ZIP — Zero-Inflated Poisson

ZIP (Zero-Inflated Poisson) handles count data with excess zeros beyond what Poisson predicts, when zeros arise from two different mechanisms: an "always zero" group (structural zeros) and a Poisson-count group (which may incidentally be zero).

When to use

Use ZIP when count data has many zeros and you believe there is a group for which "the event never happens" (e.g. cigarettes/day: non-smokers are always 0).

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Two-part mixture structure

$$P(Y_i = 0) = \pi_i + (1 - \pi_i) e^{-\mu_i}, \qquad P(Y_i = y) = (1 - \pi_i) \frac{e^{-\mu_i}\mu_i^{y}}{y!}, \; y \ge 1$$

where $\pi_i$ (probability of a structural zero) is modeled by logit/probit; $\mu_i = \exp(X_i\beta)$.

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Running in EcoLab

1. Modeling module → Count data family → ZIP.
2. Declare variables for the count part ($X$) and the inflation part (predictors of "always zero").
3. Run; compare with Poisson via the Vuong test; export the replication code.

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Replication code

Stata

* ===== ZIP — Zero-Inflated Poisson =====
* Count part: patents ~ rd_spend + firm_size
* Inflation part: inflate(small_firm)
zip patents rd_spend firm_size, inflate(small_firm) vuong

* IRR for the count part
zip patents rd_spend firm_size, inflate(small_firm) irr

* Vuong test: significant ⇒ ZIP preferred over standard Poisson
* Shown at the bottom of the output

R

# ===== ZIP — Zero-Inflated Poisson =====
library(pscl)

# Count part: patents ~ rd_spend + firm_size
# Inflation part: ~ small_firm
model <- zeroinfl(patents ~ rd_spend + firm_size | small_firm,
                  data = df,
                  dist = "poisson")

summary(model)

# IRR for the count part
exp(coef(model, "count"))

# Vuong test: ZIP vs standard Poisson
model_pois <- glm(patents ~ rd_spend + firm_size,
                   data = df, family = poisson())
vuong(model, model_pois)

Python

# ===== ZIP — Zero-Inflated Poisson =====
import statsmodels.api as sm
import numpy as np

# Prepare data
X = sm.add_constant(df[["rd_spend", "firm_size"]])  # count part
Z = sm.add_constant(df[["small_firm"]])              # inflation part
y = df["patents"]

# Estimate ZIP model
model = sm.ZeroInflatedPoisson(y, X, exog_infl=Z).fit()
print(model.summary())

# IRR for the count part
print("IRR (count part):")
print(np.exp(model.params[:X.shape[1]]))

# Compare with standard Poisson via AIC/BIC
model_pois = sm.GLM(y, X, family=sm.families.Poisson()).fit()
print(f"AIC Poisson: {model_pois.aic:.1f}")
print(f"AIC ZIP:     {model.aic:.1f}")

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Limitations

• If the count part is still overdispersed ⇒ ZINB.
• More complex interpretation (two equations); needs clear theory for the zero mechanism.

Video tutorial

Video Tutorial: Guide to running Zero-Inflated Poisson (ZIP) in EcoLab

See also

• Poisson · ZINB · Negative Binomial · Catalog