Poisson — Count regression

Poisson regression models count data (non-negative integers: number of patents, doctor visits, accidents…). It uses a log link to keep the expected value positive.

When to use

Use Poisson when $Y$ is a count. The core assumption: mean equals variance (equidispersion). If the variance exceeds the mean (overdispersion), switch to Negative Binomial.

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Model specification

$$E[Y_i \mid X_i] = \exp(\beta_0 + \beta_1 X_{1i} + \dots + \beta_k X_{ki}), \qquad Y_i \sim \text{Poisson}(\mu_i)$$

Estimated by MLE. Coefficients are interpreted via the incidence rate ratio $e^{\beta_j}$.

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Diagnostics

• Overdispersion: test whether $\text{Var}(Y) > E[Y]$ (e.g. Cameron-Trivedi test). If present ⇒ SE understated ⇒ use NegBin or Poisson with robust SE/QMLE.
• Excess zeros ⇒ ZIP.

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Running in EcoLab

1. Modeling module → Count data family → Poisson.
2. Select the count $Y$ and the $X$ variables; add an exposure/offset if needed (e.g. per population).
3. Run; read IRRs; check overdispersion; export the replication code.

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Replication code

Stata

* ===== Poisson — Count regression =====
* Estimate with robust standard errors
poisson patents rd_spend firm_size, vce(robust)

* Incidence Rate Ratios (IRR)
poisson patents rd_spend firm_size, vce(robust) irr

* Overdispersion test (Cameron-Trivedi)
* After estimation, check if Var(Y) >> E[Y]
estat gof

* Predicted counts
predict yhat, n

R

# ===== Poisson — Count regression =====
model <- glm(patents ~ rd_spend + firm_size,
             data   = df,
             family = poisson(link = "log"))

summary(model)

# Incidence Rate Ratios (IRR)
exp(coef(model))
exp(confint(model))

# Overdispersion test
library(AER)
dispersiontest(model)

# If overdispersed, use robust SE (quasi-Poisson)
model_qp <- glm(patents ~ rd_spend + firm_size,
                 data   = df,
                 family = quasipoisson())
summary(model_qp)

Python

# ===== Poisson — Count regression =====
import statsmodels.api as sm
import numpy as np

# Prepare data
X = sm.add_constant(df[["rd_spend", "firm_size"]])
y = df["patents"]

# Estimate Poisson with robust SE
model = sm.GLM(y, X, family=sm.families.Poisson()).fit(cov_type="HC1")
print(model.summary())

# Incidence Rate Ratios (IRR)
print("IRR:")
print(np.exp(model.params))

# Check overdispersion: Pearson chi² / df_resid >> 1 ⇒ overdispersed
print("Dispersion:", model.pearson_chi2 / model.df_resid)

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Limitations

• The equidispersion assumption is often violated in practice.
• Excess zeros make Poisson fit poorly ⇒ use zero-inflated/hurdle models.

Video tutorial

Video Tutorial: Guide to running Poisson regression in EcoLab

See also

• Negative Binomial · ZIP · ZINB · Catalog