Select and validate a GLM as appropriate for a business problem.

An underwriter hands you claim counts, a binary fraud flag, and a column of dollar severities, then asks for one modeling approach. Each column demands a different generalized linear model (GLM), and naming the right family plus link earns the point.

Match the family to the response. A GLM has three pieces: a random component (the response distribution), a systematic component (the linear predictor $\eta = X\beta$), and a link function connecting the mean to the linear predictor. The single most tested skill here is choosing the distribution and link that fit the data-generating process.

KEY: Two questions pick the model. What does the response look like (bounded 0/1, integer count, positive skewed)? And is the effect additive or multiplicative? Bounded and skewed responses rule out plain ordinary least squares (OLS); multiplicative effects call for a log or logit link.

Why not just use OLS everywhere? Ordinary least squares assumes a normally distributed response with constant variance and an unbounded range. Claim counts are non-negative integers. A fraud flag is bounded in $[0,1]$.

Common mistakes

• Exponentiating a logit coefficient into a probability. For binary GLMs, $e^{\beta}$ is an odds ratio, not a change in probability. A coefficient of $0.5$ gives an odds ratio of $1.65$, which is not a 65% rise in the probability of the event.
• Using plain Poisson on overdispersed counts. If the sample variance far exceeds the mean, plain Poisson understates standard errors. The tell is a dispersion estimate well above one; switch to negative binomial or quasi-Poisson.
• Reading raw residual plots for a GLM. Raw residuals have mean-dependent variance, so a fan shape is expected and uninformative. Use deviance residuals, which R plots by default.

Bottom line

• Pick the family from the response type: binary is binomial, counts are Poisson, non-negative skewed amounts are Gamma.
• The log link makes coefficients multiplicative; exp(beta) is the factor by which a one-unit rise scales the expected response.
• The logit link makes exp(beta) an odds ratio, not a probability change.
• A normal regression on log(Y) models the mean of the log; a GLM log link models the log of the mean. They differ.

Exam shortcut

Read the response column before anything else: 0/1 means binomial with logit, integer counts mean Poisson with log, positive skewed dollars mean Gamma with log. Naming the pair fast banks easy points. For any log-link coefficient, immediately write $e^{\beta}$ and read it as a multiplier; for a logit coefficient, say "odds ratio" out loud so you never call it a probability.

The full lesson (about 2,204 words, 15 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4a

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