What Quasi-GLM Inference Does Not Provide

What a Quasi-GLM Specifies

A quasi-binomial or quasi-Poisson model can estimate regression effects and quantify their uncertainty without specifying a complete probability distribution for each response. That distinction matters when choosing how to compare models or describe uncertainty: some familiar GLM outputs require a likelihood, while quasi-inference relies on the specified mean and variance relationships.

For independent responses \(Y_i\), a quasi-GLM specifies a relationship between the mean \(\mu_i=E[Y_i]\) and predictors, usually through a link function, and a variance function. In the quasi-Poisson case, the variance is \(\phi\mu_i\). In the quasi-binomial case, for a count \(Y_i\) out of \(m_i\) trials with success probability \(p_i\), the mean is \(m_i p_i\) and the variance is \(\phi m_i p_i(1-p_i)\). These statements constrain moments; they do not, by themselves, say what probability \(Y_i\) assigns to each possible value.

This is not a claim that quasi-GLMs provide no inference. Their estimating equations can yield regression estimates and, with appropriate variance estimation and assumptions, Wald tests, quasi-score tests, or approximate F tests. The limitation is narrower: without a specified probability model, they do not supply the ordinary likelihood and its likelihood-based comparisons.

Why Moments Do Not Determine a Likelihood

The essential issue is that a mean and variance need not determine a unique distribution. The following example verifies this directly for a quasi-Poisson mean–variance relationship.

The same distinction applies to quasi-binomial models. The binomial distribution is one possible full distribution, with variance \(m_i p_i(1-p_i)\); allowing an extra factor \(\phi\) changes the variance specification, but does not itself define a new probability mass function. A quasi-binomial model should therefore not be read as a complete distributional description merely because its mean resembles the binomial mean.

Quasi-Likelihood Is Not Necessarily Log-Likelihood

A quasi-likelihood criterion can be constructed from the specified variance function. For a response with mean \(\mu\), variance \(\phi V(\mu)\), fixed \(\phi>0\), and observed value \(y\), one common definition is a function \(Q(y;\mu)\) whose derivative with respect to \(\mu\) is \(\partial Q(y;\mu)/\partial\mu=(y-\mu)/\{\phi V(\mu)\}\). On an interval where \(V(t)>0\), this can be obtained by integration from any reference point \(\mu_0\) in the interval:

The derivative has the form used to construct quasi-score estimating equations. But the integration defines a criterion as a function of the mean; it does not establish that \(\exp\{Q(y;\mu)\}\) is a probability mass or density as a function of \(y\). In particular, the mean–variance specification does not provide the response probabilities needed to check that a proposed mass function sums to one or a density integrates to one. Some particular quasi-likelihood criteria can coincide with a log-likelihood up to terms that do not depend on the model parameters under additional distributional assumptions. That coincidence must be justified by the specified probability model; it is not guaranteed by the quasi-GLM specification alone.

This distinction also explains why a quasi-deviance is not automatically a likelihood-ratio statistic. Deviance-like quantities can be formed from a variance function and used to summarize lack of fit or to construct approximate tests. But the fact that a quantity is called a deviance does not make it a difference between maximized log-likelihoods. That interpretation requires an actual likelihood.

Which Likelihood-Based Quantities Are Unavailable?

For a specified likelihood \(L(\theta)\), the log-likelihood is \(\ell(\theta)=\log L(\theta)\). AIC is conventionally defined as \(-2\ell(\hat{\theta})+2k\), where \(\hat{\theta}\) maximizes the likelihood and \(k\) is the number of estimated parameters. A quasi-GLM’s mean and variance specification does not supply this maximized log-likelihood. Therefore, an ordinary AIC for that quasi-model is not defined by the quasi specification.

There are criteria sometimes called QAIC or quasi-AIC. Their definitions require additional conventions and assumptions, and they are not the ordinary AIC obtained from a specified quasi-GLM likelihood. If such a criterion is used, its definition and justification need to be stated; it should not be presented simply as the quasi-model’s AIC.

A likelihood-ratio test also requires likelihoods. For nested models fitted to the same data, its statistic is \(2\{\ell(\hat{\theta}_{\mathrm{full}})-\ell(\hat{\theta}_{\mathrm{reduced}})\}\). Under regularity conditions and the null hypothesis, this statistic has an asymptotic chi-square reference distribution with degrees of freedom given by the difference in the numbers of free parameters. Without specified likelihoods, the quasi-GLM does not provide this statistic or its likelihood-ratio reference distribution. A difference in quasi-deviances is not, solely by being a difference, a likelihood-ratio statistic.

What Quasi-Inference Does Provide

The absence of a full likelihood does not erase the inferences available from the mean model. The earlier material on quasi-score equations and covariance scaling develops how coefficient estimates and their estimated uncertainty are obtained. In suitable large samples, those estimates can support quasi-Wald tests and confidence intervals. With estimated dispersion, approximate F procedures may be used where their conditions are appropriate. These are quasi-inference procedures, not likelihood-ratio tests under another name.

The practical boundary is therefore clear. Quasi-GLMs can support inference about regression means and effects using estimating-equation methods and an appropriate covariance estimate. They do not, without extra assumptions, determine outcome probabilities, tail probabilities, or the likelihood-based quantities that depend on those probabilities. If the scientific question requires such quantities, a full response distribution must be specified and assessed.

Common Mistake

Wrong reasoning: “The software reports a deviance for my quasi-Poisson fit, so I can subtract deviances for nested models, call the result a likelihood-ratio statistic, and use the usual chi-square reference distribution.”

Correction: A deviance-like quantity can be calculated from a quasi model, but a likelihood-ratio statistic is defined using maximized log-likelihoods. A quasi mean–variance specification does not supply those likelihoods. Use a justified quasi-inference procedure, such as an appropriate Wald or approximate F procedure, or specify a full probability model if likelihood-based comparison is required. Do not label a quasi-deviance comparison an LR test unless the likelihood interpretation and its assumptions have actually been established.