Why the Number of Components Requires Judgment
A global MFA analysis can produce as many components as there are nonzero dimensions in its weighted data table, but a map with every component is rarely useful. Retaining too few can hide meaningful structure; retaining too many can make noise or weak, difficult-to-interpret directions look like findings. Choosing a number of components is therefore a decision about how to summarize the global compromise space, not a mechanical consequence of computing it.
Eigenvalues, cumulative inertia, and scree plots describe how the total global inertia is distributed across ordered components. They are useful evidence, but they answer a limited question: how much of the weighted table's inertia is represented by each direction? They do not, by themselves, tell us whether a direction is scientifically meaningful, stable, or relevant to a particular analysis.
What the Global Eigenvalues Measure
Assume the matched individuals and block preprocessing have been established, and that the MFA block weights are fixed. Let \(Z\) be the centered global table formed by placing the weighted block tables side by side. For clarity, define its global covariance matrix as \(S=Z^\mathsf{T}Z/(n-1)\), where \(n\) is the number of matched individuals. The global eigenvalues are ordered \(\lambda_1\geq\lambda_2\geq\cdots\geq 0\). They quantify inertia along the corresponding global component directions.
This definition presumes that the rows refer to the same individuals across blocks, that the chosen preprocessing and block weights are appropriate for the analysis, and that the table is centered in the PCA sense. In practice, missing values, outliers, or very small samples can make the estimated spectrum sensitive to data handling. Eigenvalues are descriptive properties of the fitted weighted table; they are not automatically tests of whether a component exists in a population.
Because the table is weighted, these percentages describe inertia in the MFA analysis space, not the unweighted variance of the original measurements. In particular, a component explaining 20% of global inertia does not necessarily explain 20% of each block's raw variance.
Calculating Component Shares and Cumulative Inertia
The arithmetic behind a cumulative-inertia rule is straightforward. Dividing the first \(K\) eigenvalues by total inertia gives the share represented by those components. The rule itself is not automatic: an analyst might use a target such as 80% as a planning guide, but no universal percentage defines the right dimension count for every MFA analysis.
The justification follows directly from the definition of total inertia. Since the first \(K\) components account for \(\sum_{k=1}^{K}\lambda_k\), dividing this amount by \(I\) gives the fraction of total inertia they account for. Since every \(\lambda_k\) is nonnegative, \(Q_K\) cannot decrease as \(K\) increases. A target percentage therefore identifies the smallest \(K\) that reaches the target, but it does not prove that this \(K\) is substantively optimal.
Worked Example: A Clear Drop and an 80% Target
Suppose a hypothetical weighted MFA table has six global eigenvalues, in descending order: 4.20, 2.10, 1.20, 0.80, 0.50, and 0.20. The total inertia is \(4.20+2.10+1.20+0.80+0.50+0.20=9.00\).
The first three component shares are \(4.20/9.00=0.467\), \(2.10/9.00=0.233\), and \(1.20/9.00=0.133\), or 46.7%, 23.3%, and 13.3%, respectively. Their cumulative proportions are:
Under the stated 80% planning target, three components are the smallest number that reaches the target. The successive drops are 2.10, 0.90, 0.40, 0.30, and 0.30. The reduction in drop size after the third component is consistent with a possible elbow near component 3. Both observations support considering three components, but neither establishes that all three are useful: the component maps and block contributions still need to be examined.
Reading a Scree Plot
A scree plot places component number on the horizontal axis and eigenvalue on the vertical axis. A sharp decline followed by a relatively flat tail can suggest that the leading components capture prominent structure while later components add smaller increments. The point where the plot changes from steep to shallow is often called an elbow. It is a visual cue, not a formal test, and plots can have no clear elbow or more than one plausible one.
One numerical aid is to inspect the successive drops \(d_k=\lambda_k-\lambda_{k+1}\). Large drops followed by much smaller drops may support an elbow near \(k\). However, the apparent pattern depends on the scale of the plot, and a small drop does not mean that the later component is irrelevant. A component with modest inertia may separate an interpretable block-specific pattern from the main shared directions.
Worked Example: When a Variance Target and Interpretation Differ
Consider a hypothetical seven-component spectrum: 2.60, 2.20, 1.90, 1.50, 0.90, 0.50, and 0.40. Its total is 10.00. The cumulative shares through components 1, 2, 3, and 4 are 26%, 48%, 67%, and 82%, respectively. Thus, a preselected 80% target would retain four components.
Suppose inspection of the global scores and partial maps shows that the first two directions reflect coherent patterns across several blocks, the third has a clear but more block-specific interpretation, and the fourth has no reproducible or scientifically useful interpretation. The drops are 0.40, 0.30, 0.40, 0.60, 0.40, and 0.10; there is no unmistakable elbow at component 4.
For an exploratory summary focused on interpretable global structure, retaining three components could be more defensible than automatically retaining four. This choice represents 67% of total inertia, so it should be reported as a trade-off rather than as a claim that the remaining 33% is unimportant. If preserving 80% is essential for a specific downstream purpose, four components may instead be retained, with the fourth clearly described as weakly interpretable.
Use Interpretability and Stability Alongside Inertia
For MFA, a component is not just an entry in the eigenvalue table. Its meaning depends on what the global scores represent and how the blocks contribute to the direction. The earlier material on global and partial coordinates, block contributions, representation quality, shared structure, and block-specific structure provides relevant checks: does the proposed component have a coherent pattern, and is that pattern supported by the blocks in a way that fits the scientific question? A component that is difficult to interpret may still be useful, but a variance percentage alone cannot supply its interpretation.
Stability is another useful diagnostic. Refit the analysis after reasonable perturbations, such as resampling individuals or making a justified alternative preprocessing choice, and compare the eigenvalue pattern and component structure. If two neighboring eigenvalues are close, their individual axes can rotate or exchange order while the combined subspace remains similar. In that case, judge stability of the subspace rather than demanding exact agreement of each axis. These checks assess sensitivity; they do not turn a descriptive choice into a formal significance test.
Check that block definitions, preprocessing, centering, and weights are appropriate and fixed before interpreting the global spectrum.
Calculate component shares and cumulative inertia, then inspect the scree plot and successive eigenvalue drops.
For plausible values of \(K\), review the component interpretations and the roles of the contributing blocks.
Assess whether the leading structure persists under reasonable perturbations, then report the selected \(K\) and the evidence behind it.
Worked Example: Checking a Candidate Count Across Resamples
Suppose a hypothetical MFA fit has five eigenvalues 3.50, 2.50, 1.20, 0.50, and 0.30, for total inertia 8.00. Its first three components account for \((3.50+2.50+1.20)/8.00=7.20/8.00=90.0\%\). This high percentage alone could make three components seem sufficient.
In two hypothetical resampled fits, the eigenvalues are (3.80, 2.20, 1.00, 0.70, 0.30) and (3.20, 2.40, 1.10, 0.80, 0.50). Each resampled total is 8.00. The first three components account for \(7.00/8.00=87.5\%\) in the first fit and \(6.70/8.00=83.75\%\) in the second. Thus, three components continue to capture a large share of inertia in these fits.
That comparison supports—but does not prove—the usefulness of a three-component summary. A complete stability check would also compare score patterns and component subspaces after aligning the resampled solutions. If the third direction changes substantially or has no stable interpretation, the original 90.0% is not sufficient reason to retain it. The example's limited resamples are illustrative, not a claim of a precise stability probability.
Alternatives, Failure Modes, and Limits
Several selection approaches emphasize different evidence. A cumulative-inertia threshold is transparent and easy to communicate, but it depends on a chosen target and may retain a weak component just to cross it. A scree plot is intuitive but subjective, especially when the spectrum declines smoothly. Interpretability emphasizes scientific usefulness, but can be influenced by expectations and should be checked against the actual component and block summaries. A parallel-analysis-style comparison with a suitable null reference can help assess whether eigenvalues exceed a noise benchmark, but it requires a defensible null construction for the matched, weighted, multi-block design; it does not replace interpretation or answer how many dimensions are useful for the scientific purpose.
Common warning signs include a choice justified only by a conventional percentage, an asserted elbow that is not apparent in the plotted spectrum, or a component count that changes sharply after minor preprocessing changes. Also check for influential individuals, poor handling of missingness, or a block whose preprocessing or weighting is inappropriate: each can alter the observed spectrum. No eigenvalue rule repairs a poorly specified analysis. The selected count should be presented as a reasoned summary choice, with the spectrum and the competing evidence reported so readers can see what is gained or omitted.
Check Your Understanding
Work through the questions, then compare your reasoning with the solutions below.
- A global MFA spectrum is 5.00, 2.00, 1.00, and 0.50. What are the total inertia and the cumulative percentage through component 2?
- Why does reaching 80% cumulative inertia not automatically establish that the retained components are scientifically useful?
- A scree plot has no clear elbow, but the first two components are stable and interpretable while the third changes under small preprocessing variations. What should be reported or checked before choosing a count?
- Why can close neighboring eigenvalues make comparison of individual component axes misleading?
Solutions. (1) Total inertia is \(5.00+2.00+1.00+0.50=8.50\). Through component 2, the cumulative share is \((5.00+2.00)/8.50=7.00/8.50=0.824\), or 82.4%. (2) The percentage describes compression of the weighted global table; it does not establish interpretability, stability, or relevance to the question. Inspect the component and block summaries and state the trade-off. (3) Report that the scree evidence is ambiguous and that the third component is sensitive. Check the source of sensitivity and compare candidate solutions, including whether the first two provide a sufficiently useful summary; do not claim the third is supported solely because it adds inertia. (4) When eigenvalues are close, small perturbations can rotate or reorder their associated directions. The axes may differ even when the subspace they span is similar, so compare the combined subspace as well as individual axes.