Why Quasi-Binomial Is Not A Full Response Distribution

What Quasi-Binomial Specifies

A binomial model makes a detailed claim: conditional on its probability and number of trials, a count has a particular probability distribution. A quasi-binomial model makes a narrower claim. It describes the response mean and its variance, but does not assign probabilities to every possible response value. That distinction matters in a generalized linear mixed model (GLMM), where a random-effects distribution alone does not supply the missing conditional response distribution.

Let \(Y_{ij}\) be the number of successes among \(m_{ij}\) trials for observation \(i\) in cluster \(j\), and let \(b_j\) be that cluster's random effect. Write the conditional success probability as \(p_{ij}\), with a logit mean model \(\operatorname{logit}(p_{ij})=x_{ij}^{T}\beta+z_{ij}^{T}b_j\). The conditional mean of the count is \(\mu_{ij}=m_{ij}p_{ij}\). The quasi-binomial specification adds a dispersion parameter \(\phi>0\) and the variance relationship below.

For a proportion \(Y_{ij}/m_{ij}\), this gives conditional variance \(\phi p_{ij}(1-p_{ij})/m_{ij}\). The binomial variance is recovered when \(\phi=1\). A value above one describes variance larger than the binomial variance for a grouped count under this specification. These equations are statements about moments; they do not state the probability of any particular count.

Why Two Moments Are Not a Likelihood

A probability distribution assigns probabilities to all possible response values, with those probabilities summing to one. Its mean and variance are only two summaries of those probabilities. Even when the response has a finite set of possible counts, different probability assignments can have the same mean and variance. Therefore, matching the quasi-binomial moments does not identify a unique response distribution.

The example also shows why setting \(\phi=1\) is not, by itself, the same as assuming a binomial likelihood. The variance matches the binomial variance, but the quasi specification still does not assign the binomial probabilities. A likelihood-based procedure needs an explicit probability model, not just a matching variance.

Estimating the Mean Without a Full Likelihood

Quasi-likelihood methods can estimate mean-model coefficients through estimating equations. For clarity, consider one grouped observation with \(m\) trials, conditional mean \(\mu=mp\), variance \(\phi mp(1-p)\), and logit linear predictor \(\eta=x^T\beta\). The derivative of the mean with respect to \(\eta\) is \(d\mu/d\eta=mp(1-p)\). A standard quasi-score contribution for \(\beta\) is the covariate vector multiplied by the residual, the derivative of the mean, and the inverse variance.

The simplification follows because \(\partial\mu_{ij}/\partial\eta_{ij}=m_{ij}p_{ij}(1-p_{ij})\), while the assumed variance is \(\phi m_{ij}p_{ij}(1-p_{ij})\); their ratio is \(1/\phi\). In this simple setting, a constant \(\phi\) scales the score but does not change the equation \(U(\beta)=0\). This explains why standard quasi-binomial GLM estimation can produce the same mean coefficient estimates as a binomial fit while adjusting their estimated uncertainty for dispersion.

This estimating equation is not the derivative of a specified binomial log-likelihood unless a binomial distribution has separately been assumed. In a GLMM, adding \(b_j\) makes the distinction more consequential: one must specify how the random effects are distributed, and a conventional likelihood calculation also needs the conditional response likelihood. The quasi-binomial moment assumptions alone provide no conditional response likelihood to combine with the random-effects distribution.

Binary Outcomes and Clustered Variation

There is an important support constraint for individual binary observations. If \(Y\) can only be 0 or 1 and has mean \(p\), then \(Y^2=Y\), so \(\operatorname{Var}(Y)=E(Y^2)-E(Y)^2=p-p^2=p(1-p)\). There is no separate individual-level variance parameter to inflate while keeping that same binary mean. Thus, applying a conditional variance \(\phi p(1-p)\) with \(\phi>1\) to a single binary response at fixed covariates and fixed random effect is incompatible with the response's binary support.

Extra-binomial variation is meaningful for grouped counts, and clustered outcomes can also exhibit dependence. But dependence among binary responses is not the same as giving each binary response an impossible variance. In a GLMM, variation in a shared random effect can induce within-cluster association while each response remains conditionally Bernoulli. The following calculation makes the distinction explicit.

What the Missing Likelihood Means in Practice

Without a full response likelihood, some familiar likelihood-based tasks are not defined in the usual way. In particular, there is no quasi-binomial likelihood to maximize, no ordinary likelihood-ratio comparison based on it, and no AIC calculated from it. A quasi-likelihood or estimating-equation analysis can still support coefficient estimation and uncertainty calculations under its assumptions, but those are not likelihood calculations.

For a GLMM, ask separately what specifies the conditional mean, the conditional variance, the random-effects distribution, and the conditional response distribution. A quasi-binomial variance formula answers only the first two (and the link connects the mean to predictors). A random-effects distribution describes the latent effects, not the response probabilities conditional on those effects. Combining these moment statements does not automatically create a joint probability model.

The appropriate approach depends on the response process and the inferential goal. A binomial GLMM supplies a binomial conditional response likelihood. A model with a different full distribution, such as a beta-binomial model for grouped extra-binomial variation, specifies response probabilities differently. These approaches are not interchangeable: they make distinct distributional assumptions, even if their means or variances resemble one another. Quasi-binomial estimation can be useful when mean-model estimation and dispersion-adjusted uncertainty are the goal, but it should not be presented as though it supplied an unobserved likelihood.

A practical review should check whether the response is an individual binary outcome or a grouped count, whether observations or clusters satisfy the assumed independence structure, whether the mean model and variance pattern are credible, and whether the requested inference requires a full likelihood. Residual or dispersion summaries can flag a mismatch between observed and fitted variation, but they cannot identify which full response distribution generated the data.

Common Mistakes and Limitations

• Mistake: “The variance is binomial, so the likelihood is binomial.” A variance does not determine all response probabilities. Specify the binomial distribution separately if a binomial likelihood is intended.
• Mistake: “A dispersion value above one gives each binary response extra variance.” At a fixed binary mean, the variance is already fixed at \(p(1-p)\). For clustered binary data, examine dependence and heterogeneity rather than assigning an impossible conditional binary variance.
• Mistake: “A random-effects distribution completes a quasi-binomial model.” It specifies a distribution for \(b_j\), not the conditional distribution of \(Y_{ij}\) given \(b_j\). A response likelihood still has to be supplied for ordinary likelihood-based GLMM inference.
• Limitation: Quasi-binomial moment assumptions do not determine tail probabilities or the probability of a particular count. They cannot, alone, support predictions that require those probabilities or likelihood-based model selection.