Negative Binomial

Concept

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Theoretically, the Negative Binomial Distribution is a discrete probability distribution used to model count data exhibiting overdispersion, where the variance exceeds the mean. It extends the Poisson distribution by introducing an additional dispersion parameter that accounts for unobserved heterogeneity. The distribution is widely used in health economics because many healthcare utilisation outcomes, such as hospital admissions, physician visits and emergency department attendances, display greater variability than can be accommodated by the Poisson model.

Mathematically, the Negative Binomial Distribution is commonly derived as a Poisson?Gamma mixture, in which the Poisson rate parameter follows a Gamma distribution. This formulation produces a count distribution with separate parameters governing the expected count and the degree of overdispersion. Parameters are typically estimated using Maximum Likelihood Estimation, and the model reduces to the Poisson distribution as the dispersion parameter approaches zero.

In practice, the Negative Binomial Distribution is applied to analyse overdispersed count outcomes in clinical and health economic studies. It is frequently used to estimate rates of hospitalisations, healthcare contacts, prescription use and adverse events while accounting for excess variability. Negative binomial regression is a standard method for modelling such outcomes and is commonly preferred over Poisson regression when overdispersion is present.

Purpose

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Used to model overdispersed count data, estimate event rates, account for unobserved heterogeneity and analyse healthcare utilisation and other count outcomes in health economic research.

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Mathematical Formulae

Primary Formula

Probability mass function:

P(X = x) = �(x + r) / [�(r)�(x + 1)] ? (r / (r + ?))? ? (? / (r + ?))?

where:

• x = observed count
• ? = expected count
• r = dispersion (shape) parameter
• �(�) = Gamma function

Supporting Formulae

Expected value:

E(X) = ?

Variance:

Var(X) = ? + ?� / r

Negative binomial regression model:

ln(??) = ?? + ??X?? + ??X?? + ? + ??X??

Related Mathematical Methods

• Negative Binomial Regression
• Poisson Distribution
• Poisson Regression
• Gamma Distribution
• Maximum Likelihood Estimation
• Generalised Linear Models
• Zero-Inflated Negative Binomial Models

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Example

A health economist analyses the annual number of hospital admissions among patients with chronic obstructive pulmonary disease.

The observed data show:

• Mean admissions = 2.4
• Variance = 5.8

Because the variance substantially exceeds the mean, a Poisson model is inappropriate.

A negative binomial regression estimates:

• Mean admission rate (?) = 2.4
• Dispersion parameter (r) = 1.7

The implied variance is:

Var(X) = 2.4 + (2.4� / 1.7)

Var(X) = 2.4 + 3.39

Var(X) = 5.79

This closely matches the observed variance, demonstrating that the Negative Binomial Distribution adequately models the overdispersion present in the hospital admission data.

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Excel Implementation

Function	Example Formula	Health Economics Application
GAMMALN	=GAMMALN(A2+B2)-GAMMALN(B2)-GAMMALN(A2+1)	Calculate components of the negative binomial probability mass function.
LN	=LN(C2)	Calculate log-likelihood contributions during parameter estimation.
EXP	=EXP(D2)	Convert linear predictors to expected event counts.
SUM	=SUM(E2:E1001)	Calculate the total log-likelihood across observations.
Solver	Maximise the log-likelihood by varying regression and dispersion parameters.	Estimate negative binomial model parameters.

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VBA (Optional)

A VBA routine can automate maximum likelihood estimation of negative binomial models, estimate dispersion parameters and generate predicted healthcare utilisation rates.

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Sources

• Hilbe JM. Negative Binomial Regression. Cambridge University Press.
• Cameron AC, Trivedi PK. Regression Analysis of Count Data. Cambridge University Press.
• McCullagh P, Nelder JA. Generalized Linear Models. Chapman & Hall.
• Greene WH. Econometric Analysis. Pearson.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Good Practice Reports.