M8 · ECONOMIC MODELLING
The simplest model there is.
Last lesson left us with a question: we know why we model, but how? We start with the plainest structure in the whole toolkit — one that fits a single decision with a handful of possible outcomes, and nothing more.
Picture a real clinical choice. A patient presents with an acute problem. You can offer a new intervention or standard care. Each path might go well or badly — and each way of going well or badly carries its own cost and its own health outcome. Lay that out as a branching map, attach numbers, and you have a decision tree: the simplest economic model there is.
It has one deliberate limitation, which we'll come to: it has no clock. But for the right kind of question, that's exactly what makes it clean.
Two kinds of node.
A tree is built from branch points, and there are two kinds — keeping them straight is the whole grammar of the model.
- A decision node (drawn as a square) is a choice you control: which arm to take — new intervention or standard care. You pick the branch.
- A chance node (drawn as a circle) is a fork nature controls: what happens to the patient — success or complication, response or no response. You don't pick; probability does.
The branches leaving a chance node must obey one rule: they are mutually exclusive and exhaustive. Exactly one of them happens, so their probabilities sum to 1. If a node splits into "success" and "complication," and success is 0.8, complication must be 0.2 — there's nowhere else for the patient to go. A dossier whose chance-node probabilities don't add to 1 has a broken model, and it's one of the first things worth checking.
Every leaf carries a payoff.
Follow any path from the root to its tip and you reach a leaf — one complete story of what happened to a patient: which arm, then which chance outcome. Every leaf carries two numbers, the payoffs for that path:
- a cost (£) — everything spent getting to that outcome, and
- a health payoff (QALYs) — the quality-adjusted life-years that outcome delivers.
So a leaf might read: "new intervention → complication → £16,000, 5.0 QALYs." These payoffs are the raw material. The model's job is to boil all the leaves of one arm down into a single expected cost and a single expected QALY — so the two arms can be compared with the ICER and net benefit from Module 7.
Rolling back the tree.
That boiling-down has a name — rolling back the tree (or "averaging out") — and it's simpler than it sounds. You work from the leaves back toward the root, and at every chance node you replace the whole fork with its expected value: multiply each branch's payoff by its probability and add them up.
Take one chance node. Two branches: success (probability 0.8, cost £6,000) and complication (probability 0.2, cost £16,000). The expected cost at that node is:
(0.8 × £6,000) + (0.2 × £16,000) = £4,800 + £3,200 = £8,000
Do the same for QALYs, and that node collapses into a single (cost, QALY) pair. Repeat outward until the whole arm is one expected cost and one expected QALY. That's it — the entire mechanics of a decision tree is weighted averaging, done repeatedly. What a modelling package does on thousands of branches, you just did on one by hand.
Build and fold a two-arm tree.
Here's a real decision — a new intervention versus standard care for an acute condition. Each arm has a chance node: the patient either does well or suffers a complication, each with its own cost and QALYs. Move the probability sliders and watch the tree fold down to an expected cost and expected QALY per arm — and an ICER for the comparison.
New intervention → expected cost £8,000 · expected QALYs 7.4
Standard care → expected cost £5,400 · expected QALYs 6.5
Incremental: ΔCost £2,600 · ΔQALY 0.9
ICER: £2,889 per QALY
COST-EFFECTIVE at £30,000
Every number below the tree is just weighted averaging rippling back to the root. Nudge the new intervention's success probability down and watch its expected QALYs fall and its ICER climb — the model responds instantly, because it's nothing but arithmetic on the branches. This is a complete economic evaluation, start to finish, in one picture.
Now you.
A single chance node for a new procedure. With probability 0.7 there's no complication, costing £5,000. With probability 0.3 there's a complication, costing £15,000.
Roll the node back: what is the expected cost? (Enter it in pounds, a plain number.)
When a tree is the right tool.
A decision tree isn't a simplified model to apologise for — for some questions it's exactly the right one. It fits best when three things are true:
- The event is acute and one-off. A single episode with a beginning and an end: emergency surgery versus conservative management, a one-time diagnostic test, treating an acute infection.
- The horizon is short. Everything that matters plays out over days, weeks, or a few months — not a lifetime of recurring events.
- The pathways are few and enumerable. You can actually draw every clinically meaningful route from decision to outcome without the diagram exploding.
When those hold, the tree's simplicity is a virtue. It's fully transparent — every path, probability, and payoff is visible on one diagram — and there's nothing hidden for an assessor to distrust. Reaching for something more elaborate here wouldn't add accuracy; it would only add places for error to hide.
Where the tree goes blind: no clock.
Now the limitation that defines the tree's edge — and sets up everything that follows. A decision tree has no concept of time.
A path runs once, left to right, and ends at a leaf. There is no "and then a year passes." No way for a patient to return to an earlier state. No recurrence, no relapse, no progression-then-remission-then-progression-again. The tree is a single throw of the dice, not a life unfolding.
For an acute event, that's fine — the story really does end. But think about a chronic or recurring disease: a cancer that can relapse, go back into remission, and relapse again; heart failure with repeated hospitalisations over years. To force that into a tree, you'd have to draw a new branch for every possible sequence of events across time — relapse, then remission, then relapse again — and the diagram grows exponentially, one fresh sub-tree per cycle, until it's unreadable. And even then it can't naturally express the one thing that matters most: the patient coming back to a state they were in before.
That's not a flaw you fix by adding branches. It's a missing dimension — time — and it's precisely the gap the next lesson's structure was invented to fill: the Markov model.
What's the core structural problem?
You're assessing a therapy for a chronic condition in which patients cycle between "stable," "relapsed," and "remission" repeatedly over a 20-year horizon, with recurring costs each time they relapse. The manufacturer has modelled it as a decision tree. What's the core structural problem?
Why this matters for HTA
Decision trees turn up constantly in dossiers — for the right problems and, more revealingly, the wrong ones. Reading them well comes down to a few reflexes:
- Check the structure before the numbers. The first question isn't "are these probabilities right?" but "does a tree even fit this disease?" A chronic, recurring condition modelled as a tree is a structural red flag, no matter how carefully the parameters are estimated.
- Confirm every chance node sums to 1. Branches that don't add up mean patients are vanishing or being double-counted. It's a basic integrity check, and it's quick.
- Simplicity is a feature when it's earned. For an acute, short-horizon decision, a clean tree is more trustworthy than an elaborate model — everything is on view. Don't mistake sophistication for rigour; the right structure is the simplest one the decision actually needs.
A decision tree shows you a decision the way a photograph shows you a moment — perfectly, and only once. The instant the story needs a sequel, you need a different kind of model.
Decision trees, in one breath.
- A decision tree is the simplest economic model: decision nodes (choices you control), chance nodes (outcomes nature controls), and leaves carrying a cost and a QALY payoff.
- Branches at a chance node are mutually exclusive and exhaustive — their probabilities sum to 1.
- You solve it by rolling back: replace each chance node with its expected value (Σ probability × payoff), working leaves-to-root, until each arm is one expected cost and one expected QALY — then compare with the ICER.
- It fits acute, one-off, short-horizon decisions. Its hard limit is that it has no time dimension — it can't represent recurrence or return to earlier states.
The tree's power and its blindness are the same fact: it runs once. For a decision that ends, that's enough. For a life that continues, it isn't.
That missing clock is exactly what the next structure adds. When patients move between health states — stable, relapsed, in remission — over many years, we need a model built on states and cycles instead of one-way branches: the Markov model.