How Duplicated Blocks Distort Joint PCA

Why Repeated Blocks Matter

Concatenated PCA treats every column as part of one feature matrix. This is consequential when columns—or whole omics blocks—are duplicated or nearly duplicated: repeated measurements contribute repeatedly to the PCA objective. A leading component can then appear to summarize a consensus across blocks even when that apparent agreement is largely the same signal counted more than once.

The key issue is not simply that a block has many columns. It is that PCA weights the total variation represented by all included columns. A second copy of a block adds its variation again, even though it adds no independent information. The algebra below makes this explicit and helps distinguish a genuine cross-block pattern from one amplified by redundancy.

Concatenated PCA and the Duplication Identity

Suppose \(B\) omics blocks measure the same \(n\) individuals. Let \(X_b\) be the \(n \times p_b\) matrix for block \(b\), with columns centered across individuals and any feature scaling specified before analysis. Assume individuals are aligned in the same row order, and that the columns are finite. The concatenated matrix is \(X=[X_1\ \cdots\ X_B]\). PCA of the centered concatenated matrix finds unit-length sample score directions \(t\) that maximize the variance of \(X^\mathsf{T}t\).

This formulation does not require the blocks to have the same number of features. It shows that each block enters through its cross-product matrix, which summarizes how its features jointly vary across the \(n\) individuals. If block \(X_r\) is copied exactly \(k\) times in the concatenation, the resulting cross-product matrix is

Thus the repeated block receives \(k\) times its original contribution to the objective. This is not a p-value or a claim about independent evidence; it is a direct consequence of the PCA optimization criterion.

Deriving the Effect on the Leading Axis

PCA selects the direction with the largest total projected sum of squares. If different blocks favor different directions, increasing the weight on one block can change which direction wins. The following two-direction argument shows the change without relying on a particular algorithm.

In more general data, the blocks need not have orthogonal leading directions. Then the summed cross-product matrix can have eigenvectors that mix directions from several blocks. Duplication still adds the same matrix again, but the resulting rotation depends on the full eigenstructure rather than a simple comparison of two numbers.

Near-Duplicates and Axis Stability

Exact equality is not necessary for distortion. For a precise illustration, let \(u,v\) be orthogonal unit centered score patterns. Suppose two one-feature blocks have unit-length columns \(u\) and \(w=\cos(\theta)u+\sin(\theta)v\), where \(0<\theta<90^\circ\). The second block is close to the first when \(\theta\) is small. Their combined cross-product matrix, expressed in the \(u,v\) basis, is

Its eigenvalues are \(1+\cos(\theta)\) and \(1-\cos(\theta)\). The leading direction is halfway between \(u\) and \(w\), at angle \(\theta/2\) from \(u\). Consequently, two highly similar blocks produce a strong leading axis close to their shared pattern, even though the alignment is not independent confirmation.

More generally, write a near-copy as \(X_b+E\), where \(E\) is a perturbation matrix. The added block contributes \((X_b+E)(X_b+E)^\mathsf{T}\), which equals \(G_b+X_bE^\mathsf{T}+EX_b^\mathsf{T}+EE^\mathsf{T}\). The cross terms and the perturbation term explain why approximate duplication adds nearly another \(G_b\), while also potentially rotating the axis. The exact change depends on the size and orientation of \(E\); small perturbations are most consequential when the leading eigenvalues are close, because then the leading direction is less well separated from alternatives.

Diagnosis, Alternatives, and Limitations

A high-variance first component alone cannot reveal whether a block has been repeated. Diagnose redundancy in the data and inspect how conclusions change when it is addressed. Useful checks include:

• Check feature and block metadata for repeated assays, reused columns, repeated transformations, or duplicated samples.
• Compare block-level patterns and leading score directions. Very high similarity may reflect shared biology, technical duplication, or both; it is a reason to investigate, not proof of duplication.
• Recompute the leading axes after removing one suspected duplicate block. Large changes in the direction or interpretation indicate sensitivity to that block’s repeated contribution.
• Report block contributions alongside a joint map. A component with several apparent supporters may still be dominated by a repeated or near-repeated pattern.

These checks have limits. Similar blocks need not be redundant: distinct assays may measure related biology, and removing one can discard useful information. Conversely, exact duplicates can arise from data handling and may be difficult to detect from correlations alone. A small eigengap also makes component directions unstable, so a rotated axis is not by itself evidence of a preprocessing error.

The main alternatives answer different questions. Separate PCA describes variation within each block without letting another block’s columns change its axes, but it does not produce one joint PCA solution. MFA, as covered in the earlier material on MFA versus separate and concatenated PCA, uses block-level scaling to limit domination from differences in block scale. That scaling does not make duplicate blocks independent: a repeated block included as another block can still add its pattern again. Removing an erroneous copy, grouping redundant measurements, or applying explicit block weights can address redundancy more directly, but each requires a defensible scientific choice.

Practical Interpretation

A joint PCA axis is a summary of the columns supplied to the analysis, not a count of independent biological confirmations. Its interpretation depends on the included features, preprocessing, and block multiplicities. State whether features were centered and scaled, identify any suspected overlap between blocks, and distinguish mathematical contribution from independent evidence. When a component changes sharply after removing a repeated block, report that sensitivity rather than presenting the original axis as an unqualified consensus.