Decision Tree

A decision tree represents alternative strategies and the uncertain pathways that can follow each choice. This page explains how decision, chance and terminal nodes are connected, how pathway probabilities and outcomes are calculated, and how rollback analysis compares expected costs and health outcomes. It also explains when a decision tree is appropriate and when a model with recurring states may be more suitable.

How a decision tree represents a decision problem

A decision tree begins with a choice between two or more mutually exclusive strategies. Each strategy can lead to uncertain events, subsequent decisions and final outcomes. The branches make the model structure visible so that readers can trace how each possible pathway contributes to the expected result.

A decision tree normally uses three node types:

Branches leaving a decision node represent the available alternatives. Branches leaving a chance node represent mutually exclusive outcomes whose probabilities must sum to one.

How pathways are constructed

Each complete route from the initial decision node to a terminal node is a pathway. A pathway should represent one logically possible and internally consistent sequence of events. Costs and health outcomes may accumulate along the pathway or be assigned at its terminal node.

A well-constructed decision tree should ensure that:

• The strategies leaving a decision node are mutually exclusive alternatives.
• The outcomes leaving a chance node are mutually exclusive and collectively exhaustive.
• Every possible pathway ends at a terminal node.
• Events appear in a clinically and temporally credible order.
• Costs and health consequences are attached once and at the correct point in the pathway.
• The same event is represented consistently across strategies when appropriate.

A decision tree can include more than one chance event along a pathway. As additional events are added, the number of branches can grow rapidly, making structural justification and validation increasingly important.

How pathway probabilities are calculated

The probability of a complete pathway is calculated by multiplying the conditional probabilities along that pathway. Probabilities leaving the same chance node must sum to one, but probabilities belonging to different stages should not be added together.

For a pathway containing two sequential events:

Pathway probability = P(Event 1) × P(Event 2 | Event 1)

If a treatment succeeds with probability 0.75 and an adverse event occurs among successful patients with probability 0.08:

Pathway probability = 0.75 × 0.08 = 0.06

Therefore, 6% of patients are expected to follow that complete pathway under the stated assumptions.

Conditional probabilities must reflect the preceding pathway. A probability estimated for the full population should not automatically be used as though it applies within every subgroup or prior outcome.

How expected costs and outcomes are calculated

Expected values combine the value attached to each terminal pathway with the probability of reaching it. Costs, health outcomes and utilities are calculated separately because they use different units and may support different decision rules.

For terminal pathways indexed by (i):

Expected cost = Σ(Pᵢ × Cᵢ)

Expected health outcome = Σ(Pᵢ × Eᵢ)

Where:

• (Pᵢ) is the probability of reaching terminal pathway (i).
• (Cᵢ) is the total cost associated with pathway (i).
• (Eᵢ) is the health outcome associated with pathway (i).
• The terminal-pathway probabilities within each strategy sum to one.

If outcomes occur at different times, costs and health outcomes may require discounting before they are combined. The applicable time horizon, perspective and outcome measure must therefore be defined before the tree is populated.

How rollback analysis compares strategies

Rollback analysis evaluates the tree from right to left. Expected values are first calculated at the chance nodes nearest the terminal outcomes and then carried backward through the tree until every strategy connected to the initial decision node has an expected cost and expected outcome.

The process is:

1. Calculate terminal pathway values. Combine all costs and outcomes accumulated along each complete pathway.
2. Calculate pathway probabilities. Multiply the relevant conditional probabilities along each pathway.
3. Evaluate chance nodes. Weight every downstream value by its probability and sum the results.
4. Move backward through the tree. Repeat the calculation until the initial decision node is reached.
5. Compare strategies. Apply the chosen decision rule to the expected costs and outcomes.

The preferred strategy cannot be identified from expected cost alone unless the strategies have equivalent outcomes. Cost-effectiveness analysis normally compares incremental costs and health outcomes or uses net benefit.

Worked example: comparing two treatments

Consider a simplified decision tree comparing a new treatment with standard care. Each strategy has three mutually exclusive terminal outcomes: treatment success, treatment failure and a serious adverse event. The values are synthetic and are included only to demonstrate the calculation.

The probabilities within each strategy sum to one.

Expected values for the new treatment

Expected cost

= (0.75 × £12,000) + (0.20 × £18,000) + (0.05 × £25,000)

= £9,000 + £3,600 + £1,250

= £13,850

Expected QALYs

= (0.75 × 4.5) + (0.20 × 3.2) + (0.05 × 2.0)

= 3.375 + 0.640 + 0.100

= 4.115 QALYs

Expected values for standard care

Expected cost

= (0.60 × £8,000) + (0.35 × £14,000) + (0.05 × £20,000)

= £4,800 + £4,900 + £1,000

= £10,700

Expected QALYs

= (0.60 × 4.0) + (0.35 × 3.0) + (0.05 × 1.8)

= 2.400 + 1.050 + 0.090

= 3.540 QALYs

Incremental comparison

Incremental cost

= £13,850 − £10,700

= £3,150

Incremental QALYs

= 4.115 − 3.540

= 0.575 QALYs

ICER

= £3,150 ÷ 0.575

= £5,478 per QALY, rounded to the nearest pound.

The example shows how a decision tree produces expected results for each strategy. Whether the new treatment is cost-effective depends on the applicable decision rule and cost-effectiveness threshold; the tree does not determine that judgment by itself.

Using net monetary benefit

Net monetary benefit converts expected health outcomes and costs into a common monetary scale. It can be calculated for each strategy after the tree has produced its expected costs and QALYs.

Net monetary benefit = (Threshold × Expected QALYs) − Expected cost

NMB = (λ × E) − C

Where:

• (λ) is the cost-effectiveness threshold.
• (E) is the expected health outcome.
• (C) is the expected cost.

The strategy with the highest expected net monetary benefit is preferred under this decision rule. Net benefit is particularly useful when comparing several strategies or performing probabilistic sensitivity analysis.

When a decision tree is appropriate

Decision trees work best when the relevant events occur over a limited period and can be represented as a manageable set of non-recurring pathways. They are commonly used for diagnostic tests, screening decisions, acute treatments and other problems in which the timing and sequence of events can be described explicitly.

A decision tree may be appropriate when:

• The decision involves a clearly defined set of alternative strategies.
• The important events occur once or over a relatively short time horizon.
• The order of events materially affects costs or outcomes.
• The number of clinically meaningful pathways remains manageable.
• Recurring health states are not central to the decision problem.

The structure should be chosen because it fits the decision problem, not simply because a tree is easy to draw.

When another model structure may be preferable

Decision trees become difficult to manage when events recur, risks change over time or patients can repeatedly move between health states. In those circumstances, a state-transition model, discrete-event simulation or another structure may represent the process more efficiently and transparently.

A hybrid model may use a decision tree for an initial diagnostic or treatment phase and a state-transition model for long-term outcomes. The connection between the model components must be explicit and internally consistent.

How uncertainty should be examined

A decision tree depends on probabilities, costs, outcomes and structural assumptions that may be uncertain. Sensitivity analysis tests whether reasonable changes to those inputs alter the expected results or preferred strategy.

Relevant approaches include:

• Deterministic sensitivity analysis changes selected inputs individually or in scenarios.
• Probabilistic sensitivity analysis assigns distributions to uncertain parameters and evaluates them jointly.
• Scenario analysis examines alternative pathways, assumptions, time horizons or model structures.
• Structural sensitivity analysis tests whether a different representation of the decision problem changes the conclusion.

Probabilities linked by a shared constraint must remain coherent when varied. For example, changing one probability in a two-outcome chance node requires the complementary probability to change so that the total remains one.

Implementing a decision tree in a spreadsheet

A small decision tree can be implemented transparently in a spreadsheet by using one row for each terminal pathway. The spreadsheet should keep inputs, pathway calculations and strategy-level outputs separate so that probabilities and accumulated values can be audited.

Spreadsheet formulas should reference labelled input cells rather than embed unexplained values. Probability checks and reconciliation tests should remain visible.

Common modelling errors

Decision-tree results can appear precise even when the underlying structure is incomplete or internally inconsistent. Validation should therefore examine the logic of the pathways as well as the numerical calculations.

Common errors include:

• Allowing probabilities leaving a chance node to sum to something other than one.
• Treating outcomes as mutually exclusive when patients can experience more than one.
• Using unconditional probabilities where conditional probabilities are required.
• Adding sequential probabilities instead of multiplying them.
• Omitting a clinically important pathway.
• Counting the same cost or health outcome at both a branch and its terminal node.
• Comparing average results without calculating incremental outcomes.
• Selecting the least costly strategy without considering differences in health outcomes.
• Using a decision tree for recurring events when another model structure would be clearer.
• Expanding the tree until its complexity can no longer be explained or validated.