Reading Global and Partial Axes Together

Reading Two Views of the Same MFA Axis

A global MFA map summarizes the compromise across matched data blocks. A partial map puts each block’s view of the same individuals into that compromise space. Reading them together helps answer a focused question: for a given individual and global axis, does a block point in the same direction as the compromise, or does it depart from it?

The comparison is meaningful only when the global and partial coordinates refer to the same fitted MFA axis, the same individuals, and compatible coordinate scaling. A partial coordinate is not a score from a separate PCA of that block. It is the block’s projection onto an axis of the global compromise. This distinction is central: independently fitted axes can point in either direction and need not represent the same orientation.

Definition: Let \(F_{i\alpha}\) be individual \(i\)’s global coordinate on global axis \(\alpha\). Let \(F_{i\alpha}^{(k)}\) be that individual’s partial coordinate for block \(k\), expressed on the same global axis and in the usual MFA barycentric scaling. Positive and negative coordinates indicate opposite directions along that axis; their absolute values indicate distances from the axis origin in coordinate units.

The Barycentric Relationship

In standard MFA, blocks are represented using a common set of individuals and scaled so that no block dominates solely because of its overall scale. For quantitative blocks, this typically involves standardizing variables within blocks and scaling each block by its first principal inertia, as in the earlier material on MFA scaling. The global analysis then uses the concatenated, scaled blocks.

To see how the partial coordinates relate to a global coordinate, write the scaled data for block \(k\) as \(Y_k\). Let \(v_{k\alpha}\) be the part of the global eigenvector for axis \(\alpha\) corresponding to block \(k\). The block’s additive projection for individual \(i\) is \(y_{ik}^{\mathsf T}v_{k\alpha}\). In the usual MFA partial-individual display, the partial coordinate is \(K\) times this projection, where \(K\) is the number of blocks. With equal block weights, the global coordinate is the sum of the block projections.

F_{i\alpha}=\sum_{k=1}^{K} y_{ik}^{\mathsf T}v_{k\alpha}, \qquad F_{i\alpha}^{(k)}=K\,y_{ik}^{\mathsf T}v_{k\alpha}, \qquad F_{i\alpha}=\frac{1}{K}\sum_{k=1}^{K}F_{i\alpha}^{(k)}.

The last equality is the barycentric relationship: for each individual and axis, the global coordinate is the average of the block-specific partial coordinates. It provides both an interpretation and a practical consistency check. In a standard equal-block MFA display, if the reported global coordinate is not the mean of the partial coordinates (up to rounding), check whether the same axis, individuals, and partial-coordinate scaling have been used.

This relationship does not say that every block must agree with the global coordinate. The average can lie between partial coordinates on opposite sides of zero. A block can therefore reinforce the global direction for an individual, oppose it, or be close to the origin while other blocks determine the average.

Interpretation rule: For an individual and a fixed global axis, partial and global coordinates with the same nonzero sign lie on the same side of the origin. Opposite signs indicate opposing directions on that axis. The size of the partial coordinate describes its position on the axis; a large departure from the global coordinate describes disagreement with the compromise for that individual and axis.

Worked Comparisons

Worked Example: Checking Agreement Across Two Blocks

Suppose an MFA of two hypothetical omics blocks gives the following partial coordinates for four matched samples on global axis 1. The global coordinate is their average, as required under the standard barycentric scaling.

Sample	Block 1 partial coordinate	Block 2 partial coordinate	Global coordinate
A	2.0	0.0	1.0
B	1.5	0.5	1.0
C	-1.0	-1.0	-1.0
D	0.5	-1.5	-0.5

For sample A, the calculation is \((2.0+0.0)/2=1.0\). Block 1 is on the positive side with the global coordinate and has a larger magnitude; Block 2 is at zero and has no directional position on this axis for this sample. For sample C, \((-1.0-1.0)/2=-1.0\), so both blocks agree with the negative global direction.

Sample D illustrates disagreement: \((0.5-1.5)/2=-0.5\). Block 1 is positive, opposite the global coordinate, while Block 2 is negative and has greater magnitude. The global point is not “wrong”; it is the compromise average of the two partial positions. These scores are hypothetical MFA output in axis-coordinate units, not measurements in the original omics units.

Worked Example: Reading Three Partial Coordinates for One Sample

For a hypothetical sample on global axis 2, suppose three blocks have partial coordinates \(1.8\), \(-0.6\), and \(0.3\). The global coordinate is

F_{i2}=\frac{1.8+(-0.6)+0.3}{3} =\frac{1.5}{3}=0.5.

The global coordinate is positive. Block 1 is also positive and is the largest partial coordinate in magnitude, so it strongly aligns with the global direction for this sample. Block 2 is negative and therefore points in the opposite direction. Block 3 is positive but closer to zero than the global coordinate, so it agrees in direction but is less displaced from the origin.

This is a statement about one sample and one axis. It does not establish that Block 1 is the most important block overall, nor does it describe agreement across all samples. Those broader questions require summaries and analyses beyond this coordinate-by-coordinate reading.

Worked Example: Correcting an Axis-Reversal Misreading

Suppose a separate PCA of a hypothetical block reports sample scores \((-2,-1,1,2)\), while the global MFA coordinates for the corresponding samples are \((2,1,-1,-2)\). It would be tempting to conclude that the block opposes the global axis for every sample because every pair of signs differs.

But the separate PCA axis can be multiplied by \(-1\) without changing the fitted one-dimensional solution. Reorienting those separate scores gives \((2,1,-1,-2)\), matching the global signs. The apparent opposition arose from an arbitrary axis reversal, not a substantive disagreement.

This separate-PCA comparison is not a substitute for partial coordinates. The partial coordinates are already projected onto the oriented global axis. If a separate analysis is used as a descriptive comparison, its axis orientation must first be aligned, and even then the two analyses may have different axes and scaling. For judging agreement with a particular MFA axis, use the partial coordinates from the same MFA fit.

How to Make the Comparison Reliably

A practical reading proceeds in a fixed order. First confirm that all coordinates refer to the same fitted global axis and matched samples. Then check the barycentric average. Finally compare signs and magnitudes, keeping the interpretation at the individual-and-axis level unless a separate across-sample summary is calculated.

Confirm the reference.
Verify that global and partial coordinates use the same axis number, the same orientation, and the same sample ordering. Partial coordinates from a global MFA fit are the relevant quantities.

Check the average.
For each sample and axis, calculate the mean of the \(K\) partial coordinates and compare it with the reported global coordinate. Small discrepancies may be due to rounding; larger ones signal a mismatch in scaling, axis, or data.

Read direction.
Matching nonzero signs mean the block’s partial point and global point lie on the same side of the origin. Opposite signs mean they lie on opposite sides. A zero coordinate has no positive or negative direction.

Read magnitude in context.
Compare absolute coordinate values and the difference \(F_{i\alpha}^{(k)}-F_{i\alpha}\). These describe position and departure along this axis; they are not original-unit changes or, by themselves, measures of overall block importance.

A difference score can make a particular comparison explicit:

\Delta_{i\alpha}^{(k)}=F_{i\alpha}^{(k)}-F_{i\alpha}.

A positive or negative difference tells whether the partial coordinate is farther in the positive or negative direction relative to the global coordinate. It does not alone say whether the block agrees with the global direction: for that, inspect both coordinates’ signs. For instance, a positive difference can occur when both coordinates are negative, if the partial coordinate is less negative.

Diagnostics, Alternatives, and Limits

Several checks help detect a faulty reading. A partial-coordinate table that does not average to the global coordinates may use a different scaling convention, an incorrect sample match, or coordinates from different axes. A scatterplot of partial against global coordinates can reveal broad alignment and unusual samples, but its appearance depends on the axis and sample set; inspect the coordinate pairs and barycentric check as well. Coordinates close to zero should be treated cautiously because a tiny sign difference can look like disagreement while indicating little displacement along that axis.

The most common alternative is to compare the global map with separate PCA maps for each block. Separate analyses can describe each block’s own leading variation, but their axes were fitted independently. Their sign and scale are not automatically aligned with the global MFA axis, and a block’s first separate axis need not correspond to global axis 1. Use them to explore block-level structure, not as if they were the partial projections from the global compromise.

Another alternative is to summarize across samples, for example by examining how partial coordinates vary with global coordinates. A single summary can be useful, but can conceal samples that point in opposing directions or lie near zero. Coordinate-by-coordinate inspection and an across-sample summary answer different questions and should not be confused.

Finally, partial/global agreement is descriptive, not causal. A block aligned with a global axis is not thereby shown to cause the pattern, and a block departing from it is not necessarily biologically irrelevant. The comparison is limited to the fitted representation, preprocessing, selected axis, and observed samples. It also does not replace later analyses of block contributions or representation quality.

Common Mistake

Common mistake: “The partial coordinate has a larger absolute value than the global coordinate, so this block must be more important overall.” This conclusion does not follow. The coordinate describes one sample’s position on one global axis, and the global coordinate is an average across blocks. A larger magnitude can indicate a stronger displacement for that sample on that axis, but it is not a general measure of block importance. Check the signs, the barycentric average, and the scope of the comparison before interpreting it.

Check Your Understanding

Use the usual equal-block barycentric scaling. Give answers in terms of the stated global axis and samples.

Two blocks have partial coordinates \(1.2\) and \(-0.4\) for one sample. What is the global coordinate, and which block opposes its direction?
For three blocks, the partial coordinates are \(-0.9\), \(-0.3\), and \(0.6\). Calculate the global coordinate and identify the block with the opposite sign.
A reported global coordinate is \(0.7\), but the three partial coordinates are \(0.8\), \(0.9\), and \(0.7\). What diagnostic issue should be checked?
Why is a block’s first axis from a separate PCA not automatically the same as the corresponding global MFA axis?

Solutions. (1) The global coordinate is \((1.2-0.4)/2=0.4\). The second block is negative and opposes the positive global direction. (2) The global coordinate is \((-0.9-0.3+0.6)/3=-0.6/3=-0.2\). The third block, at \(0.6\), has the opposite sign. (3) The partial mean is \((0.8+0.9+0.7)/3=2.4/3=0.8\), not \(0.7\). Check axis matching, sample matching, scaling convention, and rounding; the difference is too large to attribute to ordinary rounding at the displayed precision. (4) It was fitted independently, so its direction can be reversed and its variation need not align with the global axis. Partial coordinates from the MFA fit project the block onto the global axis directly.