Scheffé Simultaneous Confidence Intervals

From Scheffé's Bound to Confidence Intervals

When many linear contrasts are examined, separate confidence intervals at level $1-\alpha$ do not generally provide $1-\alpha$ coverage for the whole collection. Scheffé intervals address a stronger target: one event on which every estimable linear contrast is covered at once. Because the family includes contrasts chosen after seeing the data, the guarantee is not limited to a list fixed in advance.

The construction uses the normal linear model and the Scheffé bound for all linear contrasts. It converts that bound into a common critical value and a separate interval for each contrast. The resulting coverage statement is exact in finite samples, not merely an approximation for large samples.

Model and notation: Let $Y\in\mathbb{R}^n$ satisfy $Y=X\beta+\varepsilon$, where $X$ is a fixed $n\times p$ matrix of full column rank, $n>p$, and $\varepsilon\sim N_n(0,\sigma^2 I_n)$, with $\sigma^2>0$. Write $\hat{\beta}=(X'X)^{-1}X'Y$, $\hat{\sigma}^2=\operatorname{SSE}/\nu$, $\operatorname{SSE}=(Y-X\hat{\beta})'(Y-X\hat{\beta})$, and $\nu=n-p$. A linear contrast is $c'\beta$ for a specified nonzero vector $c\in\mathbb{R}^p$. Under this full-rank parameterization, every such contrast is estimable.

A rank-deficient design can be handled by reparameterizing the model with a full-rank basis for its mean space; the contrast must be estimable in the original model. Here we use the full-rank parameterization to keep the interval formula direct. For $0<\alpha<1$, let $F_{p,\nu;1-\alpha}$ denote the $1-\alpha$ quantile of the central $F$ distribution with numerator degrees of freedom $p$ and denominator degrees of freedom $\nu$.

Definition: The Scheffé simultaneous confidence interval for $c'\beta$ is $$ c'\hat{\beta}\ \pm\ \sqrt{pF_{p,\nu;1-\alpha}}\, \sqrt{\hat{\sigma}^2\,c'(X'X)^{-1}c}. $$ The same critical factor $\sqrt{pF_{p,\nu;1-\alpha}}$ is used for every nonzero contrast $c$. The target is simultaneous coverage over all such contrasts, rather than separate coverage for each one.

Why the Common Critical Value Works

The standard Scheffé bound says that, in this model, the largest squared standardized estimation error over all nonzero contrasts is the quadratic form $(\hat{\beta}-\beta)'X'X(\hat{\beta}-\beta)/\hat{\sigma}^2$. Its exact distribution supplies the critical value for the intervals.

Theorem: Under the stated normal linear-model assumptions, the Scheffé intervals satisfy $$ P\!\left( \left|c'(\hat{\beta}-\beta)\right| \leq \sqrt{pF_{p,\nu;1-\alpha}}\, \sqrt{\hat{\sigma}^2\,c'(X'X)^{-1}c} \text{ for every }c\ne 0 \right)=1-\alpha. $$ Thus the intervals have exact simultaneous coverage $1-\alpha$ for the entire family of estimable linear contrasts.

Proof. Let $e=\hat{\beta}-\beta$. Under the normal model, $\hat{\beta}$ is normal with mean $\beta$ and covariance matrix $\sigma^2(X'X)^{-1}$. Consequently, $Z=(X'X)^{1/2}e/\sigma$ has a $p$-dimensional standard normal distribution, so $Z'Z\sim\chi^2_p$. Also, the residual vector is the projection of $\varepsilon$ onto the orthogonal complement of the column space of $X$. That projection has rank $\nu$, giving $\operatorname{SSE}/\sigma^2\sim\chi^2_\nu$. The fitted and residual projections are orthogonal; for a multivariate normal vector, orthogonal projections are independent. Hence $Z'Z$ and $\operatorname{SSE}/\sigma^2$ are independent, and $$ \frac{e'X'X e}{p\hat{\sigma}^2} = \frac{Z'Z/p}{(\operatorname{SSE}/\sigma^2)/\nu} \sim F_{p,\nu}. $$ For any nonzero $c$, put $a=(X'X)^{-1/2}c$. Then $c'e=a'Z\sigma$, and $c'(X'X)^{-1}c=a'a$. By the Cauchy–Schwarz inequality, $$ \frac{|c'e|}{\sqrt{c'(X'X)^{-1}c}} =\sigma\frac{|a'Z|}{\sqrt{a'a}} \leq \sigma\sqrt{Z'Z} =\sqrt{e'X'X e}. $$ Equality is attained by taking $a$ proportional to $Z$ whenever $Z\ne0$, which occurs with probability one. Therefore the supremum of the squared standardized errors over all nonzero $c$ is exactly $e'X'X e/\hat{\sigma}^2$. All intervals cover simultaneously precisely when this supremum is at most $pF_{p,\nu;1-\alpha}$. The displayed $F$ distribution then gives probability $1-\alpha$. $\square$

Worked Intervals

Worked Example: Comparing Three Group Means

Suppose three independent normal groups each have five observations, with a common unknown variance. Treat the three group means as the three model parameters, so $p=3$, $n=15$, and $\nu=12$. Let the fitted group means be $(10,12,9)$ and the residual mean square be $\hat{\sigma}^2=4$. Construct a 95% Scheffé interval for the difference between the first and second means.

For $c=(1,-1,0)'$, the fitted contrast is $c'\hat{\beta}=10-12=-2$. Each group mean is based on five observations, so $X'X=5I_3$ in the cell-means parameterization. It follows that $c'(X'X)^{-1}c=(1+1)/5=0.4$. The 95% critical $F$ value is approximately $F_{3,12;0.95}=3.490$. Therefore the half-width is $$ \sqrt{3(3.490)}\sqrt{4(0.4)} =\sqrt{10.470}\sqrt{1.6} \approx 3.236(1.265) \approx 4.09. $$ The interval is $-2\pm4.09$, or approximately $[-6.09,\ 2.09]$. Its width reflects protection for all contrasts among the three means, not only this difference.

Worked Example: A Slope Contrast in an Orthogonal Design

Consider a full-rank regression design with $n=10$, $p=2$, and $X'X=10I_2$, such as an intercept and a centered predictor with sum of squares 10. Suppose $\hat{\beta}=(5,0.6)'$ and $\hat{\sigma}^2=9$. Find the 95% Scheffé interval for the slope.

The residual degrees of freedom are $\nu=10-2=8$. For the slope, $c=(0,1)'$, so $c'\hat{\beta}=0.6$ and $c'(X'X)^{-1}c=1/10=0.1$. Using $F_{2,8;0.95}\approx4.459$, the half-width is $$ \sqrt{2(4.459)}\sqrt{9(0.1)} =\sqrt{8.918}\sqrt{0.9} \approx 2.986(0.949) \approx 2.83. $$ Thus the interval is $0.6\pm2.83$, approximately $[-2.23,\ 3.43]$. Although the target is one slope, this is a Scheffé interval from the family-wide procedure; it is not the ordinary pointwise $t$ interval.

Worked Example: Checking a Scheffé Half-Width

Suppose a model has $p=3$ and $\nu=9$, and use a 95% Scheffé procedure. For a contrast $c$, assume $c'\hat{\beta}=7.3$, $c'(X'X)^{-1}c=1/2$, and $\hat{\sigma}^2=4$. With $F_{3,9;0.95}\approx3.863$, calculate the interval.

The estimated variance factor for the contrast is $\hat{\sigma}^2 c'(X'X)^{-1}c=4(1/2)=2$. Thus the half-width is $$ \sqrt{3(3.863)(2)} =2\sqrt{3(3.863)(1/2)} =2\sqrt{5.7945} \approx 2(2.407) \approx 4.814. $$ The interval is $7.3\pm4.814$, or approximately $[2.486,\ 12.114]$. The two forms of the half-width agree because $2\sqrt{3(3.863)(1/2)}=\sqrt{3(3.863)(2)}$; the variance estimate must be included when using the contrast variance factor.

How to Use and Interpret the Intervals

Specify the model and contrast.
Confirm that the normal, common-variance linear model is appropriate, and write the target as $c'\beta$.

Find the model degrees of freedom.
Use $p=\operatorname{rank}(X)$ for the numerator degrees of freedom and $\nu=n-p$ for the residual denominator degrees of freedom.

Compute the contrast estimate and its variance factor.
Calculate $c'\hat{\beta}$ and $\hat{\sigma}^2c'(X'X)^{-1}c$.

Apply the common Scheffé multiplier.
Use $\sqrt{pF_{p,\nu;1-\alpha}}$ to obtain the half-width and form the interval around $c'\hat{\beta}$.

The coverage statement concerns repeated samples under the model: with probability $1-\alpha$, the entire collection of intervals covers all corresponding true contrasts. It does not say that a particular realized interval has probability $1-\alpha$ of containing its fixed parameter. Nor does it imply that every interval is equally wide; the width depends on the contrast through $c'(X'X)^{-1}c$.

Common Mistakes

Using a pointwise critical value. A usual $t$ critical value gives an interval for one prespecified contrast, not Scheffé simultaneous coverage over all contrasts.
Using the number of listed contrasts as $p$. The numerator degrees of freedom are the model rank, not the count of intervals printed or contrasts selected for display.
Dropping the residual degrees of freedom. The critical value is $F_{p,\nu;1-\alpha}$, where $\nu=n-p$; replacing it with an infinite-degrees-of-freedom or asymptotic value changes the finite-sample procedure.
Forgetting the estimated variance in the half-width. The contrast standard error is $\sqrt{\hat{\sigma}^2c'(X'X)^{-1}c}$. The square root of $c'(X'X)^{-1}c$ alone is not the standard error when $\sigma^2$ is estimated.
Calling the intervals pointwise. The guarantee is simultaneous for the whole family, and that stronger guarantee generally produces wider intervals than pointwise intervals.

Key takeaway: For every estimable contrast, center the interval at $c'\hat{\beta}$ and use the half-width $\sqrt{pF_{p,\nu;1-\alpha}}\sqrt{\hat{\sigma}^2c'(X'X)^{-1}c}$. Under the stated normal linear-model assumptions, this one critical value gives exact finite-sample simultaneous coverage over all contrasts.

Check Your Understanding

Use the model assumptions and interval formula to answer the following.

In a model with $n=24$ observations and rank $p=5$, what are the numerator and residual degrees of freedom for the Scheffé critical value?
Why does using separate 95% pointwise $t$ intervals not generally provide 95% simultaneous coverage for all contrasts?
For a given contrast, which two quantities determine the estimated center and which factor determines its Scheffé adjustment?
Explain why the numerator degrees of freedom depend on model rank rather than on how many contrasts are reported.
What is the repeated-sampling meaning of the simultaneous coverage statement?