Interpret model coefficients, including interaction terms.

A manager hands you a GLM summary and asks what a coefficient "means" for the business. The wrong instinct is to read the number raw. In a GLM, every coefficient lives on the link scale, and interactions rewire the whole interpretation.

Coefficients live on the link scale. A GLM has three parts: a random component (the response distribution), a systematic component (the linear predictor $\eta = \beta_0 + \beta_1 x_1 + \dots$), and a link function $g$ connecting them so that $g(\mu) = \eta$. The coefficients build $\eta$. To speak about the response $\mu$, you invert the link.

KEY: The single most common interpretation error is reading a log-link or logit coefficient as an additive change in the response. Under a log link the effect is multiplicative: exponentiate first.

Continuous predictors: direction and magnitude. A full-credit interpretation states both. Direction is the sign of β. Magnitude requires the link. Under a log link, β = 0.02 means each unit multiplies the expected response by $e^{0.02} = 1.0202$, a 2.02% increase.

Common mistakes

• Skipping the link function. Reading a log-link β = 0.30 as "+0.30 units" instead of $e^{0.30} = 1.35$, a 35% multiplicative lift. The April 2025 graders required the link to be applied for full credit.
• Interpreting a factor level as numeric. Saying "each unit of CONTROL adds..." for a categorical variable. There is no per-unit meaning; state the level's effect versus the baseline.
• Using only the main effect when an interaction exists. For O'Hare the true slope is $0.85 + 0.30 = 1.15$, not 0.85. Combine main plus interaction every time.

Bottom line

• A GLM coefficient acts on the linear predictor, not the response directly. Apply the inverse link to interpret it on the outcome scale.
• Log link: exponentiate. A coefficient β means a one-unit rise multiplies the expected response by e^β.
• Logit link: e^β is an odds ratio, the multiplicative change in the odds of the event.
• Identity link (ordinary linear model): β is a plain additive change in the response per unit.

Exam shortcut

Before interpreting any coefficient, write the link first. Log means exponentiate for a multiplicative percentage, logit means $e^{\beta}$ is an odds ratio, identity means read β straight. That one habit fixes most magnitude errors. Whenever a variable shows up both alone and in an interaction, immediately write "slope = main + interaction" and plug in the level.

The full lesson (about 2,081 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4c

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