Quasi-Poisson Models for Counts

Why Use a Quasi-Poisson Model?

A Poisson regression relates a count mean to predictors and, under the Poisson distribution, sets the conditional variance equal to that mean. Count data often vary more than this assumption allows. A quasi-Poisson model keeps a count mean model but allows its variance to be a multiple of the mean. This separates the model for the expected count from a simple adjustment to its variability.

The multiplier is useful only when interpreted carefully. A quasi-Poisson specification gives a mean and a variance relationship; it does not, by itself, specify a probability distribution for each response. Also, if the moments are claimed to come from an actual count distribution, they must be mathematically possible for a nonnegative integer-valued variable.

Mean, Variance, and the Log Link

With a log link and a linear predictor, the mean is modeled as \(\log(\mu_i)=x_i^\mathsf{T}\beta\), or equivalently \(\mu_i=\exp(x_i^\mathsf{T}\beta)\). Here \(x_i\) is the predictor vector for observation \(i\), \(\beta\) is the coefficient vector, and the exponential ensures a positive fitted mean. The response remains a count, but a mean or fitted mean need not be an integer.

The ordinary Poisson mean–variance relationship is the special case \(\phi=1\). Values \(\phi>1\) describe a variance larger than the Poisson variance at the same mean; \(0<\phi<1\) describe a smaller working variance. Neither choice alone supplies a full probability model.

What the Dispersion Multiplier Changes

At a fixed conditional mean, multiplying by \(\phi\) changes the specified variance, not the value of that mean. The following calculation makes the comparison precise.

Derivation. Under the Poisson mean–variance relationship, \(\operatorname{Var}(Y_i\mid x_i)=\mu_i\). Under the quasi-Poisson specification, \(\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i\). Dividing the latter by the former gives the variance ratio \(\phi\). Taking square roots of the positive variances gives the standard-deviation ratio \(\sqrt{\phi\mu_i}/\sqrt{\mu_i}=\sqrt{\phi}\). The mean remains \(\mu_i\) in both comparisons. \(\square\)

This result concerns the variance relationship at a given mean. It does not say that multiplying \(\phi\) changes the mean equation, nor does it define a probability distribution. In an analysis, coefficient estimates and fitted means come from the specified mean model and its fitting procedure; \(\phi\) describes the variance scale in the quasi-Poisson specification.

A Feasibility Check for Actual Count Moments

For a nonnegative integer-valued response, not every positive mean and variance can occur together. In particular, a count with mean below one cannot have an arbitrarily small variance. This matters when the mean–variance equations are interpreted as actual conditional moments rather than as a working variance relationship.

Proof. For every integer value \(y\), there is no integer strictly between \(k\) and \(k+1\), so \((y-k)(y-(k+1))\geq0\). Taking expectations and expanding gives \(E[Y^2]-(2k+1)\mu+k(k+1)\geq0\). Since \(E[Y^2]=\operatorname{Var}(Y)+\mu^2\), rearrangement yields \(\operatorname{Var}(Y)\geq(2k+1)\mu-k(k+1)-\mu^2=(\mu-k)(k+1-\mu)\). This proves the bound. \(\square\)

Consequently, if the quasi-Poisson equations are claimed to be actual moments, each observation’s mean and the common \(\phi\) must at least satisfy \(\phi\mu_i\geq(\mu_i-k_i)(k_i+1-\mu_i)\), where \(k_i=\lfloor\mu_i\rfloor\). This is a necessary condition, not a complete characterization of all feasible moment pairs. In particular, a violation proves that the proposed moments cannot come from a count distribution. A working quasi-likelihood variance function need not itself be an actual distribution, but its interpretation should then remain a variance specification rather than a claim about a fully specified count law.

Interpreting a Log-Link Coefficient

Exponentiating a coefficient gives a multiplicative comparison of fitted means. This follows directly from the log-link equation. If two predictor vectors differ only by one unit in a predictor whose coefficient is \(\beta_j\), their linear predictors differ by \(\beta_j\), and exponentiation makes the ratio of their means \(\exp(\beta_j)\). This interpretation is about the mean model; it does not depend on \(\phi\).

Common Mistakes

• Treating quasi-Poisson as a complete probability distribution. The mean and variance equations do not determine the probabilities of individual counts. Do not infer a probability mass function from them alone.
• Assuming any positive \(\mu_i\) and \(\phi\) give possible count moments. If the equations are actual moments, integer-valuedness imposes constraints. Check feasibility; otherwise describe the variance as a working specification.
• Multiplying the mean by \(\phi\). The quasi-Poisson variance is \(\phi\mu_i\), but the conditional mean remains \(\mu_i\).
• Multiplying the standard deviation by \(\phi\). The variance scales by \(\phi\), so the standard deviation scales by \(\sqrt{\phi}\).
• Reading a fitted mean as an observed count. A fitted mean can be noninteger; it is the model’s expected count, not a possible individual response value.

The central distinction is simple: the log-link equation describes how the expected count depends on predictors, while the quasi-Poisson variance function scales variability around that mean. When those equations are claimed as actual moments, count-valued feasibility must also be respected.