Estimating Dispersion With Pearson Residuals

Why Estimate Dispersion From Residuals?

A quasi-binomial or quasi-Poisson model specifies a mean structure and a variance function, but it also allows a scale parameter to multiply that variance. The scale matters: it describes how variable observations are relative to the variance function, and it is estimated from the residual variation left after fitting the mean. Pearson residuals provide a direct way to summarize that variation.

The key adjustment is for the parameters already used to fit the mean. A raw sum of squared residuals tends to be smaller than the sum one would calculate using the true means. Dividing the Pearson sum by the residual degrees of freedom, rather than by the number of observations, accounts approximately for this loss of variation.

Here \(V(\mu_i)\) is the variance function without the scale factor: the quasi-model specifies \(\operatorname{Var}(Y_i)=\phi V(\mu_i)\). Thus the Pearson residual as defined above does not divide by \(\sqrt{\phi}\). The scale is what the sum of squared Pearson residuals is intended to estimate after adjusting for fitted parameters. This definition assumes independent observations without additional observation weights; the response scale used in \(V\) must match the response scale used in the residual.

Why the Residual Degrees of Freedom Appear

The adjustment can be understood by looking first at a linear approximation to the fitted means. Write \(D\) for the \(n\)-by-\(q\) matrix whose \(i\)th row is the derivative of \(\mu_i\) with respect to the mean-model parameter vector. Let \(V\) be the diagonal matrix with entries \(V(\mu_i)\). For a small parameter change, the change in the vector of means is approximately \(D\) times that change. The quasi-score equations then project the response residuals onto the fitted mean directions.

Here is the derivation, including the distinction between the linearized result and the statistic calculated from fitted values. Let \(e_i=(Y_i-\mu_i)/\sqrt{V(\mu_i)}\), and collect these into \(e\). Independence gives \(E[e]=0\) and \(\operatorname{Cov}(e)=\phi I\). Define \(G=V^{-1/2}D\), where \(V^{-1/2}\) is diagonal. The first-order change in the standardized fitted means lies in the column space of \(G\). The quasi-score linearization therefore removes the projection \(He\), where \(H=G(G^\mathsf{T}G)^{-1}G^\mathsf{T}\) when \(G\) has full column rank. If only \(q\) independent mean directions are estimable, use the corresponding rank-\(q\) projection. The linearized residual vector is \(Ae=(I-H)e\).

The last equality follows because \(H\) is a rank-\(q\) projection, so \(A=I-H\) is also a projection, with rank and trace \(n-q\). This proves the expectation statement for the linearized statistic. It is not a claim that the Pearson sum calculated from a nonlinear fitted GLM has exactly this expectation in every finite sample.

For the consistency statement, set \(u=e/\sqrt{\phi}\), so that the components of \(u\) are independent, have mean zero and variance one, and have bounded fourth moments. The linearized sum is \(\phi u^\mathsf{T}Au\). Since \(A\) is a projection, \(A^2=A\), and its rank is \(n-q\). The variance of a quadratic form in independent errors with bounded fourth moments is bounded by a constant times \(\operatorname{tr}(A^2)+\sum_i A_{ii}^2\). Both terms are at most \(n\), so this variance is \(O(n)\). Chebyshev's inequality then gives \((u^\mathsf{T}Au)/(n-q)\xrightarrow{p}1\) for fixed \(q\). The stated fitting-approximation condition transfers this result to \(X_P^2\), proving consistency of the Pearson scale estimate. \(\square\)

In practice, the design matrix and the means are estimated, and Pearson residuals use \(V(\hat\mu_i)\), not the unknown \(V(\mu_i)\). The theorem explains the first-order adjustment; it does not make the finite-sample estimate exactly unbiased. Standard regularity conditions for a well-behaved quasi-GLM fit—such as a smooth mean model, identifiable parameters, and a consistent estimator—are important to the approximation.

Calculating the Estimate in Quasi-GLMs

The variance function determines the denominator of each Pearson residual. For grouped binomial counts \(Y_i\) out of \(m_i\) trials, with success probability \(\pi_i\), the variance function is \(V(\mu_i)=m_i\pi_i(1-\pi_i)\), where \(\mu_i=m_i\pi_i\). For a quasi-Poisson count, \(V(\mu_i)=\mu_i\). In both cases, the quasi-model variance is \(\phi V(\mu_i)\); do not insert \(\phi\) into the Pearson residual denominator when calculating \(\hat\phi_P\).

Interpretation and Common Mistakes

The Pearson estimate is a residual-based estimate of a variance multiplier, not a replacement for checking the mean structure. A missing predictor, an incorrect link or mean relationship, dependence among observations, or unusual observations can all contribute to a large Pearson statistic. The statistic alone cannot distinguish these explanations. Similarly, an estimate below one is possible: it indicates less residual variation than the reference variance function suggests, rather than being an arithmetic error.

• Using \(n\) in the denominator. The mean parameters were estimated from the same observations, so the usual Pearson scale estimate divides by residual degrees of freedom \(n-q\), not \(n\).
• Including the scale in the residual denominator. If the goal is to estimate \(\phi\), calculate \(X_P^2\) using \(V(\hat\mu_i)\) alone. Dividing each residual by \(\sqrt{\hat\phi V(\hat\mu_i)}\) first would scale away the quantity being estimated.
• Using the wrong response scale. For grouped binomial data, use the variance for the count if the response is a count, or the corresponding variance for the proportion if the response is a proportion. Do not mix the two.
• Calling the result an exact chi-square test. The degrees-of-freedom adjustment comes from a projection approximation. A fitted GLM’s Pearson statistic is not generally an exact chi-square random variable, and the estimate is not, by itself, a formal test of overdispersion.

The practical value of \(\hat\phi_P\) is that it turns the residual sum into an interpretable scale estimate while accounting approximately for fitted mean parameters. It is useful to report alongside the model’s variance assumption and a check that the mean model is plausible. Its interpretation remains conditional on those modeling choices.