End-to-End Workshop on an MFA Analysis

From Assay Blocks to a Joint Map

A useful MFA analysis is more than a plot of sample points. The blocks must describe the same individuals, their scales must be justified, and the global axes must be traced back to the block-specific patterns that produce them. This workshop follows a small, entirely hypothetical dataset through those steps. It uses three quantitative blocks: metabolomics, proteomics, and transcriptomics. The calculations are deliberately small enough to check by hand; real analyses use the same logic with more features and samples.

The example uses centered feature matrices \(X_k\), one for each block \(k\). Feature scales are assumed comparable across the columns in this constructed dataset. In practice, features measured on different scales generally need appropriate variable scaling before block weighting. MFA then rescales each whole block according to its first principal component, so a block with many features or large overall variation does not automatically dominate the joint analysis.

This normalization is appropriate when the blocks are active, quantitative groups and their first principal components are defined. It does not correct mismatched individuals, unhandled missingness, poor feature scaling, or outliers. The interpretation also depends on the variables included: scaling gives blocks a defined relative influence, not an assurance that their biological contents are equally reliable or equally relevant.

Hypothetical Data and Block-Specific PCA

Consider four matched individuals, labelled A through D. Define three centered patterns across them: \(a=(1,-1,1,-1)^T\), \(b=(1,1,-1,-1)^T\), and \(c=(1,-1,-1,1)^T\). Each has squared length 4, and each is orthogonal to the other two. The six feature columns below are hypothetical centered measurements in comparable assay units.

In pattern notation, the three blocks are \(X_1=[a,\;0.5b]\), \(X_2=[0.8a,\;c]\), and \(X_3=[b,\;0.5c]\). Orthogonality makes each block’s PCA transparent: its two feature directions are orthogonal, so its eigenvalues are simply the squared column lengths.

Global Ordination and Its Justification

After normalization, the global individual structure can be computed from \(ZZ^T\), without needing to display the larger feature-space matrix \(Z^T Z\). Each block contributes \(Z_kZ_k^T\), so \(ZZ^T=\sum_k Z_kZ_k^T\). The orthogonal patterns \(a,b,c\) are eigenvectors of this sum. The corresponding unit vectors are \(u_a=a/2\), \(u_b=b/2\), and \(u_c=c/2\).

The metabolomics block contributes eigenvalues \(1\) along \(u_a\) and \(0.25\) along \(u_b\). Proteomics contributes \(0.64\) along \(u_a\) and \(1\) along \(u_c\). Transcriptomics contributes \(1\) along \(u_b\) and \(0.25\) along \(u_c\). Adding the contributions gives global eigenvalues \(1.64\), \(1.25\), and \(1.25\) along \(u_a,u_b,u_c\), respectively.

Partial Projections and Block Contributions

A partial map asks how each block relates to the global individual axes. One useful calculation is the block’s projection contribution to global individual scores. For unit global axis \(u_\ell\), define the block contribution vector as \(q_{k\ell}=Z_kZ_k^Tu_\ell\). To express it on the same score scale as the global scores \(f_\ell=\sqrt{\lambda_\ell}u_\ell\), use \(p_{k\ell}=q_{k\ell}/\sqrt{\lambda_\ell}\), where \(\lambda_\ell\) is that global eigenvalue. These contributions add exactly: \(\sum_k p_{k\ell}=f_\ell\). This is an additive projection view of partial maps; software can display partial coordinates with a different but documented normalization.

Contribution Diagnostics and Assumptions

For an axis with eigenvalue \(\lambda_\ell\), a convenient block share of its inertia is \(u_\ell^TZ_kZ_k^Tu_\ell/\lambda_\ell\). These shares are nonnegative and sum to 1 across blocks. On axis 1 they are \(1/1.64=61.0\%\) for metabolomics, \(0.64/1.64=39.0\%\) for proteomics, and 0% for transcriptomics. On axis 2 they are \(0.25/1.25=20.0\%\), 0%, and \(1/1.25=80.0\%\). These are axis-specific contributions, not overall block weights or measures of significance.

The calculation relies on several conditions that should be checked before interpretation:

• Matched rows: row \(i\) must refer to the same individual in every active block, in the same order. Otherwise the global cross-block geometry is not meaningful.
• Centering and feature scales: quantitative features should be centered, and scaling should reflect their units and measurement properties. A few extreme values can strongly affect PCA and the block’s leading singular value.
• Meaningful block definition: features should be grouped by a defensible assay or scientific domain. Splitting one assay into several blocks changes its influence in the joint analysis.
• Nondegenerate blocks: an active block needs nonzero variation. A constant block has no positive leading singular value and cannot be normalized this way.
• Missingness and preprocessing: missing values must be handled using a method compatible with the analysis; silently dropping different individuals in different blocks breaks matching.

Diagnostics include inspecting each block’s first eigenvalue and remaining spectrum before and after scaling, checking feature distributions and outliers, confirming sample alignment, and comparing global and partial maps. If one block supplies nearly all of an axis, examine whether that reflects a genuine strong pattern, an outlier, a block definition with duplicated information, or preprocessing choices. Repeating the analysis under defensible alternative feature scaling or block definitions can reveal whether the interpretation is fragile. The earlier material on sensitivity of MFA maps to block weights motivates treating such checks as part of the analysis, not as decoration.

How MFA Differs From Nearby Alternatives

Separate PCA analyzes each block on its own. It can show within-block variation clearly, but its axes need not align across blocks, so it does not directly provide one joint individual map. Concatenated PCA analyzes all columns together without correcting blocks by their leading singular values. A block with larger scale, more variables, or stronger variation can then dominate. MFA instead adjusts each active block by its leading singular value before the global PCA, making the leading block directions comparable in the specific sense that each has unit eigenvalue after normalization.

That adjustment does not make the methods interchangeable or MFA universally preferable. Separate PCA is useful when block-specific structure is the main target. Concatenated PCA may be appropriate when unweighted feature-level variation is the intended objective. MFA answers a joint-structure question under its chosen block definitions and scaling. It can still be sensitive to feature preprocessing, correlated or redundant features, small samples, and the choice of active blocks. A global map is a summary of the observed data, not a test of association or a causal analysis.

A Practical Workflow

A common mistake is to say, “MFA gives every block equal total influence because each block is divided by its first singular value.” The correction is precise: normalization makes each block’s first eigenvalue equal to 1, but total normalized inertia can differ, as the totals \(1.25,1.64,1.25\) demonstrate. Another mistake is to treat a large contribution as evidence of statistical significance. Contributions describe how the observed normalized geometry is assembled; uncertainty requires additional inferential methods and assumptions.