Global Scores and Partial Individual Coordinates

From the Compromise Map to Block-Specific Coordinates

An MFA compromise map gives one global position for each matched individual. That position summarizes information across blocks, but by itself it does not show how each block places that same individual relative to the compromise. Partial individual coordinates make this comparison possible: they place each individual separately for each block, using the global axes as the shared reference. A partial point near its global point agrees with the compromise on the displayed axes; a point far away signals disagreement on those axes.

The calculation depends on using the same individuals, the same block scaling, and the same global axes. The first-eigenvalue scaling used in MFA controls block scale before the global PCA; it does not mean that the resulting partial points are independent analyses or independent evidence. The introductory material on the MFA compromise space establishes the joint ordination. Here we derive how its global and partial individual coordinates are related.

Here \(x_{ik}\), \(z_{ik}\), and \(y_i\) denote row vectors for individual \(i\); \(X_k\), \(Z_k\), and \(Y\) collect those rows. The factor \(1/\sqrt K\) in \(Y\) is a common multiplier of the concatenated matrix. It leaves the PCA eigenvectors unchanged, while making the coordinate convention especially transparent. Equivalently, one could omit this multiplier and rescale all global scores accordingly, provided the partial-coordinate definition changes consistently. We use the stated convention throughout.

The eigenvectors are obtained from the global PCA, not from separate PCA fits for each block. A displayed map usually retains only the first \(L\) global components. The partial point for individual \(i\) and block \(k\) on that map is the vector \((f_{ik1},\ldots,f_{ikL})\); the corresponding global point is \((t_{i1},\ldots,t_{iL})\). Coordinates are in the score units induced by the scaled matrices, not in the original measurement units of any one assay.

Why the Global Point Is the Partial-Point Barycenter

The barycenter identity follows directly from the way the global matrix is concatenated. It is not an additional modeling assumption. For a fixed individual and component, the global coordinate is the projection onto the full eigenvector, whose block pieces are \(v_{k\ell}\).

The last equality uses \(f_{ik\ell}=\sqrt K\,z_{ik}^\mathsf{T}v_{k\ell}\). Thus the global coordinate is the arithmetic mean of the \(K\) partial coordinates, for each individual and each retained component. In a map, the global point is the barycenter of that individual’s \(K\) partial points. This relation is useful both for interpretation and as a numerical check on software output.

This construction presumes that blocks have been prepared consistently: they refer to the same individuals in the same order, are centered according to the analysis, and use the intended variable metric. Each \(\lambda_{1,k}\) must be positive, so a block with no variation cannot be scaled this way. Missing observations, block-specific individual sets, or different centering conventions require an explicit method for handling them; silently treating the resulting matrices as matched can invalidate the identity and the interpretation.

Distances and Block Agreement

For a map retaining \(L\) global components, the Euclidean distance from a block-specific partial point to the global point for individual \(i\) is

A small value means that block \(k\) places individual \(i\) near the compromise in the retained coordinate space. A large value means that its partial position differs more from the compromise there. This is a geometric description of agreement on the displayed axes, not a hypothesis test, a probability, or evidence of a causal effect. It also does not say that a block is globally “good” or “bad”: agreement can differ among individuals and components.

The barycenter identity also gives a useful decomposition of disagreement. For each individual and component, the deviations \(f_{ik\ell}-t_{i\ell}\) sum to zero across blocks. This follows because \(\sum_k(f_{ik\ell}-t_{i\ell})=\sum_k f_{ik\ell}-K t_{i\ell}=0\). A block on one side of the global point is therefore balanced by one or more blocks on the other side. The individual distances \(d_{ik}^{(L)}\) summarize a block’s displacement, but do not by themselves show the direction of that displacement; the signed coordinate differences do.

Diagnostics, Alternatives, and Limitations

A practical calculation should be checked before the map is interpreted. First verify that row \(i\) refers to the same individual in every block and that the block columns use the intended centering and variable scaling. Then confirm that each first eigenvalue used for scaling is positive and that the global eigenvectors are normalized. Finally, check the barycenter identity numerically: for every retained component, the average of the \(K\) partial scores should equal the global score, subject only to rounding. A substantial mismatch usually indicates a block-ordering error, an inconsistent score convention, a scaling error, or a mismatch between the global axes and the axes used for the partial projections.

A second diagnostic is to inspect signed differences as well as distances. A large distance identifies displacement, but the signed component differences show whether the block lies above or below the compromise along a particular axis. Also compare distances across more than one retained dimension when appropriate: a close match on the first axis does not guarantee agreement on later axes. Since the number of displayed components is a choice, it should be stated whenever distances are reported.

Separate PCA gives each block its own axes, so coordinates from different blocks do not automatically share a common reference for direct point-to-point comparisons. Unscaled concatenated PCA uses a common global space but can let blocks with greater total scale dominate. MFA instead scales blocks by their first principal inertia before constructing a joint PCA, and its partial coordinates project each scaled block onto the resulting common axes. These approaches answer related but distinct questions; partial-to-global distances are specific to the MFA compromise and should not be treated as generic distances between separate PCA maps.

First-eigenvalue scaling limits dominance by the leading scale of each block, but it does not make the blocks identical in dimension, noise, measurement reliability, or scientific relevance. A block can disagree with the compromise because of real block-specific structure, preprocessing choices, measurement error, or an individual observation that is unusual in that block. The distance alone cannot distinguish these explanations. Nor does a short retained map capture all block disagreement: omitted axes may matter, and the original variables may have different units or metrics. Interpret the result as a descriptive comparison conditional on the fitted MFA and displayed axes.