Why a Dominant Block Deserves a Diagnosis
In multiple factor analysis (MFA), observations are described by several data blocks, such as transcriptomic, proteomic, and metabolomic measurements. MFA weights blocks before performing a joint analysis, so one block’s dominance is not automatically a sign that the analysis has failed. A block can legitimately contribute strongly if its structure aligns especially well with a global axis. But a large contribution can also result from a preprocessing error, redundant features, or a direction that changes substantially when the data are perturbed.
The diagnosis starts by defining exactly what “dominates” means. Here, it means that one block accounts for a large fraction of the inertia of a specified global axis. This is an axis-specific statement: a block can dominate one axis but contribute little to another. There is no universal numerical cutoff at which a contribution becomes problematic. The size of the contribution is a signal to investigate, not a verdict about validity.
Block Scaling and Axis Contributions
For clarity, consider \(Q\) quantitative blocks measured on the same \(n\) observations. Let \(Z_q\) be the centered, within-block-preprocessed matrix for block \(q\). Columns might be standardized to unit variance, but the choice must suit the variables and be recorded consistently. Let \(\lambda_{q1}\) be the largest eigenvalue of \(Z_q^\mathsf{T}Z_q/n\). In the standard MFA weighting for quantitative blocks, each block is divided by \(\sqrt{\lambda_{q1}}\). The weighted block is \(Y_q=Z_q/\sqrt{\lambda_{q1}}\), and the concatenated matrix is \(Y=[Y_1,\ldots,Y_Q]\).
This scaling gives each block the same leading eigenvalue, one, before the global PCA. It does not make all blocks equally influential on every global axis: their other directions can differ, and the global axes are fitted jointly. The global PCA decomposes the variation in \(Y\); write \(u_k\) for a unit-length observation-space score direction for axis \(k\). The corresponding eigenvalue is \(\lambda_k=\|Y^\mathsf{T}u_k\|^2/n\). The block contribution to that axis is defined as follows.
This definition uses the axis’s projected inertia from each weighted block. It is not the same as asking how much total variation a block has across all axes, nor is it an individual observation’s representation quality. In particular, a large contribution does not by itself identify which feature drives the block, establish a biological mechanism, or show that the block is more important for every scientific question.
Why the Contributions Sum to One
The contribution formula has a useful internal check. Since the columns of \(Y\) are partitioned into blocks, the entries of \(Y^\mathsf{T}u_k\) are partitioned into the entries of \(Y_q^\mathsf{T}u_k\). Squared Euclidean length adds across those disjoint groups. Therefore:
Thus the block contributions to any nonzero axis should sum to one, up to numerical rounding. This identity is a justification for using contributions as shares, and a practical check on calculations or software output. The sign of an axis score direction can be reversed without changing any contribution, because the formula uses squared lengths.
Worked Example: Calculating a Dominant Block’s Share
Suppose a hypothetical MFA fit has three weighted blocks. On one global axis, their projected inertias \(\|Y_q^\mathsf{T}u_k\|^2\) are 8.4, 2.4, and 1.2, respectively. These are illustrative fitted quantities, not observations from a real study. The total projected inertia is \(8.4+2.4+1.2=12.0\).
The contributions are \(C_{1k}=8.4/12.0=0.700\), \(C_{2k}=2.4/12.0=0.200\), and \(C_{3k}=1.2/12.0=0.100\). In percentage terms, the first block accounts for 70.0% of this axis’s inertia, the second for 20.0%, and the third for 10.0%. The check is \(0.700+0.200+0.100=1.000\).
The first block is clearly dominant on this axis. The calculation alone does not say whether that dominance is scientifically meaningful or caused by a problem. The next step is to audit preprocessing and the features that define the block.
Diagnosing the Source of Dominance
First verify the construction of the weighted blocks. Check that the observations are aligned in the same order, that missing values were handled as intended, and that centering, transformations, and variable scaling match the analysis plan. Recompute each block’s leading eigenvalue from the preprocessed matrix and check that the applied weight is \(1/\sqrt{\lambda_{q1}}\) under the stated covariance convention. A missing weight, a weight applied twice, or a block scaled using a different set of variables can make comparisons misleading.
Next examine the block’s features and the reasons for including them. A group of highly redundant measurements can give one underlying direction repeated representation within a block. Leading-eigenvalue scaling controls each block’s largest preweighted eigenvalue, but it does not guarantee that all scientifically relevant directions within that block receive a balanced representation. Nor does it remove redundancy among features. Check for duplicated columns, repeated assays, derived variables that encode the same measurement, or features included at a different level of aggregation from the other blocks.
Then ask whether the direction is stable. Refit the MFA after reasonable perturbations, such as resampling observations, leaving out a small subset of features, or omitting a suspect redundant feature. Compare the contribution of each block and the alignment of the axis across fits. Axis signs can flip without changing the solution, so comparisons should allow for sign reversal. If two axes are close in eigenvalue, their directions may also rotate or exchange order; in that situation, compare the relevant subspace or interpret the individual axes cautiously rather than treating a change of axis number as proof of instability.
Worked Example: Checking a Scaling Audit
Consider a hypothetical analysis with three quantitative blocks. After the planned within-block preprocessing, their leading eigenvalues are \(\lambda_{11}=25\), \(\lambda_{21}=4\), and \(\lambda_{31}=1\). Under standard MFA weighting, their respective multipliers are \(1/\sqrt{25}=0.200\), \(1/\sqrt{4}=0.500\), and \(1/\sqrt{1}=1.000\). These multipliers are applied to the preprocessed blocks, not to raw variables before the planned preprocessing.
Suppose an initial, unweighted concatenated PCA assigned 81.0% of its first axis’s inertia to block 1. That output is not the standard MFA result: the blocks have not been balanced by their leading eigenvalues. After checking the preprocessing and refitting with the stated MFA weights, suppose the projected inertias on the first axis are 4.6, 3.1, and 2.3. Their sum is \(4.6+3.1+2.3=10.0\), so the contributions are \(4.6/10.0=0.460\), \(3.1/10.0=0.310\), and \(2.3/10.0=0.230\).
In this hypothetical calculation, the apparent 81.0% dominance does not persist under the specified MFA scaling: block 1 contributes 46.0% of the fitted MFA axis. This illustrates why the method and weighting must be verified before interpreting a contribution. It does not show that all large contributions are scaling errors; an MFA axis can remain concentrated after correctly applied weights.
Worked Example: Testing Whether Redundancy Matters
Suppose a hypothetical block contains three standardized features, two of which are exact copies of the same measurement. The covariance matrix of just those two copied columns is \(\begin{pmatrix}1&1\\1&1\end{pmatrix}\), whose eigenvalues are 2 and 0. The repeated direction therefore has twice the leading eigenvalue of a single standardized feature. In a larger block, duplication can reinforce that direction relative to other features, even though MFA scales the block by its overall leading eigenvalue.
Assume the fitted axis’s projected inertias before an audit are 6.4 for this block and 2.4 and 1.2 for the other two blocks. The total is \(6.4+2.4+1.2=10.0\), giving contributions \(0.640\), \(0.240\), and \(0.120\). After identifying and removing the exact duplicate, suppose the analysis is refitted and the projected inertias are 3.6, 3.0, and 1.4. The new total is \(3.6+3.0+1.4=8.0\), so the contributions are \(3.6/8.0=0.450\), \(3.0/8.0=0.375\), and \(1.4/8.0=0.175\).
Here, the first block’s share falls from 64.0% to 45.0% after the redundant feature is removed, which is evidence that the original dominance was sensitive to that feature. These are hypothetical fitted outputs; the contribution changes are not determined by the two-feature covariance calculation alone. A real audit would also consider whether removing the feature is scientifically justified and whether the revised analysis still represents the intended measurements.
Legitimate Signal, Instability, and Alternatives
A dominant contribution is more credible as a block-specific signal when preprocessing is correct, the features represent the intended measurements, and the axis remains interpretable under reasonable perturbations. For example, a block might show a coherent pattern across multiple nonduplicated measurements and retain a substantial contribution across resamples and modest feature changes. This supports a stable description of the data. It does not establish that the block causes changes in another block, or that the pattern will generalize beyond the sampled population.
A dominance that disappears after correcting a weight or removing an unintended duplicate points to a construction issue. A dominance that varies sharply across resamples or small feature changes is better described as unstable. These explanations are not mutually exclusive: a block may contain real signal and also have redundant measurements that exaggerate its influence. Report the sensitivity analysis rather than forcing a single explanation.
Worked Example: Quantifying Resampling Variability
Suppose a hypothetical six-resample analysis gives the dominant block contributions \(0.92, 0.91, 0.89, 0.87, 0.51,\) and \(0.48\) for a matched axis. In ascending order these are \(0.48, 0.51, 0.87, 0.89, 0.91, 0.92\). The observed range is \(0.92-0.48=0.44\). Because there are six values, the median is the average of the third and fourth ordered values: \((0.87+0.89)/2=0.88\).
The median contribution is 88.0%, but the range is wide: the block’s share varies from 48.0% to 92.0% across these resamples. A high median alone would conceal this instability. With only six resamples, the range is a descriptive diagnostic, not a precise uncertainty interval. The appropriate conclusion is that the fitted block contribution appears sensitive to resampling and warrants further investigation, not that there is an 88.0% probability the block truly dominates.
MFA is designed to balance blocks by a specified within-block criterion before joint analysis. Unweighted concatenated PCA is simpler but can let blocks with larger scales or more total variation dominate; its contributions answer a different question unless comparable weighting has been justified. Separate-block PCA can show whether a pattern is prominent within one block, but cannot establish how that pattern contributes to a shared global axis. Joint and individual decomposition methods such as JIVE address a different target by explicitly separating shared and block-specific structure. These methods are alternatives when the scientific question calls for that decomposition, not automatic corrections for an MFA axis with a large contribution.
| Approach | What it emphasizes | Relevance to a dominant MFA axis |
|---|---|---|
| MFA | Joint axes after a defined block-weighting scheme | Use block contributions to locate which blocks define each global axis |
| Unweighted concatenated PCA | Variation in the concatenated measurements without MFA block weights | Useful as a comparison, but scale and block size can affect dominance |
| Separate-block PCA | Variation within each block considered on its own | Can clarify whether a strong direction exists within a block, but not its share of a joint axis |
| JIVE | Joint and individual structure under a different decomposition | Consider when separating shared from block-specific structure is the main goal |
Common Pitfalls and Reporting
- Calling any large contribution an error. A large share can represent a real, stable block-specific direction. Check the preprocessing and robustness before judging it.
- Assuming MFA eliminates every imbalance. Leading-eigenvalue scaling controls a particular block-level quantity; it does not remove within-block redundancy, guarantee equal contributions to every axis, or make blocks equally informative.
- Confusing contribution with representation quality. A block contribution is an axis-level share. It does not say whether each observation is well represented on that axis.
- Comparing contributions without matching axes. If axes change order or rotate across fits, compare aligned directions or subspaces before interpreting a numerical difference as a change in the same axis.
- Using an arbitrary cutoff as a rule. A threshold such as 50% may help flag a result for review, but it is not a universal definition of a bad MFA axis.
A careful report names the axis and its block contributions, states the preprocessing and weighting used, and describes the checks that were performed. If the dominant share persists under reasonable checks, describe it as a stable concentration of the fitted axis in that block, with a contextual interpretation. If it is sensitive, report the sensitivity and qualify the interpretation. Neither outcome alone supports a causal conclusion.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Three blocks have projected inertias 5.0, 3.0, and 2.0 on an axis. What are their contributions?
- Why should a large MFA block contribution prompt investigation rather than an automatic rejection of the axis?
- A resampling analysis gives six contributions: 0.48, 0.51, 0.87, 0.89, 0.91, and 0.92. What is their median and range?
- What does a separate-block PCA tell you that an MFA block contribution does not?
Solutions. (1) The total projected inertia is \(5.0+3.0+2.0=10.0\); the contributions are \(5.0/10.0=0.500\), \(3.0/10.0=0.300\), and \(2.0/10.0=0.200\), summing to one. (2) Dominance may reflect legitimate structure, but can also arise from incorrect weighting, redundant features, or instability; the contribution alone cannot distinguish these explanations. (3) The ordered values are already shown. The median is \((0.87+0.89)/2=0.88\), and the range is \(0.92-0.48=0.44\). (4) Separate-block PCA describes variation within that block considered on its own; it does not give the block’s share of a jointly fitted global MFA axis.