Common Interpretation Errors in MFA and Joint PCA

Why Interpretation Requires More Than Reading a Map

MFA and joint PCA maps compress high-dimensional measurements into a small number of axes. The map is useful only if its quantities are read correctly: a point may have a global coordinate and several block-specific partial coordinates; a large contribution does not necessarily mean a point is well represented; and a high percentage of explained inertia is not a percentage of biology explained. Visual agreement between axes is also not proof that blocks measure the same biological process.

The distinctions below assume that the same \(n\) individuals are measured in each of \(G\) blocks, and that the blocks have been centered and prepared according to the analysis plan. The details of preprocessing matter: the scaling of variables and blocks changes the geometry, so interpretation must always refer to the actual fitted analysis.

Global and Partial Coordinates

Write \(Z_g\) for the centered and, where appropriate, scaled data matrix for block \(g\). In a common MFA convention, let \(\lambda_{1g}\) be the largest eigenvalue of \(Z_g^\top Z_g\), and give the block weight \(w_g=1/\lambda_{1g}\). The weighted block is \(X_g=\sqrt{w_g}Z_g\). Concatenating the weighted blocks gives \(X=[X_1,\ldots,X_G]\). Joint PCA can be expressed as PCA of such a concatenated matrix; in MFA, the block weights are chosen to limit the dominance of blocks with large first dimensions of variation.

This additive identity follows directly from matrix multiplication: the row of \(X\) for individual \(i\) is the concatenation of its block rows, and \(V_k\) is partitioned in the same way. Their inner product is the sum of the blockwise inner products. The global score therefore reflects the combined weighted blocks; it is not the score from any one block. Partial scores show how each block projects onto the feature segments of the global axis. Because scores can have opposite signs, one block may reinforce the global direction while another offsets it.

A sign reversal of an axis changes every score and loading on that axis by a factor of \(-1\), without changing the fitted subspace or distances. Thus, positive versus negative has no invariant meaning across separate fits unless signs have been aligned by an explicit convention. When eigenvalues are close, small data or preprocessing changes can also rotate axes within their nearly tied subspace. Compare subspaces or stable patterns, not just the label or sign of a single axis.

Contribution Is Not Representation Quality

Contribution and representation quality answer different questions. For centered scores on axis \(k\), the total squared score is \(\lambda_k\), the corresponding eigenvalue of \(X^\top X\). An individual’s contribution measures its share of that axis’s inertia. Its cos² measures what fraction of its total squared distance from the origin is represented on that axis.

The denominators explain the distinction. Contribution has an axis-wide denominator and identifies which individuals build that axis. Cos² has an individual-specific denominator and indicates whether the axis captures that individual’s position. The same point can be influential in forming an axis yet poorly represented by that axis, or well represented while making only a small contribution.

For points close to the origin, cos² can be unstable or uninformative because their total distance is very small. On a two-dimensional map, a point may also look near another point while having low cos² on the displayed plane: important differences may lie on omitted axes. Check cos² for the displayed axes, and avoid strong claims about individuals that are poorly represented there.

Explained Inertia Is Conditional on the Analysis

The inertia explained by an axis is its eigenvalue divided by the total inertia of the analyzed matrix. For the weighted concatenated matrix \(X\), total inertia is \(\sum_k\lambda_k=\mathrm{tr}(X^\top X)\). The proportion for axis \(k\) is \(\lambda_k/\sum_j\lambda_j\); cumulative explained inertia through axis \(K\) is \(\sum_{k=1}^K\lambda_k/\sum_j\lambda_j\).

In particular, unweighted concatenated PCA and MFA need not give the same inertia percentages or map. A large or highly variable block can dominate an unweighted concatenation. MFA’s block weighting changes the geometry to balance blocks according to its weighting rule; it does not make every block equally informative about every axis, nor guarantee that the resulting axes represent biology without bias.

Aligned Axes Do Not Prove Shared Biology

A global axis is constructed from the joint weighted data, so a visual alignment among block projections is partly a consequence of the analysis design. Even a strong correlation between block-specific scores establishes a pattern of statistical co-variation in these data, not a shared biological cause. Shared sample ordering, batch effects, treatment assignment, correlated preprocessing artifacts, or other common influences can produce aligned patterns. Biological interpretation requires external knowledge and validation appropriate to the scientific claim.

Diagnostics and Appropriate Comparisons

Before interpreting a map, check what it displays and whether the displayed points are well represented. Confirm the block weights and variable scaling; verify that each block has nonzero variation and that the same individuals are aligned across blocks. Inspect global and partial scores together, contributions for influential points, and cos² for claims about locations. Check whether conclusions persist under plausible preprocessing or weighting choices. If axes have close eigenvalues, assess whether the interpretation is stable for the corresponding subspace rather than relying on one axis alone.

These methods answer related but nonidentical questions. Separate PCA describes each block’s internal variation; joint PCA describes variation in the chosen concatenated representation; MFA describes a weighted joint compromise and supports block-specific projections onto global axes. None alone establishes causal relations or a common biological mechanism. Also, a contribution is not a significance test, and an inertia percentage is not a measure of reproducibility. Use design-aware validation and domain evidence for those claims.

Common Interpretation Errors

• Wrong: “The global coordinate is the score from the RNA block.” Correction: It is the combined score under the fitted weighting; inspect partial scores to identify block-specific direction and disagreement.
• Wrong: “A high contribution means this individual is well represented on the map.” Correction: Contribution concerns the individual’s share of axis inertia; cos² concerns representation relative to that individual’s total distance.
• Wrong: “The first two axes explain 80%, so they explain 80% of the biology.” Correction: They explain 80% of the inertia in the specified processed and weighted matrix.
• Wrong: “Aligned block axes prove the assays measure the same pathway.” Correction: Alignment describes a pattern; technical, design, or other common influences remain possible explanations.
• Wrong: “Positive scores in two separate fits have the same meaning.” Correction: Axis signs are arbitrary. Align axes before comparing signs, and account for possible rotations when eigenvalues are close.