Poisson Regression

Study guide: Exam MAS-I

Poisson Regression is the Generalized Linear Model for count data: the response is Poisson and the log Link Function makes the predictors act multiplicatively on the expected count. It is the most commonly used model for claim Frequency, with log exposure entered as an offset on a claim-count target or exposure as the weight on a frequency target.

$\ln \mu_i = \ln(e_i) + \beta_0 + \beta_1 x_{i1} + \cdots + \beta_p x_{ip}$

$\mu_i = e_i\,e^{\beta_0}\,e^{\beta_1 x_{i1}} \cdots e^{\beta_p x_{ip}}$

The log link turns each coefficient into a rating relativity: $e^{\beta_j}$ is the multiplicative effect on frequency of a one-unit increase in $x_j$ , which is exactly how rating algorithms are built
With a claim-count target, the exposure term $\ln(e_i)$ is an offset — a predictor with coefficient fixed at $1$ — required because more exposure carries the expectation of more claims; the model then predicts a rate per unit of exposure. The more common practice models frequency (claims ÷ exposure) with exposure as the weight and no offset; for the Poisson the two set-ups give identical predictions, relativities and standard errors
The Poisson variance function is $V(\mu) = \mu$ and the Dispersion Parameter is nominally $\phi = 1$ . A residual Deviance (or Pearson $\chi^2$ ) far above its degrees of freedom is first a sign of lack of fit; it can be read as overdispersion only once the systematic part of the model — its predictors and link — is trusted
Overdispersion usually comes from unmodelled heterogeneity (a Mixed Poisson Process). Remedies: estimate $\phi$ (quasi-Poisson, which inflates standard errors by $\sqrt{\hat\phi}$ but leaves $\hat\beta$ unchanged), or fit a negative binomial model
Coefficients are tested individually in Parameter Estimate Tables and in groups by analysis of deviance (Likelihood Ratio Test)
Poisson regression tolerates non-integer responses (average counts with weights) because the quasi-likelihood only needs the mean–variance relationship to be right

Predicting a Frequency from a Fitted Model A frequency GLM with log link and log-exposure offset gives $\hat\beta_0 = -2.30$ , urban $= 0.26$ , young driver $= 0.41$ . Find the expected claim count for a young urban driver with $0.5$ years of exposure.

Answer $\ln\hat\mu = \ln(0.5) - 2.30 + 0.26 + 0.41 = -0.693 - 1.63 = -2.323$ $\hat\mu = e^{-2.323} = 0.098$ Equivalently, the base rate is $e^{-2.30} = 0.100$ claims per exposure-year, multiplied by $e^{0.26} = 1.30$ for urban and $e^{0.41} = 1.51$ for young, times $0.5$ exposure: $0.100 \times 1.30 \times 1.51 \times 0.5 = 0.098$ .

Detecting Overdispersion A Poisson GLM has residual deviance $1{,}840$ on $600$ degrees of freedom. What does this indicate, and how should the results be adjusted?

Answer $\hat\phi \approx \frac{D}{n - p} = \frac{1{,}840}{600} = 3.07$ The deviance is far above its degrees of freedom. If the model's predictors and link are right — no missing variable or interaction — a dispersion near $3$ instead of $1$ means the counts are far more variable than Poisson allows: overdispersion. The estimated coefficients remain usable, but standard errors are understated by a factor of $\sqrt{3.07} = 1.75$ , so p-values are far too small. Refit as quasi-Poisson (scaling the standard errors) or as a negative binomial model, which builds the extra variance into the distribution.