Dispersion Conventions Across Response Families

Why Dispersion Conventions Matter

A response variance in a generalized linear model has two parts: a family-specific variance function and a dispersion multiplier. The multiplier describes the scale of variation around the mean, but its treatment differs across response families. In the conventional GLM specification, Bernoulli, binomial, and Poisson responses have dispersion fixed at one. Normal, gamma, and inverse-Gaussian responses instead have a common dispersion parameter that is ordinarily estimated from data.

“Fixed at one” does not mean that the response variance equals one. It means that the family’s variance function is used without an additional estimated multiplier. For example, a Bernoulli response with mean \(0.3\) has variance \(0.3(1-0.3)=0.21\), despite having unit dispersion. Keeping this distinction clear prevents confusion between a variance function and a dispersion convention.

The word “common” refers to a single dispersion value shared across the response observations in the model; it does not mean that their variances are identical. If their means differ, their variance functions can differ too. For instance, with a common \(\phi\), a gamma response with mean \(\mu_i\) has variance \(\phi\mu_i^2\), which varies with \(\mu_i\).

The Conventional Family-by-Family Comparison

The table gives the standard mean-variance forms and dispersion conventions. For a binomial count, \(m\) is its known number of trials. A Bernoulli response is the special case \(m=1\).

In each row, the displayed variance can be read as \(\phi V(\mu)\). For the first three families the conventional value is \(\phi=1\), so the variance function itself gives the variance. For the last three, \(\phi\) remains in the variance expression. These are standard modeling conventions for these response families, not a claim that no alternative model can ever introduce additional variation.

Deriving the Discrete-Family Conventions

The unit-dispersion entries follow directly from the usual probability models. A Bernoulli variable with success probability \(p\) has \(E[Y]=p\) and \(\operatorname{Var}(Y)=p(1-p)\). For a binomial count, write \(Y=\sum_{j=1}^{m}X_j\), where the \(X_j\) are independent Bernoulli variables with success probability \(p\). Linearity of expectation and independence give

Since \(\mu=mp\), we have \(p=\mu/m\), and thus

This has the listed binomial variance function with multiplier one. The Poisson distribution has \(E[Y]=\mu\) and \(\operatorname{Var}(Y)=\mu\), so its listed variance function also has multiplier one. These calculations explain the conventional entries; they do not estimate a separate dispersion parameter for any of the three models.

For a binomial proportion \(Y/m\), the mean is \(p\) and the variance is \(p(1-p)/m\). The factor \(1/m\) reflects averaging \(m\) trials. Whether the response is recorded as a count or as a proportion, the trial count is known and belongs in the variance structure; it is not a free common \(\phi\) to estimate.

Common Dispersion in Continuous Response Families

For the normal, gamma, and inverse-Gaussian families, the usual GLM parameterizations retain a scale parameter in the variance. One can see the role of that parameter by comparing standard distribution parameters with the mean-variance forms.

For a normal response, the usual notation is \(Y\sim N(\mu,\sigma^2)\). Its variance is \(\sigma^2\), so identifying \(\phi=\sigma^2\) and \(V(\mu)=1\) yields \(\operatorname{Var}(Y)=\phi V(\mu)=\phi\). For the gamma family, use a shape parameter \(\alpha=1/\phi\) and a scale parameter \(\beta=\phi\mu\). The gamma mean and variance in shape-scale form are \(E[Y]=\alpha\beta\) and \(\operatorname{Var}(Y)=\alpha\beta^2\). Substitution gives

For the inverse-Gaussian family, use mean \(\mu\) and shape parameter \(\lambda=1/\phi\). Its variance is \(\mu^3/\lambda\), and therefore

Thus the scale parameter multiplies the family-specific variance function in all three cases. The variance-function shapes \(1\), \(\mu^2\), and \(\mu^3\) are distinct from the shared multiplier \(\phi\). Estimating \(\phi\) does not mean estimating a separate dispersion for each response.

How to Read the Convention in Practice

When a model specification names a response family, separate three questions: what is the mean, what is the family’s variance function, and how is dispersion treated? For Bernoulli, binomial, and Poisson responses, the conventional answer to the third question is that dispersion is fixed at one. For normal, gamma, and inverse-Gaussian responses, the conventional answer is that a common \(\phi\) is estimated. The value of \(\phi\) is not itself the response variance unless the variance function equals one.

For example, a gamma model with \(\phi=0.2\) and mean \(5\) has variance \(5\), not \(0.2\), because its variance function is \(5^2\). Conversely, a Poisson model with mean \(4\) has variance \(4\), not one, because its unit dispersion multiplies the variance function \(V(4)=4\).

Common Mistakes

• Equating unit dispersion with unit variance. The incorrect reasoning is “the Bernoulli dispersion is one, so its variance is one.” The correction is to multiply the unit dispersion by the variance function: at mean \(0.3\), the variance is \(1(0.3)(0.7)=0.21\).
• Treating a binomial trial count as an estimated dispersion. The trial count \(m\) is part of the known binomial variance function. For a count, \(\operatorname{Var}(Y)=\mu(1-\mu/m)\); it is not an estimate of \(\phi\).
• Using the same variance function for all families. A common multiplier does not imply a common function: normal, gamma, and inverse-Gaussian variances use \(1\), \(\mu^2\), and \(\mu^3\), respectively.
• Assuming every response family estimates dispersion. Under the conventional GLM specification, Bernoulli, binomial, and Poisson dispersion is fixed at one. Adding a separately estimated scale is a change to the model specification, not the standard convention.