Scaled Deviance Is Not a Likelihood-Ratio Test

Why the Name “Deviance” Can Mislead

For a likelihood-based generalized linear model, a deviance difference has a precise interpretation: under the usual regularity conditions, it is twice a log-likelihood difference between nested models. In a quasi-likelihood analysis, however, the fitted criterion is built from a mean and a variance function, not necessarily from a probability model for the observations. A quantity may still be called a deviance, and it may still be useful, without being a likelihood-ratio statistic.

The distinction matters for calibration. Wilks’ theorem gives a chi-square reference distribution for a likelihood-ratio statistic under a correctly specified, regular likelihood model and a null hypothesis represented by nested models. The same conclusion does not follow just from subtracting two quasi-deviances. A chi-square approximation can be justified in some settings by a separate local argument, but its assumptions must be checked.

Likelihood Deviance and Quasi-Deviance

A quasi-likelihood criterion can instead be constructed from a specified mean \(\mu\) and variance function \(V(\mu)\). For one observation \(Y\), one common unscaled convention is

The criterion is useful because its derivative produces a quasi-score. But specifying \(V(\mu)\) and the mean does not, in general, specify the distribution of \(Y\), its mass or density function, or a log-likelihood. A scaled quasi-deviance is commonly formed from a contrast in this criterion, with the scale parameter \(\phi\) used in the variance model. It can measure improvement in the fitted quasi-likelihood criterion; that fact alone does not make it an exact likelihood-ratio statistic.

When a Chi-Square Approximation Can Be Justified

A local quadratic argument explains both why a chi-square approximation can sometimes work and why it is not automatic. The result below concerns a fixed-dimensional, regular nested comparison. It is an asymptotic result, not a claim of an exact finite-sample distribution.

Proof. Write \(Z_n=n^{-1/2}\nabla q_n(\theta_0)\), so \(Z_n\xrightarrow{d}N(0,B)\). For a bounded local displacement \(h\), the assumed Taylor expansion gives

The unrestricted quadratic expression is maximized at \(h=A^{-1}Z_n\), with gain \(\frac12 Z_n'A^{-1}Z_n\). Under the null, the restricted local displacements satisfy \(Ch=0\). Maximizing the same quadratic subject to this linear constraint gives the gain

Thus the unrestricted gain minus the restricted gain is

Let \(M=CA^{-1}C'\), which is positive definite because \(A\) is positive definite and \(C\) has rank \(r\). The limiting covariance of \(CA^{-1}Z_n\) is \(CA^{-1}BA^{-1}C'\). After multiplying the gain difference by \(2/\phi\), if \(B=\phi A\) this covariance is \(\phi M\). The resulting quadratic form is then the sum of squares of \(r\) independent standard normal variables, hence has a \(\chi^2_r\) distribution. If \(B\ne\phi A\), standardizing by \(M\) instead yields a quadratic form whose eigenvalues need not all equal one; its limit is generally a weighted sum of independent \(\chi^2_1\) variables. This proves the claim. \(\square\)

In a correctly specified quasi-mean and variance model with independent observations, the information equality in the theorem can hold. For example, if the variance is \(\phi V(\mu)\) and the quasi-score contribution is \((Y-\mu)(\partial\mu/\partial\theta)/V(\mu)\), its variance is \(\phi\) times the corresponding expected negative derivative of the score. This gives \(B=\phi A\) under the stated regularity conditions. If the variance model is wrong, or the dependence structure is not accounted for, the score covariance \(B\) need not match the curvature \(A\). The robust, or sandwich, covariance then reflects that mismatch; a scaled criterion difference does not automatically remove it.

Why Quasi-Likelihood Does Not Supply AIC

AIC is based on a maximized log-likelihood, conventionally written \(-2\ell(\hat\theta)+2k\) for a model with \(k\) estimated parameters under the standard formulation. A quasi-likelihood criterion is not generally a log density and does not assign probabilities to observed data. Consequently, substituting a quasi-likelihood value for \(\ell\) does not produce ordinary AIC. A criterion for comparing quasi-likelihood models would require a separately defined justification; it cannot be obtained merely by naming a quasi-deviance or adding a parameter-count penalty.

Common Pitfall: Applying Wilks’ Theorem by Vocabulary

• Wrong reasoning: “The statistic is a difference in scaled deviances, so Wilks’ theorem says it is chi-square.”
• Correction: Wilks’ theorem applies to a likelihood-ratio statistic under a regular likelihood model. For a quasi-deviance contrast, establish an appropriate local limit and check the score-variance/criterion-curvature condition, or use an inference procedure whose covariance and calibration account for the estimating equation.
• Wrong reasoning: “A quasi-deviance is a log-likelihood up to a constant, so it can be used for AIC.”
• Correction: That equivalence requires an actual likelihood representation. A mean and variance function generally do not specify a sampling distribution, so quasi-likelihood alone does not provide the likelihood needed for AIC.

The practical conclusion is not that deviance differences are useless. They summarize improvement in a fitted criterion and, under appropriate assumptions, can support approximate tests. The important distinction is between an exact likelihood identity and a statistic whose reference distribution follows from an additional asymptotic argument. State which one is being used, and do not infer likelihood-based model-selection properties from quasi-likelihood alone.