Sequential Tests for Remaining Canonical Roots

Why Test Roots Sequentially?

Canonical correlation analysis produces an ordered sequence of associations between two variable sets. A global test asks whether any population canonical correlation is nonzero. A sequential test asks a more focused question: after allowing the first few associations, is there any remaining association? This is useful when the leading canonical dimensions may capture the association adequately, while the presence or absence of further dimensions is uncertain.

The test is formulated for population canonical correlations, while its statistic is calculated from sample canonical correlations. The distinction matters: the null describes the population, and the observed roots measure how strongly the sample supports association in the dimensions under test.

Population Roots and the Sequential Null

Let the two random vectors have dimensions $p$ and $m$, with positive-definite marginal covariance matrices $\Sigma_{XX}$ and $\Sigma_{YY}$. Write $q=\min(p,m)$, and order their population canonical correlations as $1\geq \rho_1\geq\cdots\geq\rho_q\geq0$. These are the singular values of the whitened cross-covariance matrix

K=\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}.

For sample data, let $\hat\rho_1\geq\cdots\geq\hat\rho_q\geq0$ denote the corresponding sample canonical correlations, computed from the sample covariance matrices. To test the roots beginning at index $s$, where $1\leq s\leq q$, the null and alternative are

H_{0,s}:\rho_s=\rho_{s+1}=\cdots=\rho_q=0 \qquad\text{versus}\qquad H_{1,s}:\text{at least one of }\rho_s,\ldots,\rho_q\text{ is positive}.

The first $s-1$ roots are not required to be zero under this null. They are the leading dimensions that the sequential test allows to carry association. The test concerns whether any association remains beyond those dimensions.

Definition: The sequential Wilks statistic for testing $H_{0,s}$ is $$ \Lambda_s=\prod_{j=s}^{q}\left(1-\hat\rho_j^2\right). $$ It is the product over the sample roots covered by the null, not over the roots that the null leaves unrestricted.

Why the Null Is a Rank Constraint

The zero-root condition can also be stated as a constraint on the population cross-covariance. This makes clear what “no remaining canonical association” means geometrically.

Theorem: Under positive-definite $\Sigma_{XX}$ and $\Sigma_{YY}$, the null $H_{0,s}$ holds if and only if $\operatorname{rank}(K)\leq s-1$.

Proof. The canonical correlations $\rho_1,\ldots,\rho_q$ are the singular values of $K$, in decreasing order. A matrix has rank at most $s-1$ exactly when it has at most $s-1$ nonzero singular values. Since the singular values are ordered, this is equivalent to $\rho_s=\cdots=\rho_q=0$. Thus the stated null and rank constraint are equivalent. $\square$

This equivalence does not say that the first $s-1$ roots must be positive; they may themselves be zero. It says only that no more than $s-1$ canonical dimensions can carry nonzero population association.

Deriving the Remaining-Roots Product

The product form follows from a determinant identity for the whitened sample cross-covariance. Let

\widehat K=S_{XX}^{-1/2}S_{XY}S_{YY}^{-1/2},

where $S_{XX}$ and $S_{YY}$ are positive definite sample covariance matrices. The singular values of $\widehat K$ are the sample canonical correlations. The eigenvalues of $\widehat K^{\mathsf T}\widehat K$ are their squares, together with any additional zeros if the matrix has more columns than nonzero singular values. Therefore

\det\!\left(I_m-\widehat K^{\mathsf T}\widehat K\right) =\prod_{j=1}^{q}\left(1-\hat\rho_j^2\right).

The sequential statistic retains only the factors beginning at $s$. Define the cumulative tail product $\Lambda_s$ as above, and set $\Lambda_{q+1}=1$, the empty product. Then $\Lambda_s=(1-\hat\rho_s^2)\Lambda_{s+1}$. Thus each successive tail statistic removes the factor for the preceding root.

Derivation: Since the full determinant contains the factors for all $q$ sample roots, $$ \det\!\left(I_m-\widehat K^{\mathsf T}\widehat K\right) =\left\{\prod_{j=1}^{s-1}(1-\hat\rho_j^2)\right\} \left\{\prod_{j=s}^{q}(1-\hat\rho_j^2)\right\}. $$ Dividing by the contribution from the first $s-1$ roots leaves $$ \Lambda_s=\prod_{j=s}^{q}(1-\hat\rho_j^2). $$

Under the usual multivariate-normal likelihood framework, this Wilks product is the conventional likelihood-ratio statistic for the sequential rank restriction. Its calibration—how to turn an observed value into a p-value or a decision—requires a reference distribution or approximation. The product itself specifies the statistic; it does not supply a cutoff.

Worked Examples

Worked Example: Testing Only the Second Root

Suppose $p=3$, $m=2$, so $q=2$, and the observed sample canonical correlations are $\hat\rho_1=0.72$ and $\hat\rho_2=0.30$. State the null for $s=2$ and calculate its Wilks statistic.

For $s=2$, the null is $H_{0,2}:\rho_2=0$. Only the second root is tested, so

\Lambda_2=1-\hat\rho_2^2=1-(0.30)^2=1-0.09=0.91.

The first root is not included in this sequential statistic. For comparison, the global statistic is

\Lambda_1=(1-0.72^2)(1-0.30^2) =(1-0.5184)(0.91) =(0.4816)(0.91)=0.438256.

The two statistics answer different questions: $\Lambda_1$ includes both roots, whereas $\Lambda_2$ concerns only whether association remains in the second dimension.

Worked Example: Testing the Tail After the Leading Root

Suppose $p=4$, $m=3$, and the sample canonical correlations are $0.80$, $0.55$, and $0.20$. Calculate the statistic for $H_{0,2}:\rho_2=\rho_3=0$.

Here $q=3$, and the roots covered by the null are the second and third. Their factors are

1-\hat\rho_2^2=1-0.55^2=1-0.3025=0.6975, \qquad 1-\hat\rho_3^2=1-0.20^2=1-0.04=0.96.

Therefore

\Lambda_2=(0.6975)(0.96)=0.6696.

The first root, $0.80$, is deliberately excluded: the null allows the leading population root to be nonzero. Including its factor would instead give the global statistic, $0.36(0.6696)=0.241056$.

Worked Example: The Last Possible Root

Suppose $p=3$, $m=4$, so $q=3$, with sample roots $0.70$, $0.40$, and $0.20$. For the null that the third population root is zero, calculate the statistic.

This is the test with $s=3$. There is just one factor in the tail:

\Lambda_3=1-\hat\rho_3^2=1-0.20^2=0.96.

As a check, the full product is

\Lambda_1=(1-0.70^2)(1-0.40^2)(1-0.20^2) =(0.51)(0.84)(0.96)=0.411264.

The statistic for $s=3$ retains only the final factor; it is not the product of all roots merely because it is a sequential test.

Reading the Statistic Correctly

Each factor $1-\hat\rho_j^2$ lies between zero and one. A root close to one contributes a factor close to zero, while a root close to zero contributes a factor close to one. Consequently, a small $\Lambda_s$ indicates substantial sample association among the roots being tested; a value near one indicates that those roots are collectively small. This description is about the observed statistic, not by itself a formal significance conclusion.

The sequential tests are nested. For $s<q$,

\Lambda_s=(1-\hat\rho_s^2)\Lambda_{s+1}, \qquad 0\leq\Lambda_s\leq\Lambda_{s+1}\leq1.

This ordering is an algebraic consequence of the factors, not a claim that the tests have the same reference distribution or that one can compare their raw values to a common cutoff. In particular, testing one root is not the same as testing the entire remaining tail unless only one root remains.

Common Mistake

Wrong reasoning: “The test begins at root $s$, so its statistic should include the first $s-1$ roots as well.” This confuses the sequential test with the global test.
Correction: The null leaves the first $s-1$ population roots unrestricted and sets roots $s$ through $q$ to zero. Its Wilks statistic therefore multiplies only the sample factors for $j=s,\ldots,q$. The global product is the special case $s=1$.

Key takeaway: Testing that no association remains from root $s$ onward means $H_{0,s}:\rho_s=\cdots=\rho_q=0$, equivalently a whitened cross-covariance of rank at most $s-1$. The corresponding Wilks statistic is $\Lambda_s=\prod_{j=s}^{q}(1-\hat\rho_j^2)$.

Check Your Understanding

Answer each question by identifying the roots covered by the null before calculating or interpreting the statistic.

For $q=4$, write the null hypothesis tested when $s=3$.
If the sample roots are $0.60$, $0.35$, and $0.10$, what factors enter the Wilks statistic for $s=2$?
Explain why $H_{0,s}$ is equivalent to $\operatorname{rank}(K)\leq s-1$.
For any $s<q$, express $\Lambda_s$ in terms of $\Lambda_{s+1}$ and $\hat\rho_s$.
What question does the global statistic answer that the statistic for $s=2$ does not?