Adding Supplementary Blocks to an MFA Map

Why Add a Supplementary Block?

An MFA map is built from active blocks: their measured structure determines the compromise space and its global axes. A new omics block may be relevant to interpreting that map but not intended to help define it. For example, a validation assay might be measured on the same samples after the active solution has been selected. Treating that assay as supplementary lets us ask whether its variables align with the existing axes while preserving the original solution.

The distinction is substantive. If a block is appended to the active data and MFA is recomputed, it can change the global axes and the coordinates of the original samples. If it is supplementary, the active data, preprocessing choices, and axes remain fixed. The supplementary block is evaluated against that fixed reference; it does not contribute to constructing it.

Setup and Projection Coordinates

Consider an active MFA solution with \(n\) individuals and \(K\) retained global axes. Let \(f_k\) be the vector of individual scores on global axis \(k\), for \(k=1,\ldots,K\). Let \(y_j\) be one centered quantitative variable from a supplementary block, measured on those same individuals. Its coordinate on axis \(k\) is its correlation with that axis:

This coordinate lies between \(-1\) and \(1\), provided neither vector is constant. A positive value indicates that larger values of the variable tend to occur among individuals with larger scores on the axis; a negative value indicates an opposite tendency. A value near zero indicates little linear association with that axis. The coordinate is an association, not a causal effect or a regression slope.

The correlation formula applies whether the variable was standardized before projection or not: multiplying \(y_j\) by a positive constant does not change its correlation. Centering and scaling still matter for a coherent MFA preprocessing pipeline and for block-level summaries. Use the same centering and variable-scaling convention as for the active quantitative blocks; estimate any required means and scales from the supplementary block itself, without changing the active preprocessing or axes.

For an entire supplementary block with \(q\) quantitative variables, collect their coordinates on axis \(k\) as \(r_{1k},\ldots,r_{qk}\). One useful block-level summary is the sum of squared feature coordinates, \(I_k=\sum_{j=1}^q r_{jk}^2\). It measures how strongly the block’s individual variables are associated with that axis, with larger blocks potentially having larger sums simply because they contain more variables. For comparisons across blocks of different sizes, also report the mean squared coordinate \(I_k/q\). Neither summary is a new global axis or a signed block coordinate: they summarize feature associations and discard their directions.

Why the Projection Formula Works

The coordinate has a direct least-squares interpretation. Regress the centered feature \(y_j\) on the centered axis scores \(f_k\), using a single predictor and no intercept. The coefficient minimizing the sum of squared residuals is:

To see this, write the residual sum of squares as \((y_j-\beta f_k)^{\mathsf T}(y_j-\beta f_k)\). Differentiating with respect to \(\beta\) and setting the derivative to zero gives \(-2f_k^{\mathsf T}y_j+2\beta f_k^{\mathsf T}f_k=0\), yielding the expression above. The correlation coordinate is the standardized version of this slope:

Thus, projection onto a fixed axis asks how closely a supplementary feature follows that axis across individuals. It does not require the feature to have been part of the eigenvector that defined the axis. The axis is held fixed, and the feature is evaluated against it.

Respecting MFA Block Scaling

In quantitative MFA, active blocks are commonly formed by centering and scaling their variables, then dividing each block by the square root of its first principal inertia before concatenating the blocks for the global analysis. The exact variable-scaling convention should match the active analysis. For a supplementary block, the same type of block scaling can be used when reporting its scaled data or its own principal-component representation. It is essential, however, not to append that scaled block to the active matrix and recompute the global decomposition if the goal is to preserve the existing map.

Suppose a centered, variable-scaled supplementary block is \(Z_s\), and its largest eigenvalue under the covariance convention \(Z_s^{\mathsf T}Z_s/n\) is \(\lambda_{1,s}\). Its MFA-scaled version is \(Z_s/\sqrt{\lambda_{1,s}}\). This common block multiplier controls the block’s overall scale, while the individual feature correlations with fixed global axes are unchanged by that positive multiplier. The multiplier therefore does not create a new relationship with an axis; it preserves the block-weighting convention for quantities that depend on the scaled block’s magnitude.

Summarizing a Block Across Axes

A block may be represented differently on different global axes. For each retained axis, report feature correlations and a clearly labelled block summary. The sum of squared coordinates \(I_k\) is useful for describing the total squared feature association on axis \(k\); the mean squared coordinate \(I_k/q\) facilitates comparisons when blocks have different numbers of variables. Reporting the signed feature coordinates alongside the aggregate avoids hiding whether features align in the same or opposing directions.

If the scientific question is whether the block, considered jointly, predicts an axis, a different summary is available. Regress the centered axis scores on all centered variables in the block and report the coefficient of determination \(R^2\), the proportion of axis-score variation explained by their linear span. This multivariable summary accounts for redundancy among features, unlike the sum of squared marginal correlations. It can be unstable when the block has many correlated features or more variables than individuals, so its estimation requires care. State explicitly which summary is used; a sum of squared correlations and a multivariable \(R^2\) are not interchangeable.

Assumptions, Diagnostics, and Alternatives

The projection is interpretable only if the supplementary observations correspond to the same individuals, in the same order, as the active score vectors. The variables should be quantitative for the correlation formulas above, adequately measured, and processed using defensible centering and scaling. The axes must be the fixed global axes from the active MFA solution, not axes estimated using the supplementary block. These are prerequisites for calculation and interpretation, not guarantees that an observed association is meaningful beyond the analyzed sample.

Before interpreting coordinates, check that row identifiers align exactly; a permutation of sample order can produce plausible-looking but meaningless correlations. Confirm that no feature is constant, that missing values have not caused different individuals to be used silently for different calculations, and that outliers are not driving the associations. Compare feature coordinates with scatterplots of each feature against the relevant axis scores. Strong curvature or clusters can yield a small correlation despite a clear non-linear pattern. Also examine the number of features and their redundancy: a large block can produce a large sum of squared coordinates partly because it contains many opportunities for alignment.

The main alternatives answer different questions. Refitting MFA with the new block makes it active and asks how the compromise changes when that block participates. Separate PCA describes the new block’s internal variation, but does not by itself establish its alignment with the existing global axes; correlations between its component scores and the fixed axes can provide that comparison. Concatenated PCA is another joint ordination, but it does not apply MFA’s balancing of active blocks in the same way. None of these alternatives is a substitute for a fixed-axis supplementary projection when the goal is specifically to preserve the established MFA map.

A supplementary projection is descriptive. It does not show that the active axes generalize to new individuals, and it does not provide a significance test. If inferential claims are needed, the sampling design and an appropriate inferential procedure must be considered separately. The projection is also sensitive to the choice of active blocks, preprocessing, and retained axes: a new block can look weakly associated simply because the relevant shared pattern lies beyond the displayed dimensions.

Common Pitfall

A common error is to append the new block to the active matrix, rerun MFA, and then call the resulting coordinates supplementary. That reasoning is wrong because the new block has helped determine the axes; the map is no longer the original reference solution. To keep a block supplementary, retain the original global scores and calculate its feature correlations against those scores. Conversely, if the scientific question is how the full joint solution changes after including the block, then refitting is appropriate—but the block is active, not supplementary.