Interpret the extended linear model output from statistical software, such as parameter estimate tables and ANOVA tables.

A pricing analyst runs a generalized linear model (GLM) in R and stares at three blocks of output: a coefficient table, an analysis-of-deviance table, and a row of fit statistics. Knowing what each number means, and which one answers which question, is the entire skill MAS-I tests here.

The extended linear model family. Extended linear models cover GLMs (Poisson, binomial, gamma, inverse Gaussian), models with offsets, models with categorical factors, and ordinary least squares (OLS) as the Gaussian-identity special case. Software output looks similar across all of them: a fitted linear predictor $\eta =X\beta$, a link function $g$ relating $\eta$ to the mean $\mu =g^{-1}(\eta )$, and a variance function tied to the chosen family.

The parameter estimate table. Every row is one coefficient. Columns are usually labeled Estimate, Std. Error, z value (or t value), and Pr(>|z|) (or Pr(>|t|)).

$$z=\frac{{\hat{\beta }}_j}{\text{SE}({\hat{\beta }}_j)},p=2\mathrm{Pr}(|Z|>|z|)$$

KEY: The intercept row tests whether the baseline link-scale mean equals zero, which is rarely interesting. Focus on the slope rows.

Common mistakes

• Interpreting log-link coefficients as additive effects on the mean. A coefficient of 0.30 with log link means a 35% multiplicative increase ($e^{0.30}=1.35$), not an absolute increase of 0.30 in the mean.
• Using Chi-square when dispersion is estimated. For Gaussian and gamma fits, the correct test is F. Using ${\chi }^2$ gives a p-value that is too small.
• Reading Type I as if it were Type III. Sequential ANOVA depends on term order. A term marginally significant when entered second may be non-significant when entered first.

Bottom line

• Parameter table columns: estimate, standard error, test statistic ($z$ or $t$), p-value. The statistic equals the estimate divided by the standard error.
• GLM coefficients live on the link scale. For a log link, $\exp (\hat{\beta })$ is the multiplicative effect on the mean; for a logit link, $\exp (\hat{\beta })$ is an odds ratio.
• A categorical factor with k levels generates k-1 contrast rows against the reference; reconstruct a cell mean by adding the relevant contrast to the reference on the link scale, then back-transforming.
• Offsets log-transform exposure and lock the coefficient at 1, so they never appear in the parameter table; without an offset a frequency model estimates counts, not rates.

Exam shortcut

When a question shows a Poisson coefficient and asks for the effect on the mean, exponentiate first, then subtract 1 to read off the percentage change. To compare two nested GLMs, take twice the log-likelihood difference (or the deviance difference) and test against chi-square with df equal to the parameter-count difference.

The full lesson (about 2,272 words, 15 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C4

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