Gamma Model

Concept

Theoretically, a Gamma Model is a statistical regression model that assumes the response variable follows a gamma distribution and is therefore appropriate for modelling positive, continuous and right-skewed outcomes. It is most commonly implemented as a generalised linear model (GLM) with a gamma family and a log link. In health economics, gamma models are widely used to analyse healthcare costs, resource utilisation and other positive continuous outcomes that violate the assumptions of ordinary linear regression.

Mathematically, a gamma model specifies that the response variable follows a gamma distribution with its mean related to explanatory variables through a link function, most commonly the natural logarithm. Regression coefficients are estimated using maximum likelihood estimation, allowing the expected outcome to vary multiplicatively with the covariates while accommodating heteroscedasticity inherent in skewed cost data.

In practice, gamma models are fitted to patient-level cost or utilisation data using statistical software. Model adequacy is assessed using residual diagnostics, goodness-of-fit measures and comparison with alternative models. Gamma regression is routinely recommended for analysing healthcare costs in economic evaluations because it accommodates positive skewness without requiring transformation of the outcome variable.

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Purpose

Used to model positive, right-skewed continuous outcomes, particularly healthcare costs and resource utilisation, supporting statistical analysis and economic evaluation.

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Mathematical Formulae

Primary Formula

g(??) = x???

For the commonly used log-link gamma model:

ln(??) = ?? + ??x? + ??x? + ? + ??x?

where:

• ?? = expected outcome
• x? = vector of explanatory variables
• ? = regression coefficients

Supporting Formulae

Distributional assumptions:

Y? ~ Gamma(??, �)

Variance function:

Var(Y?) = �??�

Model parameters are estimated by:

?? = arg max L(?)

Related Mathematical Methods

• Generalised linear models
• Gamma distribution
• Maximum likelihood estimation
• Log-link regression
• Deviance residuals
• Akaike Information Criterion (AIC)

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Example

A health economic study evaluates annual healthcare costs for patients with chronic obstructive pulmonary disease. Because the cost data are strictly positive and highly right-skewed, a gamma generalised linear model with a log link is fitted. The model estimates that a new intervention reduces expected annual costs by 12% after adjustment for age, disease severity and comorbidities.

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Excel Implementation

Function	Example Formula	Health Economics Application
LN	=LN(A2)	Calculate the log-transformed mean for exploratory gamma regression.
EXP	=EXP(B2)	Convert predicted log means to expected healthcare costs.
LINEST	=LINEST(LN(Y_range),X_range,TRUE,TRUE)	Perform an approximate exploratory regression before fitting a formal gamma GLM.
SUMPRODUCT	=SUMPRODUCT(CoefficientRange,CovariateRange)	Calculate the linear predictor for expected costs.

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VBA (Optional)

Automate preparation of healthcare cost datasets, calculation of fitted values and comparison of gamma regression models across alternative covariate specifications.

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Sources

• McCullagh P, Nelder JA. Generalized Linear Models.
• Manning WG, Basu A, Mullahy J. Generalised modelling approaches to risk adjustment of skewed healthcare cost data.
• 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.
• NICE. Health Technology Evaluation Manual.