Regression & Correlation

Poisson Regression

Generalized linear model for count data with log link function and Poisson distribution.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Regression > Poisson Regression > Fit Poisson Model

Description

Poisson regression models count data (non-negative integer response) via a generalized linear model with canonical log link function. Coefficients are estimated using Iteratively Reweighted Least Squares (IRLS). Output includes coefficients, standard errors, z-values, p-values, rate ratios with confidence intervals, deviance, null deviance, AIC, BIC, log-likelihood and pseudo-R² measures (McFadden, Cox-Snell, Nagelkerke). An overdispersion check via Pearson χ²/df is provided.

Formulas

Log-Link (kanonisch)

lo g (μ_{i}) = x_{i}^{'} β

Canonical link function: the log of the expected value is modelled as a linear combination of predictors

Varianzfunktion

Var (Y_{i}) = μ_{i}

Poisson variance: the variance equals the mean (equidispersion)

IRLS-Gewichte

w_{i} = μ_{i}

Iterative weights for the IRLS algorithm

Working Response

z_{i} = η_{i} + \frac{y _{i} - μ _{i}}{μ _{i}}

Linearised working variable in the IRLS step

Log-Likelihood

ℓ = i = 1 \sum n [y_{i} lo g (μ_{i}) - μ_{i} - lo g (y_{i}!)]

Poisson log-likelihood function

Rate Ratio

RR_{j} = exp (\hat{β}_{j})

Rate ratio: exponentiated coefficient, interpretable as multiplicative effect on the rate

Deviance

D = 2 i = 1 \sum n [y_{i} lo g (\frac{y _{i}}{μ ^ _{i}}) - (y_{i} - \overset{μ}{^}_{i})]

Deviance: goodness-of-fit measure (comparison with saturated model)

Überdispersion (Pearson χ²/df)

\frac{χ _{Pearson}^{2}}{df} = \frac{\sum _{i = 1}^{n} ( y _{i} - μ ^ _{i} ) ^{2} / μ ^ _{i}}{n - p}

Pearson χ² statistic divided by degrees of freedom: values substantially > 1 indicate overdispersion

Assumptions

Response is count data (non-negative integers)
Equidispersion: Var(Y) = E(Y) = μ (mean equals variance)
Independence of observations
Log-linear relationship between log(μ) and predictors
No perfect multicollinearity among predictors

Limitations

Cannot handle overdispersion (use negative binomial or quasi-Poisson instead)
Not suitable for zero-inflated data
Standard errors are underestimated when overdispersion is present
Large counts may cause numerical instability

References

Cameron, A. C., Trivedi, P. K. (2013), Regression Analysis of Count Data, 2nd Ed., Cambridge University Press
McCullagh, P., Nelder, J. A. (1989), Generalized Linear Models, 2nd Ed., Chapman & Hall
Hilbe, J. M. (2014), Modeling Count Data, Cambridge University Press