Generalized Linear Models: Poisson and Gamma Regression
Study GLMs with Poisson and gamma response distributions for Exam SRM.
GLM Framework
A generalized linear model extends ordinary linear regression by allowing the response variable to follow any distribution from the exponential family, not just the normal distribution. Every GLM has three components: (1) a random component specifying the response distribution (normal, Bernoulli, Poisson, gamma, inverse Gaussian, etc.), (2) a systematic component consisting of the linear predictor eta = beta_0 + beta_1*X1 + ... + beta_p*Xp, and (3) a link function g that connects the mean of the response to the linear predictor: g(mu) = eta.
This framework unifies many seemingly different models. Linear regression is a GLM with a normal response and identity link (g(mu) = mu). Logistic regression uses a Bernoulli response and logit link (g(mu) = ln(mu/(1-mu))). Poisson regression uses a Poisson response and log link. Recognizing this unity helps you understand common principles of estimation, inference, and diagnostics that apply across all GLMs.
Parameters are estimated by maximum likelihood using iteratively reweighted least squares (IRLS). The deviance D = 2 * (l_saturated - l_fitted) measures goodness of fit, where the saturated model has one parameter per observation. The deviance plays the same role as the residual sum of squares in linear regression.
The Exponential Family
A distribution belongs to the exponential family if its density can be written as f(y | theta, phi) = exp((y*theta - b(theta)) / a(phi) + c(y, phi)), where theta is the natural parameter, phi is the dispersion parameter, and b(theta) determines the mean and variance. The mean is mu = b'(theta) and the variance is Var(Y) = b''(theta) * a(phi).
For the Poisson distribution: theta = ln(mu), b(theta) = exp(theta) = mu, a(phi) = 1, so Var(Y) = mu. For the gamma: theta = -1/mu, b(theta) = -ln(-theta), a(phi) = phi, so Var(Y) = phi * mu^2. Understanding this canonical form helps you derive the mean-variance relationship for each distribution, which is crucial for choosing the right GLM.
Each distribution has a canonical link function derived from the natural parameter. For the Poisson, the canonical link is the log; for the gamma, the inverse (1/mu); for the normal, the identity; and for the Bernoulli, the logit. Using the canonical link has computational advantages but is not always required.
Poisson Regression for Claim Frequency
For count data such as the number of insurance claims, the Poisson GLM with log link is the standard starting point: ln(mu) = beta_0 + beta_1*X1 + ... + beta_p*Xp, so mu = exp(eta). The log link ensures the predicted count is always positive.
Interpretation: a one-unit increase in X_j multiplies the expected count by exp(beta_j). If beta_j = 0.15, the expected count increases by a factor of exp(0.15) = 1.162, or 16.2%. This multiplicative interpretation makes the log link natural for insurance rating, where relativities (multipliers) are the standard way to express the effect of rating factors.
An offset term is essential when observations have different exposure periods. If policyholder i has exposure e_i (measured in policy-years), the model becomes ln(mu_i) = ln(e_i) + beta_0 + beta_1*X1_i + ... The offset ln(e_i) is included as a predictor with a fixed coefficient of 1, converting the model from counts to rates: mu_i / e_i = exp(beta_0 + beta_1*X1_i + ...).
The Poisson model assumes the mean equals the variance (equidispersion). In practice, insurance claim data often exhibits overdispersion (variance exceeding the mean), which occurs when there is unobserved heterogeneity among policyholders. Two common remedies are the negative binomial model (which adds a parameter for overdispersion) and the quasi-Poisson approach (which scales the variance by a dispersion parameter phi, estimated from the data as the Pearson chi-squared statistic divided by the residual degrees of freedom).
Gamma Regression for Claim Severity
For positive continuous data such as individual claim amounts, a Gamma GLM with log link is widely used: ln(mu) = eta. The gamma distribution is well-suited to severity modeling because it allows the variance to be proportional to the square of the mean: Var(Y) = phi * mu^2. This constant coefficient of variation property matches the empirical observation that larger claims tend to have proportionally more variability.
The inverse link g(mu) = 1/mu is the canonical link for the gamma distribution, but the log link is far more commonly used in actuarial practice. The log link provides multiplicative interpretation (just like Poisson regression) and ensures positive predicted values. The canonical inverse link can produce negative predicted values for extreme predictor combinations, which is nonsensical for claim amounts.
A worked example: suppose the fitted gamma GLM for claim severity gives ln(mu) = 7.5 + 0.30 * Urban + 0.02 * VehicleAge. For an urban policyholder with a vehicle age of 5 years: ln(mu) = 7.5 + 0.30(1) + 0.02(5) = 7.90, so mu = exp(7.90) = 2,697. The expected claim severity is $2,697. The Urban coefficient indicates that urban claims are exp(0.30) = 1.35 times larger than rural claims, holding vehicle age constant.
Model Comparison and Diagnostics
Within the GLM framework, several tools are available for model comparison and assessment. The deviance statistic tests whether the model fits adequately: D / phi should be approximately chi-squared with n - p degrees of freedom. Comparing two nested models uses the difference in deviances: Delta_D = D_reduced - D_full, which follows a chi-squared distribution with degrees of freedom equal to the difference in parameters (the likelihood ratio test).
AIC = -2l + 2k and BIC = -2l + k*ln(n) allow comparison of non-nested models. Lower values are preferred. AIC tends to select more complex models; BIC tends to select simpler ones.
Residual diagnostics for GLMs use deviance residuals or Pearson residuals rather than raw residuals. Deviance residuals are defined so that their sum of squares equals the deviance. Plot them against fitted values and each predictor to check for patterns. Quantile residuals (randomized for discrete responses) provide a more reliable diagnostic for Poisson and binomial models.
Choosing Between Distribution and Link
Selecting the right distribution depends on the nature of the response variable. For counts, start with Poisson. For positive continuous values with right skew, consider gamma or inverse Gaussian. For proportions, use binomial. The mean-variance relationship is a key guide: constant variance suggests normal, variance proportional to the mean suggests Poisson, and variance proportional to the mean squared suggests gamma.
The link function is somewhat more flexible. While canonical links have theoretical advantages, practical considerations often dominate. The log link is popular for both Poisson and gamma models because of its multiplicative interpretation and guarantee of positive fitted values. The identity link is occasionally used for gamma models when additive effects are more appropriate, though care must be taken to avoid negative predictions.
Study Tips for Exam SRM
For Exam SRM, master these core concepts about GLMs. First, be able to identify the three components (random, systematic, link) for any given GLM. Second, understand the mean-variance relationship for each distribution and how it guides model selection. Third, practice interpreting coefficients under the log link as multiplicative factors. Fourth, know how to use the deviance and likelihood ratio test for comparing nested models. Fifth, understand the role of offsets in rate models. Common exam questions present output from a fitted GLM and ask you to interpret coefficients, compute predicted values, or compare models using AIC or deviance. Practice these calculations until they become routine.