Distinguish among categorical, ordinal, and continuous predictors and their interactions, and how these relate to their usage in an extended linear model.

A pricing analyst tosses driver age, vehicle class, and a five-point body-style rating into the same GLM and watches the design matrix swell from 8 columns to 47. Predictor type drives that explosion, and choosing the wrong type silently breaks coefficients, inference, and forecasts.

The three predictor types. An extended linear model (ELM) writes a linear predictor

$$\eta = \beta_0 + \sum_{j=1}^{p} \beta_j x_j, \qquad g(\mu) = \eta$$

The columns $x_j$ are what you control. How a raw variable becomes a column depends on whether it is continuous, categorical, or ordinal.

Continuous predictors. A continuous predictor (driver age in years, vehicle weight in kilograms, prior claim count treated as a number) enters as a single numeric column. The fitted coefficient $\beta_j$ is the change in the linear predictor per one-unit increase in $x_j$, holding everything else fixed. One column, one slope, one degree of freedom.

The implicit assumption is linearity on the link scale. If the true effect of age on log-frequency is U-shaped, a single slope averages the rise and the fall and reports...

Common mistakes

• Including all $k$ dummies plus the intercept. Creates perfect collinearity; the design matrix is singular. Software drops a column you did not pick, scrambling the contrast interpretation. Always use $k-1$ dummies.
• Treating credit tier, satisfaction score, or policy band as continuous without checking equal spacing. A 0.17-per-step slope smooths over a 0.55 jump at the worst tier, underpricing the riskiest segment by double digits.
• Dropping a main effect while keeping its interaction. Violates the hierarchy principle. The interaction coefficient now mixes in the missing main effect's mean shift, and contrasts no longer mean what their labels say.

Bottom line

• Continuous predictors enter as one column with one slope; the slope is the change in $\eta$ per unit increase in $x$, assumed linear on the link scale unless transformed.
• Categorical (nominal) predictors with $k$ levels enter as $k-1$ dummy columns plus a baseline absorbed into the intercept; each coefficient is the contrast versus the reference level.
• Ordinal predictors carry order but not spacing; treat as continuous (1 df) only if equal-spacing is defensible, otherwise use dummies ($k-1$ df) or a monotone basis.
• Interaction = effect of one predictor depends on another, measuring departure from additivity; build it as a product column (continuous-continuous), slope-shift dummies (categorical-continuous), or a cross-tab of dummies (categorical-categorical).

Exam shortcut

When a problem hands you a predictor and asks how it enters the design matrix, count levels first: continuous gives 1 column, categorical with $k$ levels gives $k-1$, and ordinal forces an encoding decision you should justify in one sentence. For any interaction question, compute df as the product of the parts: continuous-by-continuous is 1, categorical-by-continuous is $k-1$, categorical-by-categorical is $(k_1-1)(k_2-1)$.

The full lesson (about 3,227 words, 22 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C5

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