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Count outcomes like protest events or arrests are often overdispersed—more variable than Poisson assumes. Negative binomial regression accommodates overdispersion. Zero-inflated models address excess zeros. Proper model selection improves inference and prevents bias.
You already know that ordinary linear regression assumes a continuous, normally distributed outcome. But many social science outcomes are counts — the number of protests in a country in a year, arrests per month, bills introduced in a legislative session, or war casualties in a conflict. Counts are non-negative integers, and their distribution tends to be highly skewed: many observations near zero, a long tail of large values. Fitting OLS to count data can produce nonsensical predictions (negative counts, fractional events) and incorrect standard errors. The solution is a family of regression models built specifically for count data.
The starting point is Poisson regression. The Poisson distribution — which you've studied — has one defining property: its mean equals its variance. Poisson regression models the log of the expected count as a linear function of predictors, which guarantees non-negative predictions and has a natural interpretation: coefficients are log-incident rate ratios, and exponentiated coefficients are multiplicative effects on the expected count. If a coefficient is 0.5, e0.5 ≈ 1.65, meaning a one-unit increase in that predictor multiplies the expected count by 1.65.
The critical limitation of Poisson is its mean-equals-variance constraint. Real count data is almost always overdispersed — the variance exceeds the mean, often dramatically. This happens when outcomes are clustered (protests cluster in time and space), when unobserved heterogeneity exists across units, or when events follow a contagion process. If you force Poisson on overdispersed data, the model is misspecified: standard errors are underestimated, test statistics are inflated, and you will declare spurious significance. Negative binomial regression relaxes the constraint by adding a dispersion parameter that captures extra-Poisson variation. Think of it as a Poisson model where each observation has its own underlying rate drawn from a gamma distribution — the resulting mixture is the negative binomial. In practice, negative binomial fits are nearly always preferred when overdispersion tests flag a problem.
A further complication is excess zeros — outcomes where the count is zero far more often than Poisson or negative binomial predicts. This arises when two distinct processes generate the data: one process determines whether any events occur at all (a logistic-type "always-zero" mechanism), and a second process governs how many occur when they do (a count mechanism). Zero-inflated Poisson and zero-inflated negative binomial models estimate both processes simultaneously. Model selection between these options typically uses the Vuong test, AIC/BIC comparison, and rootograms (graphical comparison of observed vs. predicted count frequencies) to diagnose where the plain Poisson fails.
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