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Negative Binomial Regression

Negative Binomial Regression

Definition

Core Statement

Negative Binomial Regression is an extension of Poisson Regression used when count data exhibits overdispersion (Variance > Mean). It introduces an extra parameter to model the heterogeneity in event rates across observations.

Purpose

  1. Provide valid inference for count data when Poisson's equidispersion assumption fails.
  2. Model counts where unobserved heterogeneity causes extra variance (e.g., some customers are inherently more active than others).

When to Use

Use Negative Binomial When...

  • Outcome is a count.
  • The Poisson Regression model shows overdispersion (Dispersion statistic > 1.5).
  • Data has unobserved heterogeneity in rates.

Do NOT Use NegBin When...

  • Equidispersion holds (Poisson is simpler and sufficient).
  • Excess zeros are the main problem (use Zero-Inflated Models).

Theoretical Background

Mean-Variance Relationship

Model Variance Implication
Poisson Var(Y)=μ Mean = Variance
Negative Binomial Var(Y)=μ+αμ2 Variance > Mean

The parameter α (dispersion parameter) controls the extra variance. When α=0, NegBin reduces to Poisson.

The Model

Like Poisson, uses a log link:

ln⁡(μ)=β0+β1X1+…

Coefficients are interpreted as Rate Ratios (eβ), same as Poisson.

Assumptions

Limitations

Pitfalls

  1. Still sensitive to excess zeros. If you have structural zeros (e.g., people who will never use a product), consider Zero-Inflated Models.
  2. Model selection: Use likelihood ratio tests or AIC to compare Poisson vs NegBin.

Python Implementation

import statsmodels.api as sm
import statsmodels.formula.api as smf

# Fit Negative Binomial (uses NegativeBinomial family in GLM)
# Alternative: statsmodels.discrete.discrete_model.NegativeBinomial

model_nb = smf.negativebinomial("count ~ age + income", data=df).fit()
print(model_nb.summary())

# Rate Ratios
import numpy as np
print("\n--- Rate Ratios ---")
print(np.exp(model_nb.params))

R Implementation

library(MASS)

# Fit Negative Binomial
model_nb <- glm.nb(count ~ age + income, data = df)
summary(model_nb)

# Check Theta (Dispersion Parameter)
# Smaller theta = more overdispersion
cat("Theta:", model_nb$theta, "\n")

# Rate Ratios
exp(coef(model_nb))
exp(confint(model_nb))

# Compare to Poisson via LRT
model_pois <- glm(count ~ age + income, data = df, family = poisson)
anova(model_pois, model_nb, test = "LRT")

Worked Numerical Example

Hospital Visits: Poisson vs Negative Binomial

Outcome: Number of hospital visits per year (count data)
Predictor: Chronic illness severity (0-10 scale)

Poisson Results:

  • β_severity = 0.22, RR = e^0.22 = 1.25
  • Dispersion = Residual Deviance/df = 156/98 = 1.59 (overdispersion!)
  • Many p-values < 0.001 (likely inflated)

Negative Binomial Results:

  • β_severity = 0.18, RR = e^0.18 = 1.20
  • α (dispersion parameter) = 2.3
  • Variance = μ + 2.3μ² (much larger than Poisson's Var = μ)
  • Same predictors, but p-values more conservative (some now p = 0.04 instead of p < 0.001)
  • AIC: Poisson = 892, NegBin = 765 (NegBin much better)

Interpretation:

  • Severity increases visits by 20% per unit increase
  • Poisson underestimated SEs → inflated significance
  • NegBin accounts for extra variability (unobserved heterogeneity)

Interpretation Guide

Output Example Interpretation Edge Case Notes
α (dispersion) 2.5 High overdispersion. Var = μ + 2.5μ². If α→0, NegBin converges to Poisson.
α (dispersion) 0.1 Mild overdispersion. NegBin barely differs from Poisson. May not need NegBin; Poisson sufficient.
RR (predictor) 1.15 Each unit increase multiplies count by 1.15. Same interpretation as Poisson RR.
AIC_NB << AIC_Poisson NegBin fits much better. Overdispersion confirmed. Difference > 10 is substantial. Use NegBin.
AIC_NB ≈ AIC_Poisson Models similar. Overdispersion marginal. May stick with simpler Poisson model.
LRT p < 0.05 NegBin is significantly better than Poisson; overdispersion is real.

Common Pitfall Example

Ignoring Overdispersion and Reporting Poisson

Scenario: Analyzing number of defects in manufacturing

Analyst's Mistake:

  1. Fits Poisson model
  2. Sees dispersion = 3.2 (severe overdispersion!)
  3. Ignores it, reports Poisson results anyway
  4. Conclusion: "Quality training significantly reduces defects (p < 0.001)"

Problem:

  • Standard errors are too small (underestimated by ~√3.2 = 1.8×)
  • p-values are artificially low
  • True p-value might be 0.08 (not significant!)

Correct Practice:

  • Always check dispersion: Residual Deviance/df
  • If > 1.5, use Negative Binomial
  • Or at minimum, report quasi-Poisson with corrected SEs

Real Result with NegBin:

  • Same β coefficient
  • SE increased by 1.8×
  • p = 0.07 (no longer significant at α=0.05)
  • Conclusion changes: "Marginally significant trend, needs more data"