Concept Architecture
Concept
Theoretically, a Linear Pool is a method for aggregating multiple probability distributions or expert judgements into a single combined distribution by calculating a weighted arithmetic average of the component distributions. It is founded on opinion-pooling theory and exists to synthesise differing probabilistic assessments while preserving the contribution of each source. In health economics, linear pooling is used to combine expert-elicited distributions when empirical evidence is limited or several experts provide distinct estimates for uncertain model parameters.
Mathematically, the pooled probability density or cumulative distribution function is constructed as a convex combination of the individual distributions. Each expert or evidence source is assigned a non-negative weight, and the weights sum to one. The combined distribution therefore represents a finite mixture in which the weights determine the relative influence of each component distribution.
In practice, analysts elicit probability distributions from multiple experts, assess the experts? performance or credibility where appropriate, assign equal or performance-based weights and calculate the pooled distribution. Samples, moments or quantiles from the pooled distribution may then be incorporated into probabilistic sensitivity analysis, decision models and value-of-information analysis.
Purpose
Used to combine multiple expert or evidence-based probability distributions into a single distribution for representing uncertain parameters in health economic models.
Mathematical Formulae
Primary Formula
f?(?) = ????� w?f?(?)
subject to:
w? � 0
????� w? = 1
Supporting Formulae
Pooled cumulative distribution function:
F?(?) = ????� w?F?(?)
Pooled expectation:
E?(?) = ????� w?E?(?)
Pooled variance:
Var?(?) = ????� w?[Var?(?) + ??�] ? ??�
where:
?? = ????� w???
Related Mathematical Methods
- Expert Elicitation
- Probability Distribution Mixtures
- Equal-Weight Pooling
- Performance-Weighted Pooling
- Bayesian Model Averaging
- Probabilistic Sensitivity Analysis
Example
Three clinical experts estimate the probability of treatment discontinuation as 0.10, 0.16 and 0.22. Performance-based weights of 0.50, 0.30 and 0.20 are assigned.
Pooled expected probability:
E?(?) = (0.50 ? 0.10) + (0.30 ? 0.16) + (0.20 ? 0.22)
E?(?) = 0.050 + 0.048 + 0.044
E?(?) = 0.142
The pooled expected discontinuation probability is 0.142. The complete pooled distribution, rather than only its mean, is used as an input to probabilistic sensitivity analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B4,C2:C4) | Calculate a weighted pooled mean from expert estimates |
| SUM | =SUM(C2:C4) | Confirm that expert weights sum to one |
| IF | =IF(ABS(SUM(C2:C4)-1)<0.000001,"Valid","Review Weights") | Validate pooling weights |
| RAND | =RAND() | Generate a random value for mixture-component selection |
| XLOOKUP | =XLOOKUP(RAND(),CumulativeWeights,ExpertID,,1) | Select an expert distribution according to its pooling weight |
VBA (Optional)
A VBA procedure can select component distributions according to their assigned weights, generate samples from the pooled distribution and transfer them into a probabilistic sensitivity analysis.
Sources
- Stone M. The Opinion Pool. Annals of Mathematical Statistics. 1961.
- Genest C, Zidek JV. Combining Probability Distributions: A Critique and an Annotated Bibliography. Statistical Science. 1986.
- O?Hagan A, Buck CE, Daneshkhah A, et al. Uncertain Judgements: Eliciting Experts? Probabilities.
- Cooke RM. Experts in Uncertainty: Opinion and Subjective Probability in Science.
- Bojke L, Soares M, Claxton K, et al. Developing a Reference Protocol for Structured Expert Elicitation in Health-Care Decision-Making. Health Technology Assessment. 2021.
Related Concepts (2)
Library
Publications
1
Value of Information Analysis for Research Decisions — An Introduction: Report 1 of the ISPOR Value of Information Analysis Emerging Good Practices Task Force — Fenwick, Steuten, Knies, Ghabri, Basu, Murray, Koffijberg, Strong, Sanders Schmidler & Rothery, Vol. 23, No. 2 ed., 2020 (Value in Health)
The introductory ISPOR good-practice report on value-of-information (VOI) analysis, explaining how VOI quantifies the value of reducing decision uncertainty through further research and where it fits in resource-allocation decisions.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the linear pool?
A method combining multiple experts' probability distributions by taking a weighted arithmetic average, producing a single pooled distribution.
Source: Cooke 1991
What does the linear pool produce when experts disagree?
The linear pool combines experts' probability distributions by taking a weighted arithmetic average of them, so the pooled distribution is a blend that keeps a presence wherever any expert placed weight. When experts disagree, this produces a wide, sometimes multi-peaked distribution that preserves the full spread of their views rather than forcing a consensus. This faithfully reflects genuine disagreement, though it can leave the pooled uncertainty large. It averages the distributions themselves. O'Hagan and colleagues (2006) describe this method.
Source: O'Hagan et al. 2006
How does the linear pool combine distributions?
The linear pool combines distributions by forming a weighted arithmetic average: at each value, the pooled probability density is the sum of the experts' densities each multiplied by its weight, with the weights summing to one. The result is a mixture distribution containing all the experts' distributions in proportion to their weights. This means the pooled distribution can be multimodal, retaining peaks from the individual experts, and its spread reflects both each expert's uncertainty and the disagreement among them, since differing views widen the mixture.
Source: O'Hagan et al. 2006
What are the properties of the linear pool?
The linear pool produces a mixture distribution, so it preserves features of the individual distributions, and can be multimodal if experts disagree, with the spread reflecting both individual uncertainty and between-expert disagreement. It is simple to compute and interpret and satisfies certain desirable properties, such as giving zero probability to an outcome only if all experts do. However, it is not externally Bayesian, meaning updating the pool with data differs from updating each expert first. These properties characterise how the linear pool represents combined judgement.
Source: Cooke 1991
How does the linear pool differ from the logarithmic pool?
The linear pool takes a weighted arithmetic average of the experts' distributions, producing a mixture that can be multimodal and that reflects disagreement as a wider spread, whereas the logarithmic pool takes a weighted geometric average, producing a distribution that is typically unimodal and tighter, tending to a compromise that is narrower where experts agree. The linear pool retains individual peaks and assigns positive probability where any expert does, while the logarithmic pool is more concentrated and assigns low probability where any expert does. So they aggregate disagreement differently.
Source: Cooke 1991
What are the limitations of the linear pool?
The linear pool, being a simple mixture, can produce a wide or multimodal distribution when experts disagree, which may overstate uncertainty and be awkward to use, and it is not externally Bayesian, so combining then updating differs from updating then combining. The choice of weights affects the result and involves judgement. It treats disagreement by spreading probability rather than reconciling views. These limitations mean the linear pool is applied with attention to the weights and to how disagreement is represented, with the logarithmic pool or facilitated consensus considered as alternatives.
Source: O'Hagan et al. 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 31 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/linear-pool
- Term code
- HE-EM-VI-018
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