Response Variance Versus The Variance Function

Three Quantities That Are Easy to Confuse

In a generalized linear model, a statement about “the variance” can refer to three different things. The response variance, \(\operatorname{Var}(Y)\), is the actual variance of the random response under the specified model. The variance function, \(V(\mu)\), describes how variance depends on the mean \(\mu=E[Y]\) for a chosen response-family convention. The dispersion parameter, \(\phi\), is a scale factor that multiplies the variance function.

For a response family using the usual mean-variance convention, these quantities are related by

This equation is the conversion rule, not a claim that its three components are interchangeable. In particular, \(V(\mu)\) is a function evaluated at the mean; it is not generally the variance itself. And \(\phi\) is not automatically the response variance: it is the multiplier of the variance function under the chosen convention.

The distinction matters because different response families use different functions, and a change in how the response is recorded can change the appropriate variance function. We will use a Gaussian response with constant variance, binomial counts and proportions, and a gamma response to make the conversions explicit.

Read the Mean-Variance Law in the Right Direction

When \(V(\mu)\) and \(\phi\) are given, multiply them to obtain the response variance. If the response variance and variance function are known and \(V(\mu)\ne 0\), divide to obtain the dispersion:

The division formula is only valid where the variance function is nonzero. If \(V(\mu)=0\), the mean-variance law gives zero response variance for finite \(\phi\), but it does not allow \(\phi\) to be recovered by division. In ordinary nondegenerate examples, the mean is in the interior of the family’s mean domain and \(V(\mu)>0\).

A variance function is defined relative to a particular response convention. For example, recording a binomial outcome as a count or as a proportion changes its numerical variance, as well as the corresponding variance-function representation. It is therefore important to identify the response before substituting into the formula.

Binomial Responses: Counts and Proportions

Binomial examples make the role of response convention especially visible. Let \(K\) be a count of successes in \(m\) independent Bernoulli trials, each with success probability \(p\). Then \(E[K]=mp\) and \(\operatorname{Var}(K)=mp(1-p)\). If instead the recorded response is the proportion \(Y=K/m\), its mean is \(p\), and its variance is \(p(1-p)/m\).

For a binomial count, write its mean as \(\mu=mp\). Then \(p=\mu/m\), and the count variance becomes

One convenient count convention is therefore \(V_m(\mu)=\mu(1-\mu/m)\) and \(\phi=1\). For the proportion \(Y=K/m\), the mean is \(\mu=p\), and \(V(\mu)=\mu(1-\mu)\) with \(\phi=1/m\). Both give the actual variance for their stated response. In the ordinary binomial model, dispersion is fixed by the binomial sampling convention rather than freely estimated as an additional family parameter.

Gamma Responses: A Mean-Squared Variance Function

For a gamma response under the usual mean-variance convention, \(V(\mu)=\mu^2\), so the response variance is \(\phi\mu^2\). The mean alone does not determine the variance unless the dispersion is also specified. A larger \(\phi\) means a larger variance at the same mean.

Two Useful Consequences of the Distinction

The first consequence is that changing the measurement scale of the response changes its variance function. Let \(Z=a+bY\), where \(a\) and \(b\) are fixed real constants and \(b\ne 0\). Write \(\mu=E[Y]\). Then \(E[Z]=a+b\mu\). Suppose \(Y\) follows the mean-variance law \(\operatorname{Var}(Y)=\phi V(\mu)\).

Proof. Linearity of expectation gives \(E[Z]=a+bE[Y]=a+b\mu\). Since adding a constant does not change variance and multiplying a random variable by \(b\) multiplies its variance by \(b^2\),

From \(\mu_Z=a+b\mu\), we have \(\mu=(\mu_Z-a)/b\). Substituting this expression yields \(\operatorname{Var}(Z)=\phi b^2V((\mu_Z-a)/b)\), as claimed. \(\square\)

The second consequence is that the split into a dispersion and a variance function depends on the stated convention. The product determines the response variance, but without a normalization of \(V\), the factors are not uniquely determined.

Proof. Direct multiplication gives

Thus the data-generating variance is unchanged by this rescaling. A convention for the variance function is needed before \(\phi\) has a particular numerical interpretation. For example, the Gaussian convention sets \(V(\mu)=1\), making \(\phi\) equal to the Gaussian response variance. The gamma convention \(V(\mu)=\mu^2\) gives a different interpretation of \(\phi\). \(\square\)

Common Mistake: Calling the Variance Function “The Variance”

A frequent error is to calculate \(V(\mu)\) and report it as \(\operatorname{Var}(Y)\), implicitly treating \(\phi\) as one. That is only correct when the chosen convention and model actually specify \(\phi=1\). For the Gaussian example above, reporting variance one because \(V(\mu)=1\) is wrong: the response variance is \(9(1)=9\). For the gamma example, reporting variance 144 because \(V(12)=144\) is also wrong: the response variance is \(0.25(144)=36\).

Another error is to compare binomial variances without first checking whether the response is a count or a proportion. The count has mean \(mp\), while the proportion has mean \(p\); their variance functions and dispersion multipliers are not interchangeable. A reliable calculation names the response, states the convention, identifies its mean, and then applies \(\operatorname{Var}(Y)=\phi V(\mu)\).