Quasi-Poisson Versus Negative-Binomial Inference

Why the Variance Assumption Changes Inference

For count regression, the fitted mean describes how the expected count changes with predictors. But estimates of uncertainty and tests of effects also depend on how the variability around that mean is modeled. Quasi-Poisson and negative-binomial regression can both accommodate variance larger than the mean, yet they do so differently: quasi-Poisson scales the Poisson variance by a constant, while the common NB2 model lets the variance grow quadratically with the mean.

That difference has two consequences. With a log link, quasi-Poisson and Poisson regression have the same coefficient estimates, though their standard errors can differ. NB2 regression generally changes the coefficient estimates as well as their uncertainty. In addition, a fully specified negative-binomial model has a likelihood; a quasi-Poisson mean-and-variance specification alone does not.

Definition: For independent counts $Y_i$ with covariate vector $x_i$, let $\mu_i=E[Y_i\mid x_i]$ and use the log-mean model $\log\mu_i=x_i^\mathsf{T}\beta+o_i$, where $o_i$ is a known offset. A quasi-Poisson specification assumes $\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i$, with $\phi>0$. The NB2 specification assumes $\operatorname{Var}(Y_i\mid x_i)=\mu_i+\alpha\mu_i^2$, with $\alpha>0$, and specifies a negative-binomial distribution as well as this mean and variance.

These assumptions concern conditional variation given the predictors. The quasi-Poisson dispersion $\phi$ is constant across observations; NB2’s extra variance, $\alpha\mu_i^2$, increases more rapidly at larger means. The comparison below concerns NB2, not every possible negative-binomial variance parameterization.

Mean Estimates: The Special Equivalence

For a model with design matrix $X$, the quasi-score for $\beta$ is formed by weighting each mean residual by the derivative of the mean and dividing by its variance. With a log link, $\partial\mu_i/\partial\beta=\mu_i x_i$. Thus, under the quasi-Poisson variance, the score is a constant multiple of the Poisson score.

Result: If the log-link model has a finite solution to its estimating equations, quasi-Poisson and Poisson regression give the same fitted coefficients and means. This equality does not generally extend to NB2 regression.

To see the distinction, the NB2 score weights a residual by $1/(1+\alpha\mu_i)$. Observations with larger fitted means therefore contribute differently from their contributions to the Poisson or quasi-Poisson score. Unless the model has a special structure, the solution for $\beta$ changes.

Worked Example: Comparing Scores at a Candidate Fit

Consider three observations with $x_i=0,1,2$, counts $y_i=1,2,8$, and a log-linear candidate mean $\mu_i=1,2,4$. These means correspond to $\beta_0=0$ and $\beta_1=\log 2$. Compare the scores at this candidate under Poisson, quasi-Poisson, and NB2 with $\alpha=0.5$.

The residuals are $y_i-\mu_i=0,0,4$. For Poisson regression with a log link, the score is $X^\mathsf{T}(y-\mu)$. The intercept and slope components are therefore $0+0+4=4$ and $0(0)+1(0)+2(4)=8$.

For quasi-Poisson, the score is divided by $\phi$, so its components are $4/\phi$ and $8/\phi$. The zero of this score is unchanged when $\phi>0$.

For NB2, the residual for observation 3 is divided by $1+\alpha\mu_3=1+0.5(4)=3$. Its contribution is $4/3$, giving score components $4/3$ and $2(4/3)=8/3$. The NB2 score is not just a constant multiple of the Poisson score across observations. These values show why the same candidate need not solve the two models’ estimating equations; they are not themselves fitted estimates.

Standard Errors and Tests

A variance function determines the usual model-based covariance calculation for the regression coefficients. For quasi-Poisson, the covariance estimate is the Poisson covariance multiplied by an estimate of $\phi$. For NB2 with known $\alpha$, the information gives less weight to observations with large $\mu_i$ than Poisson information does. When $\alpha$ is estimated, its uncertainty also matters for coefficient inference.

Result: Under independent observations and a full-rank design, the usual large-sample, model-based covariance estimates are $$ \widehat{\operatorname{Var}}(\hat\beta)_{\mathrm{QP}} =\hat\phi\left\{X^\mathsf{T}\operatorname{diag}(1/\hat\mu_i)X\right\}^{-1} $$ and, for NB2 with $\alpha$ treated as known, $$ \widehat{\operatorname{Var}}(\hat\beta)_{\mathrm{NB2}} =\left\{X^\mathsf{T}\operatorname{diag}\left(\frac{\hat\mu_i}{1+\alpha\hat\mu_i}\right)X\right\}^{-1}. $$ These are model-based approximations, not finite-sample guarantees. If NB2 dispersion is estimated, standard errors should account for its estimation using the fitted joint likelihood.

Form the quasi-score.
For independent observations, $U_{\mathrm{QP}}(\beta)=\sum_i x_i(y_i-\mu_i)/\phi$. Since $\phi$ is a positive constant, setting this score to zero is equivalent to setting the Poisson score $\sum_i x_i(y_i-\mu_i)$ to zero.

Obtain quasi-Poisson uncertainty.
The score’s variance is $\phi X^\mathsf{T}\operatorname{diag}(\mu_i)X$, while the negative expected derivative of the score is $X^\mathsf{T}\operatorname{diag}(\mu_i)X$. The resulting covariance is $\phi\{X^\mathsf{T}\operatorname{diag}(1/\mu_i)X\}^{-1}$; replacing unknown quantities by estimates gives the displayed formula.

Obtain NB2 information.
The NB2 score contribution for $\beta$ is $x_i(y_i-\mu_i)/(1+\alpha\mu_i)$. Its conditional variance is $x_ix_i^\mathsf{T}\mu_i/(1+\alpha\mu_i)$, because $\operatorname{Var}(Y_i\mid x_i)=\mu_i(1+\alpha\mu_i)$. Summing these variances gives the expected information for $\beta$ when $\alpha$ is fixed, whose inverse yields the stated covariance.

For quasi-Poisson, a common estimate is the Pearson dispersion estimate $\hat\phi=\sum_i (y_i-\hat\mu_i)^2/\hat\mu_i\,/\,(n-p)$, where $p$ is the number of regression coefficients and $n>p$. This estimate is not automatically exact in finite samples. Wald tests based on it are approximate; some implementations use a $t$ or $F$ reference with residual degrees of freedom, while large-sample inference often uses normal or chi-square approximations.

For NB2, Wald tests can use the fitted likelihood’s coefficient covariance. Likelihood-ratio tests and likelihood-based confidence intervals are also available for hypotheses about parameters in a specified negative-binomial model. A quasi-Poisson specification does not define an ordinary likelihood, so ordinary likelihood-ratio tests and likelihood-based criteria such as AIC are not available from it. A quasi-likelihood criterion is not the same thing as a full probability likelihood.

Worked Example: Dispersion and Standard Errors in Two Groups

Suppose a saturated two-group log-link model is fitted to counts $0,1,2$ in group A and $2,4,6$ in group B. Each group mean is a separate model parameter.

Means. The fitted means are $\hat\mu_A=(0+1+2)/3=1$ and $\hat\mu_B=(2+4+6)/3=4$. In this saturated mean model, the NB2 score for each group is proportional to $\sum(y_i-\mu)/(1+\alpha\mu)$; within a group the denominator is constant, so its zero also gives the sample mean. Thus the fitted group means coincide here even though this is not the general result for NB2 regression.

Quasi-Poisson dispersion. The Pearson contributions for group A sum to $(1+0+1)/1=2$. For group B they sum to $(4+0+4)/4=2$. There are $n-p=6-2=4$ residual degrees of freedom, so $\hat\phi=(2+2)/4=1$.

Quasi-Poisson standard errors. The estimated standard errors of the group means are $\sqrt{\hat\phi\hat\mu_A/3}=\sqrt{1/3}\approx0.577$ and $\sqrt{\hat\phi\hat\mu_B/3}=\sqrt{4/3}\approx1.155$.

NB2 comparison. For illustration, take $\alpha=0.5$ as known. The variance of a group mean is $(\mu+\alpha\mu^2)/3$. Its standard error is therefore $\sqrt{(1+0.5)/3}=\sqrt{0.5}\approx0.707$ for A and $\sqrt{(4+0.5(16))/3}=\sqrt{4}=2$ for B. These NB2 standard errors condition on the specified $\alpha$; estimating $\alpha$ requires accounting for its uncertainty.

Likelihood Inference Is Specific to the Negative-Binomial Model

For NB2, one common parameterization sets $r=1/\alpha$. Its probability mass function is $$ P(Y_i=y_i)=\frac{\Gamma(y_i+r)}{\Gamma(r)y_i!} \left(\frac{r}{r+\mu_i}\right)^r \left(\frac{\mu_i}{r+\mu_i}\right)^{y_i}, \qquad y_i=0,1,2,\ldots . $$ Specifying this distribution supplies a log-likelihood, which can be maximized over $\beta$ and $\alpha$. A quasi-Poisson variance assumption by itself supplies no corresponding mass function: many distributions can share the same mean and variance.

The NB2 score and information used above follow from this likelihood. Holding $\alpha$ fixed, differentiating the log probability with respect to $\log\mu_i$ gives $(y_i-\mu_i)/(1+\alpha\mu_i)$. Multiplying by $\partial\log\mu_i/\partial\beta=x_i$ gives the regression score contribution. Its variance gives the expected information derived earlier. If $\alpha$ is also estimated, the likelihood is optimized over both parameters, and the coefficient covariance should reflect the joint fit rather than treating the fitted dispersion as certainly known.

Worked Example: A Negative-Binomial Likelihood Contribution

Take an NB2 observation with $y=3$, fitted mean $\mu=2$, and $\alpha=0.5$. Then $r=1/\alpha=2$, and $r/(r+\mu)=2/4=1/2$, as is $\mu/(r+\mu)$. The probability is $$ P(Y=3)=\frac{\Gamma(5)}{\Gamma(2)3!}\left(\frac12\right)^2\left(\frac12\right)^3 =\frac{24}{1\cdot6}\cdot\frac{1}{32} =4\cdot\frac{1}{32} =\frac18. $$

The log-likelihood contribution is $\log(1/8)\approx-2.0794$. This contribution can be added to those from other observations to form the NB2 log-likelihood and support likelihood-based inference. A quasi-Poisson mean and variance do not determine this probability or log-likelihood contribution.

Choosing and Interpreting the Inference

The distinction is not simply “robust” versus “parametric.” Quasi-Poisson specifies a mean relationship and a variance proportional to that mean, then uses estimating equations and a dispersion adjustment. NB2 makes a fuller distributional assumption, with a particular quadratic variance function. If that distribution is a suitable description, likelihood-based inference can be efficient and supports likelihood comparisons. If the distributional specification is doubtful but the mean and variance structure is useful, quasi-Poisson may be preferable, with inference interpreted under its estimating assumptions.

A common mistake is to treat a quasi-Poisson fit as an ordinary Poisson fit whose standard errors alone have been multiplied by a correction, and then use the Poisson likelihood for a likelihood-ratio test or AIC. The coefficient equality under a log link does not create a Poisson likelihood for the quasi model. Conversely, it is also a mistake to assume that a negative-binomial fit must have different coefficient estimates: saturated group means, as in the example, can coincide, even though the models’ general estimating equations differ.

Key takeaway: Quasi-Poisson keeps the log-link Poisson coefficient estimates and adjusts their uncertainty for a constant dispersion factor. NB2 changes the variance pattern, generally changes the fitted coefficients, and—because it specifies a full distribution—provides likelihood-based inference. The method used for a test must match the model that supplied its estimates and uncertainty.

Check Your Understanding

Use the assumptions and calculations in this tutorial to answer the following.

Why do quasi-Poisson and Poisson regression have the same coefficient estimates with a log link?
What feature of the NB2 variance function can make its regression estimates differ from quasi-Poisson estimates?
In the two-group example, why did NB2 and quasi-Poisson give the same fitted group means despite having different variance assumptions?
Which uncertainty is omitted when NB2 standard errors are calculated as if an estimated dispersion parameter were known?
Why does a quasi-Poisson variance specification not, by itself, justify an ordinary likelihood-ratio test?