Fitted Means and Uncertainty Under Quasi-Likelihood

Fitted Means and Two Kinds of Uncertainty

A generalized linear model produces a fitted mean for a specified covariate pattern. Under quasi-likelihood, that fitted mean is still obtained from the model’s mean structure, but uncertainty calculations must account for the estimated dispersion. A further distinction matters in applications: uncertainty about the unknown mean is not the same as the variation of a future response around that mean.

For an observation or covariate pattern with design vector \(x_0\), let \(\beta\) be the coefficient vector and \(g\) the link function. The linear predictor is \(x_0^\top\beta\), and the response mean is \(\mu_0=g^{-1}(x_0^\top\beta)\). Replacing \(\beta\) by its estimate gives the fitted mean \(\hat\mu_0=g^{-1}(x_0^\top\hat\beta)\). This is an estimate of the conditional mean, not a prediction of the exact response that will occur.

Link-Scale Confidence Intervals

The standard Wald approach first constructs an interval for the linear predictor. Write \(\widehat{\operatorname{Var}}(\hat\beta)\) for the dispersion-scaled estimated covariance matrix from the quasi-GLM fit. Then

When dispersion is estimated, a commonly used approximate interval is \(\hat\eta_0\pm t_{1-\alpha/2,\nu}\widehat{\operatorname{SE}}(\hat\eta_0)\), where \(\nu\) is the residual degrees of freedom used by the analysis. The \(t\) critical value is an inferential approximation, not an exact consequence of a quasi-likelihood: a quasi-model specifies moment relationships, not a full response distribution.

Transform the two endpoints using \(g^{-1}\). If the inverse link is increasing, the transformed lower endpoint remains the lower response-scale endpoint. For a decreasing inverse link, the transformed endpoints must be put in increasing order. For example, the inverse logit is increasing, as is the exponential inverse link for the log link.

To verify the claim, if \(g^{-1}\) is increasing, \(\eta_0\in[L,U]\) holds exactly when \(g^{-1}(\eta_0)\) lies between \(g^{-1}(L)\) and \(g^{-1}(U)\). If \(g^{-1}\) is decreasing, the inequalities reverse, but the value \(g^{-1}(\eta_0)\) still lies between the two transformed endpoints. Thus the two coverage events are identical. \(\square\)

Approximating Uncertainty on the Response Scale

An interval obtained by transforming link-scale endpoints is often preferable to a symmetric interval around \(\hat\mu_0\), particularly when the response mean is positive or constrained to lie between zero and one. A response-scale standard error is also useful for describing local uncertainty. The delta method gives it by multiplying the link-scale variance by the squared derivative of the inverse link.

For justification, apply a first-order Taylor expansion of \(h(\eta)=g^{-1}(\eta)\) around \(\eta_0\):

Under the stated approximate normality and consistency conditions, the leading random term is \(h'(\eta_0)(\hat\eta_0-\eta_0)\). Its variance is \(h'(\eta_0)^2 x_0^\top\operatorname{Var}(\hat\beta)x_0\). Substituting estimates for the unknown quantities gives the displayed approximation. The Taylor remainder is negligible relative to the leading estimation error when the usual first-order delta-method conditions hold, including differentiability near \(\eta_0\) and \(\hat\eta_0\) converging to \(\eta_0\). \(\square\)

For a log link, \((g^{-1})'(\eta)=\exp(\eta)=\mu\), so the approximate response-scale standard error is \(\hat\mu_0\widehat{\operatorname{SE}}(\hat\eta_0)\). For a logit link, \((g^{-1})'(\eta)=\mu(1-\mu)\), giving \(\hat\mu_0(1-\hat\mu_0)\widehat{\operatorname{SE}}(\hat\eta_0)\). These are local standard-error approximations; the transformed link-scale interval need not be symmetric around the fitted mean.

Future-Response Variation Is Different

The quasi-binomial and quasi-Poisson assumptions specify conditional means and variances. For a future quasi-binomial response recorded as a proportion from \(m_0\) trials, the model variance is \(\phi\mu_0(1-\mu_0)/m_0\). For a future quasi-Poisson count, it is \(\phi\mu_0\). These quantities describe response variation around the true mean, not uncertainty in estimating that mean.

If the future response is independent of the data used to estimate the model, a first-order variance approximation for its prediction error adds the two contributions:

The addition follows because independence makes the covariance between \(Y_0-\mu_0\) and \(\hat\mu_0-\mu_0\) zero. This formula gives a variance approximation, not by itself a prediction interval. A quasi-model does not specify the full distribution of \(Y_0\), so it cannot determine exact predictive quantiles. Any approximate prediction interval needs an additional approximation or distributional assumption, and should be described as such.

Common Mistakes and Practical Interpretation

• Calling a mean confidence interval a prediction interval. A confidence interval covers uncertainty in the conditional mean. A future response also varies around that mean, so its uncertainty is generally larger.
• Forgetting dispersion in coefficient uncertainty. In a quasi-GLM, the covariance matrix used for the fitted linear predictor must include the estimated dispersion, not just the unscaled model-based covariance.
• Reversing endpoints for a decreasing inverse link. Transform both endpoints, then order the resulting values from smaller to larger.
• Treating quasi-likelihood as a full probability model. Quasi-binomial and quasi-Poisson specifications provide mean and variance relationships, but do not determine exact predictive probabilities or quantiles.
• Assuming the response-scale interval must be symmetric. Transforming a symmetric link-scale interval generally produces an asymmetric response-scale interval. That asymmetry can be appropriate, especially near a response boundary.

In practice, report the fitted response mean and its confidence interval on the response scale, while stating that the interval concerns the mean. If the scientific question instead concerns a future count or trial proportion, include the relevant response variance and explain what additional assumptions are being used to construct any predictive interval.