Why Dispersion Ratios Vary Under the Null

Why a Correct Model Does Not Give a Dispersion Ratio of Exactly One

A Pearson dispersion ratio or a deviance dispersion ratio is a diagnostic summary, not a constraint imposed by a correctly specified model. Even when the mean structure and response distribution are right, a particular sample will usually have more or less residual variation than its average reference level. The ratios therefore fluctuate around one. Their approximate spread follows from the reference distribution of the statistics in their numerators.

The chi-square reference is an approximation, not a universal finite-sample identity. Its quality depends on the sampling regime. In this tutorial, the derivation is for a fixed number \(m\) of independent response cells, with adequate information in each cell—for example, grouped binomial cell sizes or Poisson expected counts increasing—while the number \(p\) of fitted regression parameters remains fixed. The true parameter is assumed to be an interior point, the model is identifiable and regular, and the design has full column rank. These conditions rule out boundary estimates and sparse-cell regimes in which the usual approximation may be poor.

For Poisson and binomial GLMs, the conventional null model has unit dispersion. In models with a known common dispersion \(\phi\), the Pearson statistic is instead scaled by \(\phi\), and the corresponding null comparison is \(X_P^2/\phi\) against its degrees of freedom. Below, the unit-dispersion case makes the reasoning most direct.

How Fitting Parameters Changes the Pearson Statistic

If the means were known rather than estimated, standardized residuals would, under suitable large-information conditions, behave approximately like independent standard normal variables. Their squared sum would then have approximately \(m\) degrees of freedom. Estimating \(p\) parameters uses information in the response to fit the means. To first order, this removes \(p\) directions of variation. The residual quadratic form therefore has \(m-p\), rather than \(m\), degrees of freedom.

Proof. Write \(\mu\) for the vector of true means, \(\Sigma=\operatorname{diag}\{V(\mu_1),\ldots,V(\mu_m)\}\), and \(D\) for the \(m\)-by-\(p\) matrix whose \(i\)th row is the derivative of \(\mu_i\) with respect to the parameter vector. Let \(z=\Sigma^{-1/2}(Y-\mu)\). By assumption, \(z\xrightarrow{d}N(0,I_m)\).

A first-order expansion of the score equation around the true parameter gives \(\hat\beta-\beta \approx (D'\Sigma^{-1}D)^{-1}D'\Sigma^{-1}(Y-\mu)\). This expression is the first-order adjustment to the parameter estimate needed to make the fitted residuals satisfy the score equations. Consequently, the fitted standardized residual vector is approximately \(z-A(A'A)^{-1}A'z\), where \(A=\Sigma^{-1/2}D\). The matrix \(P=A(A'A)^{-1}A'\) is the orthogonal projection onto the column space of \(A\). Its rank is \(p\), by full column rank of the design derivatives.

The Pearson statistic is the squared length of the fitted standardized residual vector, up to a term tending to zero from evaluating the variances at fitted rather than true means. Thus \(X_P^2\) is asymptotically \(z'(I_m-P)z\). The matrix \(I_m-P\) is also an orthogonal projection; its rank is \(m-p\). Orthogonally diagonalizing it transforms \(z\) into a vector that still converges to \(N(0,I_m)\), and the quadratic form becomes the sum of squares of \(m-p\) standard normal coordinates. That sum has a \(\chi^2_{m-p}\) distribution. Therefore \(X_P^2\xrightarrow{d}\chi^2_{m-p}\). \(\square\)

This derivation explains the degrees of freedom, but also its limitations. It relies on a normal approximation for standardized errors and on a regular first-order fit. A large number of individual Bernoulli or Poisson observations does not by itself guarantee the stated regime: if each observation is its own cell and expected counts stay small, the fixed-cell, high-information argument does not apply.

Why Deviance Has the Same Approximate Reference

The residual deviance compares the fitted model with the cell-by-cell saturated model. When the model is regular and the true parameter is interior, Wilks' theorem says that twice the log-likelihood difference between nested models is asymptotically chi-square, with degrees of freedom equal to the difference in their parameter dimensions. Here the saturated model has \(m\) cell means and the fitted model has \(p\) parameters, so the dimension difference is \(m-p\).

The local expansion can be seen cell by cell. For a Poisson response with fitted mean \(\mu\) and observed value \(y=\mu+\delta\), its deviance contribution is \(2\{y\log(y/\mu)-(y-\mu)\}\). Writing \(x=\delta/\mu\), this is \(2\mu\{(1+x)\log(1+x)-x\}\). The Taylor expansion \((1+x)\log(1+x)-x=x^2/2-x^3/6+O(x^4)\) gives a leading contribution \(\delta^2/\mu\), the Poisson Pearson contribution. For grouped binomial data, expansion of the two-category log likelihood around its fitted success probability gives the leading term \((y-np)^2/\{np(1-p)\}\), also the Pearson contribution. When the model is fitted, the same parameter-estimation adjustment acts on these local quadratic terms. The chi-square result for deviance also follows directly from Wilks' theorem; the expansion explains why it tracks Pearson's statistic near the fitted model.

Why the Ratios Fluctuate Around One

If \(Q\) has a \(\chi^2_\nu\) distribution, it is a sum of squares of \(\nu\) independent standard normal variables. Each squared standard normal has mean 1 and variance 2. Hence \(E[Q]=\nu\), \(\operatorname{Var}(Q)=2\nu\), and for \(R=Q/\nu\), \[ E[R]=1,\qquad \operatorname{Var}(R)=\frac{2}{\nu},\qquad \operatorname{SD}(R)=\sqrt{\frac{2}{\nu}}. \]

These identities give the basic explanation: a statistic with \(\nu\) residual degrees of freedom averages \(\nu\) units of squared standardized error, but that average remains random. For example, the approximate standard deviation is \(\sqrt{2/8}=0.5\) with eight degrees of freedom and \(\sqrt{2/50}=0.2\) with fifty. More degrees of freedom usually make the ratio less variable relative to one. They do not force a finite-sample ratio to equal one.

The expectation statements for an actual GLM statistic are approximate, not automatically exact. Convergence in distribution alone does not guarantee convergence of expectations; an additional moment condition, such as uniform integrability of the relevant statistics, is needed to infer convergence of their means. The chi-square mean and variance are exact for the reference distribution itself. In routine use, they describe the approximate null center and spread when the approximation is adequate.

Common Mistakes and Practical Meaning

• Treating one as a hard cutoff. A ratio of 1.2 is not automatically evidence against the model. Its interpretation depends partly on the residual degrees of freedom and on whether the chi-square approximation is credible.
• Forgetting that parameters are estimated. Comparing a residual sum of squares to \(m\) degrees of freedom ignores the fitted directions. In the regular full-rank setting, the residual reference uses \(m-p\).
• Assuming Pearson and deviance must match. They share a leading quadratic approximation, not an exact identity. Their individual contributions can differ, particularly when a response is far from its fitted mean.
• Using the approximation with sparse cells without checking. The normal and likelihood approximations can be poor when expected counts are small, estimates approach a boundary, or the design is nearly rank deficient. A ratio near one does not repair a weak reference approximation.

The practical value of this derivation is calibration: it prevents ordinary sampling fluctuation from being mistaken for a defect in the model. A dispersion ratio is evidence to interpret in context, not a direct estimate that must equal one under the null. Its approximate chi-square reference explains both its null center and why the amount of variation depends on residual degrees of freedom.