Zero-Inflated Models

Definition

Core Statement

Zero-Inflated Models are used for count data that has an excess of zero counts beyond what standard count models (Poisson, Negative Binomial) predict. They assume zeros come from two sources: a structural process (permanent zeros) and a count process (sampling zeros).

Purpose

Model count data where many observations are structural zeros (e.g., people who never smoke, websites with no traffic).
Separate the factors influencing "whether an event occurs at all" from "how many times it occurs."

When to Use

Use Zero-Inflated Models When...

Outcome is a count with many more zeros than Poisson/NegBin predicts.
There are two types of zeros:
- Structural Zeros: Zeros due to a characteristic (e.g., non-drinkers reporting 0 drinks).
- Sampling Zeros: Zeros due to chance (e.g., drinkers who happened not to drink during the observation period).

Diagnosing Excess Zeros

Fit a standard Poisson model.
Compare observed vs expected zeros.
If Observed >> Expected, consider zero-inflation.

Theoretical Background

Two-Part Structure

Inflation (Logit/Probit) Model: Predicts the probability of being a structural zero (from a degenerate distribution at 0).
Count (Poisson/NegBin) Model: Predicts counts for those not structural zeros.

Zero-Inflated Poisson (ZIP)

P (Y = 0) = π + (1 - π) e^{- λ}

P (Y = k) = (1 - π) \frac{λ^{k} e^{- λ}}{k!}, k \geq 1

$π$ : Probability of being a structural zero.
$λ$ : Expected count for the count component.

ZIP vs ZINB

Model	Count Component	When to Use
ZIP	Poisson	Equidispersion in count component
ZINB	Negative Binomial	Overdispersion in count component

Assumptions

Two Sources of Zeros: Conceptually valid distinction between structural and sampling zeros.
Correct Link Functions: Logit for inflation, Log for count.
Independence.

Limitations

Pitfalls

Conceptual Justification Needed: You must have a theoretical reason for two zero-generating processes. Otherwise, results are hard to interpret.
Convergence Issues: More complex models may fail to converge with small samples.
Alternative: Hurdle Models are similar but assume all zeros come from a single process (inflation model), and counts > 0 from a truncated count model.

Python Implementation

import statsmodels.api as sm
from statsmodels.discrete.count_model import ZeroInflatedPoisson

# Fit ZIP
# exog: predictors for count model
# exog_infl: predictors for inflation model (can be same or different)
model_zip = ZeroInflatedPoisson(endog=df['count'], exog=df[['x1', 'x2']],
                                 exog_infl=df[['x1']]).fit()
print(model_zip.summary())

# ZINB is similar: ZeroInflatedNegativeBinomialP

R Implementation

library(pscl)

# Fit ZIP
# Formula: count ~ X1 + X2 | inflation predictors
model_zip <- zeroinfl(count ~ x1 + x2 | 1, data = df, dist = "poisson")
summary(model_zip)

# Fit ZINB
model_zinb <- zeroinfl(count ~ x1 + x2 | 1, data = df, dist = "negbin")
summary(model_zinb)

# Compare models with Vuong test
vuong(model_zip, model_zinb)

Interpretation Guide

Component	Coef	Interpretation
Count: X1	0.3	For non-zero group, 1-unit increase in X1 increases expected count by ~35%.
Inflate: (Intercept)	-1.5	Baseline log-odds of being a structural zero is -1.5 (low probability of structural zero).
Inflate: X1	0.5	Higher X1 increases probability of being a structural zero.

Poisson Regression
Negative Binomial Regression
Hurdle Models - Alternative for excess zeros.