Interpreting Shared Structure Across Blocks

What Counts as Shared Structure?

A global MFA map summarizes matched individuals across several data blocks, but a point on that map does not by itself show whether the blocks agree. A sample may have an extreme global score because several blocks place it in the same direction, or because one block contributes a strong signal while the others contribute little or point elsewhere. Interpreting shared structure therefore requires reading the global score alongside the aligned partial coordinates, block contributions, and feature loadings.

The goal here is descriptive: identify whether a visible sample pattern is supported across blocks and describe the features associated with it. A shared pattern is evidence of concordant representation in the analyzed data; it is not, by itself, evidence that one molecular layer causes changes in another, or that the pattern is statistically significant.

Definition: Let \(Z_k\) be the centered, MFA-scaled data matrix for block \(k\), with the same \(n\) matched individuals in every block. The concatenated matrix is \(Z=[Z_1,\ldots,Z_K]\). For global component \(\alpha\), let \(v_\alpha\) be its unit-length loading vector, partitioned by block as \(v_\alpha=(v_{1\alpha}^{T},\ldots,v_{K\alpha}^{T})^{T}\). The global score for individual \(i\) is \(F_{i\alpha}=(Zv_\alpha)_i\). The partial coordinate from block \(k\) is \(F_{ik\alpha}=(Z_kv_{k\alpha})_i\). With this convention, the global score is the sum of the block partial coordinates.

F_{i\alpha}=\sum_{k=1}^{K}F_{ik\alpha}

This equality is useful because it makes the components of an individual’s global position explicit. A large positive global score can result from several positive partial coordinates, or from one large positive coordinate partly offset by negative coordinates from other blocks. The plotted MFA output may apply a display convention to coordinates; when checking the equality, use partial coordinates and global scores on the same scale.

For an axis with a unit-length loading vector, define the loading-based block contribution as the share of squared loading length assigned to that block:

C_{k\alpha}=\sum_{j\in k}v_{j\alpha}^{2}, \qquad \sum_{k=1}^{K}C_{k\alpha}=1

Thus, \(C_{k\alpha}=0.40\) means that 40% of the squared loading length for component \(\alpha\) lies in block \(k\), under the stated scaling and loading convention. It is a summary of how the axis direction is allocated across blocks, not the percentage of an individual’s score explained by that block. A block contribution can be substantial while a particular sample’s partial coordinate from that block is small.

A feature loading describes how an individual feature is associated with a global component. When features are represented by their correlations with the global scores, a positive loading means that higher feature values tend to occur with higher scores on that component; a negative loading means that higher values tend to occur with lower scores. If a software output instead reports projection coefficients, interpret those according to their scale rather than treating them automatically as correlations.

A Practical Reading Sequence

A convincing shared interpretation needs agreement at more than one level. First locate individuals with interesting global scores. Then check whether their partial coordinates have similar signs and reasonably comparable magnitudes across blocks. Next ask whether block contributions show that the axis draws appreciably on multiple blocks. Finally, inspect feature loadings to describe what high and low scores correspond to in each block.

Orient and identify the axis.
Use the global map to identify the component and the individuals whose positions need interpretation. Remember that the sign of a PCA axis is arbitrary.

Compare partial coordinates.
For each individual of interest, check the signs and relative sizes of the block partial coordinates. Similar signs indicate that the blocks place the individual in a concordant direction on that axis.

Check block contributions.
Use the axis-specific contributions to see whether multiple blocks participate in defining the direction. Contributions describe the axis as a whole, not an individual sample.

Describe the feature pattern.
Use feature loadings to identify which measured features are associated with each end of the axis, and state the direction of the association.

These pieces of evidence answer different questions. Partial coordinates show where a particular individual lies according to each block. Block contributions summarize how the axis direction is distributed across blocks. Feature loadings help explain the meaning of the direction. They should be interpreted together rather than substituted for one another.

Worked Example: Concordant Positions Across Three Blocks

Suppose a hypothetical MFA of transcript, protein, and metabolite blocks produces the following partial coordinates on one global component. Coordinates are on the same MFA scale, and the global scores equal their sums.

Sample	Transcript partial coordinate	Protein partial coordinate	Metabolite partial coordinate	Global score
A	0.50	0.45	0.45	1.40
B	-0.45	-0.50	-0.35	-1.30
C	0.15	-0.10	0.05	0.10

For sample A, the arithmetic is \(0.50+0.45+0.45=1.40\). All three blocks place A on the positive side of the component, with similar-sized contributions to its score. For sample B, \(-0.45-0.50-0.35=-1.30\); again, all blocks place the sample on the same side. Sample C is near the origin globally because its small positive and negative partial coordinates partly cancel: \(0.15-0.10+0.05=0.10\).

Suppose the block contributions to this component are 34%, 33%, and 33%, respectively. The axis is not concentrated in a single block, which is consistent with a multi-block interpretation. Suppose, in addition, that several transcript and protein features have positive loadings, while several other features have negative loadings. That would support describing A as having relatively high values for the positively associated features and relatively low values for the negatively associated features. The values are hypothetical; this interpretation does not establish a biological mechanism.

Derivation: Why Partial Coordinates Add to the Global Score

The additive relation follows from partitioning the concatenated data and loading vector by block. For one individual, write its row of the scaled data matrix as \(z_i=(z_{i1},\ldots,z_{iK})\), where \(z_{ik}\) is the row for block \(k\). The global score is the inner product of this row with the global loading vector. Partitioning that inner product gives:

F_{i\alpha} =z_i v_\alpha =\sum_{k=1}^{K}z_{ik}v_{k\alpha} =\sum_{k=1}^{K}F_{ik\alpha}

This is an algebraic identity, not a claim that the blocks are independent or that each block contributes equally. The partial coordinates can have opposite signs, so they may reinforce or offset one another. The contribution percentages \(C_{k\alpha}\) also sum to one because the loading vector has unit length and is partitioned into disjoint blocks:

\sum_{k=1}^{K}C_{k\alpha} =\sum_{k=1}^{K}\sum_{j\in k}v_{j\alpha}^{2} =\sum_jv_{j\alpha}^{2} =1

The component’s sign can be reversed without changing the underlying PCA solution: multiplying \(v_\alpha\), the global scores, the partial coordinates, and the feature loadings by \(-1\) describes the same axis. Therefore, interpret the two ends of an axis relative to one another, not as inherently positive or negative biological states.

Worked Example: A Global Pattern Led by One Block

In a hypothetical three-block analysis, sample D has partial coordinates \(1.20\), \(0.05\), and \(-0.10\) for transcript, protein, and metabolite blocks, respectively. Its global score is:

F_{D\alpha}=1.20+0.05-0.10=1.15

The global score is positive, but the transcript block supplies nearly all of the positive displacement. The protein coordinate is close to zero, and the metabolite coordinate points in the opposite direction. Suppose the loading-based block contributions are 62%, 25%, and 13%. This axis is more heavily weighted toward the transcript block than toward the other two.

Together, these results are weaker evidence for a broadly shared pattern than the first example. They do not show that the protein and metabolite blocks have no relevant structure; they show that this particular sample’s position on this particular axis is not concordantly supported across all three blocks. If transcript features with large positive loadings are associated with the positive end of the axis, that can describe the transcript-associated pattern, but it does not make the other blocks concordant.

Feature Loadings and the Meaning of a Shared Pattern

Partial-coordinate agreement shows that blocks place individuals similarly; it does not say what measured features distinguish the ends of the axis. Feature loadings supply that description. A useful interpretation names the block, identifies features with relatively large loadings, and states which end of the component they are associated with. Avoid treating a loading as a causal effect or as proof that the feature is uniquely responsible for an individual’s score.

A common pattern across blocks is more interpretable when the feature-level evidence is also coherent. For example, a positive global direction might align with positive loadings for a group of transcripts, proteins, and metabolites, with opposite-signed features defining the other end. This is a descriptive cross-block pattern. It need not mean that the same individual features are measured in each block; the evidence is that different features in the matched blocks align with the same sample ordering.

Worked Example: Combining Sample Positions and Feature Loadings

Consider six hypothetical samples on a component with block contributions of 40% for transcripts, 35% for proteins, and 25% for metabolites. For sample E, the partial coordinates are \(0.70\), \(0.60\), and \(0.50\). For sample F, they are \(-0.60\), \(-0.50\), and \(-0.70\). Their global scores are:

F_{E\alpha}=0.70+0.60+0.50=1.80, \qquad F_{F\alpha}=-0.60-0.50-0.70=-1.80

The partial coordinates indicate concordant positions: E lies on the positive side in all three blocks, and F lies on the negative side in all three. Contributions are distributed across the blocks rather than concentrated in one. Now suppose transcript features T1 and T2 have loadings \(0.78\) and \(0.71\), protein features P1 and P2 have loadings \(0.74\) and \(0.68\), and metabolite M1 has loading \(0.69\), with each loading interpreted as a correlation with the global score. These features are positively associated with the positive end of the axis. Features with negative loadings would be associated with the opposite end.

A defensible summary is that the component describes a sample pattern supported by all three blocks, with the listed features associated with its positive end. The contribution percentages alone would not establish that E and F are concordant, and the loadings alone would not show how each block places those samples. The conclusion relies on combining the three kinds of evidence.

Diagnostics, Comparisons, and Limits

Before giving a shared-structure interpretation, check that the underlying comparison is meaningful. The individuals must be correctly matched across blocks, the features must have been centered and scaled according to the intended MFA analysis, and the partial coordinates must be read in a common orientation and on compatible scales. A mismatch in sample order, inconsistent preprocessing, or comparing differently scaled map coordinates can produce apparent disagreement or agreement that is only a display artifact.

Several patterns should prompt caution:

One block dominates the direction. Large contribution from one block, together with large partial coordinates there and near-zero or opposing coordinates elsewhere, supports a block-led interpretation rather than a strong shared one.
Partial coordinates cancel. A modest global score can hide larger block-specific positions in opposite directions. Inspect the partial coordinates rather than concluding that the sample is unstructured.
Features have mixed associations. A global axis may summarize a sample ordering even when the high-loading features within a block do not form a simple biological theme. Report the feature-level pattern cautiously.
Representation is weak. A point or feature may be poorly represented on the displayed component. The earlier material on cosine squared and representation quality in MFA provides the relevant check; do not overinterpret a two-dimensional projection when the representation on those axes is low.

MFA is useful here because it constructs a joint compromise while limiting the influence of blocks through its scaling. Separate PCA can show structure within each block, but its axes are fitted separately and their directions do not automatically provide a common coordinate system. Concatenated PCA without block scaling can allow a large or highly variable block to dominate. CCA and PLS instead target forms of cross-block association; their objectives differ from MFA’s joint ordination, so their axes and interpretations are not interchangeable.

Finally, the evidence in a map is descriptive and depends on preprocessing, retained components, and the chosen display. Similar partial coordinates do not prove that the same underlying mechanism generated each block’s measurements. Nor do contributions or loadings provide a p-value. Formal uncertainty or association testing requires an inferential procedure suited to the question and design.

Common Mistake

Common mistake: “The global score is large, so every omics block supports this sample pattern.” A global score is the sum of block partial coordinates, so one large coordinate can outweigh small or opposing coordinates from other blocks. Check the partial coordinates for the individual, the block contributions for the axis, and the feature loadings for its interpretation before calling the pattern shared.

Key takeaway: Evidence for shared structure comes from converging descriptions: individuals have concordant partial coordinates, the axis draws on multiple blocks, and feature loadings give a coherent cross-block account of the pattern. No single map statistic establishes all three.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

An individual has partial coordinates \(0.80\), \(-0.10\), and \(0.20\) on an axis. What is the global score, and how concordant is the pattern?
Three blocks contribute 20%, 35%, and 45% to an axis. Does this show that a particular sample has concordant partial coordinates? Explain.
A feature has a correlation loading of \(-0.72\) with a global component. What does this say about higher feature values and the component scores?
All partial coordinates and feature loadings for a component are multiplied by \(-1\). Has the interpretation changed?
Why is a large global score alone insufficient evidence that an axis captures a shared pattern?

Solutions. (1) The global score is \(0.80-0.10+0.20=0.90\). The sample is positive globally, but the protein block points slightly in the opposite direction, so support is not fully concordant. (2) No. Contributions describe how the axis loading direction is distributed across blocks; they do not give the partial coordinates of a specific sample. (3) Higher values of that feature tend to be associated with lower scores on the component, in the analyzed data. This is an association, not a causal effect. (4) No. Reversing all signs changes which end is called positive but leaves the same axis and relative pattern. (5) The score is a sum of partial coordinates, so one block can produce a large score while other blocks contribute little or oppose it. Inspect the partial coordinates and contributions before describing the pattern as shared.