Quasi-Score Tests for Nested Mean Models

Why Test at the Restricted Fit?

Suppose a mean model has several coefficients, and the scientific question concerns a set of linear restrictions on them. One could fit the unrestricted model and use a Wald test, as in the earlier material on robust Wald tests for quasi-likelihood contrasts. A quasi-score test takes a different route: fit the model under the null hypothesis, then measure how strongly the quasi-score points away from that restricted fit.

The route matters because quasi-likelihood specifies a mean and a variance function, not necessarily a full probability distribution. The quasi-score may still have a useful large-sample distribution, but its variability must be estimated appropriately. In particular, the sensitivity of the estimating equation and the variability of its score need not agree. A valid reference distribution therefore depends on consistent estimation of the covariance used to scale the restricted-fit score.

Model, Restriction, and Quasi-Score

Let \(Y_1,\ldots,Y_n\) be independent observations with mean \(\mu_i(\beta)\), where \(\beta\in\mathbb{R}^p\). Assume the mean model is correctly specified at the parameter value under consideration, and let \(V_i(\beta)\) be the positive working variance used to form the quasi-score. For a differentiable mean model, write \(D_i(\beta)=\partial\mu_i(\beta)/\partial\beta\), a column vector. A common quasi-score is

For example, \(V_i(\beta)\) may include a dispersion factor. The argument below does not require a full likelihood. It does require regular estimating-equation behavior: differentiability near the null, a nonsingular sensitivity for the nuisance parameters, a central limit theorem for the score contributions, and consistent covariance estimation.

Assume \(R\) is a \(q\times p\) matrix of rank \(q\). The null model is nested in the full mean model: it restricts \(q\) combinations of the full-model parameters while leaving \(p-q\) nuisance directions free. Choose a vector \(\beta_r\) satisfying \(R\beta_r=r\), and matrices \(T_1\) and \(T_2\) whose columns span the null space of \(R\) and a complementary space, respectively, with \(RT_2=I_q\). Then write

Here \(\eta\) is the nuisance parameter under the null and \(\delta\) is the \(q\)-vector of tested departures. The transformed scores are \(U_\eta=T_1'U_n\) and \(U_\delta=T_2'U_n\). The restricted estimate \(\tilde\eta\) solves \(U_\eta(\tilde\eta,0)=0\).

Nuisance Adjustment and the Asymptotic Test

Define the sensitivity matrix and score variability per observation at the null parameter \((\eta_0,0)\) by

Assume these limits exist, \(A_{\eta\eta}\) is nonsingular, and \(B\) is finite. Also assume \(U_n(\eta_0,0)/\sqrt n\) converges in distribution to a mean-zero normal vector with covariance \(B\), the restricted estimator is consistent and root-\(n\) consistent, and the score has a locally valid first-order Taylor expansion. These are standard regularity conditions for the estimating-equation expansion used here.

The matrix \(C\) removes, to first order, the part of the tested score attributable to estimating the nuisance parameter. Its covariance is \(B_{\mathrm{eff}}\), not automatically a block of \(B\). The covariance estimate in the statistic must be on the scale of the unscaled score; asymptotically it estimates \(nB_{\mathrm{eff}}\).

Proof. Use the normalized scores \(U_\eta/\sqrt n\) and \(U_\delta/\sqrt n\), evaluated at the true null parameter. A first-order expansion of the restricted nuisance equation gives

Solving this equation, using nonsingularity of \(A_{\eta\eta}\), yields

Expand the tested score at the same restricted estimate. Substituting the nuisance expansion gives

The right-hand side is \(C(U_\eta',U_\delta')'/\sqrt n+o_p(1)\). By the assumed joint central limit theorem and the continuous mapping theorem, it converges to a normal vector with covariance \(CBC'=B_{\mathrm{eff}}\). This proves the first assertion.

For the second assertion, let \(\widehat B_{\mathrm{eff},n}/n\xrightarrow{p}B_{\mathrm{eff}}\), equivalently expressing that the unscaled covariance estimate consistently estimates \(nB_{\mathrm{eff}}\). Then the quadratic statistic can be written using the normalized score and this normalized covariance. By Slutsky's theorem, it converges to \(Z'B_{\mathrm{eff}}^{-1}Z\), where \(Z\sim N_q(0,B_{\mathrm{eff}})\). Writing \(Z=B_{\mathrm{eff}}^{1/2}W\) with \(W\sim N_q(0,I_q)\) shows that this quadratic form is \(W'W\), which has a \(\chi^2_q\) distribution. \(\square\)

A common construction estimates \(B\) with the empirical covariance of individual score contributions and estimates \(A\) with the derivative of the estimating equation. Substituting those estimates into \(C\widehat B C'\) gives an estimate of the efficient-score variability; multiplying by \(n\) puts it on the unscaled-score scale. Under a correctly specified variance model, the model-based covariance may also be consistent. Under variance misspecification, the sandwich variability is generally needed.

Common Pitfall: Treating Sensitivity as Variability

A frequent error is to use the sensitivity matrix as the covariance of the score simply because the quasi-score resembles a likelihood score. In a correctly specified likelihood, an information identity often equates these quantities. A quasi-score does not, by itself, guarantee that identity. Under a correctly specified variance function and suitable conditions, model-based sensitivity can provide the required covariance. If the variance is misspecified, the score's variability can differ, so a sandwich estimate is needed for the chi-square calibration.

The correction is to identify the target covariance explicitly. First adjust the tested score for nuisance estimation; then estimate the variability of that adjusted score consistently. A quadratic form using an inconsistent covariance need not converge to \(\chi^2_q\), even if the score itself is asymptotically normal.