What Feature Loadings Can—and Cannot—Tell You
An MFA map can show that several features point in similar directions, or that analogous features from different omics blocks align with the same global axis. These patterns help describe how measured features vary with the sample structure represented by that axis. But the interpretation depends on which scores the features are associated with: the global score, assembled from all blocks, or a block-specific partial score.
This tutorial focuses on interpreting those two types of feature association. It assumes the blocks have been prepared for MFA and their leading singular values used for block scaling. The central distinction is straightforward: a global loading describes association with a component built from all blocks; a partial loading describes association with that component as expressed through one block. Neither is a causal effect, and similar loadings do not by themselves establish a direct relationship between two features.
Reading Global and Partial Loadings
For a standardized feature \(x_{jk}\) in block \(k\), its global loading on component \(l\) is \(\operatorname{corr}(x_{jk},F_l)\). Its partial loading is \(\operatorname{corr}(x_{jk},F_{kl})\). The same feature can have different global and partial loadings because its global association includes the score contributions of all blocks, whereas its partial association uses only its own block’s contribution.
With sample covariance computed using divisor \(n-1\), a standardized feature has sample variance 1. Therefore, whenever the score has positive variance,
The analogous formula for a partial loading replaces \(F_l\) by \(F_{kl}\). These are correlations, so they are dimensionless and range from \(-1\) to \(1\). A positive loading indicates that higher feature values tend to occur with higher scores on the specified axis; a negative loading indicates that higher feature values tend to occur with lower scores. The axis itself has no inherent direction: reversing its sign reverses all associated scores and loadings without changing the PCA solution.
The relation \(F_l=\sum_k F_{kl}\) gives a useful interpretation of the difference between global and partial association:
Thus a feature may load strongly on the global component even if its association with its own block’s partial score is modest. Its global association can also reflect how that feature varies alongside partial scores from other blocks. This is a description of covariance in the observed data, not evidence that one block influences another.
Worked Example: Finding a Global Feature Loading
In a hypothetical MFA analysis of 120 individuals, a standardized metabolite \(x\) has sample covariance \(0.72\) with global score \(F_1\). The sample variance of \(F_1\) is \(1.20\). Calculate and interpret the global loading.
Because \(x\) is standardized, its sample variance is 1. The correlation is
The metabolite has a positive global loading of about \(0.657\): individuals with higher values of this metabolite tend, in this dataset, to have higher scores on global component 1. This does not mean that the metabolite causes the score or that component 1 causes a change in the metabolite.
Interpreting Patterns Within a Block
Within one block, features with loadings of similar sign and substantial magnitude are associated with the same side of an axis. For example, several positively loading transcripts may jointly characterize individuals with high global scores on a component. Features with opposite signs distinguish opposing ends of that axis. When the goal is to describe a pattern, it is often helpful to report the features, their block, the component, whether the values are global or partial loadings, and the sign convention used for the axis.
Similar loadings do not guarantee a high pairwise correlation between the features. Both features can correlate with the same score for different reasons, and their remaining variation may differ. If the scientific question concerns whether two features are correlated with one another, inspect or calculate their pairwise correlation directly. A loading plot is not a substitute for that calculation.
Worked Example: Separating Global and Partial Associations
For a hypothetical standardized feature \(x\) in block A, suppose the partial score \(F_{A1}\) has variance \(0.25\), with \(\operatorname{cov}(x,F_{A1})=0.20\). The global score \(F_1\) has variance \(1.20\), with \(\operatorname{cov}(x,F_1)=0.72\). Calculate both loadings and explain the difference.
The partial loading is
The global loading is
The global association is larger than the partial association in this example. Since \(F_1=F_{A1}+\sum_{m\ne A}F_{m1}\), the covariance contributed by the other blocks is \(0.72-0.20=0.52\). The feature’s global loading therefore reflects its association with the global score, including covariation with other blocks’ partial scores. It would be incorrect to describe \(0.657\) as the feature’s within-block association; that is the partial loading, \(0.40\).
Comparing Analogous Features Across Blocks
When features represent analogous measurements in different blocks—for example, a metabolite and a transcript in a hypothesized pathway—compare their global loading profiles and their partial loadings. Similar global loadings indicate that both features are associated with the same global direction. Similar partial loadings indicate that each is associated with its own block’s contribution in a similar direction. These comparisons can suggest coordinated structure worth investigating, but they do not establish that the features are directly correlated, functionally linked, or causally connected.
A useful interpretation sequence is to first identify the component and its sign orientation, then inspect global loadings to describe the all-block pattern, and finally inspect partial loadings to see how each block expresses that pattern. If the scientific claim concerns a direct cross-block relationship, calculate an appropriate cross-block association separately and account for the study design and any relevant confounding.
Worked Example: Comparing Analogous Features Across Blocks
In a hypothetical analysis, a transcript in block A has global loading \(0.76\) and partial loading \(0.58\). An analogous metabolite in block B has global loading \(0.73\) and partial loading \(0.62\), both on global component 1. Interpret the comparison.
The two global loadings are both positive and similar: \(0.76\) and \(0.73\). Both features therefore tend to be higher among individuals with higher global component 1 scores. Their partial loadings are also positive and similar, \(0.58\) and \(0.62\), indicating that each feature is positively associated with its own block’s partial score.
This is evidence of a similar loading pattern on the chosen component, not proof that the transcript and metabolite have a pairwise correlation of \(0.73\), or that one regulates the other. To answer whether the two measurements are correlated across individuals, their direct cross-block correlation must be calculated from the paired observations. To make a causal claim would require a design and assumptions that support causal inference; MFA loadings alone do not provide them.
Derivation and Practical Checks
The loading formula follows directly from the definition of correlation. For any nonconstant feature \(x\) and score \(F\), sample correlation is sample covariance divided by the product of their sample standard deviations:
Because the feature columns in \(X_k\) are standardized, \(s_x=1\). Also \(s_F=\sqrt{\operatorname{var}(F)}\). Substitution gives \(r_{x,F}=s_{x,F}/\sqrt{\operatorname{var}(F)}\), the stated loading formula. If the score has zero variance, its correlation with a feature is undefined; it should not be interpreted as a zero loading.
Before drawing conclusions, check that the loadings are being interpreted on the intended scale and against the intended scores. In particular, confirm that they are feature-score correlations rather than entries of a PCA direction vector. These quantities are related but are not generally interchangeable. Also check whether reported coordinates are global or partial and whether the axis sign has been flipped for display.
Patterns can be unreliable when loadings are small, when a component is weak, or when nearby components have similar eigenvalues and their directions may shift across resamples. Compare the loading pattern across plausible resamples or preprocessing choices if stability matters. A feature can also appear prominent because of preprocessing, outliers, or a small number of influential individuals; inspect the data and the relevant score-feature relationships rather than relying only on a static map.
| Question | Useful quantity | What it supports |
|---|---|---|
| Which features vary with the all-block axis? | Global feature loadings | Description of association with the global score |
| How does a feature relate to its own block’s contribution? | Partial feature loadings | Description of within-block association with the partial score |
| Are two particular features correlated? | Their direct pairwise correlation | Description of their observed pairwise association |
| Does one feature cause another to change? | A suitable causal design and analysis | Causal interpretation only under its assumptions |
A correlation matrix or a direct cross-block correlation is more appropriate than loadings when the target is a particular pairwise association. Separate PCA can help describe structure within each block, but its axes need not align across blocks. MFA loadings are useful when the target is association with shared, weighted global components and their block-specific contributions. These approaches answer different questions; one does not make the others unnecessary.
The main limitations are interpretive, not just computational. Loadings summarize linear association with selected scores, depend on the data representation and component direction, and can conceal variation not represented by the displayed components. Similarity of loadings is not a test of equality, and a loading is not an effect size for a causal intervention. Keep the claim at the level supported by the quantity calculated.
Common Mistake
Wrong reasoning: “Two features have large positive loadings on the same MFA axis, so they must be strongly correlated and one may drive the other.”
Correction: Each loading describes association between one feature and a score. The two features may both align with that score without having a strong direct pairwise correlation. Calculate their pairwise association to assess that claim; use a suitable causal design and assumptions for causal conclusions. Shared direction in an MFA map is descriptive evidence of structure, not a causal result.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A standardized feature has covariance \(0.30\) with a score whose variance is \(0.64\). What is its loading?
- A feature has partial loading \(0.25\) and global loading \(0.70\). Does this show that its association with its own block contribution is stronger than its association with the global score?
- Two features have positive global loadings of \(0.80\) and \(0.78\). What can you conclude, and what can you not conclude, about their pairwise correlation?
- Why does reversing the sign of a component not change the substantive PCA solution?
Solutions. 1. The loading is \(0.30/\sqrt{0.64}=0.30/0.80=0.375\). 2. No. The partial loading is the association with the feature’s own block partial score; the global loading is its association with the all-block score. Here the global association is stronger, but the two quantities answer different questions. 3. Both features are positively associated with the same global direction. Their pairwise correlation cannot be determined from these two loadings; calculate it directly from the paired feature values. 4. Reversing the component direction changes every score and loading sign together but leaves fitted relationships and the underlying axis unchanged. The sign is a convention, not a substantive property.