Concept Architecture
Concept
Theoretically, Generalised Linear Model (GLM) is a unified statistical modelling framework that extends ordinary linear regression to outcomes that follow probability distributions other than the normal distribution. It is based on exponential family theory and consists of three components: a random component describing the outcome distribution, a systematic component formed by the linear predictor and a link function connecting the expected outcome to the predictor. In health economics, GLMs are widely used to model healthcare costs, resource utilisation, binary outcomes, counts and rates.
Mathematically, a GLM relates the expected value of the response variable to a linear predictor through a specified link function while assuming that the response follows a member of the exponential family of distributions. Regression coefficients are typically estimated using maximum likelihood estimation through iterative algorithms such as Iteratively Reweighted Least Squares (IRLS). The variance is modelled as a function of the mean according to the selected probability distribution.
In practice, GLMs are implemented by selecting an appropriate outcome distribution and link function based on the characteristics of the data. In health economics, gamma models with a log link are commonly used for skewed cost data, Poisson or negative binomial models for healthcare utilisation and logistic regression for binary outcomes. Model fit is assessed using deviance, information criteria and residual diagnostics.
Purpose
Used to estimate associations between explanatory variables and non-normally distributed health outcomes while accommodating appropriate probability distributions and link functions.
Mathematical Formulae
Primary Formula
g(E(Y)) = X?
where:
- g(�) = link function
- E(Y) = expected value of the outcome
- X = design matrix
- ? = vector of regression coefficients
Supporting Formulae
Linear predictor:
? = X?
Mean response:
? = g??(?)
Variance function:
Var(Y) = ?V(?)
Log-likelihood:
?(?) = ? ln[f(y? | ??, ?)]
where:
- ? = linear predictor
- ? = expected outcome
- ? = dispersion parameter
- V(?) = variance function
- f(�) = probability density or mass function
Related Mathematical Methods
- Maximum Likelihood Estimation
- Iteratively Reweighted Least Squares
- Logistic Regression
- Poisson Regression
- Gamma Regression
- Negative Binomial Regression
- Generalised Estimating Equations
- Quasi-Likelihood Models
Example
A health economist models annual healthcare costs using a gamma GLM with a log link because the cost data are positively skewed. The estimated treatment coefficient is ?0.18.
exp(?0.18) = 0.835
The intervention is therefore associated with approximately 16.5% lower expected healthcare costs after adjustment for patient characteristics.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Convert log-link coefficients to multiplicative effects |
| LN | =LN(A2) | Apply logarithmic link transformations |
| SUMPRODUCT | =SUMPRODUCT(B2:B101,C2:C101) | Calculate linear predictors |
| MMULT | =MMULT(A2:D5,F2:F5) | Matrix multiplication during regression calculations |
| Solver | Maximise log-likelihood | Estimate GLM regression coefficients by maximum likelihood |
VBA (Optional)
Automate GLM estimation by iteratively fitting regression coefficients using maximum likelihood, calculating fitted values and producing diagnostic statistics.
Sources
- Nelder JA, Wedderburn RWM. Generalized Linear Models. Journal of the Royal Statistical Society Series A. 1972;135(3):370-384.
- McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed.
- Dobson AJ, Barnett AG. An Introduction to Generalized Linear Models. 4th ed.
- Hardin JW, Hilbe JM. Generalized Linear Models and Extensions.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
Related Concepts (3)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a generalized linear model?
A modelling framework extending linear regression to non-normal outcomes, such as binary or count data, via a specified link function.
Source: Nelder & Wedderburn 1972
How does a generalized linear model extend ordinary regression?
A generalized linear model extends ordinary linear regression so that it can handle outcomes that are not continuous and normally distributed, such as yes or no results or counts. It does this through a link function, which connects the predictors to a transformed version of the outcome's expected value, letting a linear combination of predictors describe probabilities or rates. Logistic regression for binary outcomes and Poisson regression for counts are members of this family. Bringing regression to non-normal outcomes is its purpose. Kirkwood and Sterne (2003) describe this framework.
Source: Kirkwood & Sterne 2003
How does a generalized linear model work?
A generalized linear model works by specifying three components: a distribution for the outcome from the exponential family, a linear predictor combining the covariates, and a link function connecting the mean of the outcome to the linear predictor. The link transforms the mean so that it relates linearly to the predictors, and the parameters are estimated by maximum likelihood. So a generalized linear model works by modelling a transformed mean of the outcome as a linear function of the predictors under an appropriate distribution, which allows the framework to handle binary, count, and other outcomes by choosing the distribution and link suited to the data.
Source: Nelder & Wedderburn 1972
What are examples of generalized linear models?
Examples of generalized linear models include logistic regression, for binary outcomes, using a logit link and the binomial distribution; Poisson regression, for count data, using a log link and the Poisson distribution; and ordinary linear regression, which is the special case with a normal distribution and identity link. So generalized linear models encompass a family of familiar regression methods, unified by the common structure of a distribution, a linear predictor, and a link function, which is why logistic, Poisson, and linear regression can all be understood as particular cases, chosen according to the type of outcome being modelled.
Source: Nelder & Wedderburn 1972
What is the role of the link function in a generalized linear model?
The link function in a generalized linear model connects the mean of the outcome to the linear predictor, transforming the mean so that it relates linearly to the covariates and, for many outcomes, keeping the modelled quantity within its valid range, such as probabilities between zero and one for binary data. So the link function is central to a generalized linear model, since it determines how the predictors relate to the outcome's mean and ensures the model respects the outcome's constraints, with different links suiting different outcomes, such as the logit for binary data and the log for counts, which is why choosing an appropriate link is part of specifying the model.
Source: Nelder & Wedderburn 1972
Why are generalized linear models useful?
Generalized linear models are useful because they extend the familiar and interpretable structure of linear regression to a wide range of outcome types within a single framework, allowing binary, count, and other data to be modelled with appropriate distributions and links while retaining the interpretability of regression coefficients. So generalized linear models are valuable for their generality and coherence, providing a unified approach to regression across many kinds of outcomes, which simplifies analysis and interpretation and is why they are among the most widely used tools in statistics, underpinning methods such as logistic and Poisson regression used throughout health research.
Source: Nelder & Wedderburn 1972
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-071
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