Worked Comparison of Bonferroni and Scheffé Bounds

Two Ways to Protect a Family of Contrasts

Suppose a study measures a \(p\)-dimensional Gaussian response and asks for several scientifically chosen linear combinations of its mean. A simultaneous confidence procedure aims to cover all the requested contrasts at once, rather than giving each interval a separate nominal confidence level and treating those levels as a joint guarantee. Bonferroni and Scheffé bounds both provide familywise coverage, but they protect different scopes of contrasts.

Bonferroni calibrates intervals for a finite, prespecified list. Scheffé starts from a confidence region for the entire mean vector and thereby protects every linear contrast, including ones not listed in advance. That broader guarantee can cost width. The comparison is clearest when both methods are applied to the same data, confidence level, and contrast family.

For a nonzero fixed vector \(c\) and \(n\ge2\), the usual Gaussian one-sample result gives \((c'\bar X-c'\mu)/\sqrt{c'Sc/n}\sim t_{n-1}\). The positive-definite covariance assumption ensures the denominator is positive with probability one. A zero contrast is different: its target and estimate are both zero, so its interval is simply \([0,0]\); the ratio above is undefined and no \(t\) distribution is asserted for it.

Bonferroni Bounds for a Prespecified List

Let \(c_1,\ldots,c_m\) be a fixed finite list of nonzero contrasts, specified independently of the observed data. For a desired familywise error level \(\alpha\in(0,1)\), define \(t_{\nu,q}\) as the \(q\)-quantile of a \(t\) distribution with \(\nu\) degrees of freedom. Allocate error \(\alpha/m\) to each two-sided interval. The resulting intervals are

If the list also includes zero contrasts, their intervals are \([0,0]\). They are covered with probability one. They may be omitted before defining \(m\), which avoids spending any of the error allocation on intervals that cannot fail.

Proof. For each \(j\), the marginal \(t_{n-1}\) result implies that the probability its two-sided interval misses \(c_j'\mu\) is \(\alpha/m\). Let \(A_j\) denote that miss event. The event of at least one miss is \(\bigcup_{j=1}^m A_j\). The union bound gives

Taking complements proves the stated simultaneous coverage. This argument uses the individual error probabilities, not independence; correlation among the contrasts can affect the actual joint coverage but does not invalidate the bound. \(\square\)

Scheffé Bounds for All Linear Contrasts

Scheffé's method uses the full \(p\)-dimensional mean vector, not just the \(m\) listed contrasts. Assume \(n>p\), so the usual Hotelling statistic is defined with an invertible sample covariance matrix with probability one. Let \(F_{p,n-p;q}\) denote the \(q\)-quantile of the \(F\) distribution with \(p\) and \(n-p\) degrees of freedom. Define

For any contrast vector \(c\), Scheffé's interval is

Derivation. The exact Hotelling \(T^2\) result for this model states that

Therefore, the event \(T^2\le \frac{p(n-1)}{n-p}F_{p,n-p;1-\alpha}=K_{\mathrm{Sch}}^2\) has probability \(1-\alpha\). On this event, put \(d=\bar X-\mu\). For every \(c\), Cauchy–Schwarz applied to \(S^{1/2}c\) and \(S^{-1/2}d\) gives

The last inequality follows from \(n d'S^{-1}d=T^2\le K_{\mathrm{Sch}}^2\). Taking square roots yields the claimed bound for every \(c\) on the same event. This is why the guarantee covers contrasts chosen after the data are observed as well as contrasts specified beforehand: the event is a single simultaneous statement over all \(c\), not a collection of separately calibrated intervals. \(\square\)

Worked Comparison on One Data Set

For a given contrast, both methods use the same estimated standard error. Their interval widths differ only through their critical multipliers: \(t_{n-1,1-\alpha/(2m)}\) for Bonferroni and \(K_{\mathrm{Sch}}\) for Scheffé. The following example uses the same two contrasts for both procedures.

Interpreting Width and Scope

For a fixed nonzero contrast \(c\), both interval widths equal twice the method's multiplier times \(\sqrt{c'Sc/n}\). Therefore, comparing widths for the same data and contrast is a direct comparison of critical values. Bonferroni is designed for the stated finite list; Scheffé pays for the stronger protection of the whole \(p\)-dimensional contrast space. Neither multiplier is guaranteed to be smaller for every possible choice of \(p,n,m,\alpha\), so the critical values should be compared rather than relying on a slogan.

Bonferroni can be attractive when the scientific questions define a modest prespecified family. Scheffé is useful when many contrasts may be examined or when the family is effectively all linear combinations. Its guarantee does not mean every interval will be equally informative, and it does not remove the need to define meaningful contrasts. Both methods are exact finite-sample procedures under the stated Gaussian model; their coverage statements are lower bounds, so actual simultaneous coverage can be larger than \(1-\alpha\).

Common Pitfall

Wrong reasoning: “Every contrast has a \(t_{n-1}\) statistic, so I can apply that statement even to a zero contrast and treat all interval errors as independent.” The first clause fails for a zero vector: its numerator and standard error are both zero, so the ratio is undefined. The independence claim is also unnecessary and generally false for correlated contrasts.

Correction: State the marginal \(t\) result only for nonzero contrasts and \(n\ge2\); cover a zero contrast trivially by \([0,0]\). Then use the union bound for Bonferroni, which needs no independence. For Scheffé, use the simultaneous Hotelling ellipsoid argument, which supplies a guarantee over all contrasts at once.