Link Functions and Deviance in Generalized Linear Models

Link Functions

The link function g connects the expected response to the linear predictor: g(mu_i) = eta_i = x_i'*beta. The identity link (g(mu) = mu) is used for Normal responses. The log link (g(mu) = ln(mu)) ensures positivity and is used for Poisson and Gamma models. The logit link (g(mu) = ln(mu/(1-mu))) maps probabilities to the real line for Binomial models. Non-canonical links are sometimes preferred: the log link with Gamma responses is common in insurance severity modeling because coefficients represent multiplicative effects rather than the reciprocal relationship implied by the canonical link.

Deviance and Model Assessment

The deviance D = 2*(ln(L_saturated) minus ln(L_model)) measures how well the model fits compared to a perfect fit. For Normal models, deviance equals the residual sum of squares. For Poisson, D = 2*sum(y_i*ln(y_i/mu-hat_i) minus (y_i minus mu-hat_i)). The scaled deviance D/phi follows approximately chi-squared with n-p degrees of freedom. Deviance residuals are signed square roots of individual deviance contributions. Comparing deviances of nested models tests the significance of removed terms. Exam MAS-I tests link function selection, deviance computation, and model comparison using deviance.