Identifying Block-Specific Structure in Partial Maps

Why Compare Partial and Global Coordinates?

A global MFA map summarizes the compromise among matched data blocks. It can show where samples lie in the shared space, but a global position alone does not show whether every block supports that position. A sample may have a distinctive profile in one block that is averaged with different profiles in the others. Conversely, an apparent separation on a plotted axis may be a poor representation of the sample’s full profile rather than a meaningful block-specific pattern.

Partial maps make this comparison possible: they show where each sample would lie when its position is projected from a particular block onto the global axes. The central diagnostic is the relationship between a block’s partial position and the sample’s global position. A departure can flag a block-specific pattern, but it is not by itself evidence of a biological mechanism, a causal effect, or a statistically significant finding.

The interpretation depends on comparing coordinates from the same samples, the same global axes, and the same MFA fit. The earlier material on global scores and partial individual coordinates established how to read these maps; here the emphasis is on diagnosing departures and deciding when they merit attention.

Coordinates and the Global Barycenter

Suppose \(n\) individuals are observed in \(K\) matched blocks. Let \(Z^{(k)}\) be the preprocessed, column-centered data matrix for block \(k\), with one row per individual. In MFA, block \(k\) is given a weight \(a_k\), conventionally based on the inverse square root of its first principal inertia. Write \(Y^{(k)}=a_k Z^{(k)}\) for the weighted block. Concatenating these weighted blocks gives the matrix whose PCA defines the global axes.

Let \(v_\alpha^{(k)}\) be the part of global axis \(\alpha\)'s loading vector corresponding to block \(k\). The block’s contribution to individual \(i\)'s global score on that axis is \(q_{i\alpha}^{(k)}\). The global score is the sum of those contributions. To make the partial maps directly comparable as positions around the global score, define the rescaled partial coordinate \(p_{i\alpha}^{(k)}=Kq_{i\alpha}^{(k)}\). With this convention, the average of the \(K\) partial coordinates equals the global coordinate.

Here \(f_{i\alpha}\) is the global score and \(p_{i\alpha}^{(k)}\) is the rescaled partial coordinate, both on axis \(\alpha\). The block-specific departure is \(d_{i\alpha}^{(k)}=p_{i\alpha}^{(k)}-f_{i\alpha}\). Across \(r\) displayed axes, the distance \(D_i^{(k)}=\sqrt{\sum_{\alpha=1}^{r}(d_{i\alpha}^{(k)})^2}\) summarizes the departure on those axes. These are distances in the MFA coordinate scale, not physical units such as gene expression units.

Why the Partial Coordinates Average to the Global Score

This identity is useful because it makes the comparison precise. The global PCA uses the concatenation of the weighted blocks, so the score for a row on a global axis is the inner product of that row with the corresponding loading vector. Both the row and loading vector are concatenated blockwise; the inner product is therefore the sum of the blockwise inner products. Rescaling each contribution by \(K\) makes the average of the partial coordinates recover the global score:

Consequently, for each individual and axis, the partial departures across blocks sum to zero: \(\sum_k d_{i\alpha}^{(k)}=0\). One block’s partial position cannot depart from the global score without the other blocks collectively balancing that departure. This is a geometric identity, not an assumption that blocks are independent or that their departures are random errors.

Interpreting the Shape and Size of Departures

Read partial maps as comparisons, not as a search for the largest plotted distance. A departure in one direction on one axis may indicate a block-specific contrast along that axis. Departures across several axes may indicate a broader difference in profile. When examining multiple axes, use the distance over the axes being interpreted and state which axes were included. A distance on the first two axes does not measure disagreement in dimensions that were not plotted.

The direction of a departure is also informative. If a sample’s partial position lies farther along an axis than its global position, that block reinforces the global direction more strongly than the other blocks do, on that axis. If it lies on the opposite side of the origin, its contribution may oppose the consensus. Axis signs can be reversed without changing the PCA solution, so interpret signs relative to the plotted global axis and the other samples, not as intrinsically positive or negative biological effects.

A sizable departure becomes more persuasive when the sample is well represented in the displayed space, the pattern is not created by one extreme feature or sample, and the direction and relative size persist under reasonable perturbations of the analysis. This last check matters because PCA axes can be unstable, particularly when eigenvalues are close. When axes are unstable or nearly tied, apparent movement between them may not mean the underlying subspace has changed. In that situation, avoid over-interpreting individual axis directions.

Distinguishing a Reproducible Pattern from Noise

A partial-map departure is descriptive. MFA does not, by itself, turn the distance between a partial point and the global point into a hypothesis test. No universal cutoff for \(D_i^{(k)}\) determines that a sample is genuinely block-specific: the coordinate scale, number of displayed axes, sample size, preprocessing, and feature correlations all affect what distances are typical.

Useful diagnostics begin with the data and the map. Confirm that rows refer to the same individuals in every block and that the intended preprocessing and block weights were used. Check whether a large departure is driven by one unusual sample or a small set of features. Inspect representation quality for both the global and partial positions, and compare the pattern across additional axes where appropriate. If inference about stability is needed, resampling individuals or otherwise perturbing the analysis can show whether the departure persists. Axes must be aligned across refits, for example by matching their directions or comparing their subspaces, because PCA signs can flip and close axes can rotate.

Assumptions, Alternatives, and Limits

The coordinate comparison is interpretable when the block rows are correctly matched; preprocessing and weights are defined consistently; the partial and global coordinates come from the same MFA fit; and the plotted axes are the same global axes for every block. To interpret a plotted departure as a useful summary of the sample, the selected axes must also represent the sample adequately. These conditions support a geometric diagnosis, not causal interpretation or a test of a null hypothesis.

Separate PCA is useful when the question concerns each block’s own structure and axes. However, separate maps can use different directions and scales, so comparing their coordinates directly does not give the same common-axis diagnostic as MFA partial maps. Concatenated PCA provides a joint map, but without MFA’s block scaling a high-inertia block can dominate the global solution. MFA partial maps retain a shared coordinate system while showing how individual blocks relate to the compromise.

These approaches answer related but different questions. A large partial departure is not a substitute for partitioning global inertia, choosing the number of components, or fitting a model designed to separate joint and individual structure. Nor does a small departure prove that blocks are biologically identical: their differences may be weak on the displayed axes, masked by preprocessing, or present in dimensions not retained for the map.

A Practical Diagnostic Sequence

A disciplined reading keeps geometric description separate from scientific interpretation.