Block Contributions to Global Components

What a Block Contribution Measures

A global component in Multiple Factor Analysis (MFA) is a direction in the joint, block-scaled feature space. Its score for an individual combines information from features in all active blocks. A block contribution summarizes how much of that direction’s feature-loading weight belongs to one block. It helps answer which blocks define this axis? It does not, by itself, say whether the axis explains a large amount of total data variation or whether a particular individual is well represented.

The earlier material on the MFA compromise space established the use of blocks scaled by their first principal inertia before joint PCA. For the formulas here, assume that the active data matrix for each block has been centered, its features have been put on the chosen within-block scale, and the block has been divided by the square root of its first principal inertia. The global PCA is then performed on the concatenated scaled features, using the same individuals in every block and the same active-feature weights throughout. These assumptions matter: the contribution refers to the scaled features actually used in that PCA, not to an arbitrary unscaled representation.

Here, “scaled feature loading” means the coefficient in the unit-length global eigenvector of the scaled data. Some software uses “loading” for a feature’s correlation with a component instead. Those quantities are related, but are not interchangeable without accounting for feature scaling and the component eigenvalue. The distinction is especially important in MFA because block scaling generally means that the scaled feature columns do not all have variance one.

Thus, each component’s contributions partition 100% of the squared length of its loading direction across the active blocks. A block with a large contribution supplies a large share of the direction’s squared coefficient weight. The calculation squares coefficients, so positive and negative feature coefficients both contribute; their signs do not cancel.

From Correlation Loadings to Contributions

The definition can be used directly when the scaled eigenvector coefficients are available. If a report instead provides correlations, convert them using the actual scaling. Suppose original within-block standardized feature \(x_j\) has variance one, and its block is divided by \(\sqrt{L_g}\), where \(L_g\) is that block’s first principal inertia. The scaled feature is \(z_j=x_j/\sqrt{L_g}\), so its variance is \(1/L_g\). Let \(t_k=Zv_k\) be the global score, with variance \(\lambda_k\), and let \(r_{jk}=\operatorname{Corr}(z_j,t_k)\). From the eigenvector equation, \(\operatorname{Cov}(z_j,t_k)=\lambda_kv_{jk}\). Also, by the definition of correlation, that covariance is \(r_{jk}\sqrt{\lambda_k/L_g}\). Equating the expressions gives:

This conversion assumes that the original features were standardized to variance one and that the block’s sole subsequent scaling was division by \(\sqrt{L_g}\). If features have other weights or variances, use the corresponding feature variances in the covariance-to-coefficient conversion, or obtain the eigenvector coefficients from the fitted analysis. Do not apply this correlation formula blindly.

Why the Contribution Formula Works

The derivation is short but clarifies both the interpretation and its limits. Global PCA gives a unit eigenvector for each component. Partition that vector into subvectors \(v_{1k},\ldots,v_{Gk}\), one per block. Since these subvectors occupy disjoint coordinates, the squared length of the full vector is the sum of their squared lengths:

The block contribution is therefore the squared length of that block’s part of the global direction. Nonnegativity follows because every term is a square, and the sum-to-one property follows from the unit length of \(v_k\). This also explains why reversing the sign convention for a component has no effect: replacing \(v_k\) by \(-v_k\) changes every coefficient’s sign but leaves every squared coefficient unchanged.

Contributions do not partition the component eigenvalue into independent block-specific pieces. In the scaled data, features from different blocks can covary. The variance of a global score includes cross-products between feature columns, so it cannot generally be allocated to blocks by simply summing within-block variances. Contribution is a measure of loading direction, not a decomposition of score variance.

Interpreting, Comparing, and Diagnosing Contributions

Contribution is most useful when reported for a specified component and interpreted alongside the feature coefficients and the component’s overall context. A high value identifies a block whose features collectively receive substantial squared coefficient weight. A low value means the block contributes little to that direction under the fitted scaling. Neither result alone tells whether the component is substantively important, how well individuals are represented on it, or whether the pattern is stable under resampling.

This measure is distinct from two common alternatives. Separate PCA finds directions within each block independently; it can describe block-specific variation but does not give that block’s loading share in the global component. Concatenated PCA without MFA block scaling uses a different feature geometry, so a block with many or high-variance features can dominate more readily. MFA’s contribution calculation is based on its scaled global eigenvector. These approaches answer different questions and their numerical contributions should not be treated as interchangeable.

Several checks make the interpretation more reliable:

• Check the total. Contributions across all active blocks for a component should sum to one, apart from rounding. A substantial discrepancy can indicate that some active features were omitted, the wrong loading quantities were squared, or the reported values are not the eigenvector coefficients.
• Check the scaling. Verify that coefficients correspond to the MFA-scaled active matrix, rather than the original unscaled feature matrix or a supplementary projection.
• Inspect within-block concentration. A high block contribution may be spread over many features or concentrated in one feature. The block total alone does not reveal which situation holds.
• Compare components separately. The contributions sum to one separately for each component. A block’s contribution can be high on one axis and low on another; percentages should not be added across axes as though they were one partition.
• Consider uncertainty and stability. If small data perturbations materially change the loadings or contributions, avoid presenting the block ranking as robust. Contribution calculations describe a fitted solution; they do not provide a sampling uncertainty interval by themselves.

Feature count deserves particular care. A block’s contribution is a sum over its features, so it can be high because several features each contribute moderately, or because a few carry large coefficients. MFA scaling limits the dominance associated with a block’s overall first principal inertia, but it does not force equal contribution on every global component. Nor does a contribution automatically adjust for interpretive redundancy among correlated features within a block.