Quasi-Binomial and Quasi-Poisson Variance Functions

Why Specify a Quasi Variance?

A generalized linear model has a mean structure and a variance structure. The mean structure describes how the expected response changes with predictors. The variance structure describes how much responses vary around those expectations. In applications, observations may be more variable than the standard binomial or Poisson model allows, even when the chosen mean relationship remains useful.

Quasi-binomial and quasi-Poisson specifications address this by multiplying a familiar variance function by a dispersion parameter \(\phi\). This changes the prescribed variance at a given mean; it does not, by itself, change the mean model or provide a complete probability distribution. That distinction is central to interpreting these specifications correctly.

The Two Variance Functions

The factors \(\mu(1-\mu)\) and \(\mu\) are the reference variance functions for a Bernoulli response and a Poisson response, respectively. In the corresponding ordinary models, the dispersion is fixed at \(\phi=1\). A quasi specification allows the variance to be multiplied by another positive value. When \(\phi>1\), the variance exceeds the reference variance at the same mean; when \(0<\phi<1\), it is smaller.

For a literal Bernoulli random variable, the variance is not free to vary this way. If \(Y\) takes only the values zero and one, then \(Y^2=Y\), so \(E[Y^2]=E[Y]=\mu\). Therefore \( \operatorname{Var}(Y)=E[Y^2]-E[Y]^2=\mu-\mu^2=\mu(1-\mu) \). For \(0<\mu<1\), a single literal Bernoulli response can satisfy the quasi-binomial variance equation only when \(\phi=1\). Thus, with \(\phi\ne1\), “quasi-binomial” names a variance specification based on the binomial variance function; it does not mean that each response is literally Bernoulli with that enlarged or reduced variance.

Likewise, a quasi-Poisson variance specification does not assert that the response has a Poisson distribution. A Poisson variable has variance equal to its mean, so the ordinary Poisson case corresponds to \(\phi=1\). With other values, the quasi-Poisson model specifies a different variance for a given mean, without defining all probabilities of the possible count values.

How Dispersion Scales Variability

To verify this, let \(V_0(\mu)\) denote either reference variance function: \(V_0(\mu)=\mu(1-\mu)\) for quasi-binomial or \(V_0(\mu)=\mu\) for quasi-Poisson. The quasi variance is \(V_\phi(\mu)=\phi V_0(\mu)\). Since standard deviation is the nonnegative square root of variance and \(\phi>0\),

provided \(V_0(\mu)>0\), as it is in the stated interior ranges of the two means. For example, \(\phi=4\) means four times the reference variance but twice the reference standard deviation. Confusing these two scale factors leads to incorrect descriptions of how much extra variability the model permits.

Mean and Variance Are Separate Model Choices

In a GLM, a mean model can be written as \(\mu_i=g^{-1}(\eta_i)\), where \(g\) is the chosen link and \(\eta_i=x_i^\mathsf{T}\beta\) is the linear predictor. For either quasi family, the variance is then specified as a function of that mean:

This makes the role of \(\phi\) precise: it appears in the variance equation, not in the stated equation for \(\mu_i\). The following consequence is immediate but important.

For a direct verification, take any two positive dispersion values, \(\phi_1\) and \(\phi_2\), and hold \(g\), \(x_i\), and \(\beta\) fixed. Both specifications give the same value \( \mu_i=g^{-1}(x_i^\mathsf{T}\beta) \), because none of these quantities depends on \(\phi\) in the mean equation. Their variances are \(\phi_1V_0(\mu_i)\) and \(\phi_2V_0(\mu_i)\), respectively, so the second variance divided by the first is \(\phi_2/\phi_1\) when \(V_0(\mu_i)>0\). This is a statement about the model specification at fixed coefficients; it does not claim that estimates from a particular data analysis must remain identical under every fitting procedure. \(\square\)

Interpretation and Common Pitfalls

A dispersion parameter is useful because the ordinary binomial and Poisson variance relationships can be too restrictive for observed data. Quasi variance functions let the analyst retain a selected mean–predictor relationship while specifying a different amount of variation around that mean. This separation is also why the quasi specifications are not full probability models: a mean and variance do not determine all probabilities, and the quasi variance equations do not provide a likelihood for the response.

• Pitfall: treating \(\phi\) as a multiplier of the standard deviation. The variance is multiplied by \(\phi\), but the standard deviation is multiplied by \(\sqrt{\phi}\). For \(\phi=2.5\), for instance, the standard deviation factor is about \(1.581\), not \(2.5\).
• Pitfall: claiming that \(\phi\) changes the fitted mean equation. In the stated mean model, \(\phi\) is in the variance function. With the same link and coefficients, changing it leaves the specified mean curve alone.
• Pitfall: calling a quasi-binomial response a Bernoulli variable when \(\phi\ne1\). A literal Bernoulli response has variance \(\mu(1-\mu)\). A different quasi variance is a moment specification, not a Bernoulli distribution with a freely adjustable variance.
• Pitfall: interpreting quasi-Poisson as a new named count distribution. The formula \(\operatorname{Var}(Y)=\phi\mu\) specifies the first two moments, not the complete probability mass function.

A careful description therefore states both components: the chosen mean model and the quasi variance function. For example, one might specify a logistic mean relationship with quasi-binomial variance \(\phi\mu_i(1-\mu_i)\), or a log-linked count mean relationship with quasi-Poisson variance \(\phi\mu_i\). Naming the variance function does not replace stating the mean model.