Generalized linear models and link functions

A basic GLM specifies conditionally independent responses from an exponential dispersion family, a linear predictor $\eta_i=x_i^{\mathsf T}\beta$, and a one-to-one link $g(\mu_i)=\eta_i$ for the conditional mean μᵢ. A common density or mass representation uses natural parameter θᵢ, cumulant function b, dispersion φ, and known positive scaling weights aᵢ. The base-measure term may also depend on that known row weight. Differentiating the cumulant gives the mean and variance. Merely specifying a nonlinear mean curve without a response distribution is not the complete likelihood model.

A canonical link equates the linear predictor with the natural parameter. For Bernoulli outcomes this produces the logit link, while Poisson counts use the logarithm. Other links are possible when their domains are compatible with the mean. The link acts on the conditional mean, not on each observed response: taking a logarithm of a zero count or a logit of an observed zero is neither required nor a valid description of GLM fitting.

In logistic regression, the mean is a Bernoulli success probability. Its odds are the probability divided by one minus the probability, and its log odds are linear in the predictors. Increasing one predictor by one unit while holding the others fixed multiplies the modeled odds by the exponential of its coefficient. It does not add a constant to the probability: the same coefficient produces different probability changes at different starting probabilities. Conditional association also does not establish a causal intervention effect.

For a Poisson count, the variance equals the mean under the model. A log link makes the mean positive. If observations have different known exposure durations, a log-exposure offset with coefficient fixed at one models a rate rather than conflating duration with risk. Doubling exposure then doubles the expected count at fixed rate. Overdispersion, clustering, or excess zeros can contradict the simple Poisson distribution even when a log-linked mean remains useful; changing uncertainty or the response family requires a stated alternative model.

Likelihood score equations for a GLM are generally nonlinear in β and are often solved by iteratively reweighted least squares. The working response and weights change with the current mean estimate; they are computational devices, not newly observed independent Gaussian data. A design can have full column rank while logistic likelihood still lacks a finite maximizer, as occurs under complete separation. Convergence messages, coefficient magnitudes, and the data geometry deserve inspection rather than assuming an invertible design settles existence.

Wald, score, and likelihood-ratio comparisons usually rely on large-sample regularity rather than the exact regression t and F distributions. Small samples, rare events, separation, estimated dispersion, and response-driven selection can undermine routine approximations. Examine calibration, residual patterns appropriate to the response family, and validation on the target population. A fitted probability curve can be smooth and plausible while being systematically wrong; the response family, independence structure, link, and sampling process all contribute to the inferential claim.