Why Direction Matters
A confidence statement about a contrast can have a direction. For example, a researcher may need evidence that a treatment effect exceeds a clinically meaningful threshold, or may need an upper limit on a possible adverse effect. These are not the same request as estimating the contrast within a two-sided interval. The direction determines which tail probability controls the bound and, consequently, which critical value is appropriate.
Here we consider a prespecified finite family of linear contrasts of a multivariate Gaussian mean. Each contrast receives either a lower bound, an upper bound, or both. The goal is to control the probability that any of the requested statements fails, rather than controlling the error separately for each statement.
The positive-definiteness assumption and \(c_j\ne 0\) ensure that the contrast has positive variance. Under the stated Gaussian sampling model, its studentized error has an exact \(t\) distribution:
To see why, set \(Y_i=c_j'X_i\). Then the \(Y_i\) are independent normal observations with mean \(\theta_j\) and variance \(\tau_j^2=c_j'\Sigma c_j>0\). Their sample mean is \(\hat\theta_j\), and their sample variance is \(c_j'Sc_j\). For a normal sample, \((\bar Y-\theta_j)/(\tau_j/\sqrt n)\) is standard normal, \((n-1)c_j'Sc_j/\tau_j^2\) is chi-square with \(n-1\) degrees of freedom, and these two quantities are independent. Their ratio, after division by the square root of the scaled chi-square variable, has the stated \(t_\nu\) distribution.
One-Sided Bounds and Simultaneous Coverage
For a single contrast and a specified error probability \(a\), where \(0<a<1\), the exact one-sided lower and upper bounds are
The lower bound satisfies \(P(\theta_j\ge L_j(a))=1-a\), and the upper bound satisfies \(P(\theta_j\le U_j(a))=1-a\). Both use the \(1-a\) quantile, but the sign of the critical-value term changes. A lower bound subtracts the critical margin; an upper bound adds it.
For a family of contrasts, assign a separate error allowance to each requested directional statement. Let \(J_-\) be the set of contrasts for which lower bounds are requested, and \(J_+\) the set for which upper bounds are requested. Assign allowances \(a_j^-\) for \(j\in J_-\) and \(a_j^+\) for \(j\in J_+\). Require every assigned allowance to be strictly between zero and one, and require their sum to be no greater than the desired familywise error rate \(\alpha\).
Proof. For each requested lower statement, the marginal \(t_\nu\) result gives \(P(\theta_j<L_j)=a_j^-\). For each requested upper statement, it gives \(P(\theta_j>U_j)=a_j^+\). Failure of the joint event means at least one of these individual failures occurs. By the union bound, the probability of that union is no greater than the sum of its probabilities:
Taking complements proves the simultaneous coverage claim. This argument does not require the contrast estimates to be independent. \(\square\)
Choosing and Allocating the Error Budget
The simplest construction is equal Bonferroni allocation: divide \(\alpha\) by the number of directional statements. If there are \(m\) requested one-sided bounds, assign \(a=\alpha/m\) to each. The critical value is then \(t_{\nu,1-\alpha/m}\), not \(t_{\nu,1-\alpha/(2m)}\). The latter convention splits each contrast’s error between two tails and is appropriate when constructing two-sided intervals, not when making only one directional statement per contrast.
Allocations need not be equal. A statement considered especially important can receive a larger allowance, which gives it a smaller critical value and a tighter bound, while other statements receive smaller allowances and wider bounds. The total allowance must still be at most \(\alpha\). Any directional claim assigned zero allowance would require an infinite one-sided critical value, so the finite-bound procedure here requires strictly positive allowances.
A two-sided interval is different: it makes both a lower and an upper statement about the same parameter. For a single contrast with total error probability \(a\), the equal-tail interval is \([\hat\theta_j-t_{\nu,1-a/2}s_j,\ \hat\theta_j+t_{\nu,1-a/2}s_j]\). Its critical value is larger than the one-sided value \(t_{\nu,1-a}\), because each endpoint must protect a separate tail. More generally, unequal tail allocation uses a lower-tail allowance \(a^-\) and an upper-tail allowance \(a^+\), with \(a^-+a^+\) included in the family’s error budget.
Worked Example: Repeated Contrast Statements
Suppose \(n=20\), so \(\nu=19\), and a contrast has estimate \(\hat\theta=8\) and standard error \(s=\sqrt{4/20}=\sqrt{0.2}\approx0.4472\). A family contains two copies of the same lower-bound statement, and the desired familywise error rate is \(0.05\). What is the exact simultaneous lower bound?
The two statements concern the identical parameter and identical estimate. They either both cover or both fail; their joint coverage is therefore exactly the coverage of one bound. The exact one-sided critical value is \(t_{19,0.95}\approx1.7291\), giving
Thus the simultaneous lower bound is approximately 7.23. Treating the two identical statements as distinct errors and splitting \(0.05\) between them would still give a valid, but unnecessarily conservative, Bonferroni bound. The exact result here follows because the repeated statements have identical failure events.
Worked Example: A Mixed Family of Lower and Upper Bounds
For a sample of size \(20\), consider two prespecified contrasts. The first has estimate \(8\) and standard error \(\sqrt{0.2}\); request a lower bound. The second has estimate \(3\) and standard error \(0.5\); request an upper bound. Use familywise error \(\alpha=0.05\), split equally between these two directional statements.
Each allowance is \(0.025\), so both use \(t_{19,0.975}\approx2.093\). For the first contrast, the lower bound is
For the second contrast, the upper bound is
The simultaneous statement is that the first parameter is at least approximately \(7.064\) and the second is at most approximately \(4.047\). The union-bound theorem guarantees coverage of both statements together of at least \(0.95\), regardless of dependence between the two contrast estimates.
Worked Example: Unequal Allowances with Known Covariance
Suppose instead that the covariance matrix is known, so each contrast estimate has a known standard error and its standardized error is standard normal. A study requests a lower bound for a contrast estimated as \(5\), with standard error \(1\), and an upper bound for another contrast estimated as \(2\), with standard error \(2\). Allocate \(0.01\) to the lower statement and \(0.04\) to the upper statement.
The allowances add to \(0.05\). For a standard normal quantile \(z_q\), the lower and upper bounds are respectively \(\hat\theta-z_{1-a^-}s\) and \(\hat\theta+z_{1-a^+}s\). Using \(z_{0.99}\approx2.32635\) and \(z_{0.96}\approx1.75069\), the lower bound is
The upper bound is
The familywise coverage is at least \(1-(0.01+0.04)=0.95\). The unequal allocation makes the lower statement more stringent in its tail probability and gives it a larger critical value; the upper statement receives a larger allowance and a smaller critical value. The calculation illustrates the allocation principle without substituting normal critical values into the unknown-covariance \(t\) setting.
Practical Interpretation and Common Pitfalls
A simultaneous one-sided bound is useful when the scientific decision is inherently directional: establishing that an effect exceeds a threshold, or ruling out values above a safety limit. Its guarantee concerns the complete declared family. It does not say that each bound has error probability \(\alpha\); rather, each individual allowance is part of a total error budget, and simultaneous coverage is protected by controlling that budget.
- Using a two-sided critical value for a one-sided claim. A one-sided bound with allowance \(a\) uses \(t_{\nu,1-a}\). The two-sided equal-tail interval uses \(t_{\nu,1-a/2}\). These answer different questions.
- Allocating error per contrast when both directions are reported. If a contrast receives both a lower and an upper endpoint, both directional statements count toward the familywise budget. A two-sided interval with total allowance \(a\) uses \(a/2\) in each tail when allocated equally.
- Ignoring dependence in the wrong way. Bonferroni does not require independence, so it remains valid for correlated contrasts. Dependence may make it conservative, but it does not invalidate the guarantee.
- Allowing a zero allocation while displaying a finite bound. A zero tail allowance corresponds to a limiting quantile at probability one, not a finite critical value. Use strictly positive allowances for finite bounds.
- Rounding before the final subtraction or addition. Keep enough precision in the standard error and critical value, then round the endpoint. For example, \(8-t_{19,0.95}\sqrt{0.2}\approx7.2267\), which rounds to \(7.23\), not \(7.22\).
When the contrast family contains repeated or linearly redundant statements, Bonferroni can count multiple nominal claims even when some failures are identical or otherwise strongly dependent. Exact treatment can exploit the joint structure when its distribution is available; otherwise the Bonferroni construction remains a transparent guaranteed-coverage method. The key practice is to specify the directional statements first, then allocate error across those statements and select the matching one-sided critical values.
Check Your Understanding
For each question, identify the direction of the bound and the error allowance that governs it.
- A family requests four lower bounds and no upper bounds, with total error rate \(0.08\). What equal allowance and Student’s \(t\) quantile should be used for each bound?
- Why does a lower bound have the form \(\hat\theta-t_{\nu,1-a}s\), whereas an upper bound has a plus sign?
- A contrast receives both endpoints of an equal-tail interval with total error probability \(0.04\). What tail probability determines each endpoint’s critical value?
- Two requested statements are exact copies of the same lower-bound statement. Why can their joint coverage be evaluated using the single-bound coverage?
- Can Bonferroni simultaneous coverage be guaranteed when contrast estimates are dependent? State the reason.