Mean–Variance Specification for Quasi-Likelihood

Why Specify Moments Without a Full Distribution?

A generalized linear model often begins with a probability distribution for each response. But in practice, the conditional mean may be scientifically credible while the familiar distributional variance is too small, or the response law may not be known well enough to justify a full distributional assumption. A quasi-likelihood model makes a narrower commitment: it specifies how the conditional mean depends on predictors and how the conditional variance depends on that mean.

This distinction is useful when assessing overdispersion. A variance larger than the Poisson or binomial variance does not, by itself, identify a particular alternative distribution. A mean–variance specification can represent the extra variability without claiming more about the response than the available assumptions support.

The Mean–Variance Specification

The mean may be related to predictors through a regression structure, for example \(\mu_i=g^{-1}(\mathbf{x}_i^{T}\boldsymbol{\beta})\), where \(g\) is a link function. The central object here is the conditional variance relationship. The function \(V\) describes how the variance changes as the mean changes; \(\phi\) multiplies that variance relationship by a common scale factor. A conventional normalization of \(V\) matters: replacing \(V\) by \(cV\) and \(\phi\) by \(\phi/c\), for \(c>0\), leaves the product unchanged. Thus the variance function is normally fixed by the chosen model family or convention.

For a baseline Poisson model, \(V(\mu)=\mu\) and \(\phi=1\). Writing \(\phi V(\mu)\) allows a Poisson-type mean–variance relationship with a different variance scale. For a binomial count with a fixed number of trials \(m\), the baseline variance function in terms of the count mean is \(V(\mu)=\mu(1-\mu/m)\). The specified function must be compatible with the response and its possible means; for example, a binomial count mean must lie between 0 and \(m\).

The conditional formulation is important. It describes the variation remaining after conditioning on the stated predictors and other information in \(\mathcal C_i\). If important predictors or dependence structure are omitted, the resulting conditional moments need not describe the process of interest. The specification also does not imply that different responses are independent; conditional independence is an additional assumption when needed.

Two Consequences of the Specification

Derivation. Conditional linearity of expectation gives \(E[\bar Y\mid\mathcal C]=n^{-1}\sum_i E[Y_i\mid\mathcal C]=n^{-1}\sum_i\mu_i\). For the variance, conditional independence makes every conditional covariance between distinct responses zero. Therefore,

If the responses are not conditionally independent, covariance terms must be included, so the displayed variance formula does not follow from the individual mean–variance specifications alone. \(\square\)

Verification. In the first distribution, assign probabilities \(1/2\) to each of 0 and 4. Its mean is \((0+4)/2=2\), and its variance is \[ \frac{1}{2}(0-2)^2+\frac{1}{2}(4-2)^2=4. \] In the second distribution, assign probabilities \(1/3\), \(1/2\), and \(1/6\) to 0, 2, and 6, respectively. These probabilities sum to 1, and its mean is \[ 0\left(\frac13\right)+2\left(\frac12\right)+6\left(\frac16\right)=2. \] Its second moment is \(0^2(1/3)+2^2(1/2)+6^2(1/6)=8\), so its variance is \(8-2^2=4\). Yet the first distribution assigns probability \(1/2\) to zero, while the second assigns probability \(1/3\). Thus the distributions differ despite having the same mean and variance. \(\square\)

What the Dispersion Parameter Does—and Does Not Do

For fixed \(V\) and \(\mu_i\), multiplying \(\phi\) by a constant multiplies each specified conditional variance by that same constant, without changing the specified conditional mean. When \(\phi=1\), the variance follows the chosen baseline function \(V\); \(\phi>1\) gives a larger variance than that baseline, while \(0<\phi<1\) gives a smaller one. The terms “overdispersion” and “underdispersion” therefore depend on which baseline variance function is being used.

A single common \(\phi\) makes a particular claim: all conditional variances share the same multiplicative adjustment. It does not allow an arbitrary variance at each mean. If the observed variance pattern changes shape rather than simply changing scale, the chosen \(V\) may not be adequate. Nor does a scalar dispersion parameter correct for dependence between responses.

The moment specification also does not automatically supply every feature needed for distribution-based calculations. It does not determine tail probabilities, the probability of a zero, or the shape of a sampling distribution. These require further assumptions or methods. The strength of a quasi-likelihood approach is precisely that it can use a specified mean and variance without pretending those moments identify a complete response law.

Common Mistake

A related modeling error is to attribute every variance excess to \(\phi>1\). A misspecified mean structure or dependence among observations can also produce extra variation relative to a proposed baseline. The conditional mean, variance function, and any dependence assumptions must be considered together.