaged results of each strategy. The ISPOR-SMDM good practice report on model conceptualisation notes that decision trees can be simple to conceptualise, create and modify, and are most suitable when the outcome set is small and defined, the time horizon is short, or the consequences of a decision are known with some certainty. Chance nodes also appear in longer-horizon models, as a later section shows, but a short tree is the clearest place to see how they work.

The probability rules at a chance node

The branches leaving a chance node must describe outcomes that cannot occur together and that between them cover every possibility. When that holds, the branch probabilities form a complete distribution for the event at that point in the pathway. Written formally, the rule is as follows.

$$\sum_{j=1}^{k} p_{j \mid h} = 1$$

where $k$ is the number of branches leaving the node, $p_{j \mid h}$ is the probability of branch $j$, and $h$ is the history of events on the pathway that leads to the node. The notation stresses that a branch probability is conditional. A probability of a complication placed after a "treatment fails" branch is the probability of a complication among patients whose treatment has failed, not the average across all patients.

The probability of reaching a given terminal node is the product of the conditional branch probabilities along its pathway.

$$\pi_t = p_{1} \times p_{2 \mid 1} \times \cdots \times p_{m \mid 1, \ldots, m-1}$$

where $\pi_t$ is the probability of reaching terminal node $t$, $m$ is the number of chance nodes on the pathway, $p_{1}$ is the probability of the branch taken at the first chance node, and each later term is the probability of the branch taken at that node given the branches taken before it. This is the multiplication rule for a joint probability. Within one strategy the values of $\pi_t$ over all terminal nodes sum to one, which gives a simple check on any tree. Probabilities of successive events along a pathway are multiplied, not added; probabilities of mutually exclusive pathways may be summed.

Evaluating a chance node through expected costs and QALYs

When a tree is evaluated, each chance node is replaced by the probability-weighted average of the values on its branches, which is its expected value. In an economic evaluation this is done separately for costs and for health outcomes, because they are measured in different units. For a chance node $n$ the expected values are as follows.

$$\bar{C}_n = \sum_{j=1}^{k} p_{j} C_{j}, \qquad \bar{E}_n = \sum_{j=1}^{k} p_{j} E_{j}$$

where $\bar{C}_n$ and $\bar{E}_n$ are the expected cost and expected health outcome at the node, $p_{j}$ is the conditional probability of branch $j$, and $C_{j}$ and $E_{j}$ are the cost and health outcome at the end of branch $j$. If branch $j$ ends at a terminal node, $C_{j}$ and $E_{j}$ are that pathway's values; if it leads to another chance node, they are the expected values already calculated there. In a cost-utility analysis the health outcome is usually measured in QALYs.

This right-to-left calculation is called rolling back, or folding back, the tree. Because expectation is linear, net monetary benefit can also be rolled back at a stated threshold, and the expected net monetary benefit at a node equals the threshold times the expected QALYs minus the expected cost. A ratio cannot be rolled back in the same way. The ICER between two strategies is formed from their expected costs and QALYs after the tree has been evaluated, and averaging subgroup ratios at a chance node does not, in general, give the ratio of expected values.

Where branch probabilities come from and how they are conditioned

Branch probabilities in health economic models come from trial event rates, disease prevalence, measures of diagnostic accuracy, registries and life tables. The evidence rarely arrives in the exact conditional form the tree needs, so part of building a chance node is converting reported figures into probabilities conditional on the pathway leading to it. Test accuracy is a common case. Sensitivity and specificity are probabilities of a test result given true disease status, so a tree that places a disease-status node before a test-result node can use them directly.

The same problem can be drawn with the test result first and disease status second. The second chance node then needs the probability of disease given the test result, which after a positive result is the positive predictive value, and these probabilities follow from prevalence, sensitivity and specificity through Bayes theorem. Both orderings give the same pathway probabilities and expected values when the conditional probabilities are consistent, so the order can follow the clinical sequence or the form of the evidence.

Worked example of a test-and-treat decision with nested chance nodes

All figures in this example are illustrative. A health service compares three strategies for a condition with a prevalence of 0.20. The strategies are to treat nobody, test everyone and treat those who test positive, or treat everyone without testing. The test has a sensitivity of 0.90 and a specificity of 0.85 and costs £300; treatment costs £2,000; untreated disease leads to later care costing £2,500, while treated disease incurs no later-care cost. Discounted lifetime QALYs are 8.0 for treated disease, 6.0 for untreated disease, 9.8 for a person without the disease who is treated unnecessarily and 10.0 for a person without the disease who is not treated.

1. Calculate the pathway probabilities for the test strategy. The first chance node is disease status and the second is the test result, conditional on disease status. A positive result costs £300 for the test plus £2,000 for treatment, and a false negative costs £300 plus £2,500 for later care.

The four pathway probabilities sum to 1.00, which confirms that the branches at each chance node are complete.

2. Roll back the test-result chance nodes. Among people with the disease, the expected cost is £2,350 (0.90 × 2,300 + 0.10 × 2,800 = 2,350) and the expected QALYs are 7.8 (0.90 × 8.0 + 0.10 × 6.0 = 7.8). Among people without the disease, the expected cost is £600 (0.15 × 2,300 + 0.85 × 300 = 600) and the expected QALYs are 9.97 (0.15 × 9.8 + 0.85 × 10.0 = 9.97).

3. Roll back the disease-status chance node. The test strategy has an expected cost of £950 (0.20 × 2,350 + 0.80 × 600 = 950) and expected QALYs of 9.536 (0.20 × 7.8 + 0.80 × 9.97 = 9.536). The same totals follow from multiplying each pathway in the table by its cost or QALYs and summing.

4. Evaluate the other strategies. Each has a single chance node for disease status. Treating nobody gives an expected cost of £500 (0.20 × 2,500 = 500) and expected QALYs of 9.2 (0.20 × 6.0 + 0.80 × 10.0 = 9.2). Treating everyone gives an expected cost of £2,000 and expected QALYs of 9.44 (0.20 × 8.0 + 0.80 × 9.8 = 9.44).

5. Compare the strategies at the decision node. Treating everyone costs more than testing and yields fewer QALYs, so it is dominated. Moving from treating nobody to testing adds £450 and 0.336 QALYs, as shown below.

$$ICER = \frac{950 - 500}{9.536 - 9.2} = \frac{450}{0.336} = 1339.29$$

where $ICER$ is the incremental cost in pounds per QALY gained by testing compared with treating nobody, here about £1,339. At an illustrative threshold of £20,000 per QALY, this ICER is below the threshold. The net monetary benefits are £183,500 for treating nobody (20,000 × 9.2 − 500), £189,770 for testing (20,000 × 9.536 − 950) and £186,800 for treating everyone (20,000 × 9.44 − 2,000), so testing is preferred under both rules.

The value of testing comes from the conditional structure of the chance nodes. It avoids the QALY loss from untreated disease in true positives while sparing most people without the disease the cost and harm of unnecessary treatment. The expected values describe the average person in the cohort; no individual has 9.536 QALYs.

Branch probabilities under uncertainty

Two kinds of uncertainty meet at a chance node and are easy to confuse. The ISPOR-SMDM report on parameter estimation and uncertainty distinguishes stochastic, or first-order, uncertainty, the random variation in outcomes between identical patients, from parameter uncertainty, or second-order uncertainty, the uncertainty in the estimate of a parameter. In a cohort tree the branch probabilities act as the proportions of the cohort following each branch, and the report notes that in cohort models parameter uncertainty can be addressed without concern for stochastic uncertainty. Sensitivity analysis examines the second kind, namely how far the branch probabilities themselves might differ from their point estimates.

In a one-way sensitivity analysis of a two-branch node, the varied probability $p$ must carry its complement $(1 - p)$ with it. A node with three or more branches is harder, because raising one branch probability requires the others to fall, and an arbitrary adjustment can leave probabilities that do not sum to one. Briggs, Ades and Price note that a common recommendation has been to use only two branches per chance node for this reason, and propose the Dirichlet distribution, the multivariate equivalent of the beta distribution, for multi-branch probabilities in probabilistic sensitivity analysis, so that each sampled set of probabilities sums to one. For a two-branch node, the ISPOR-SMDM uncertainty report describes beta distributions as a natural match for binomial data.

Chance nodes in Markov models and microsimulation

Chance nodes also sit inside longer-horizon models. The ISPOR-SMDM report on state-transition modelling describes how events within a cycle can be modelled with a Markov cycle tree, a series of chance nodes representing those events. It describes an up-front decision tree as an efficient and transparent way to model initial immediate or short-term events before patients enter a state-transition model, unless there are justified reasons for representing them within the states.

The same report states that a state-transition model should be used, rather than a simpler model with limited ability to reflect time such as a decision tree, if the problem requires time-dependent parameters, time to an event or repeated events. A tree with chance nodes for a second or third recurrence grows quickly and hides the timing of events, which is where a Markov model or microsimulation usually serves better. In a cohort model, chance-node probabilities divide the cohort into proportions; the conceptualisation report notes that microsimulation instead represents each patient individually and calculates outcomes by aggregating across individuals, so the same probabilities are used to sample each simulated patient's path.

Errors that arise at chance nodes

Several common errors in decision models start at a chance node, and they can produce results that look plausible. Checking the branches of every chance node, rather than only the final ICER, catches them before they reach the comparison of strategies.

• Branches that overlap or leave gaps. "Treatment failure" and "serious adverse event" are not mutually exclusive if a patient can have both, and a node that omits a possible outcome misstates expected costs and QALYs.
• Unconditional probabilities used downstream. A complication rate estimated across all treated patients is placed after a "treatment fails" branch, although the rate among patients whose treatment failed may differ.
• Costs counted twice. A cost attached to a branch and also included in the terminal value of the same pathway inflates expected costs.

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. doi:10.1177/0272989X03255922
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
• 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/
• 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.
• 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://pmc.ncbi.nlm.nih.gov/articles/PMC4207095/
• Siebert U, Alagoz O, Bayoumi AM, Jahn B, Owens DK, Cohen DJ, Kuntz KM. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812-820. https://www.ispor.org/docs/default-source/resources/outcomes-research-guidelines-index/state-transition_modeling-3.pdf