Three analyses, three questions
When several omics blocks are measured on the same individuals, a principal-component map depends on how the blocks enter the analysis. Separate PCA asks what patterns summarize each block on its own. Unweighted concatenated PCA asks what patterns explain the greatest total variation across all features combined. Multiple factor analysis (MFA) first scales each block by its leading principal inertia, then finds a common compromise map. These are related methods, but their first axes need not tell the same scientific story.
A useful comparison holds the individuals, feature preprocessing, and centering convention fixed. The methods then differ primarily in their treatment of blocks. If those choices are not made explicit, a dominant axis can be mistaken for a consensus across assays, or separate block-specific directions can be compared as though they were a shared coordinate system.
Here, inertia means the sum of feature variances under the stated \(1/n\) convention. Changing all covariance matrices to use \(1/(n-1)\) multiplies their eigenvalues by the same factor and does not change PCA directions or the relative MFA block weights. Centering means subtracting each feature’s mean across individuals. Features may also be standardized to unit variance, but this is a substantive choice: it changes the geometry before any of the three methods is applied.
PCA is a descriptive linear method. It does not require independent features or a particular probability distribution to calculate its axes. Meaningful interpretation does require suitable measurements, appropriate preprocessing, and a defensible choice of features. To interpret the map beyond the observed individuals, the sampling design and any relevant dependence or batch structure also matter.
What each method prioritizes
| Method | Primary question | What can dominate? | What the leading axes mean |
|---|---|---|---|
| Separate PCA | What are the main patterns within each block? | High-variance features within that block | Block-specific directions; axes from different blocks are not automatically shared |
| Unweighted concatenated PCA | What explains the most total feature variation across the combined matrix? | A block with large overall variance, many features, or both | Directions maximizing total variance in the unweighted feature set |
| MFA | What compromise directions emerge after each block’s leading inertia is normalized? | A block can still matter through its lower axes, feature structure, and alignment with other blocks | Global directions in the block-weighted compromise space |
In separate PCA, each block gets its own axes. The first component of one block and the first component of another are not necessarily aligned, and their signs can be reversed without changing the PCA solution. Concatenated PCA does create one shared set of axes, but every feature enters at its chosen scale with no additional block-level correction. MFA also creates a shared set of axes, but each block is first divided by the square root of its own leading eigenvalue. This makes the leading eigenvalue of each scaled block equal to one under the definition above. It does not make all the block’s eigenvalues equal, nor does it guarantee that each block contributes equally to every global component.
Worked Example: The Same Individuals, Different Leading Directions
Consider four hypothetical individuals and two centered, two-feature blocks. Let \(u=(1,-1,1,-1)^\mathsf{T}\) and \(v=(1,1,-1,-1)^\mathsf{T}\). Both vectors have mean zero, \(u^\mathsf{T}u/4=v^\mathsf{T}v/4=1\), and \(u^\mathsf{T}v=0\). Define block A’s columns as \(10u\) and \(0.9v\), and block B’s columns as \(0.1u\) and \(3v\). Thus the same two individual-level patterns are available to all three analyses, but their amplitudes differ across blocks.
Separate PCA. In block A, the feature variances are \(10^2=100\) and \(0.9^2=0.81\), so its first component follows \(u\). In block B, the variances are \(0.1^2=0.01\) and \(3^2=9\), so its first component follows \(v\). Separate PCA therefore highlights different leading patterns in the two blocks.
Unweighted concatenated PCA. Because \(u\) and \(v\) are orthogonal, the combined feature covariance has one eigenvalue for each pattern. Along \(u\), the total variance is \(100+0.01=100.01\). Along \(v\), it is \(0.81+9=9.81\). The first concatenated component therefore follows \(u\), which is driven almost entirely by block A.
MFA. The first eigenvalues of blocks A and B are \(100\) and \(9\), respectively. Their MFA divisors are \(\sqrt{100}=10\) and \(\sqrt{9}=3\). After weighting, the feature amplitudes along \(u\) are \(10/10=1\) in block A and \(0.1/3\) in block B. The resulting variance along \(u\) is \(1^2+(0.1/3)^2=1.00111\). Along \(v\), the weighted amplitudes are \(0.9/10=0.09\) and \(3/3=1\), giving variance \(0.09^2+1^2=1.00810\). MFA’s first global component therefore follows \(v\), although the margin over \(u\) is small. The methods disagree because they answer different questions: the unweighted total-variance question favors \(u\), while the weighted compromise favors the direction most prominent in block B.
This calculation assumes the four rows are matched and ordered identically, both blocks are centered, and the feature scaling shown is intentional. Since the patterns are orthogonal, their variances add without cross-terms. If the two patterns were not orthogonal, the leading directions would generally be mixtures rather than exactly \(u\) or \(v\).
Why MFA’s scaling changes the comparison
The MFA weighting has a direct algebraic justification. Write the eigenvalues of block \(k\)’s covariance matrix \(X_k^\mathsf{T}X_k/n\) as \(\lambda_{1,k}\geq \lambda_{2,k}\geq\cdots\geq 0\). Scaling the block by \(1/\sqrt{\lambda_{1,k}}\) scales its covariance matrix by \(1/\lambda_{1,k}\). Therefore, its largest eigenvalue after scaling is \(\lambda_{1,k}/\lambda_{1,k}=1\). That is the precise sense in which MFA limits dominance by a block whose first eigenvalue is much larger than the others’ first eigenvalues.
The global MFA axes still maximize total variance in the weighted concatenated matrix. Consequently, a block can contribute substantially through several directions, and a global axis can be poorly represented by some blocks. The weighting is a targeted correction to the leading block-scale imbalance, not a guarantee of equal block influence or evidence that blocks share biological signal.
Worked Example: How Unequal Raw Inertia Is Rebalanced
Use the same hypothetical blocks from the previous example. The unweighted inertia of block A is \(100+0.81=100.81\); block B’s is \(0.01+9=9.01\). Across both blocks the total is \(109.82\), so block A accounts for \(100.81/109.82\approx 0.9180\), or 91.80%, of the raw inertia. This makes it unsurprising that unweighted concatenated PCA is strongly oriented toward \(u\).
After MFA scaling, block A’s inertia is \(100.81/100=1.00810\), and block B’s is \(9.01/9=1.00111\). Their weighted total inertias are close, but not identical. In particular, the MFA first component from the calculation above has variance \(1.00810\) along \(v\), of which \(1\) comes from block B and \(0.00810\) from block A. Thus block B supplies \(1/1.00810\approx 0.9920\), or 99.20%, of that component’s variance. A global component can still be almost entirely associated with one block even when the blocks’ overall weighted inertias are similar.
These percentages describe the specified hypothetical matrices, not a biological finding. They also show why a near balance of total block inertia should not be interpreted as proof that every global axis is a balanced consensus.
A practical comparison workflow
Before interpreting differences among maps, make explicit what was held fixed and what each analysis is intended to answer. A disciplined comparison separates preprocessing differences from method differences.
Confirm that each row represents the same individual in every block and that the row order agrees. Resolve missingness or exclusions before comparing methods; otherwise, the analyses may not describe the same individuals.
Center features and document any transformations, filtering, and within-block scaling. A variance-based PCA on unstandardized measurements and one on unit-variance features answer different questions.
Use separate PCA for within-block structure, concatenated PCA for total unweighted feature variance, or MFA for a block-weighted compromise. Do not describe these as interchangeable ways to find a single “true” map.
Compare each block’s initial and weighted inertia, then examine which blocks and features align with the global directions. A block’s influence on the first component may differ from its share of total inertia.
Look for outliers, batch effects, poorly represented individuals or blocks, and sensitivity to defensible preprocessing choices. Interpret an axis cautiously when its eigenvalue is close to a competing one.
The numerical comparison above has a narrow eigenvalue gap in MFA: \(1.00810-1.00111=0.00699\). That small separation warns that the leading direction may be sensitive to small changes in the data. With real data, a plot alone cannot establish stability; resampling or other sensitivity analyses may be useful, provided they reflect the sampling design.
Do not compare component coordinates across separate PCAs as if they were common scores. The axes were calculated independently; even when two axes appear similar, their signs, scales, and ordering can differ. In contrast, concatenated PCA and MFA each define a shared global coordinate system, but those coordinate systems are based on different feature weights. Their maps can be compared only with that difference in mind.
Worked Example: Choosing a Method for a Scientific Question
Suppose a hypothetical study measures transcript abundance and metabolite concentrations on the same 60 individuals. After centering, the transcript block has first eigenvalue \(64\), while the metabolite block has first eigenvalue \(4\). A researcher wants to know whether each assay has a dominant pattern and whether a joint representation gives the two blocks a less unequal starting scale.
For the first part, fit separate PCA. Each assay has its own leading direction and its own explained variance. The values \(64\) and \(4\) show that the transcript block’s leading variance is 16 times the metabolite block’s leading variance under the stated preprocessing, but do not by themselves establish why that difference occurs.
For a joint map that prioritizes the largest total variance across all features at their current scales, use unweighted concatenated PCA. Its objective will include the 16-fold difference in first eigenvalues, so the transcript block may have greater influence. This is appropriate only if those feature scales and the resulting weighting match the scientific objective.
For a compromise map that first equalizes the leading eigenvalues, use MFA. The block divisors are \(\sqrt{64}=8\) for transcripts and \(\sqrt{4}=2\) for metabolites. After these divisions, each block’s leading eigenvalue is \(1\). This reduces the initial imbalance, but does not establish equal influence on every global axis, nor does a shared MFA direction prove that the assays measure a common causal mechanism.
The assumptions for this comparison are that the measurements are correctly matched to the same 60 individuals, the blocks are centered, and any feature transformations are justified and consistently applied. If the blocks have different missing individuals or the metabolite features were transformed differently without documentation, the comparison would not isolate the intended difference between methods.
Diagnostics, alternatives, and limits
Useful checks begin before interpreting component labels. Confirm the individual matching, inspect feature distributions and outliers, and verify that centering and scaling were applied as intended. Compare block-level inertia before and after MFA weighting. Then inspect global and partial representations: an axis that appears global in the score plot may be driven primarily by one block. An unusually large block, redundant features, strong within-block correlation, or batch structure can also shape the result. Block weights do not remove confounding, correct poor measurement, or turn association into causation.
The main alternatives are useful for different aims. Separate PCA is the clearest choice when the main objective is to describe each assay on its own, but it does not produce one shared set of global axes. Unweighted concatenated PCA is direct when each feature’s current variance should count in the combined objective, but it can give disproportionate influence to high-inertia blocks. MFA is appropriate when the analysis calls for a common map after equalizing each block’s leading eigenvalue, but that weighting is a design choice rather than a universally correct correction.
None of these methods explicitly decomposes variation into shared and block-specific components. They also do not identify a biological mechanism or, on their own, support formal inferential claims. The choice among them should therefore follow the scientific question, be accompanied by transparent preprocessing, and be reported with enough detail to make the block weighting clear.
Common mistake
A common mistake is to call the first MFA axis “the pattern shared equally by all assays” because MFA scales each block. The incorrect reasoning is that equal leading eigenvalues imply equal contribution to every component. In fact, scaling makes each block’s largest eigenvalue one; its other eigenvalues can be much smaller, and its features may align weakly with a particular global axis. The correction is to inspect block and feature representation for the axis in question, and to describe MFA as a weighted compromise rather than guaranteed equal consensus.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A matched two-block data set has first eigenvalues 25 and 1 before MFA scaling. What are the MFA divisors, and what are the scaled first eigenvalues?
- Why can unweighted concatenated PCA be dominated by one block even when every feature has been centered?
- Separate PCA gives the first axis \(a\) for one block and \(b\) for another. What must be established before claiming these are the same pattern?
- Does equalizing blocks’ first eigenvalues in MFA ensure that both blocks contribute equally to the first global component? Explain.
- Which method would you choose to summarize each assay independently, and which would you choose for a joint map that maximizes total variance at the current feature scales?
Solutions. (1) The divisors are \(\sqrt{25}=5\) and \(\sqrt{1}=1\). Dividing each block by its divisor divides its covariance eigenvalues by 25 and 1, respectively, so both scaled first eigenvalues are 1. (2) Centering removes feature means, not differences in feature variance. A high-inertia block can still supply most of the combined variance. (3) Since separate PCAs have independently estimated axes, assess whether the directions are aligned using a common representation and the original feature or individual patterns; matching component numbers alone is insufficient, and axis signs are arbitrary. (4) No. MFA equalizes the leading eigenvalues before the global analysis, but the blocks can have different lower eigenvalues and different alignment with the first global direction. (5) Use separate PCA for independent within-assay summaries and unweighted concatenated PCA for the joint total-variance objective at the chosen feature scales.