Why Estimated Dispersion Changes an Interval
A regression contrast is a quantity such as a difference between two fitted effects or a log rate ratio. Its interval depends on both the estimated contrast and its uncertainty. In quasi-likelihood models, that uncertainty often includes a common dispersion parameter: estimating the mean does not, by itself, determine the scale of the response variation. Replacing the unknown dispersion with an estimate makes the standard error random as well.
There are two different inferential ideas to keep separate. In large samples, a consistent estimated standard error leads to a normal-calibrated interval by asymptotic normality and Slutsky's theorem. A Studentized interval uses a \(t\) critical value. It is exactly calibrated only under special distributional assumptions; in general quasi-likelihood models it is a finite-sample approximation, not a consequence of estimating dispersion alone.
Large-Sample Normal Calibration
Suppose independent observations contribute estimating equations \(U_i(\beta)\), and \(\hat{\beta}\) solves \(\sum_{i=1}^n U_i(\hat{\beta})=0\). Under regularity conditions for the estimating-equation estimator—identifiability, a nonsingular sensitivity matrix, suitable finite moments, and a central limit theorem—the estimator is asymptotically normal. Write this as
where \(\Sigma\) is positive definite on the parameter directions under consideration. For a fixed nonzero contrast \(c\), require \(c^T\Sigma c>0\). These conditions imply a standard normal limit for the studentized contrast. The interval below therefore has asymptotic, not generally exact finite-sample, coverage.
Proof. Set \(Z_n=\sqrt{n}(\hat{\beta}-\beta)\) and \(\hat{\Sigma}_n=n\,\widehat{\operatorname{Var}}(\hat{\beta})\). The numerator of the displayed statistic, after multiplying numerator and denominator by \(\sqrt n\), is \(c^T Z_n\). By the assumed multivariate limit, \(c^T Z_n\xrightarrow{d}N(0,c^T\Sigma c)\). Covariance consistency gives \(c^T\hat{\Sigma}_n c\xrightarrow{p}c^T\Sigma c\), whose limit is strictly positive. Slutsky's theorem therefore gives the stated standard normal limit. The probability that the statistic lies between \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\) tends to \(1-\alpha\), since the standard normal distribution is continuous at both endpoints. Rearranging that event yields the stated interval. \(\square\)
A common dispersion model has \(\operatorname{Var}(Y_i)=\phi V_i\), where the variance-function shape \(V_i\) is determined by the mean model and \(\phi>0\) is common across observations. When the covariance approximation has the form \(\operatorname{Var}(\hat{\beta})\approx\phi C_0\), one may estimate it by \(\hat{\phi}\hat{C}_0\). The contrast standard error is then \(\sqrt{\hat{\phi}\,c^T\hat{C}_0c}\). This multiplication is appropriate only when \(\hat{C}_0\) estimates the covariance factor with the dispersion removed. If a sandwich covariance estimate already incorporates the response variability, multiplying it by \(\hat{\phi}\) again would generally count the scale twice.
Worked Example: A Quasi-Poisson Log Rate-Ratio Interval
Suppose a fitted quasi-Poisson model estimates a log rate ratio as \(c^T\hat{\beta}=0.30\). The estimated covariance factor with dispersion removed gives \(c^T\hat{C}_0c=0.0144\), and the estimated common dispersion is \(\hat{\phi}=1.44\). Construct a 95% normal-calibrated interval for the log rate ratio, then transform it to a rate-ratio interval.
The estimated variance of the contrast is $$ \hat{\phi}\,c^T\hat{C}_0c =1.44(0.0144)=0.020736. $$ Its standard error is \(\sqrt{0.020736}=0.144\). Using \(z_{0.975}=1.96\), the interval on the log scale is $$ 0.30\mathbin{\pm}1.96(0.144) =0.30\mathbin{\pm}0.28224 =[0.01776,\ 0.58224]. $$ Exponentiation is monotone, so the corresponding rate-ratio interval is $$ [\exp(0.01776),\exp(0.58224)] \approx[1.018,\ 1.790]. $$ This is an asymptotic interval: the calculation does not make the quasi-Poisson estimator's finite-sample distribution exactly normal.
What Studentization Does—and Does Not—Guarantee
Studentization means dividing an estimation error by an estimated standard error. It often improves a normal approximation because the scale is no longer treated as known. But the name “Studentized” does not imply that the resulting statistic has a Student \(t\) distribution. A \(t\) reference law requires additional structure, such as an independent normal numerator and a suitable chi-square estimate of its variance. Quasi-likelihood specifies a mean-variance relationship, not a full response distribution, so it does not supply those properties by itself.
A residual degrees-of-freedom choice such as \(n-p\) is sometimes used for a \(t\)-calibrated quasi-likelihood interval. This can be a useful finite-sample convention, but its validity depends on the model and the scale estimator; it does not follow merely because \(\phi\) was estimated. When the degrees of freedom are large, \(t\) and normal critical values are close. When they are modest, the \(t\) interval is wider, but that extra width alone does not establish correct coverage.
Worked Example: Comparing Normal and \(t\) Critical Values
An estimated contrast is \(0.25\). Its dispersion-free variance estimate is \(0.0100\), and \(\hat{\phi}=1.50\). Compare a 95% normal interval with a \(t_{18}\)-calibrated interval.
The estimated standard error is $$ \sqrt{1.50(0.0100)}=\sqrt{0.015}\approx0.12247. $$ The normal margin is \(1.96(0.12247)\approx0.2400\), giving $$ [0.25-0.2400,\ 0.25+0.2400]\approx[0.0100,\ 0.4900]. $$ For 18 degrees of freedom, \(t_{0.975,18}\approx2.101\). The \(t\) margin is \(2.101(0.12247)\approx0.2573\), so the \(t\)-calibrated interval is $$ [0.25-0.2573,\ 0.25+0.2573]\approx[-0.0073,\ 0.5073]. $$ The \(t\) interval is wider. Unless the sampling model and scale estimator justify a \(t_{18}\) law, however, it should be described as an approximate calibration rather than an exact confidence interval.
When a \(t\) Interval Is Exact
An exact Student \(t\) result is available in the Gaussian linear model. State the assumptions carefully: the design matrix \(X\) is fixed, has full column rank \(p\), and \(n>p\); the responses satisfy \(Y=X\beta+\varepsilon\), with \(\varepsilon\sim N(0,\sigma^2 I_n)\) for \(\sigma^2>0\). Let \(\hat{\beta}=(X^TX)^{-1}X^TY\), let \(s^2=\|Y-X\hat{\beta}\|^2/(n-p)\), and take any fixed nonzero contrast vector \(c\).
Proof. Write \(P=X(X^TX)^{-1}X^T\), the orthogonal projection onto the column space of \(X\). The contrast error is \(c^T(\hat{\beta}-\beta)=c^T(X^TX)^{-1}X^T\varepsilon\), which is normal with mean zero and variance \(\sigma^2 q\), where \(q=c^T(X^TX)^{-1}c>0\). The residual vector is \((I-P)\varepsilon\). Its squared length divided by \(\sigma^2\) has a chi-square distribution with \(n-p\) degrees of freedom, because \(I-P\) is an orthogonal projection of rank \(n-p\) and the errors are Gaussian. The contrast error is independent of the residual vector: their covariance is zero, since \(X^T(I-P)=0\), and jointly normal random variables with zero covariance are independent. Thus \(s^2/\sigma^2=\|(I-P)\varepsilon\|^2/((n-p)\sigma^2)\) is an independent chi-square variable divided by \(n-p\). The defining ratio of an independent standard normal variable and the square root of an independent chi-square variable divided by its degrees of freedom has a \(t_{n-p}\) distribution. This proves the result. \(\square\)
Worked Example: An Exact Gaussian Contrast Interval
In the Gaussian linear model, suppose \(n=12\), \(p=2\), \(c^T\hat{\beta}=0.80\), \(s=0.40\), and \(c^T(X^TX)^{-1}c=0.25\) for a fixed nonzero contrast \(c\). Find its exact 95% interval.
The estimated standard error is $$ s\sqrt{c^T(X^TX)^{-1}c} =0.40\sqrt{0.25} =0.40(0.50)=0.20. $$ The degrees of freedom are \(n-p=10\), and \(t_{0.975,10}\approx2.228\). Hence the exact interval is $$ 0.80\mathbin{\pm}2.228(0.20) =0.80\mathbin{\pm}0.4456 =[0.3544,\ 1.2456]. $$ The exactness here rests on Gaussian errors, fixed full-rank design, and the residual variance estimate specified above. The same conclusion cannot be transferred automatically to a quasi-likelihood fit.
Sandwich Covariance and a Common Pitfall
If the variance-function model is misspecified or the observations have more complicated variability, a sandwich covariance estimate may be appropriate. In that case the estimated covariance is constructed to reflect variability in the estimating equations. Use the contrast standard error directly from that covariance estimate, \(\sqrt{c^T\widehat{\operatorname{Var}}(\hat{\beta})c}\). Do not presume it has the form “estimated dispersion times a dispersion-free covariance” unless that factorization has actually been established.
Worked Example: A Contrast From a Sandwich Covariance
Suppose \(\hat{\beta}=(0.40,-0.20)^T\), the contrast is \(c=(1,1)^T\), and the estimated sandwich covariance matrix is $$ \widehat{\operatorname{Var}}(\hat{\beta}) =\begin{pmatrix}0.040&0.012\\0.012&0.025\end{pmatrix}. $$ Calculate the contrast estimate and a 95% normal interval.
The estimate is \(c^T\hat{\beta}=0.40-0.20=0.20\). The estimated contrast variance is $$ c^T\widehat{\operatorname{Var}}(\hat{\beta})c =0.040+0.012+0.012+0.025=0.089, $$ so the standard error is \(\sqrt{0.089}\approx0.2983\). The normal margin is \(1.96(0.2983)\approx0.5847\), giving $$ [0.20-0.5847,\ 0.20+0.5847]\approx[-0.3847,\ 0.7847]. $$ The stated covariance is already the sandwich estimate used for the contrast; multiplying its quadratic form by another dispersion estimate would require a separate justification.
Common mistake: “Dispersion was estimated, so the contrast statistic must follow a \(t_{n-p}\) distribution.” This reasoning confuses a random standard error with the assumptions that produce a \(t\) law. Estimating a scale does not establish normality of the numerator, independence between numerator and scale estimate, or a chi-square distribution for the scale estimate. The correction is to use a normal interval when the large-sample conditions and covariance consistency support it, and to call a \(t\)-calibrated quasi-likelihood interval approximate unless a finite-sample result specifically justifies that reference law.
Check Your Understanding
Use the assumptions stated in the tutorial when deciding which calibration is justified.
- What condition on \(c\) and \(\Sigma\) ensures that the asymptotic contrast standard error has a positive limiting variance?
- If \(\operatorname{Var}(\hat{\beta})\approx\phi C_0\), how does \(\hat{\phi}\) enter the estimated contrast standard error?
- Why does estimating dispersion alone not imply a \(t\) distribution for a quasi-likelihood contrast statistic?
- Which assumptions in the Gaussian linear model yield the exact \(t_{n-p}\) result?
- A sandwich covariance estimate already includes the variability of \(\hat{\beta}\). What should be checked before multiplying it by an additional dispersion estimate?