Sensitivity of MFA Maps to Block Weights

Why Examine Weight Sensitivity?

Multiple Factor Analysis (MFA) uses block weights to keep a block with large within-block variation from automatically dominating the global analysis. The usual reference weighting divides each preprocessed block by its largest singular value. But that weighting is a choice with a purpose, not a guarantee that every scientific conclusion is insensitive to reasonable alternatives. If modest changes to the weights substantially alter the sample map or the contributions of blocks, an interpretation based on one map alone may be fragile.

A sensitivity analysis recomputes the global solution under specified alternative weights while holding the individuals, feature preprocessing, and block definitions fixed. It asks how much the resulting axes, sample coordinates, and block contributions change. It does not estimate sampling uncertainty, prove that one weighting is correct, or turn an association in a map into a causal claim.

Reference and Alternative Weighting

Definition: Let $X_k$ be the preprocessed data matrix for block $k$, with the same individuals in each block, and let $\delta_k$ be its largest singular value. The reference MFA block is $Z_k=X_k/\delta_k$. For a set of positive multipliers $r_1,\ldots,r_K$, the alternative weighted matrix is $$ Z(r)=\left[r_1 Z_1\;\;r_2 Z_2\;\;\cdots\;\;r_K Z_K\right]. $$ Reference MFA corresponds to $r_k=1$ for every block. An alternative weighting changes the relative influence of blocks; it does not change the within-block preprocessing or the observed individuals.

For a chosen weighting, compute the singular value decomposition $Z(r)=U D V^\mathsf{T}$. Write $\lambda_\ell$ for the $\ell$th eigenvalue of $Z(r)^\mathsf{T}Z(r)$, and $v_\ell$ for its unit eigenvector. The sample coordinates on component $\ell$ are $f_\ell=Z(r)v_\ell$, with $f_\ell=U_\ell\sqrt{\lambda_\ell}$. These conventions use the unscaled cross-product matrix, so the eigenvalues and coordinates are not divided by the number of individuals.

The comparison is meaningful only if the data being compared are genuinely held fixed apart from the weights. In particular, keep the same individuals, feature filtering, missing-data handling, centering or scaling choices, block membership, and retained dimensions. If one of these changes too, the observed difference cannot be attributed to weights alone. Multipliers should be positive; setting a block's multiplier to zero removes it rather than merely downweighting it.

Comparing Axes, Coordinates, and Contributions

Eigenvectors have arbitrary signs: $v_\ell$ and $-v_\ell$ describe the same axis, but reverse all coordinates on it. Before comparing two solutions, orient a matched pair of axes to have a nonnegative inner product, or equivalently a nonnegative correlation between their sample coordinates. If eigenvalues change rank, match axes by their similarity rather than assuming that “component 1” in one fit corresponds to “component 1” in the other.

For axes that have been matched and sign-aligned, a simple coordinate-change summary is the root mean squared displacement across the $n$ individuals:

\operatorname{RMSD}_\ell= \sqrt{\frac{1}{n}\sum_{i=1}^{n} \left(f_{i\ell}^{\,\mathrm{alt}}-f_{i\ell}^{\,\mathrm{ref}}\right)^2}.

This is in the units of the sample factor coordinates (here, the weighted, dimensionless data scale). It should be interpreted alongside the scale of those coordinates and the eigenvalues. A small displacement can still matter if it changes a key sample ordering; a larger displacement may simply reflect a uniform rescaling.

When two eigenvalues are close, the corresponding axes can rotate considerably under small perturbations even if the plane they span changes little. In that case, comparing individual axes can be misleading. Align the two-dimensional coordinate configurations by an orthogonal Procrustes rotation, or compare the projectors onto the two retained subspaces. A stable subspace does not imply that each axis within it has a stable interpretation.

Definition: For a nonzero component $\ell$, the contribution of block $k$ to that component, using the weighted columns in $Z_k(r)=r_kZ_k$, is $$ \operatorname{ctr}_{k\ell} =\frac{\left\|Z_k(r)^\mathsf{T}f_\ell\right\|^2}{\lambda_\ell^2}. $$ It is the fraction of the component's squared feature associations attributable to features in block $k$. The contributions sum to one across blocks for each component.

To verify the sum, let $v_{k\ell}$ be the entries of $v_\ell$ belonging to block $k$. Since $Z(r)^\mathsf{T}f_\ell=\lambda_\ell v_\ell$, the block part is $Z_k(r)^\mathsf{T}f_\ell=\lambda_\ell v_{k\ell}$. Therefore $\operatorname{ctr}_{k\ell}=\|v_{k\ell}\|^2$, and summing over blocks gives $\sum_k\|v_{k\ell}\|^2=\|v_\ell\|^2=1$. A contribution is not the fraction of total inertia in the block, nor does it measure the block's causal importance.

Worked Example: A Leading Axis Rotates Under a Lower Block Weight

Consider two hypothetical one-feature blocks on three individuals. The preprocessed columns are $x_1=(1,-1,0)^\mathsf{T}/\sqrt{2}$ and $x_2=(1,0,-1)^\mathsf{T}/\sqrt{2}$. Each has singular value 1, so these are also the reference MFA columns. Their inner product is $x_1^\mathsf{T}x_2=0.5$. Compare reference weights $(1,1)$ with alternative weights $(1,0.5)$.

Under the reference weights, the cross-product matrix for the two columns is $$ \begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}. $$ Its leading eigenvalue is $1.5$, with unit eigenvector $(1,1)^\mathsf{T}/\sqrt{2}$. Thus the leading sample coordinates are $f_1=(x_1+x_2)/\sqrt{2}=(1,-0.5,-0.5)^\mathsf{T}$.

Under alternative weights, the columns are $x_1$ and $0.5x_2$, giving $$ \begin{pmatrix}1&0.25\\0.25&0.25\end{pmatrix}. $$ The leading eigenvalue is $\{1.25+\sqrt{0.8125}\}/2=1.0757$. If the leading eigenvector is $(\cos\theta,\sin\theta)^\mathsf{T}$, then $\tan(2\theta)=2(0.25)/(1-0.25)=2/3$, so $\theta\approx16.85^\circ$. The alternative leading coordinates are $\cos(\theta)x_1+0.5\sin(\theta)x_2\approx(0.779,-0.677,-0.102)^\mathsf{T}$.

The signs already agree, so no sign reversal is needed. The RMS displacement is $\sqrt{\{(0.779-1)^2+(-0.677+0.5)^2+(-0.102+0.5)^2\}/3}\approx0.282$. The sample map has changed appreciably: the third individual's coordinate moved from $-0.500$ to about $-0.102$. This calculation describes a change in the fitted map under the specified weighting, not uncertainty about an individual or a parameter.

Worked Example: Tracking Block Contributions Across Weights

Use the same hypothetical two-block data and fits. Under reference weights, the leading eigenvector has squared entries $0.5$ and $0.5$. By the contribution definition, block 1 contributes $0.5$ and block 2 contributes $0.5$ to the leading component.

For the alternative weighting, the leading eigenvector has angle $\theta\approx16.85^\circ$. Its squared entries are $\cos^2\theta\approx0.9160$ and $\sin^2\theta\approx0.0840$. Hence the alternative component contributions are approximately $0.9160$ for block 1 and $0.0840$ for block 2; they sum to 1. The reduction of block 2's contribution is consistent with its downweighting, but the contribution is not simply equal to the multiplier 0.5. It depends on the joint eigenvector as well as the weights.

This comparison gives a concrete sensitivity result: block 2 supplies half the leading component's feature association under reference weighting, but less than one tenth under the alternative. Any claim that both blocks contribute comparably to this particular axis is therefore weight-dependent.

When Conclusions Are and Are Not Robust

Worked Example: A Stable Plane but an Unstable First Axis

Suppose two hypothetical one-feature blocks have orthogonal unit-norm columns $x=(1,-1,0)^\mathsf{T}/\sqrt{2}$ and $y=(1,1,-2)^\mathsf{T}/\sqrt{6}$. Their inner product is zero. Compare weights $(1,0.8)$ with $(0.8,1)$.

For weights $(1,0.8)$, the cross-product matrix is diagonal with entries $1$ and $0.64$. The leading axis is $x$, and the second is $y$. Swapping the weights gives diagonal entries $0.64$ and $1$, so the leading axis is now $y$. For example, individual 3 has coordinate $0$ on the first axis under the first weighting, but coordinate $-2/\sqrt{6}\approx-0.8165$ on the first axis under the second.

The conclusion “individual 3 is especially low on the leading axis” is not robust to this weighting change. However, the two-dimensional subspace spanned by $x$ and $y$ is identical in both fits. A conclusion based on the full two-dimensional configuration can therefore be robust even though a one-axis interpretation is not. The leading-axis switch is especially important to report; it should not be hidden by matching components solely by their rank.

A useful sensitivity design specifies plausible alternatives before inspecting the resulting maps. For example, vary one block multiplier at a time around 1, then examine a small grid that includes simultaneous changes when those are scientifically plausible. Report the actual multiplier values and compare eigenvalues, matched coordinates, and block contributions. No universal cutoff makes a coordinate displacement “large”; judge it against the map's scale and the scientific distinctions being interpreted.

Diagnostics, Alternatives, and Limitations

Several patterns signal that a map interpretation may be fragile:

A leading eigenvalue approaches the next eigenvalue, or their order changes across weight choices. This warns that axis directions may rotate or exchange rank.
Sample positions, rankings, or apparent group separations change after sign and axis alignment. Inspect whether the change is concentrated in a few individuals or widespread.
Block contributions move sharply, particularly when an interpretation depends on one block being dominant or on several blocks sharing an axis.
Different plausible weights support different interpretations. Report this dependence rather than selecting the weighting that produces the clearest desired pattern.

The primary alternative is to report only the reference MFA solution. That is simpler and retains the standard equalization based on each block's leading singular value, but does not show whether conclusions depend on that choice. An unweighted concatenated PCA answers a different question because blocks with more columns or greater total variation can exert more influence. Separate PCA describes within-block structure rather than a shared global map. These alternatives can be informative, but none is a substitute for stating which weighting defines the analysis.

Sensitivity analysis has limits. It evaluates the selected range of weights, not every possible weighting, and it does not resolve uncertainty caused by sampling, measurement error, missingness, or block construction. Nor can stability establish biological validity: the same pattern may persist under several weights and still reflect confounding or a preprocessing artifact. Keep weight sensitivity distinct from bootstrap or other sampling-uncertainty analyses.

Common Pitfall

Common mistake: “The alternative map has a different first axis, so the biological structure disappeared.” This conclusion can be wrong because axes may change sign, swap order, or rotate within a nearly tied subspace. First align and match axes; if eigenvalues are close, compare the retained subspace and the sample configuration within it. Conversely, a stable-looking plot under one alignment does not establish robustness: also compare coordinates and block contributions across the stated weights.

Fix the analysis inputs.
Keep preprocessing, individuals, blocks, and retained features unchanged; state the reference and alternative multipliers.

Refit and align.
Recompute each weighted global PCA, match components by similarity, and account for sign reversals or near-tied axes.

Quantify sensitivity.
Compare eigenvalues, aligned sample coordinates or subspaces, and block contributions; explain which substantive conclusions persist.

Check Your Understanding

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

What must be held fixed to attribute a difference between two MFA maps to block weights?
Why should the signs of matched component coordinates be aligned before calculating an RMS displacement?
Two leading eigenvalues are nearly equal, and their individual axes rotate under small weight changes. What comparison is more informative?
If block contributions to one component are 0.72 and 0.28, what does that say, and what does it not say?

Solutions. (1) Hold fixed the individuals, feature preprocessing, missing-data handling, block membership, and retained features; change only the specified weights. (2) An eigenvector and its negative represent the same axis, so an uncorrected sign reversal would create an artificial coordinate difference. (3) Compare the retained subspace, for example by aligning the coordinate configurations with an orthogonal Procrustes rotation or comparing subspace projectors. (4) Under that weighting, the first block accounts for 72% of the component's squared feature associations under the stated contribution definition. It does not establish causal importance, total block inertia, or that the same contribution holds under another weighting.