Concept Architecture
Concept
Theoretically, Linear Model Health is a mathematical model in which the relationship between an outcome and one or more explanatory variables is assumed to be linear. The method is grounded in linear algebra and statistical regression theory and is used to quantify associations, predict outcomes and estimate the effects of healthcare interventions or patient characteristics. In health economics, linear models are widely applied to analyse healthcare costs, resource utilisation, health outcomes and determinants of economic endpoints when linearity assumptions are appropriate.
Mathematically, a Linear Model expresses the dependent variable as a linear combination of explanatory variables and unknown parameters. Model coefficients are typically estimated by ordinary least squares (OLS), which minimises the sum of squared residuals between observed and predicted values. The resulting model estimates the expected change in the outcome associated with a one-unit change in each explanatory variable while holding other variables constant.
In practice, Linear Models are fitted using patient-level trial data, observational datasets, administrative records or registry data. Analysts assess model assumptions, including linearity, homoscedasticity, independence and normality of residuals, before interpreting estimated coefficients. In health economics, linear models are commonly used to estimate healthcare costs, productivity losses, resource use and health-related quality-of-life outcomes and to adjust for baseline covariates in economic evaluations.
Purpose
Used to estimate and predict linear relationships between explanatory variables and economic or clinical outcomes, supporting statistical inference, adjustment for confounding and prediction within health economic analyses.
Mathematical Formulae
Primary Formula
y = ?? + ??x? + ??x? + ? + ??x? + �
where:
- y = dependent variable
- ?? = intercept
- ?? = regression coefficients
- x? = explanatory variables
- � = random error term
Supporting Formulae
Ordinary least squares estimator:
?? = (X?X)??X?y
Residual:
e? = y? ? ??
Related Mathematical Methods
- Ordinary least squares regression
- Multiple linear regression
- General linear models
- Weighted least squares
- Analysis of covariance
- Matrix algebra
- Regression diagnostics
Example
A health economist models annual healthcare expenditure using age, sex and disease severity.
The fitted model is:
Cost? = 520 + 18(Age) + 1,250(Disease Severity)
For a 60-year-old patient with disease severity coded as 2:
Cost? = 520 + 18(60) + 1,250(2) = �4,100
The model predicts annual healthcare costs of approximately �4,100 for a patient with these characteristics.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B101,C2:E101,TRUE,TRUE) | Estimate regression coefficients for healthcare costs or outcomes. |
| TREND | =TREND(B2:B101,C2:E101,C102:E102) | Predict economic outcomes for new observations. |
| FORECAST.LINEAR | =FORECAST.LINEAR(A102,B2:B101,A2:A101) | Predict values from a simple linear model. |
| RSQ | =RSQ(B2:B101,C2:C101) | Assess model goodness-of-fit. |
VBA (Optional)
Automate regression estimation, diagnostic testing and generation of prediction reports for multiple health economic datasets.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models.
- Wooldridge JM. Introductory Econometrics: A Modern Approach.
- ISPOR Good Practices for Outcomes Research.
Related Concepts (3)
Library
Publications
1
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What is a linear model?
A statistical or decision-analytic model in which inputs and outputs are related by a simple linear function, such as a constant per-unit cost.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
When does assuming linearity in a model break down?
A linear relationship assumes that a change in an input produces a proportional change in the output, so that doubling the input doubles the effect, as when each unit of a resource adds the same cost. This holds well over small ranges but often fails over large ones, where effects level off, accelerate, or reverse. Treating a genuinely curved relationship as linear then misstates outcomes away from the range where the assumption was fitted. Recognising this limit guides where linear approximations are safe. Briggs and colleagues (2006) note this caution.
Source: Briggs et al. 2006
What does linearity mean in a model?
Linearity in a model means that the relationship between an input and an output is a straight-line one: the output is proportional to the input plus a constant, so each additional unit of input adds the same amount to the output regardless of the level. There are no interactions between inputs and no curvature, such as increasing or diminishing effects. A constant per-unit cost is a linear relationship, since total cost rises in direct proportion to the quantity.
Source: Briggs, Claxton & Sculpher 2006
Where are linear relationships used in models?
Linear relationships are used in models wherever a quantity is proportional to another, such as a constant cost per unit of service, a fixed effect per dose, or a total that is the sum of proportional components. They are common in costing, where total cost is often the sum of quantities multiplied by unit costs, and in simple representations of effects. Their simplicity makes them a natural default where proportionality is a reasonable approximation of the underlying relationship.
Source: Briggs, Claxton & Sculpher 2006
What are the advantages of linear models?
Linear models are simple, transparent, and easy to compute and interpret, since the effect of each input is constant and additive, making the model's behaviour clear. They require little data to specify, being defined by constant rates, and they are easy to check. Where the true relationship is approximately proportional, a linear model captures it adequately with minimal complexity. These features make linearity a convenient and common choice in modelling, particularly for costs and simple effects.
Source: Briggs, Claxton & Sculpher 2006
What are the limitations of linear models?
Linear models can misrepresent relationships that are not proportional, such as effects that diminish or increase with scale, thresholds, or interactions between inputs, since they assume a constant, additive effect. Where the true relationship is non-linear, a linear approximation may be adequate over a small range but wrong over a wider one. Using linearity where it does not hold biases results, so the assumption is appropriate only when proportionality reasonably describes the relationship being modelled.
Source: Briggs, Claxton & Sculpher 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 1 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/linear-model-health
- Term code
- HE-EM-DM-049
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