NB1 and NB2 Variance Structures

Why the Variance Structure Matters

Two count models can make the same prediction for the conditional mean and still imply different amounts of uncertainty around that prediction. This distinction matters when counts are more variable than a Poisson model allows: a variance specification determines how the extra variation is expected to change as the fitted mean changes. Negative binomial models are often described as “NB1” or “NB2” precisely to distinguish these variance patterns.

Both specifications can use a log-linked conditional mean, such as \(\mu_i=\exp(x_i^\top\beta)\), where \(Y_i\) is a nonnegative count and \(x_i\) is a vector of observed predictors. The comparison here concerns the conditional variance given those predictors. It is not a comparison of two different mean links.

These equations define the variance restrictions, not a complete account of every distributional assumption that may be used for likelihood-based estimation. For the comparisons below, assume the conditional mean is specified as stated, the conditional variance follows the NB1 or NB2 function, and observations are conditionally independent given their predictors. A full negative binomial likelihood also specifies a probability distribution for the counts; matching a variance function alone does not establish that the entire distribution is correct.

How the Two Variance Functions Change

The clearest comparison comes from examining both the variance itself and its ratio to the conditional mean. For NB1, the extra-Poisson component is proportional to the mean. For NB2, it is proportional to the square of the mean.

This gives a compact derivation of the distinct patterns. Dividing the NB1 variance by \(\mu\) yields \((\mu+\alpha\mu)/\mu=1+\alpha\), which does not depend on \(\mu\). Dividing the NB2 variance by \(\mu\) yields \((\mu+\alpha\mu^2)/\mu=1+\alpha\mu\), which increases linearly with \(\mu\) when \(\alpha>0\). Differentiating the variance functions shows that NB1 variance increases at a constant rate, whereas the rate of increase under NB2 grows with the mean.

The ratio \(V(\mu)/\mu\) is the conditional variance-to-mean ratio, also called the Fano factor. A Poisson model has ratio 1. Under NB1 the ratio is constant and greater than 1 when \(\alpha>0\). Under NB2 it rises with the fitted mean: low-mean observations can have modest extra variation, while high-mean observations can have substantially more.

Reading Conditional Variability in Data

In principle, grouped observations with similar predictors can reveal how conditional variance changes across different fitted means. One can compare the mean and variance of counts within groups, or inspect residual variation across ranges of fitted means. The relevant question is whether the extra-Poisson variability looks roughly proportional to the mean (NB1) or grows more rapidly as the mean increases (NB2).

The following hypothetical summaries illustrate the comparison. Suppose three groups of observations have similar predictor values within each group. Their fitted means are 2, 4, and 8 counts, and their observed within-group variances are 3.2, 6.5, and 13.4 counts squared. These are sample summaries and therefore subject to sampling variation; they need not fall exactly on any theoretical variance curve.

The reverse pattern can point toward NB2: the variance-to-mean ratio increases with the fitted mean, while the quantity \((V(\mu)/\mu-1)/\mu\) is roughly stable. This follows directly from the formulas. Under NB1, \(V(\mu)/\mu-1=\alpha\) is constant; under NB2, dividing that excess ratio by \(\mu\) gives \(\alpha\).

What the Comparison Can—and Cannot—Tell You

NB1 and NB2 are useful alternatives because overdispersion need not behave identically across the range of fitted means. NB1 imposes a constant multiplicative inflation of the Poisson variance: \(V_1(\mu)=(1+\alpha)\mu\). NB2 allows the inflation to increase with the mean: \(V_2(\mu)=\mu(1+\alpha\mu)\). Consequently, evidence of overdispersion at one mean does not by itself determine which structure is more suitable.

It is important to compare variability conditional on predictors or fitted means. A raw variance calculated across observations with widely different expected counts combines within-group variability with differences in the group means. In particular, the law of total variance gives \(\operatorname{Var}(Y)=E[\operatorname{Var}(Y\mid X)]+\operatorname{Var}(E[Y\mid X])\). The second term reflects variation in conditional means across \(X\), not just the count variance function. A large overall variance therefore cannot, on its own, identify NB1 or NB2.

Neither variance pattern repairs a poorly specified conditional mean. If the mean model misses important predictors or has the wrong functional form, apparent residual patterns may reflect that misspecification rather than the choice between NB1 and NB2. Likewise, matching a variance function is not evidence that the data have a particular pattern of zeros or that every aspect of a negative binomial distribution is appropriate.

Common Mistake

• Wrong reasoning: “Both models allow overdispersion, so they are interchangeable once the variance exceeds the mean.”
• Correction: They impose different changes in overdispersion across the mean range. Under NB1, \(V(\mu)/\mu=1+\alpha\) is constant; under NB2, \(V(\mu)/\mu=1+\alpha\mu\) increases with \(\mu\). Compare conditional variability across fitted means rather than checking only whether a single variance exceeds its mean.