Negative Binomial Distribution

The Negative Binomial (NB) you’re using (as implemented by MASS::glm.nb()) is parameterized by two quantities:

• $μ$ = expected count (mean)
• $θ$ (often called size or dispersion parameter) > 0

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🔢 Probability Mass Function (PMF)

For $y = 0,1,2,\dots$, the NB‑2 PMF (MASS’s “size–mu” parameterization) is

• $\Gamma(\cdot)$ = gamma function
• $θ/(θ+μ)$ = probability of “failure”
• $μ/(θ+μ)$ = probability of “success”

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📈 Mean & Variance

Because $\frac{μ^2}{θ} ≥ 0$, Var(Y) ≥ μ, making NB inherently a over‑dispersed generalization of Poisson.

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📊 Role of θ

θ (size)	Interpretation	Var(Y) relative to μ
→∞	Gamma mixing variance → 0	Var ≈ μ (Poisson limit)
large (≫μ)	Weak overdispersion	Var slightly > μ
small (≪μ)	Strong overdispersion	Var ≫ μ

• $θ$ is the shape parameter of the Gamma distribution in the Poisson–Gamma mixture view:
  1. Draw $\lambda \sim \mathrm{Gamma}(\text{shape}=θ,\;\text{scale}=μ/θ)$
  2. Then $Y \mid \lambda \sim \mathrm{Poisson}(\lambda)$

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⚠️ Why $θ>0$?

• Gamma(shape=$θ$) is only defined for $θ>0$
• Ensures PMF integrates to 1
• Guarantees Var(Y) ≥ μ (no underdispersion)

If $θ\to∞$, NB “degenerates” to Poisson; if $θ$ is very small, variance blows up (extreme overdispersion).