Decision tree rollback

Compare choices with a finite sequence of chance outcomes and terminal costs and QALYs.

Method lesson · Updated October 6, 2026

Concepts and reasoning

Separate choices from uncertain outcomes

A decision tree describes an ordered sequence of choices and events. At a decision node, someone selects an available action, such as testing or treating. At a chance node, an uncertain outcome occurs, such as a positive test or a treatment complication. A terminal node records the consequences after following one complete path.

These distinctions determine the arithmetic. The values at chance branches are probability-weighted. The values at decision branches are compared according to a stated objective. Averaging two available treatments merely because they are both drawn in the tree would confuse a choice with random allocation.

Branch probabilities are conditional

Suppose a test has a sensitivity of 0.9. This means the probability of a positive result among people with the condition, not the probability of the condition among people with a positive result. A testing tree must combine prevalence, sensitivity, specificity, and downstream management in the correct order.

Each set of branches leaving a chance node should represent mutually exclusive and exhaustive outcomes. Their conditional probabilities must sum to 1. The probability of a complete path is the product of the conditional branch probabilities along that path. Branches that omit a complication or use overlapping categories can produce an attractive but incoherent answer.

Trees work particularly well for short pathways with a manageable sequence of events. Repeating the same events over many years can make a tree enormous. State-transition models or discrete-event simulation may then describe the problem more naturally. The economic-modeling tutorial compares these decision structures.

Mathematical foundations

On a small screen, scroll wide equations and tables horizontally.

Expected value and rollback

For terminal outcomes reached with probabilities , the expected value at a chance node is

For cost-effectiveness decisions, one possible terminal value is . At a decision node, choose the branch with the largest expected value under that objective. Begin at terminal nodes and work backward until the initial decision has a value. This procedure is called rollback.

Cost and health can also be rolled back separately, then combined through net benefit. With a fixed threshold the two calculations agree because expectation is linear. In contrast, averaging branch-specific ICERs generally does not recover the ICER of the expected cost and effect increments; ratios are nonlinear.

For a testing example, the probability of a true positive is prevalence multiplied by sensitivity. With prevalence and specificity , false-positive probability is . These are unconditional path probabilities, whereas sensitivity and specificity are conditional branch probabilities.

Worked example

Hypothetical data for learning; independent of your analysis inputs.

Compare two short treatment pathways

Consider a hypothetical one-year decision with two treatments. Treatment A costs 1000. It succeeds with probability 0.8, producing 0.9 QALYs, and fails with probability 0.2, producing 0.5 QALYs and an extra 4000 cost. Treatment B costs 1800. It succeeds with probability 0.9 and has the same health outcomes and failure cost.

For A, expected QALYs are . Expected cost is . For B, expected QALYs are 0.86 and expected cost is 2200. The initial treatment cost is paid on every path; adding it only to the success branch would undercount cost.

Strategy	Expected cost	Expected QALYs	Net benefit at 50,000 per QALY
A	1800.000	0.820	39200.000
B	2200.000	0.860	40800.000

B adds 400 in cost and 0.04 QALYs. Its INMB is 1600, and its ICER is 10,000 per QALY. The decision is conditional on the assumed success rates and consequences.

For a separate probability check, prevalence 0.10, sensitivity 0.90, and specificity 0.80 produce true-positive probability 0.09 and false-positive probability 0.18. The probability of disease given a positive test is , not 0.90. This illustrates why tree order and conditioning matter.

Interpreting the results

Roll expectations back from outcomes to choices

A chance node combines its children's outcomes using their conditional probabilities. A decision node selects among available alternatives using the stated decision criterion. These operations differ: averaging over choices would describe a mixture of policies rather than the best modeled choice.

Expected cost and expected quality-adjusted life-years summarize all paths below a node plus costs and outcomes incurred there. They are population averages over the modeled uncertainty, not a prediction that every person experiences the average path. A rare expensive outcome may materially influence the expectation.

At a fixed willingness-to-pay threshold, the preferred child has the largest expected net monetary benefit. That selected branch can change with the threshold or input probabilities. Report the expected costs and health outcomes of alternatives, not only the selected node, so the trade-off remains visible.

Assumptions and limitations

Make each branch conditional and complete

Branches leaving a chance node must be mutually exclusive and exhaustive conditional on reaching that node. Their probabilities sum to one locally. A downstream probability is usually conditional on its preceding events; substituting an unconditional population probability can double-count or misrepresent a pathway.

A decision tree represents a finite sequence without cycles. Recurrent events or long histories may create an unwieldy tree and can require a state-transition or individual model. Events should occur in a clinically coherent order, with the same perspective and time horizon across choices.

Specify whether rewards occur at a node or are already included in terminal totals. Including the same treatment cost at both locations counts it twice. Discounting, if appropriate for different event times, must be handled consistently before comparing total outcomes.

Diagnostics and sensitivity analysis

Audit paths before accepting the preferred option

Trace at least one path by hand: multiply its conditional branch probabilities and add its node-level costs and health outcomes. Check that all terminal path probabilities beneath a chance-only subtree sum to one. Verify that every node is reachable and that no node points back to an ancestor.

Set probabilities to limiting values such as zero and one to confirm that outcomes reduce to the expected pathway. Remove duplicate rewards and compare identical alternatives as a consistency check.

Vary uncertain branch probabilities, high-cost consequences, and the willingness-to-pay threshold. Examine whether the selected child changes and why. A deterministic rollback provides no uncertainty interval by itself; a stable ranking under one set of inputs should not be confused with certainty about the decision.

References and other material

Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.

Cost-effectiveness analysis and decision modelling: a tutorial for clinicians — An introduction to economic decisions, trees and uncertainty.