GLM Exercises in R: 20 Logistic, Poisson, Gamma Practice Problems

Explanation: The default link for family = binomial is logit, so coefficients are on the log-odds scale. A negative wt coefficient means heavier cars are less likely to have manual transmission. The Z statistic plus its p-value comes from the Wald test, which is fine for large samples but unreliable near zero events. AIC and residual deviance let you compare against nested alternatives later.

Explanation: Exponentiating a logit coefficient gives an odds ratio: each extra $1,000 borrowed multiplies the odds of default by about 1.21, holding income fixed. Report odds ratios to stakeholders rather than raw logits, because business owners reason in multiplicative risk, not log-odds. Always state the unit ("per $1,000") and the holding-constant assumption to avoid misinterpretation.

Explanation: type = "response" runs the linear predictor through the inverse link (here, the logistic CDF plogis) so the output is a probability in [0, 1]. Without that argument you would get the link-scale value, which is the log-odds and ranges over the real line. Newcomers often forget the type argument and report log-odds as probabilities by accident.

Explanation: The likelihood ratio test compares twice the difference in log-likelihoods to a chi-square with degrees of freedom equal to the number of dropped parameters. Here p = 0.158, so mpg does not add significant explanatory power beyond wt. Prefer LRT over the Wald p-value from summary() when comparing nested GLMs, especially with small samples like 32 observations.

Explanation: Poisson regression uses a log link by default, so coefficients are on the log-rate scale. exp(-0.206) for woolB means wool B has about 81% as many breaks as wool A at the same tension. The dispersion parameter is fixed at 1; if the residual deviance is much larger than the residual degrees of freedom (here 210 vs 50) you have overdispersion and should consider quasi-Poisson or negative binomial.

Explanation: An offset enters the linear predictor with coefficient fixed at 1, so log(rate) = log(visits/hours) = beta0 + beta1 staff. Without the offset, longer shifts would absorb their own dummy effect and you'd lose the rate interpretation. Offsets are standard in epidemiology (person-years), insurance (policy duration) and call-centre analytics (open hours).

Explanation: Exponentiating a Poisson coefficient yields an IRR: a multiplicative effect on the expected count. The Wald CI is exp(estimate plus or minus 1.96 SE); for tighter coverage with small samples use profile likelihood via confint(). IRRs are easier to communicate than log-rates: "tension H cuts breaks to 60% of tension L" beats "the log-rate falls by 0.518".

Explanation: A deviance/df ratio of 4.2 is far above 1, signalling that the Poisson variance assumption underestimates uncertainty. Quasi-Poisson keeps the same coefficient estimates but inflates standard errors by sqrt(dispersion), so Wald tests become more conservative. If the dispersion is also large, negative binomial (next section) is often a better choice because it has a proper likelihood and AIC.

Explanation: Negative binomial adds a second parameter (theta) that absorbs extra variance, so unlike quasi-Poisson it produces a real likelihood and a comparable AIC. A drop of 84 AIC units is overwhelming evidence in favour of NB. Always log theta and check whether it's large (close to Poisson) or small (heavy overdispersion); here theta of 9.9 is moderate.

Explanation: Gamma with a log link is the natural GLM for strictly positive, right-skewed continuous outcomes like claim cost, hospital LOS or component lifetimes. The constant coefficient of variation built into the Gamma matches actuarial intuition that larger expected claims have proportionally larger spread. Avoid the canonical inverse link in practice: it can yield negative fitted values during iteration.

Explanation: Logit and probit are nearly indistinguishable in fitted probabilities because the logistic and normal CDFs agree closely in the middle range. Logit dominates because coefficients double as log-odds (interpretable), while probit is preferred in econometrics for latent-variable narratives. Coefficient magnitudes differ by roughly the ratio of the link slopes at 0, about 1.6 to 1.8.

Explanation: Null deviance measures fit of an intercept-only model, residual deviance measures fit of the full model, and their difference is the deviance explained. AIC penalises that fit by 2 * k parameters, which lets you compare non-nested models. Always read both deviance and AIC: a model with great AIC but tiny deviance drop relative to null might be capturing noise.

Explanation: confint() on a glm object profiles the log-likelihood across each coefficient, giving CIs that respect the curvature of the likelihood, not just a Wald approximation. With n = 32, Wald and profile intervals can differ noticeably; profile is the safer default. Use confint.default() if you specifically want Wald (estimate +/- 1.96 SE) for reporting consistency with summary().

Explanation: Pearson residuals standardise the raw difference by the model-implied standard deviation, so absolute values above 2 are unusually large under the model. A cluster of large Pearson residuals often signals overdispersion (Section 2.4) rather than individual outliers. Pair with cooks.distance() for leverage, and inspect flagged rows for data-entry errors before discarding any.

Explanation: Pseudo-R^2 statistics for GLMs do not match the proportion-of-variance interpretation from OLS, so the same numeric value carries different meaning. McFadden values above 0.2 are usually considered an excellent fit for cross-sectional logistic models. Report it alongside AIC and the Hosmer-Lemeshow test rather than as a single goodness-of-fit headline.

Explanation: The link-scale prediction eta is the linear combination Xb, returned by default. Applying the inverse link plogis(eta) recovers prob. Always reason about uncertainty (standard errors, CIs) on the link scale where coverage is symmetric, then back-transform endpoints to probabilities; CIs computed on the response scale can run below 0 or above 1.

Explanation: Logistic coefficients alone don't give a probability change because the slope of the sigmoid varies across the input. The average marginal effect (AME) integrates dP/dX over the empirical distribution, producing a single interpretable number on the probability scale. For more complex GLMs and interactions, the marginaleffects package automates this with proper standard errors via the delta method.

Explanation: Holding mpg at its mean isolates the effect of wt but hides interactions. For a richer picture, facet by binned mpg or use a partial-dependence plot that integrates over the empirical distribution of mpg. Jittering the raw outcomes vertically by a small amount keeps the 0/1 anchors readable without misleading the eye about probability values.

Explanation: Confidence bands constructed in the linear-predictor space and back-transformed remain inside [0, 1] and are asymmetric near the boundaries, which is what you want for a probability. Bands built directly on the response scale (estimate +/- 1.96 SE_p) can extend past 1, which embarrasses your plot. This is the same pattern as Poisson, Gamma and survival GLMs: do uncertainty work on the link scale.

Explanation: A multinomial logit fits K-1 simultaneous binary logits against a reference category (here setosa, alphabetically first). Convergence can be fragile with perfectly separable classes (iris setosa is linearly separable on petal length), so check warning messages. For ordered outcomes use proportional-odds logistic regression instead, which is more parsimonious and respects the ordering.

Explanation: type = "probs" returns a row vector that sums to 1 across the K categories. type = "class" returns the modal class, which is convenient but throws away calibration information. For decision making, expose the full probability vector so downstream code can apply asymmetric loss (the cost of confusing setosa and virginica may differ from confusing the two non-setosa species).

Explanation: A proportional-odds model fits one slope per predictor and K-1 thresholds for the K-level outcome, assuming the effect of each predictor on the cumulative log-odds is constant across cutpoints. Check that assumption with the Brant test (brant::brant(ex_6_3)); if it fails, use a partial-proportional-odds model via VGAM::vglm() or fall back to multinomial logistic, which gives up parsimony for flexibility.