Poisson Regression

Concept

---

Theoretically, Poisson Regression is a generalised linear modelling (GLM) technique used to model count outcomes as a function of one or more explanatory variables. It assumes that the response variable follows a Poisson distribution and that the logarithm of the expected event count is a linear combination of predictor variables. The method is particularly appropriate when modelling rare events, incidence rates, healthcare utilisation, and other count data frequently encountered in health economics and epidemiology.

Mathematically, Poisson Regression belongs to the exponential family of generalised linear models and employs a logarithmic link function to ensure that predicted event rates remain positive. Regression coefficients quantify multiplicative effects on the expected event rate, with exponentiated coefficients interpreted as incidence rate ratios (IRRs). Model parameters are estimated using maximum likelihood estimation, and goodness of fit is commonly assessed using deviance statistics, likelihood ratio tests, information criteria, and residual diagnostics.

In practice, Poisson Regression is used to estimate associations between patient characteristics, treatments, or interventions and counts of healthcare events such as hospital admissions, outpatient visits, prescriptions, adverse events, or disease incidence. In health economic evaluations it is frequently applied to model healthcare resource utilisation and event frequencies that contribute to cost estimation. Where observed variance substantially exceeds the mean, alternative models such as Negative Binomial Regression are generally preferred.

Purpose

---

Used to model count outcomes and event rates while estimating the effects of explanatory variables on healthcare utilisation, disease incidence, adverse events, and resource consumption within health economic and clinical studies.

---

Mathematical Formulae

Primary Formula

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

or equivalently

?? = exp(?? + ??x?? + ??x?? + ? + ??x??)

where:

• ?? = expected event count
• ?? = intercept
• ?? ? ?? = regression coefficients
• x? ? x? = explanatory variables

Supporting Formulae

Y? ~ Poisson(??)

Var(Y?) = ??

IRR = e?

Log-likelihood:

ln(L) = ? [y? ln(??) ? ?? ? ln(y?!)]

For models including exposure time:

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

where t? is the exposure offset.

Related Mathematical Methods

• Generalised Linear Models
• Maximum Likelihood Estimation
• Poisson Distribution
• Incidence Rate Ratio
• Likelihood Ratio Test
• Wald Test
• Deviance Analysis
• Negative Binomial Regression
• Quasi-Poisson Regression

---

Example

A health economist models annual emergency department attendances using age, treatment group, and comorbidity score.

The fitted model is:

ln(?) = ?0.85 + 0.028(Age) + 0.41(Comorbidity) ? 0.32(Treatment)

For a patient aged 70 years with a comorbidity score of 2 receiving treatment:

ln(?) = ?0.85 + (0.028 ? 70) + (0.41 ? 2) ? (0.32 ? 1)

ln(?) = 1.61

? = e?�?? = 5.00

The model predicts approximately five emergency department attendances during the observation period. The treatment incidence rate ratio is:

IRR = e??�?� = 0.73

indicating a 27% reduction in the expected event rate after adjusting for other variables.

---

Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=EXP(B2)	Convert the linear predictor into the expected event count.
LN	=LN(A2)	Calculate logarithmic offsets or transform event rates.
POISSON.DIST	=POISSON.DIST(A2,B2,FALSE)	Calculate probabilities from the fitted Poisson model.
LINEST	=LINEST(LN(ExpectedCounts),PredictorRange,TRUE,TRUE)	Approximate estimation of regression coefficients for educational purposes.
SUMPRODUCT	=SUMPRODUCT(CoefficientRange,VariableRange)	Calculate the linear predictor prior to exponentiation.

VBA (Optional)

Automate estimation of Poisson regression models, calculate incidence rate ratios, and generate regression summary tables for healthcare utilisation analyses.

---

Sources

• McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed.
• Cameron AC, Trivedi PK. Regression Analysis of Count Data.
• Hilbe JM. Modeling Count Data.
• Dobson AJ, Barnett AG. An Introduction to Generalized Linear Models.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
• NICE Health Technology Evaluation Manual.
• ISPOR Good Practice Reports.