Upper-Bound F Calibration for Roy's Root

Why Roy’s Root Needs a Calibration

Roy’s largest root focuses on the strongest hypothesis direction: among the generalized roots of the hypothesis and error matrices, it retains only the largest. That focus makes the criterion sensitive to a concentrated effect, but it also means its null distribution is not generally an ordinary $F$ distribution. The classical upper-bound $F$ calibration gives a convenient way to report a reference tail probability while accounting for the dimensions of the hypothesis and response.

The calibration is useful only if its scale and degrees of freedom are matched to the design. A root that looks numerically small or large has no direct $F$-scale interpretation until that transformation is applied. The resulting p-value is an upper-bound approximation in standard MANOVA settings, not an assertion that the largest root itself has an exact $F$ law.

Definition: Let $p$ be the number of response variables, $q$ the hypothesis degrees of freedom, and $v$ the error degrees of freedom. Let $\theta_1$ be the largest generalized root of the hypothesis matrix relative to the error matrix. Define $s=\min(p,q)$. Roy’s upper-bound $F$ calibration is $$ F_R=\frac{v-p+s}{\max(p,q)}\theta_1, \qquad \nu_1=\max(p,q), \qquad \nu_2=v-p+s. $$ The reference distribution is $F_{\nu_1,\nu_2}$; its upper-tail probability is $P(F_{\nu_1,\nu_2}\geq F_R)$.

Here $F_{\nu_1,\nu_2}$ denotes an $F$ random variable with numerator degrees of freedom $\nu_1$ and denominator degrees of freedom $\nu_2$. In the usual full-rank MANOVA setting, $p,q$ are positive integers, the error matrix is positive definite, and the error degrees of freedom are sufficient for that matrix to be nonsingular. The formula then has positive degrees of freedom. The largest root is nonnegative, and a larger observed root produces a larger $F_R$ and a smaller upper-tail probability.

Where the Scaling and Degrees of Freedom Come From

The formula is compact, but its dimensions can be easier to check by using $s$ and $m=(|p-q|-1)/2$. The identities $\max(p,q)=2m+s+1$ and $v-p+s=v-p+s$ give the degrees of freedom directly. Equivalently, the standard parameter $n=(v-p-1)/2$ gives $\nu_2=2n+s+1$. These forms are bookkeeping devices: the calibration still uses the response dimension $p$, the hypothesis dimension $q$, and the error degrees of freedom $v$.

The multiplier is $\nu_2/\nu_1$, so $F_R=(\nu_2/\nu_1)\theta_1$. This choice makes the transformation agree with the exact $F$ reduction in the one-dimensional boundary cases. For a rank-one hypothesis, $q=1$, so $s=1$, $\nu_1=p$, and $\nu_2=v-p+1$. The exact scalar-root result is $\{(v-p+1)/p\}\theta_1\sim F_{p,v-p+1}$ under the central null. For a scalar response, $p=1$, one instead has $\nu_1=q$, $\nu_2=v$, and the exact ratio is $F_{q,v}$. Thus the formula matches both rank-one limits.

Derivation: An $F_{\nu_1,\nu_2}$ variable has the form $(U/\nu_1)/(V/\nu_2)$, where $U$ and $V$ are independent chi-square variables with $\nu_1$ and $\nu_2$ degrees of freedom. Its natural scale is therefore $\nu_2/\nu_1$. Roy’s calibration applies that scale to $\theta_1$, using $\nu_1=\max(p,q)$ and $\nu_2=v-p+s$. In the exact cases $q=1$ or $p=1$, those choices reduce to the corresponding exact $F$ ratios. When both dimensions exceed one, the same construction supplies the classical upper-bound approximation; it does not make the general largest-root law exactly $F$.

The dimension $s$ is the smaller of the two design dimensions. It is the number of potentially nonzero generalized roots; the degrees of freedom use the larger dimension for $\nu_1$. Keeping these roles separate prevents a frequent error: using $s$ itself as the numerator degrees of freedom. The multiplier must use the same $\nu_1$ and $\nu_2$ that define the reference $F$ distribution.

Worked Calibrations

Worked Example: More Responses Than Hypothesis Degrees of Freedom

Suppose a MANOVA has $p=3$ responses, $q=2$ hypothesis degrees of freedom, $v=20$ error degrees of freedom, and observed largest root $\theta_1=0.45$. Here $s=\min(3,2)=2$, $\nu_1=\max(3,2)=3$, and $\nu_2=20-3+2=19$.

The scale factor is $\nu_2/\nu_1=19/3$. Therefore $$ F_R=\frac{19}{3}(0.45)=\frac{8.55}{3}=2.85. $$

The upper-bound calibration reports the upper-tail probability $P(F_{3,19}\geq 2.85)$. The statistic is $2.85$, not $0.45$; the reference distribution is $F_{3,19}$, not $F_{2,19}$. A numerical p-value can be obtained from the $F$ distribution function.

Worked Example: More Hypothesis Degrees of Freedom Than Responses

Now take $p=2$, $q=4$, and $v=18$, with $\theta_1=0.50$. Then $s=2$, $\nu_1=4$, and $\nu_2=18-2+2=18$. The statistic is $$ F_R=\frac{18}{4}(0.50)=4.5(0.50)=2.25. $$

The calibrated tail is $P(F_{4,18}\geq 2.25)$. In particular, although the response dimension is only two, the numerator degrees of freedom are four because $\max(p,q)=4$. The denominator degrees of freedom remain $18$ in this design because $v-p+s=18-2+2$.

Worked Example: Checking the Rank-One Exact Case

Let $p=3$, $q=1$, $v=12$, and $\theta_1=0.60$. Since $s=1$, the formula gives $\nu_1=3$, $\nu_2=12-3+1=10$, and $$ F_R=\frac{10}{3}(0.60)=2.00. $$

This is not merely the general approximation: when $q=1$, the exact null reduction is $F_R\sim F_{3,10}$. The p-value is $P(F_{3,10}\geq 2.00)$. The equality of the calibration and the exact law in this case is a useful check on both the multiplier and the degrees of freedom.

Why the P-Value Is Called an Upper Bound

The largest root is selected after considering the hypothesis directions, rather than being the root for one direction fixed in advance. Maximizing over directions creates a more extreme statistic under the null than a single-direction comparison would. A calibration that ignored this selection could make the upper tail too small and the evidence against the null look too strong.

Roy’s upper-bound calibration is designed to provide a conservative upper-tail assessment in the standard central MANOVA setting: the reported F-tail probability is used as an upper-bound approximation to the significance level, rather than as an exact probability from the largest-root distribution. In practical terms, conservatism means that this calibration is on the cautious side: it can report a larger p-value than a more exact largest-root calculation, making rejection harder, not easier. This is the direction of the bound that matters when interpreting the result.

There is an important qualification. Except in the exact special cases, the transformed root does not have an exact $F_{\nu_1,\nu_2}$ distribution. The formula is a calibration, not an identity that can be used to derive the full null law. Its upper-bound interpretation belongs to the standard null calibration and should not be extended automatically to nonstandard covariance structures, noncentral alternatives, or a different statistic. Where the distinction matters, report the method used and use an appropriate exact or numerical null calculation if one is available.

Theorem: For fixed $p,q,v$, the upper-tail probability from Roy’s calibration is nonincreasing in the observed largest root. More precisely, if $0\leq\theta_a\leq\theta_b$, then $$ P\left(F_{\nu_1,\nu_2}\geq\frac{\nu_2}{\nu_1}\theta_b\right) \leq P\left(F_{\nu_1,\nu_2}\geq\frac{\nu_2}{\nu_1}\theta_a\right). $$ Proof: The degrees of freedom are fixed by $p,q,v$, and the scale $\nu_2/\nu_1$ is positive. Hence $(\nu_2/\nu_1)\theta_a\leq(\nu_2/\nu_1)\theta_b$. For any random variable, the event that it is at least the larger threshold is contained in the event that it is at least the smaller threshold. Taking probabilities under $F_{\nu_1,\nu_2}$ proves the inequality. $\square$

Common Mistakes and Practical Checks

A tempting but incorrect shortcut is to treat the largest root itself as the F statistic, or to scale it by the reciprocal of the required factor. That reverses the calibration: because the statistic must increase with the root, the correct transformation is $F_R=(\nu_2/\nu_1)\theta_1$, not $(\nu_1/\nu_2)\theta_1$. Check that $\nu_1$ and $\nu_2$ in the scale are exactly the degrees of freedom in the reference distribution.

A second mistake is to interchange $\min(p,q)$ and $\max(p,q)$. Use $s=\min(p,q)$ in $\nu_2=v-p+s$, and use $\max(p,q)$ in $\nu_1$. Then check the boundary cases: $q=1$ must yield $F_{p,v-p+1}$, and $p=1$ must yield $F_{q,v}$. These are effective arithmetic checks, not permission to treat a general multiple-root case as exact.

Finally, “conservative” does not mean that the reported p-value is exact, nor does it mean every approximation is guaranteed to be conservative under every possible model. It describes the intended upper-bound direction in the standard setting. The distinction is especially important when comparing results from different MANOVA criteria: each criterion has its own statistic and calibration.

Check Your Understanding

Work through these checks using the definitions and formula above.

For $p=4$, $q=2$, and $v=24$, find $s$, both F degrees of freedom, and the multiplier applied to $\theta_1$.
For $p=2$, $q=5$, and $v=16$, explain why the numerator degrees of freedom are five and calculate the denominator degrees of freedom.
Show that setting $q=1$ in the calibration gives the exact rank-one degrees of freedom and scale.
Why does a conservative upper-bound p-value make rejection harder than an exact p-value that is smaller?
What is wrong with using $(\nu_1/\nu_2)\theta_1$ as the calibrated statistic?