MFA With Quantitative and Categorical Blocks

Why Mixed-Data MFA Needs Careful Representation

A multi-omics study may contain continuous measurements, such as metabolite abundances, alongside categorical annotations, such as genotype class or clinical subtype. A global MFA map can represent both kinds of information, but only after each block is encoded in a way that respects its measurement scale. Treating category labels as numbers, or simply appending indicator columns to standardized measurements, can make the geometry depend on arbitrary coding choices.

The construction below combines two established ideas: quantitative variables are centered and scaled as in correlation-based PCA, while categorical variables are represented through complete disjunctive coding and the MCA chi-square metric. Each block is then balanced by its own first principal inertia before the global PCA. The earlier material on the MFA compromise space establishes how this block scaling produces the shared individual map; here the focus is on what the mixed representations mean and how to interpret their plotted variables and categories.

Representing and Balancing the Blocks

For a quantitative variable \(x_j\) with nonzero variance, define its standardized value for individual \(i\) by \(z_{ij}=(x_{ij}-\bar{x}_j)/s_j\), where \(\bar{x}_j\) is its sample mean and \(s_j=\sqrt{\frac{1}{n}\sum_{i=1}^n(x_{ij}-\bar{x}_j)^2}\). Thus each variable is centered and has variance 1 under the \(1/n\) covariance convention. This makes the analysis depend on relative patterns rather than measurement units. A constant variable cannot be standardized and must be removed or handled separately.

For a categorical variable with categories \(1,\ldots,q\), let \(d_{ik}=1\) if individual \(i\) is in category \(k\), and 0 otherwise. Let \(p_k=n_k/n\) be the observed proportion in category \(k\). The MCA-scaled column is \(y_{ik}=(d_{ik}-p_k)/\sqrt{p_k}\). This centers the indicator and gives it the MCA chi-square weighting: rare categories receive a larger column scale than common ones. Every observed category has \(p_k>0\); an unobserved category contributes no information and is omitted. The columns for one variable are linearly dependent after centering, so the dependence is expected, not a computational error.

For each transformed block \(Y_g\), calculate its first principal inertia \(\lambda_{g1}\), the largest eigenvalue of \(Y_g^\mathsf{T}Y_g/n\). The global data matrix concatenates \(Y_g/\sqrt{\lambda_{g1}}\) across blocks. The definition of \(\lambda_{g1}\) must match the chosen row weights and variance convention; software conventions can differ, so use the same convention throughout.

Interpreting Quantitative Variables and Categories

Global individual scores are coordinates in the compromise space. A quantitative variable is commonly shown by its correlation with each global component. If \(F_\ell\) is the score on component \(\ell\), the plotted coordinate for original quantitative variable \(x_j\) on that axis is \(\operatorname{cor}(x_j,F_\ell)\). A large absolute correlation indicates a strong linear association with that component; its sign indicates direction relative to the chosen component orientation. Because eigenvector signs can be reversed without changing the solution, interpretations should focus on relative directions and relationships.

For categorical variables, the usual category coordinates in the global individual map are category barycentres: the mean global score among individuals in that category. If \(I_k\) is the set of individuals in category \(k\), then the coordinate on component \(\ell\) is \(n_k^{-1}\sum_{i\in I_k}F_{i\ell}\). Categories far apart have different average profiles on the displayed components. A category's distance from the origin is not itself a test of significance, and the map's Euclidean distances are meaningful only as approximations within the displayed dimensions.

These two kinds of points have different meanings. Quantitative coordinates are correlations with global axes; category coordinates are means of individual scores. They can be shown together in a global map, but they should not be described as if they were the same statistic or had the same scale.

Why Category Coordinates Are Barycentres

The barycentre interpretation follows directly from averaging scores within a category. For component scores \(F_{i\ell}\), the category mean is the conditional sample mean of those scores among members of category \(k\). In terms of the indicator, this can also be written as a covariance-like expression. Since the grand mean of the centered component scores is zero,

The first equality uses \(p_k=n_k/n\). The second holds because \(\sum_i F_{i\ell}=0\), so subtracting \(p_k\) from each indicator does not change the sum. This explains why category position reflects association between membership and the global score. MCA scaling affects the geometry used to fit the global axes; when plotting category barycentres in the individual map, the mean-score definition remains the same.

Diagnostics, Alternatives, and Limitations

Check the analysis before assigning biological meaning to the map. Confirm that quantitative variables have been centered and scaled as intended, categorical indicators use the observed category proportions, and every block's first inertia is computed using the same row weights and convention. Inspect category counts: a very rare level can have an unstable position and can exert substantial influence through MCA scaling. Check whether the first global components represent the blocks adequately; a point may appear close to the origin simply because its information lies on omitted dimensions. Use representation quality and block contributions alongside the map rather than judging importance by visual distance alone.

The main alternatives answer different questions. Separate PCA or MCA preserves each block's own geometry but does not give one compromise map. Concatenating standardized quantitative variables and unweighted indicator columns is simpler, but blocks with more columns or different coding can dominate. Mixed-data MFA retains the common map while balancing blocks by first inertia; it is not a guarantee that every block contributes equally to every axis, nor does it remove confounding, dependence among individuals, missing-data problems, or measurement error.

Results can also depend on category definitions. Combining or splitting levels changes the indicator table and may change the fitted axes. A category coordinate is descriptive, not a causal effect or a formal significance test. Likewise, a quantitative correlation with an axis is an association in the fitted sample, not evidence of a causal mechanism. Check sensitivity to defensible preprocessing and category choices, especially when the sample is small or levels are sparse.

• Incorrect: “The category labels are coded 1, 2, and 3, so they can be treated as a quantitative variable.”
  Correction: Those values are labels unless their numerical spacing has a real measurement interpretation. Use indicator coding and the categorical-block metric.
• Incorrect: “First-eigenvalue scaling makes all blocks equally influential.”
  Correction: It makes each block's leading eigenvalue equal to 1 before the global PCA. Contributions to a particular global component can still differ.
• Incorrect: “A category far from the origin has a statistically significant effect.”
  Correction: The location is descriptive in the displayed map. It is not a hypothesis test, and its reliability depends on sample size, representation quality, and omitted dimensions.