Folding Back

Folding Back: Concept Architecture

Orientation and learning roadmap

The page follows a decision tree from its terminal outcomes back to the first choice. It shows how to average chance branches, choose at decision nodes using a consistent objective, and handle a later decision that depends on observed information. A worked health-economic example and audit checks make the calculation and its limits visible.

Read a tree from its outcomes toward its root

Each complete path through a decision tree ends with outcomes such as costs, health effects, or a utility-valued payoff. Folding back replaces the rightmost chance and decision nodes with their calculated values, then repeats until each initial strategy has an expected result. The order matters because a node's value depends on the values of branches that follow it.

At a chance node, the mutually exclusive, exhaustive outgoing events have conditional probabilities that sum to one. The value is a probability-weighted mean of its successor values; add any cost or benefit incurred at the current node according to the model's accounting convention. At a decision node, select an available action that maximises the prespecified objective, or minimises it if the objective has explicitly been defined as a loss.

$$V_{\mathrm{chance}}=r+\sum_j p_jV_j,\qquad \sum_j p_j=1; \qquad V_{\mathrm{decision}}=r+\max_{a\in\mathcal A}V_a,$$

where $r$ is an immediate payoff on the chosen scale, $p_j$ is a probability conditional on reaching the chance node, and $\mathcal A$ is the set of actions feasible with the information then available. A different action may be optimal at a later decision node reached after an observed test result. The maximum does not apply to chance branches, which cannot be selected at will.

Keep cost and health outcomes aligned

Health-economic trees often carry both monetary costs and health outcomes, such as quality-adjusted life-years (QALYs), through every path. Compute the expected cost and QALYs under each strategy using the same branch probabilities, time horizon, and perspective. If a decision node needs a single scalar objective, expected net monetary benefit (NMB) at a stated value $\lambda$ per QALY can be used: $\operatorname{NMB}=\lambda Q-C$.

Maximising expected NMB is equivalent to comparing $\lambda\operatorname{E}(Q)-\operatorname{E}(C)$ when $\lambda$ is fixed and the specified decision criterion is risk neutral in this net-benefit scale. Taking the smallest expected cost branch and the largest expected QALY branch separately would combine different, possibly incompatible actions. An incremental cost-effectiveness ratio can be reported after strategy outcomes are calculated; it is not an appropriate value to average over chance branches.

Work through a test-and-treat decision

Suppose standard care has expected cost £1,000 and expected health of 0.70 QALYs over a stated horizon. An alternative testing strategy costs £100 for the test, reveals a positive result with probability 0.30 and a negative result with probability 0.70, and permits a treatment choice after the result. The following treatment costs exclude the test charge; each listed QALY figure is the full-horizon outcome conditional on the observed result and chosen action.

At the positive-result decision node, treatment has the higher NMB; at the negative-result node, no treatment has the higher NMB. Those actions form one contingent testing policy because the result is observed before the action is chosen. The result probabilities should be applied once to the chosen action for each result, not once to every row of the table.

The testing policy has expected cost $100+0.30(2{,}000)+0.70(500)=£1{,}050$ and expected QALYs $0.30(0.90)+0.70(0.72)=0.774$. Its expected NMB is $10{,}000(0.774)-1{,}050=£6{,}690$, compared with $10{,}000(0.70)-1{,}000=£6{,}000$ for standard care. At this illustrative threshold and with these assumed inputs, folding back selects the test-and-treat policy; the example is arithmetic, not a real screening recommendation.

Respect what the decision maker knows

A later decision can depend on a result only if the result has become available before the choice. Probabilities at a chance node may be conditional on the earlier pathway, so a branch probability copied from another population or treatment arm can produce an invalid expected value. Shared patients, competing events, and repeated outcomes also require a tree structure that neither omits nor double counts paths.

Folding back is a calculation applied to the tree that has been specified; it does not verify whether that tree represents actual care well. A long-term condition with recurrent events or time-varying risks may be better represented with another model or with a tree linked to later modelling. The chosen structure should follow the decision problem rather than the convenience of a spreadsheet diagram.

Recalculate under uncertainty and audit the tree

The base-case rollback uses specified probabilities, costs, and outcomes, each of which may be uncertain. Deterministic scenarios and probabilistic sensitivity analysis can rerun the folding-back calculation under alternative coherent input sets. When a result changes the preferred action at a downstream decision node, that action must be reoptimised in each relevant scenario if the analysis assumes the decision maker can respond to that information.

• Check every chance-node probability sum, branch label, and conditioning population.
• Verify each terminal path's cumulative cost and health outcome before folding it into earlier nodes.
• Compare a hand-calculated small tree with the model's node values and root outputs.
• State the objective and threshold before choosing branches at decision nodes.
• Report the selected contingent policy and the expected cost and effect for every initial strategy.
• Explore structural and parameter uncertainty rather than presenting one rollback as a certainty.

Sources and technical basis

Methodological publications on decision-analytic modelling describe probability-weighted chance nodes and backward evaluation of decision trees in economic analysis. NICE guidance places a simple decision tree among possible structures for health-economic questions when it adequately represents the pathway. The numerical test-and-treat values above are original and solely illustrate the arithmetic and information timing.

• BMJ, Economic evaluation using decision analytical modelling: design, conduct, analysis, and reporting.
• Decision Analysis and Cost-effectiveness Analysis, methodological article.
• NICE, Developing NICE guidelines: the manual, incorporating economic evaluation.
• Clinical decision analysis, worked method article.