Inference for Linear Contrasts

Why Linear Contrasts Need the Full Covariance Matrix

Many scientific questions concern a combination of regression coefficients rather than one coefficient in isolation. A contrast such as the difference between two treatment effects, or the sum of two slope coefficients, is estimated by combining coefficient estimates. Its uncertainty depends not only on the uncertainty of each coefficient but also on how those estimates vary together. The sandwich covariance matrix provides the ingredients for this calculation when quasi-estimating equations are used.

The focus here is inference for one prespecified linear contrast. The resulting confidence interval is an asymptotic, large-sample procedure. It is not automatically an exact finite-sample interval, and it does not require a likelihood for the response distribution.

Write \(\widehat C\) for an estimated covariance matrix of \(\hat{\beta}\), using the sandwich covariance or an appropriate dispersion-scaled covariance. The estimated variance and standard error of the contrast are

The off-diagonal entries of \(\widehat C\) matter whenever the contrast combines more than one coefficient. In particular, a difference can be more or less precise depending on whether the two coefficient estimates tend to move together.

Deriving the Contrast Variance and Interval

Suppose a quasi-estimator satisfies the asymptotic normality result \(\sqrt{n}(\hat{\beta}-\beta)\xrightarrow{d}N(0,V)\), where \(V\) is a finite, positive semidefinite \(p\times p\) matrix. For a fixed vector \(c\), the corresponding contrast has asymptotic variance \(c^\mathsf{T}Vc/n\). In practice, an estimated covariance \(\widehat C\) estimates this covariance for \(\hat{\beta}\) itself.

Proof. The map \(x\mapsto c^\mathsf{T}x\) is linear and continuous. The continuous mapping theorem therefore gives \(\sqrt{n}c^\mathsf{T}(\hat{\beta}-\beta)\xrightarrow{d}N(0,c^\mathsf{T}Vc)\). Since \(c^\mathsf{T}Vc>0\), dividing by \(\sqrt{c^\mathsf{T}Vc}\) gives a standard normal limit. Consistency of \(\widehat C\) for \(V/n\) implies \(n c^\mathsf{T}\widehat Cc\xrightarrow{p}c^\mathsf{T}Vc\), so the estimated standard deviation divided by \(\sqrt{c^\mathsf{T}Vc/n}\) converges in probability to one. Slutsky’s theorem gives the stated studentized limit. \(\square\)

Let \(z_{1-\alpha/2}\) denote the \(1-\alpha/2\) quantile of the standard normal distribution. A two-sided, approximate \(100(1-\alpha)\%\) confidence interval is

This interval has asymptotic coverage \(1-\alpha\) under the theorem’s conditions. The conclusion follows because the standardized statistic converges to a continuous standard normal distribution, whose probability between \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\) is \(1-\alpha\). It is an asymptotic result: the sandwich estimate’s consistency and the normal approximation do not guarantee exact coverage at a particular finite sample size.

Where Dispersion Enters

In quasi-likelihood settings, dispersion describes the scale of the response variance relative to the specified mean–variance relationship. Under a correctly specified variance function with scalar dispersion \(\phi\), the usual quasi-score covariance has the form \(\phi M^{-1}\), where \(M\) is the information-like matrix formed from the mean-model derivatives and working weights. If \(\phi\) is estimated, the plug-in covariance uses \(\hat{\phi}M^{-1}\), and the contrast variance is \( \hat{\phi}c^\mathsf{T}M^{-1}c\).

A robust sandwich covariance is instead built from an estimated sensitivity (“bread”) and variability of the estimating contributions (“meat”). When the meat is computed from residual contributions, the scale of their variability is already represented in that matrix. One should not multiply such an already dispersion-inclusive sandwich covariance by \(\hat{\phi}\) again. Always identify how the covariance estimate was constructed before applying a dispersion factor.

Estimating \(\phi\) rather than knowing it does not ordinarily change the first-order large-sample interval argument when \(\hat{\phi}\xrightarrow{p}\phi\): replacing \(\phi\) by a consistent estimate changes the standard error by a factor converging in probability to one. This is a plug-in, asymptotic justification; it does not make a normal interval an exact small-sample procedure.

Why the Off-Diagonal Terms Matter

For a two-coefficient contrast \(c^\mathsf{T}\beta=a\beta_1+b\beta_2\), its variance is \[ a^2\operatorname{Var}(\hat{\beta}_1) +b^2\operatorname{Var}(\hat{\beta}_2) +2ab\operatorname{Cov}(\hat{\beta}_1,\hat{\beta}_2). \] The covariance term can increase or decrease the uncertainty, depending on the signs of \(a\), \(b\), and the covariance. Treating the coefficient estimates as independent without justification drops this term and generally gives the wrong standard error.

For a single contrast, the essential inferential steps are to state the coefficient vector \(c\), compute the estimate \(c^\mathsf{T}\hat{\beta}\), identify the correct covariance estimate, and evaluate its quadratic form. The substantive interpretation then belongs to the original parameterization: for example, a contrast may represent an adjusted difference between groups or a change in a specified linear predictor. The interval quantifies uncertainty about that particular scalar quantity, not about all coefficients simultaneously.