Common Errors in Canonical Correlation Analysis

Introduction

Canonical correlation analysis (CCA) summarizes association between two sets of variables by forming paired linear combinations. Its output can look deceptively simple: a list of canonical correlations, weights, loadings, and test results. The difficulty is that these quantities answer different questions. Confusing them can turn a valid numerical result into an incorrect scientific interpretation.

This tutorial focuses on five recurring errors: treating weights as variable–variate correlations, assigning substantive meaning to arbitrary signs, overinterpreting repeated roots, reading significance tests as stronger evidence than they provide, and calculating redundancy incorrectly. The goal is not merely to recognize a mistake, but to replace it with a precise reading of the output.

Weights, Loadings, and Cross-Loadings Are Not Interchangeable

Weights specify how variables are combined to form a canonical variate. A weight is not the correlation of its variable with that variate. Structure loadings directly describe those correlations, while cross-loadings describe correlations with the paired variate from the other set. These summaries can tell different stories, especially when variables within a set are correlated.

Weights also depend on the scale and coordinate system used to represent the variables. Standardized weights remove units but do not become structure loadings: they remain coefficients in a linear combination. A large coefficient does not, by itself, establish that the corresponding variable has the strongest marginal association with the canonical variate. For assessing which original variables are associated with a variate, inspect the loadings alongside the weights.

For a canonical pair with variates standardized to variance 1, the cross-loading of \(X_k\) on \(V_j\) is related to its structure loading on \(U_j\) by \(\operatorname{Corr}(X_k,V_j)=\rho_j\operatorname{Corr}(X_k,U_j)\); the corresponding relation holds for variables in \(Y\). This makes cross-loadings useful for describing how strongly variables in one set relate to the canonical variate in the other set. But they are still a distinct summary, not the weights used to construct that other variate.

Signs Are a Convention, Not a Substantive Finding

Canonical variates have an unavoidable sign indeterminacy. If both variates in a pair are multiplied by \(-1\), the pair represents the same association: both sets of loadings change signs, while the canonical correlation remains positive and unchanged. If only one variate is reversed, the signed correlation changes sign, but the magnitude of the association does not.

To verify this, write \(U_j=a_j^\mathsf{T}X\) and \(V_j=b_j^\mathsf{T}Y\). Under the joint reversal, the new variates are \(-U_j\) and \(-V_j\). Their covariance is \(\operatorname{Cov}(-U_j,-V_j)=\operatorname{Cov}(U_j,V_j)\); their variances are unchanged. Therefore their correlation is unchanged. For any \(X_k\), \(\operatorname{Corr}(X_k,-U_j)=-\operatorname{Corr}(X_k,U_j)\), and likewise for variables in \(Y\). Under a reversal of \(U_j\) alone, the covariance with \(V_j\) changes sign while both variances remain fixed, so the correlation changes sign. Since canonical correlations are conventionally reported as nonnegative magnitudes, either orientation can be chosen for the pair. \(\square\)

Repeated Roots Identify Directions Only Up to a Subspace

A repeated canonical correlation creates another interpretation trap. If two or more roots are equal, individual canonical directions within their shared space need not be uniquely determined. Small numerical differences in software output, or a different valid choice of basis, can produce different-looking weights and loadings even when the underlying repeated-root subspace is the same.

A simple example shows why. Let \(X=(X_1,X_2)^\mathsf{T}\) and \(Y=(Y_1,Y_2)^\mathsf{T}\) each have identity covariance matrix, and let their cross-covariance matrix be \(0.6I_2\). The joint covariance matrix is valid: its eigenvalues are \(1.6\) and \(0.4\), each with multiplicity two, so it is positive definite. For the coordinate directions, the paired canonical correlations are both \(0.6\). But any common orthogonal rotation of the two coordinates gives another valid pair of canonical directions with the same two correlations.

Indeed, for any orthogonal matrix \(Q\), define \(U=Q^\mathsf{T}X\) and \(V=Q^\mathsf{T}Y\). Both within-set covariance matrices remain \(I_2\), and \(\operatorname{Cov}(U,V)=Q^\mathsf{T}(0.6I_2)Q=0.6I_2\). Thus the two rotated pairs remain uncorrelated across indices and each paired correlation is \(0.6\). There is no unique pair of axes singled out by the data in this example. The repeated-root subspace is the stable object to describe; individual directions within it require an additional convention or external scientific rationale. \(\square\)

Significance Tests Answer Specific Null Questions

For sample canonical correlations \(\hat\rho_1,\ldots,\hat\rho_m\), the global null concerns whether all population canonical correlations are zero. A sequential test after the first \(s-1\) roots concerns the remaining roots. The usual Wilks statistic for the latter null is \(\Lambda_s=\prod_{j=s}^{m}(1-\hat\rho_j^2)\). The global test is the special case \(s=1\). A small p-value for the global test is evidence against the claim that every canonical correlation is zero; it does not show that every root is nonzero.

Sequential tests can help assess how many dimensions have evidence of association, but their conclusions depend on the specified testing procedure, its assumptions, and the chosen error rate. In particular, a nonsignificant test is not proof that the remaining population roots are exactly zero. Nor does statistical significance alone show that an association is large enough to matter scientifically or that either set causes the other.

Redundancy Is Not the Square of a Canonical Correlation

A canonical correlation measures association between a paired pair of variates. It does not by itself measure how much of the original variables in one set is represented by the other set. The redundancy index combines the canonical correlation with squared structure loadings, so it reflects both the strength of the paired association and how well a canonical variate represents its own variable set.

The factor \(\rho_j^2\) matters because \(U_j\) and \(V_j\) are correlated, not identical. If a canonical variate represents its own set well but its paired correlation is small, it contributes little redundancy across sets. Conversely, a high canonical correlation does not guarantee high redundancy if the variate has weak loadings on most of the original variables.

Common Mistakes and a Better Reading Routine

Most errors arise from treating every CCA output column as if it had the same meaning. A careful interpretation keeps the constructions, associations, and tests separate.

• Using weights as variable–variate correlations. Weights define linear combinations; use structure loadings to describe correlations with those combinations.
• Assigning meaning to a sign in isolation. A simultaneous reversal of a canonical pair changes loading signs but not the association. State the sign convention and focus on relative patterns.
• Giving unique labels to axes with repeated roots. Equal roots permit rotations within the repeated-root space. Interpret the subspace unless an additional convention identifies axes.
• Taking a significant global test to mean every root matters. The global test addresses the joint null of all roots being zero; sequential tests address specified remaining-root nulls.
• Calling redundancy the squared canonical correlation. Redundancy also weights each squared correlation by the average squared structure loading for the set being described.
• Equating statistical evidence with practical importance or causation. CCA summarizes linear association between sets; the tests do not establish causal direction or practical value by themselves.