Concept Architecture
Concept
Theoretically, Expected Value is the probability-weighted average of all possible outcomes of a random variable and represents its long-run average value over repeated observations. In health economics, expected value provides the theoretical basis for decision analysis under uncertainty by combining the value of each possible outcome with its probability of occurrence. It is fundamental to decision trees, probabilistic models and economic evaluation because it quantifies the average costs, health outcomes or net benefits expected from competing interventions.
Mathematically, Expected Value is calculated as the weighted sum of all possible outcomes, with each outcome multiplied by its probability. This framework estimates the average value of uncertain quantities, including costs, quality-adjusted life-years, utilities and net monetary benefits. In continuous settings, Expected Value is represented by integration over the probability distribution of the random variable.
In practice, Expected Value is estimated using observed probabilities from clinical studies, epidemiological data, expert elicitation or probabilistic models. Health economists calculate expected costs and expected health outcomes at chance nodes within decision trees and other decision-analytic models to compare alternative interventions and determine the strategy with the greatest expected value.
Purpose
Used to estimate the average costs, health outcomes or economic value associated with uncertain events, providing the basis for decision-making under uncertainty in health economic evaluation.
Mathematical Formulae
Primary Formula
E(X) = ????� x?P(x?)
where:
- E(X) = expected value
- x? = possible outcome
- P(x?) = probability of outcome i
Supporting Formulae
For a continuous random variable:
E(X) = ??�^� x f(x) dx
Expected cost:
E(C) = ????� C?P?
Expected health outcome:
E(H) = ????� H?P?
Related Mathematical Methods
- Decision tree analysis
- Probability theory
- Bayesian decision analysis
- Markov modelling
- Monte Carlo simulation
- Probabilistic sensitivity analysis
- Expected Net Monetary Benefit analysis
Example
A treatment has two possible outcomes.
| Outcome | Probability | Cost (�) |
|---|---|---|
| Recovery | 0.80 | 2,000 |
| Complication | 0.20 | 12,000 |
Expected cost:
E(C) = 2,000(0.80) + 12,000(0.20)
E(C) = 1,600 + 2,400 = �4,000
The intervention is therefore expected to cost �4,000 per patient when uncertainty is considered.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B3,C2:C3) | Calculate expected costs, QALYs or utilities from probabilities and outcomes. |
| SUM | =SUM(D2:D10) | Aggregate expected values across decision branches. |
| IF | =IF(ExpectedCost<Comparator,""Preferred"",""Not Preferred"") | Compare expected outcomes between interventions. |
| RAND | =RAND() | Generate random values for simulation of uncertain events. |
VBA (Optional)
Automate repeated expected value calculations across multiple decision models, scenarios or probabilistic sensitivity analyses.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Raiffa H, Schlaifer R. Applied Statistical Decision Theory.
- Briggs AH, Weinstein MC, Fenwick EAL, et al. ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
- NICE. Health Technology Evaluation Manual.
Related Concepts (3)
Library
Publications
1
Conceptualizing a Model: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2 — Roberts, Russell, Paltiel, Chambers, McEwan & Krahn, Task Force Report 2 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on model conceptualisation — defining the decision problem, scoping, and choosing an appropriate model structure before implementation.
Journal ArticleView source →
Frequently Asked Questions (6)
What is expected value?
The probability-weighted average of all possible outcomes of an uncertain event, found by multiplying each outcome by its probability and summing.
Source: von Neumann & Morgenstern 1944
Why is expected value used to compare uncertain options?
When the outcome of a choice is uncertain, no single result can represent it, so decision analysis summarises each option by the average result it would give over many repetitions, weighting each possible outcome by its probability. Comparing options on this probability-weighted average provides a consistent rule for choosing under uncertainty and is the basis of the expected value criterion. It assumes the decision-maker is content to be guided by the long-run average rather than by the worst or best case. Hunink and colleagues (2014) explain this rationale.
Source: Hunink et al. 2014
How is expected value calculated?
Expected value is calculated by listing all the possible outcomes of an uncertain event, multiplying each outcome by its probability, and adding these products together. For example, the expected value of a treatment's cost is the sum, over all possible cost outcomes, of each cost multiplied by its probability. The result is a single number summarising the uncertain situation by its probability-weighted average, which represents the outcome expected on average across the range of possibilities.
Source: von Neumann & Morgenstern 1944
How is expected value used in decision analysis?
In decision analysis, expected value is used to summarise and compare uncertain options: the expected cost and expected effect of each strategy are computed by weighting the outcomes of its possible paths by their probabilities, and the strategies are compared on these expected values. Folding back a decision tree computes expected values at chance nodes and chooses the best at decision nodes. Choosing the option with the best expected value is the standard criterion for decisions under uncertainty in economic evaluation.
Source: von Neumann & Morgenstern 1944
How does expected value relate to expected utility?
Expected value is the probability-weighted average of outcomes measured in their natural units, such as money or health, while expected utility is the probability-weighted average of the utility, or value, of those outcomes. Von Neumann and Morgenstern showed that a rational decision maker under risk should maximise expected utility, not expected value, because utility captures attitudes to risk. When outcomes are valued linearly, the two coincide, but where risk aversion matters, expected utility, not expected value, is the appropriate criterion.
Source: von Neumann & Morgenstern 1944
What are the limitations of using expected value?
Using expected value alone ignores the spread and risk of outcomes, treating a certain outcome and a risky one with the same average as equivalent, which may not match a decision maker's preferences when large losses or risk aversion are involved. It also depends on the probabilities and outcomes being correctly specified, which may be uncertain. For decisions where risk matters, expected utility is preferred, and expected value is best seen as a summary that must be complemented by attention to the uncertainty around it.
Source: von Neumann & Morgenstern 1944
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 30 Sep 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/expected-value
- Term code
- HE-EM-DM-033
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