Why model event counts with Poisson regression instead of ordinary linear regression?

## What a count outcome is A count outcome records how many times something happened in a fixed window of observation: support tickets filed by a customer this quarter, accidents at an intersection this year, page views in a session. Its values are whole numbers, they cannot go below zero, and there is no fixed upper bound. That shape is different from the continuous, roughly symmetric outcome that ordinary least squares was designed around. ## What ordinary least squares assumes, and where counts break it A linear model states `E[Y|x] = b0 + b1*x1 + ...` and, for its usual inference, that the scatter of observations around that line is roughly constant in size and roughly symmetric. Three things go wrong for counts. **Predictions can go negative.** A straight line is unbounded downward. If most units have small counts, the fitted line will dip below zero somewhere in the predictor range, and it will do so exactly in the region you often care about — the low-activity units. **The spread is not constant.** In count data the observations with a high expected value are also the ones that vary the most in absolute terms; units expected to see 2 events cluster tightly, units expected to see 200 swing by tens. Least squares treats all observations as equally noisy, so it under-weights the quiet, informative units and over-weights the loud ones, and the reported standard errors do not reflect the true pattern. **Additive effects are usually the wrong story.** A least-squares coefficient says: one more unit of the predictor adds `b` events. Real count processes rarely work that way. Doubling exposure, moving to a busier product surface, or serving a heavier-usage segment tends to scale events, not shift them. An additive model forced onto a multiplicative process produces a coefficient that is a blend of the effect at high and low baselines, and it fits neither. ## What Poisson regression does instead Poisson regression is a generalized linear model: the outcome is assumed to follow a count distribution, and a **link function** connects the linear predictor to the mean. With the standard log link, ``` log(E[Y|x]) = b0 + b1*x1 + b2*x2 + ... ``` which is equivalent to ``` E[Y|x] = exp(b0 + b1*x1 + b2*x2 + ...) ``` Three consequences follow immediately. 1. **The fitted mean is always positive.** The exponential of any real number is positive, so no combination of predictors ever produces a negative expected count. 2. **Effects are multiplicative.** Adding to the log of the mean multiplies the mean itself. A coefficient of `b` means the expected count is multiplied by `exp(b)` for each one-unit increase in that predictor, holding the others fixed. 3. **Predictors combine multiplicatively too.** On the log scale their contributions add, so on the count scale they multiply. Two factors each raising expected counts by 20% together give about `1.2 * 1.2 = 1.44`, a 44% increase, not 40%. The model is fitted by maximum likelihood rather than by minimising squared error. That fitting automatically gives more weight to observations whose expected count is small — precisely the ones whose variability is small and whose information about the low end is best. ## Why not just take logs of the outcome? A common alternative is to regress `log(y)` with least squares. Two problems. First, `log(0)` is undefined, and counts are full of zeros; the usual patch is `log(y + 1)`, whose results depend on a constant nobody can justify. Second, it changes the estimand: it models the mean of the log rather than the log of the mean. Exponentiating back gives something closer to a median or geometric mean than to the expected count, which is normally the quantity a stakeholder is asking about. Poisson regression models `log(E[Y])` directly, handles zeros without any shift, and returns expected counts. ## What Poisson regression does not assume It does not assume normally distributed errors. It does not require the predictors to be transformed. It does not require the outcome to be small. What it does assume is that, given the predictors, the outcome behaves like a count process whose variability is tied to its mean, and that the log-linear mean is correctly specified. When the variability turns out to be much larger than that assumption allows, the mean model can still be fine while the reported uncertainty is not — a separate issue diagnosed and repaired on its own terms. ## When least squares is still fine If every count is large and far from zero, the distribution is close to symmetric, negative predictions are impossible in the observed range, and an average additive difference is genuinely the quantity of interest, a linear fit can be a reasonable working approximation. That is a deliberate, defensible simplification — quite different from reaching for it by default.