Generalized Linear Models

Ordinary regression assumes a continuous outcome with normal, constant-variance errors. Counts, yes/no outcomes, and rates break those rules. Rather than invent a brand-new method for each, generalized linear models (GLMs) stretch one elegant framework to cover them all — and you've already met two members of the family without knowing it.

Two ingredients

Every GLM is built from the same two pieces:

• A linear predictor: the familiar straight line, η = b₀ + b₁x.
• A link function that connects that line to the mean of the outcome, g(μ) = η, or equivalently μ = g⁻¹(η).

The link is the clever bit. It keeps the predicted mean in a sensible range, so a straight line on the "link scale" becomes whatever shape the data needs on the response scale.

The family you already know

• Continuous outcome → identity link → ordinary regression. The link does nothing; the line stays a line.
• Yes/no outcome → logit link → logistic regression. The line becomes an S-curve trapped between 0 and 1, and the slope is a log-odds (so e^b₁ is an odds ratio).
• Count outcome → log link → Poisson regression. The line becomes an exponential curve that can't go negative, and e^b₁ is a multiplicative rate ratio.

Fitting and assumptions

GLMs are fit by maximum likelihood rather than least squares, and each family carries its own mean–variance relationship (for Poisson, the variance equals the mean; watch for overdispersion when real counts vary more than that). But the modeling workflow (pick predictors, fit, check, interpret) is exactly the one you've been practicing all along.

Counts are not rates

Poisson regression models a count, and a count only means something once you know how long you watched. Two clinics report falls: clinic A logs 40 over 500 patient-days, clinic B logs 60 over 1,500. Feed those raw counts to the model and it dutifully reports that B has 1.5 times A's rate, because 60/40 = 1.5. Per patient-day, A is running at 0.08 and B at 0.04, so B is half as bad. Same data, opposite finding.

The repair is an offset: a predictor forced into the model with its coefficient nailed to 1. Put log(patient-days) in that slot and the log link does the rest, since log(count) − log(days) is log(rate). The model still predicts counts, but the coefficients now describe rates. Fitted on the two clinics above, the offset version returns b₁ = log(0.5) = −0.693 and a rate ratio of exactly 0.50, while the same model without it returns 0.405 and 1.50. In R it is one argument, offset = log(days); in SPSS it is the Offset variable box in Generalized Linear Models. Skipping it is the single most common way a count model answers a question nobody asked. Offset and deviance both turn up in software output before anyone defines them, so the glossary carries both.

Unequal exposure is also a common hidden source of the overdispersion the section above warns about, because counts collected over wildly different windows vary more than any single Poisson rate can explain. Which raises the practical question: how do you tell?

The standard check runs on the deviance, the number a GLM minimizes in place of the sum of squared residuals. It measures how far your fitted model sits from a hypothetical one that reproduces every observation exactly, and for a well-specified Poisson model it should land near its degrees of freedom. So divide: a deviance/df ratio near 1 is what equidispersion looks like, and anything much above it says the counts are more variable than Poisson allows. The 412.6 on 178 df in the practice problem gives 2.32, comfortably into negative-binomial territory. The counts in the widget above are drawn from a true Poisson, so the ratio sits near 1 by construction: refitting that generator 20,000 times averages 1.07. It is a noisy number at these sample sizes, though, with 90% of those 32-point fits landing anywhere between 0.67 and 1.55, so treat a mild excess as a shrug and a 2 as a finding.

Problem 35 of the practice problems is a family-and-link drill on five hospital outcomes at once (counts with many zeros, a yes/no readmission, right-skewed euros, a score out of 20, and blood pressure), then asks you to read one fitted Poisson model back on the rate scale.