Regression & Correlation

Negative Binomial Regression

Regression model for overdispersed count data using a log link and negative binomial distribution.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Regression > Poisson Regression (with overdispersion test)

Description

Negative binomial regression extends Poisson regression by adding a dispersion parameter θ that models overdispersion (Var(Y) > E(Y)). The variance function V(Y) = μ + μ²/θ includes a quadratic term capturing extra-Poisson variation. Coefficients β are estimated via maximum likelihood (IRLS), while θ is simultaneously optimized through profile likelihood. The exponentiated coefficients exp(βⱼ) are interpretable as rate ratios: an RR of 1.5 means a one-unit increase in the predictor raises the expected rate by 50%.

Formulas

Link

lo g (μ) = X^{'} β

Log link function: the natural logarithm of the expected value is a linear combination of predictors

Varianz

V (Y) = μ + \frac{μ ^{2}}{θ}

Quadratic variance function: variance grows faster than the mean. As θ → ∞ it converges to the Poisson case V(Y) = μ

PMF

P (Y = y) = \frac{Γ ( y + θ )}{Γ ( θ ) y !} (\frac{θ}{θ + μ})^{θ} (\frac{μ}{θ + μ})^{y}

Probability mass function of the negative binomial distribution (NB2 parameterization with mean μ and dispersion parameter θ)

Log-Likelihood

ℓ = i = 1 \sum n [lo g Γ (y_{i} + θ) - lo g Γ (θ) - lo g (y_{i}!) + θ lo g (\frac{θ}{θ + μ _{i}}) + y_{i} lo g (\frac{μ _{i}}{θ + μ _{i}})]

Log-likelihood function of the negative binomial distribution, jointly maximized over β and θ

IRLS-Gewichte

w_{i} = \frac{μ _{i}}{1 + μ _{i} / θ}

Weights for Iteratively Reweighted Least Squares: combination of expected count rate and dispersion correction

θ-Schätzung

S (θ) = i = 1 \sum n [ψ (y_{i} + θ) - ψ (θ) + lo g (\frac{θ}{θ + μ _{i}}) + \frac{y _{i} - μ _{i}}{θ + μ _{i}}] = 0

Score function for θ: solved via profile likelihood using the digamma function ψ

Rate Ratio

R R_{j} = exp (\hat{β}_{j})

Rate ratio: the factor by which the expected rate changes for a one-unit increase in the j-th predictor

Dispersion

ϕ = 1

The scale parameter φ is fixed at 1 — all overdispersion is captured by θ, no additional scaling

Assumptions

Response is count data (non-negative integers)
Overdispersion present: Var(Y) > E(Y)
Independence of observations
Log-linear relationship between log(μ) and predictors
No perfect multicollinearity

Limitations

θ estimation via profile likelihood adds computational cost (outer optimization loop)
Small samples may give unstable θ estimates
Not appropriate for zero-inflated data — consider a ZINB model instead
Convergence may be slower than Poisson due to joint β/θ estimation

References

Hilbe, J.M., Negative Binomial Regression, 2nd Ed., Cambridge University Press
Cameron, A.C., Trivedi, P.K., Regression Analysis of Count Data, 2nd Ed., Cambridge University Press
Lawless, J.F., Negative Binomial and Mixed Poisson Regression, Canadian Journal of Statistics, 1987