Partitioning Global Inertia Across Blocks

Why Partition Global Inertia?

In multiple factor analysis (MFA), blocks are first scaled by their leading principal inertia and then concatenated for a joint analysis. The resulting global components describe variation in that scaled concatenated matrix. To interpret those components, it helps to distinguish three quantities: the total inertia in the joint matrix, the amount supplied by each block, and each block’s contribution to a particular global component. These quantities are related, but they answer different questions.

Here, inertia means the sum of squared entries of a centered data matrix, equivalently its squared Frobenius norm. This convention uses unnormalized cross-products, so inertia and eigenvalues are sums of squared deviations rather than averages per individual. Dividing every inertia and eigenvalue by the same sample-size factor would change their numerical scale, but not their proportions.

The same definitions apply if a block’s columns have been standardized before its principal inertia is calculated. The preprocessing convention must be fixed first: the \(\lambda_{1k}\) used for scaling must belong to the actual centered block being analyzed. The setup assumes matched rows, finite entries, and positive leading inertia in each block. It does not require blocks to be independent. If a block has zero inertia, it contains no variation to scale and needs separate treatment rather than division by zero.

How Block Inertias Add

Concatenation places the blocks side by side; it does not combine or average their columns. Consequently, the squared entries of the concatenated matrix are exactly the squared entries of all the blocks taken together. The trace calculation makes the partition explicit.

If the eigenvalues of \(X_kX_k^\mathsf{T}\), including zero eigenvalues if necessary, are \(\lambda_{1k},\lambda_{2k},\ldots\), then scaling divides every eigenvalue by \(\lambda_{1k}\). Thus the block’s scaled inertia is \(I_k=\sum_j\lambda_{jk}/\lambda_{1k}\). The leading scaled eigenvalue in each block equals one, while the other eigenvalues express that block’s remaining inertia relative to its own leading direction.

Partitioning Inertia Across Global Components

Let \(C=ZZ^\mathsf{T}=\sum_k Z_kZ_k^\mathsf{T}\). Write its eigenvalues in descending order as \(\mu_1,\mu_2,\ldots\), with corresponding orthonormal eigenvectors \(u_1,u_2,\ldots\). These are the global eigenvalues and individual-space axes of the concatenated analysis. For global component \(\ell\), define the inertia supplied by block \(k\) as \(c_{k\ell}=u_\ell^\mathsf{T}Z_kZ_k^\mathsf{T}u_\ell\). This is a nonnegative quantity: it equals the squared length of that block’s projection onto the unit vector \(u_\ell\).

For the first identity, use \(Cu_\ell=\mu_\ell u_\ell\) and \(u_\ell^\mathsf{T}u_\ell=1\). Then \(u_\ell^\mathsf{T}Cu_\ell=\mu_\ell\), while substituting \(C=\sum_k Z_kZ_k^\mathsf{T}\) gives \(\sum_k u_\ell^\mathsf{T}Z_kZ_k^\mathsf{T}u_\ell=\sum_k c_{k\ell}\). For the second identity, sum over an orthonormal eigenbasis: \(\sum_\ell c_{k\ell}=\sum_\ell u_\ell^\mathsf{T}Z_kZ_k^\mathsf{T}u_\ell=\operatorname{tr}(Z_kZ_k^\mathsf{T})=I_k\). The trace identity holds even when some eigenvalues are zero; zero-inertia directions add nothing.

A component’s block share is \(c_{k\ell}/\mu_\ell\), provided \(\mu_\ell>0\). It describes the fraction of that component’s inertia associated with the block. In contrast, \(I_k/I\) is the block’s share of total inertia across all components. The first is component-specific; the second is not. Neither by itself establishes a causal or biological mechanism.

Reading the Partition and Checking the Calculation

In the example, each block supplies half of each global eigenvalue, because their component contributions are equal. Each also supplies half of the total inertia. These equalities happen because of the chosen data; they are not guaranteed by first-eigenvalue scaling. Scaling gives every block a leading eigenvalue of one, but blocks can retain different amounts of inertia in their remaining directions, and their axes can align or disagree. Thus equal leading eigenvalues do not imply equal contributions to every global component.

Several checks help identify calculation or interpretation problems. The scaled block inertias must add to total inertia; the global eigenvalues must also add to that same total. Each \(c_{k\ell}\) must be nonnegative, and contributions across blocks must add to the corresponding global eigenvalue. Finally, summing a block’s contributions across a complete set of axes must recover its inertia. Small discrepancies can result from rounding, but a substantial mismatch usually signals inconsistent scaling, omitted axes, or a matrix/eigenvalue convention that has not been applied consistently.

A frequent mistake is to divide a block’s global component contribution by its raw inertia when interpreting its share of the scaled global component. That mixes scales: the relevant denominator for a component share is \(\mu_\ell\), and the relevant block numerator is \(c_{k\ell}\), both calculated from the scaled matrix. Another mistake is to treat the block’s total inertia share \(I_k/I\) as its share of every axis. Component-specific shares can vary across axes even when total block shares are fixed.

How This Partition Differs From Other Summaries

Separate PCA reports each block’s eigenvalues on its own scale; it does not, by itself, provide the global component contributions \(c_{k\ell}\). PCA of an unscaled concatenation instead lets blocks with large raw inertia have correspondingly large influence. MFA’s first-eigenvalue scaling changes the scale before concatenation, and the partition identities above describe the resulting joint analysis. They do not imply that each block has equal total inertia or equal influence on every axis.

The partition is an accounting identity, not a test of whether a block is biologically important. It depends on the observed blocks, the preprocessing and scaling choices, and the axes retained in a reported summary. In particular, if only a subset of axes is shown, the displayed contributions for a block sum to no more than its total inertia; the difference is the block’s inertia on omitted axes. Interpret the figures alongside the analysis design and data quality rather than treating a large share as proof of a substantive mechanism.