Rollback

Concept

Theoretically, Rollback is a mathematical evaluation procedure used in decision analysis to determine the optimal sequence of decisions in a decision tree by solving the tree from its terminal nodes back to the initial decision node. It is founded on the principles of expected utility and expected value theory, whereby chance events are evaluated before decisions are made. The concept exists to identify the decision pathway that maximises expected benefit or minimises expected cost under uncertainty.

Mathematically, rollback is performed using backward induction. Expected values are calculated at each chance node as the probability-weighted average of subsequent outcomes, while decision nodes retain the branch associated with the optimal expected value. The recursive evaluation continues until the root node is reached, producing the optimal decision strategy.

In practice, rollback is implemented after a complete decision tree has been constructed with all probabilities, costs and outcomes assigned. In health economics it is applied to evaluate diagnostic strategies, treatment pathways, screening programmes and health technology assessments by identifying the intervention with the highest expected net benefit or lowest expected expected cost.

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Purpose

Used to evaluate decision trees by calculating expected values through backward induction, thereby identifying the optimal healthcare decision under uncertainty.

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Mathematical Formulae

Primary Formula

Expected value at a chance node:

EV = ????� p?V?

where:

• EV = expected value
• p? = probability of outcome i
• V? = value (cost, utility, net benefit or payoff) associated with outcome i

Decision rule at a decision node:

V = max{V?, V?, ?, V?}*

or

V = min{V?, V?, ?, V?}*

depending on whether the objective is to maximise benefit or minimise cost.

Supporting Formulae

Expected Net Monetary Benefit:

NMB = ?E ? C

where:

• ? = willingness-to-pay threshold
• E = expected health effect
• C = expected cost

Related Mathematical Methods

• Backward induction
• Expected value analysis
• Decision tree analysis
• Expected utility theory
• Net monetary benefit analysis
• Expected value of perfect information

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Example

A decision tree compares two treatment options.

Treatment A

• 80% probability of recovery with a payoff of �100,000
• 20% probability of treatment failure with a payoff of �20,000

Expected value:

EV? = 0.80(100,000) + 0.20(20,000) = 84,000

Treatment B

Expected value:

EV? = 76,000

During rollback, the chance nodes are evaluated first and the decision node retains Treatment A because

84,000 > 76,000.

Thus, Treatment A is selected as the optimal strategy.

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Excel Implementation

Function	Example Formula	Health Economics Application
SUMPRODUCT	=SUMPRODUCT(B2:B4,C2:C4)	Calculates expected values at chance nodes.
MAX	=MAX(D2:D5)	Selects the optimal strategy at decision nodes when maximising benefit.
MIN	=MIN(D2:D5)	Selects the least-cost strategy where appropriate.
IF	=IF(D2=MAX($D$2:$D$5),""Optimal"","""")	Identifies the preferred branch during rollback analysis.

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VBA (Optional)

A VBA procedure can automate rollback by evaluating decision trees recursively from the terminal nodes to the root node and identifying the optimal decision pathway.

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Sources

• Raiffa H. Decision Analysis: Introductory Lectures on Choices Under Uncertainty. Addison-Wesley.
• TreeAge Software. TreeAge Pro User Guide.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
• NICE. Health Technology Evaluation Manual.
• ISPOR Good Practice Reports on decision-analytic modelling.