Diagnostic Plots for Common and Specific Structure

Why Diagnose a Compromise Map?

An MFA global map compresses information from several data blocks into a shared low-dimensional display. Its points summarize a compromise: they do not guarantee that every block tells the same story about every individual. A point can appear central globally while one block places that individual far away; conversely, an apparent extreme may be driven mainly by one block. Diagnostic plots make these distinctions visible.

This tutorial focuses on four complementary views: the global-individual map, partial-coordinate plots, contribution plots, and residual summaries. They are descriptive diagnostics, not hypothesis tests. Their usefulness depends on the fitted representation, the scaling used for the blocks, and the quality of the underlying data.

Definition: Let $K$ denote the number of blocks and $n$ the number of individuals. In MFA, $F_{ia}$ denotes individual $i$'s global coordinate on axis $a$, and $F_{ika}$ denotes that individual's partial coordinate from block $k$ on the same axis. In a plot using barycentric scaling, the global coordinate is the weighted compromise of the partial coordinates: $$ F_{ia}=\sum_{k=1}^{K}\omega_k F_{ika}, \qquad \sum_{k=1}^{K}\omega_k=1, $$ where $\omega_k$ are the block weights defined by the MFA scaling. Coordinates should be compared only when they use compatible axes and scaling.

A global-individual map plots $F_{i1}$ against $F_{i2}$, usually labeling points by individual. It shows which observations are separated along the first two global axes. A partial-coordinate plot overlays or juxtaposes $F_{ik1},F_{ik2}$ for each block with the global coordinates. The visual comparison asks whether blocks place the same individual in similar directions and at similar distances. The interpretation is conditional on the displayed axes: agreement on the first two axes does not imply agreement in dimensions omitted from the plot.

Reading Global and Partial Coordinates Together

In a global map, nearby points have similar coordinates on the displayed axes, while points far apart have different coordinates there. An individual far from the origin has an extreme position on those axes, but whether that position is well represented depends on how much of its information lies in the displayed dimensions. A low representation quality warns against interpreting the plotted location as a complete summary.

Partial coordinates answer a more specific question: which block places this individual where it is? A small distance between a block's partial point and the global point indicates relative agreement with the compromise in the displayed plane. Large displacement indicates that the block's view differs from the global position. It does not, by itself, show that the block is erroneous. It may reflect genuine block-specific structure, noise, or an unusual individual.

Worked Example: Detecting Disagreement in a Partial-Coordinate Plot

Suppose three omics blocks have equal weights in a two-axis MFA display. For one hypothetical sample, the partial coordinates are $(2,1)$ for block 1 and $(0,3)$ for block 2. For this illustration, consider just these two blocks.

The global coordinate is their equal-weight average: $$ (F_{i1},F_{i2})=\tfrac12(2,1)+\tfrac12(0,3)=(1,2). $$

The block 1 displacement from the global point is $(2-1,1-2)=(1,-1)$, with Euclidean length $\sqrt{1^2+(-1)^2}=\sqrt{2}\approx1.414$. The block 2 displacement is $(0-1,3-2)=(-1,1)$, also of length $1.414$. The two blocks lie on opposite sides of the compromise. In particular, their first-axis coordinates differ by $2-0=2$ coordinate units.

The plot therefore diagnoses disagreement for this sample on the displayed axes. The global point $(1,2)$ is not evidence that each block independently places the sample there. Before calling this block-specific biology, an analyst would check the block scaling, data quality, and whether the pattern occurs for other individuals.

A useful visual cue is the direction of displacement, not only its length. If many partial points from one block are displaced in a similar direction, the block may contain a systematic pattern not captured by the compromise. If only one sample has a large displacement, it may be an individual-specific discrepancy. Always inspect the axis labels, scaling convention, and point overlap; a crowded map can hide important differences.

Contribution Plots: Which Individuals Shape an Axis?

A contribution plot ranks individuals by how much they help define a global axis. With equal individual weights, the contribution of individual $i$ to axis $a$ is its squared coordinate divided by the sum of squared coordinates on that axis. If individuals have weights $q_i$, use the weighted form below. Contributions are nonnegative and sum to 100% across individuals for an axis. They are not the same as representation quality: a point may contribute little to an axis yet still be well represented by the plotted dimensions.

Definition: For individual weights $q_i$, the contribution of individual $i$ to global axis $a$, expressed as a percentage, is $$ \operatorname{Ctr}_{ia} =100\frac{q_iF_{ia}^{2}}{\sum_{r=1}^{n}q_rF_{ra}^{2}}. $$ With equal weights, the common factor $q_i$ cancels. A large contribution means the individual is influential in defining that axis; it does not mean the individual is accurately represented or scientifically important.

The formula follows directly from partitioning the axis inertia. The weighted inertia of the individual coordinates on axis $a$ is $\sum_r q_rF_{ra}^2$. Individual $i$'s share of that total is $q_iF_{ia}^2/\sum_rq_rF_{ra}^2$; multiplying by 100 expresses the share as a percentage. Summing those shares over $i$ gives $100\sum_iq_iF_{ia}^2/\sum_rq_rF_{ra}^2=100\%$.

Worked Example: Calculating Axis Contributions

Consider three equally weighted hypothetical samples with coordinates $2,-1,$ and $1$ on a global axis. Their squared coordinates are $4,1,$ and $1$, so the total is $4+1+1=6$.

Their contributions are $$ 100\frac{2^2}{6}=66.67\%,\qquad 100\frac{(-1)^2}{6}=16.67\%,\qquad 100\frac{1^2}{6}=16.67\%. $$

The first sample supplies two-thirds of the displayed axis inertia and merits inspection in a contribution plot. Its large contribution does not establish that it is an outlier: the coordinate may be a valid extreme, and the axis may capture meaningful structure. Check the sample's partial coordinates, its measurement quality, and its contribution to other axes before drawing a substantive conclusion.

A contribution plot can also be calculated for variables or blocks under their corresponding MFA definitions, but those quantities answer different questions. Here the individual contribution identifies which observations shape an axis. Use partial maps to locate block disagreement and representation quality to judge how faithfully the axes summarize an observation; do not substitute one diagnostic for another.

Residual Summaries for Joint and Individual Structure

For JIVE, write the centered data block $X_k$ as the sum of a joint component $J_k$, a block-specific individual component $A_k$, and a residual $E_k$: $$ X_k=J_k+A_k+E_k. $$ The joint component captures structure shared across blocks, the individual component captures structure specific to block $k$, and the residual is what the fitted components do not reconstruct. A residual summary therefore complements the MFA maps: it asks what remains unexplained after the fitted common and specific structure are taken into account.

For individual $i$ in block $k$, a convenient summary is the root mean squared residual across that block's $p_k$ features: $$ \operatorname{RMS}_{ik} =\sqrt{\frac{1}{p_k}\sum_{j=1}^{p_k}E_{kij}^{2}}. $$ This is in the scale of the analyzed features. If features were standardized before fitting, it is in standardized units. To compare overall residual burden across blocks with different total variation, also report a residual sum-of-squares proportion such as $\lVert E_k\rVert_F^2/\lVert X_k\rVert_F^2$, where $\lVert\cdot\rVert_F$ is the square root of the sum of squared matrix entries. Neither summary says whether the residual is random; residual patterns across individuals or features still need inspection.

Worked Example: Comparing Residual Summaries

Suppose a hypothetical standardized block has four features for each sample. For sample A, the fitted JIVE residuals are $(0.2,-0.1,0.3,0.0)$. Its residual mean square is $$ \frac{0.2^2+(-0.1)^2+0.3^2+0.0^2}{4} =\frac{0.04+0.01+0.09+0}{4}=0.035, $$ so $\operatorname{RMS}_{A}=\sqrt{0.035}\approx0.1871$ standardized units.

For sample B, let the residuals be $(0.5,0.4,0.3,0.2)$. Then $$ \operatorname{RMS}_{B} =\sqrt{\frac{0.25+0.16+0.09+0.04}{4}} =\sqrt{0.135}\approx0.3674 $$ standardized units. Sample B has almost twice sample A's residual RMS, suggesting that the fitted joint-plus-individual components reconstruct B less closely in this block.

This comparison is meaningful because both samples use the same four standardized features. It does not prove that sample B is an outlier or that the model is wrong: its residual may reflect noise or unmodeled signal. Inspect the residuals across blocks, features, and samples, and compare the residual to the scale of the original data.

Diagnostics, Failure Modes, and Alternatives

The main diagnostics are patterns, not isolated visual impressions. Large partial-to-global displacements concentrated in one block suggest block-specific behavior; many high contributions concentrated in a few individuals suggest that an axis may be strongly shaped by a small number of observations; large or structured JIVE residuals suggest that the chosen joint and individual ranks may not capture all relevant variation. Since rank selection is a separate modeling decision, these plots can flag a concern but do not determine the rank by themselves.

Check scale first. Partial coordinates must be on compatible axes and scaling. Different standardization or weighting choices can change distances and apparent influence.
Inspect more than two axes. A sample may look close to the global point in the first two dimensions while disagreeing on a later axis. Compare additional axis pairs when they carry meaningful structure.
Look for repeated patterns. A single distant partial point is not enough to establish a systematic block effect. Check whether the same block repeatedly departs from the compromise.
Do not equate residual size with model failure. Measurement noise is expected, and residual magnitude depends on feature scale and the amount of original variation.
Check influential observations. Refit or inspect the analysis with a high-contribution observation set aside as a sensitivity check; do not remove it solely because its contribution is large.

These displays complement, rather than replace, numerical summaries. Partial-coordinate plots are most direct for visualizing block agreement around a compromise. Contribution plots identify influential observations on particular axes. Residual summaries are useful for assessing reconstruction after a JIVE fit, but they do not explain the source of residual structure. Separate block analyses can reveal patterns hidden by a compromise, while numerical residual checks are less dependent on visual judgment; each alternative answers a different question.

The limitations are important. Two-dimensional plots omit higher-dimensional information, and apparent clusters or distances depend on chosen scaling and axes. Contributions measure geometric influence, not causal importance or biological relevance. Residuals combine noise, measurement error, and omitted structure. None of these diagnostics alone establishes that a pattern is reproducible, statistically significant, or caused by a particular biological process.

Common pitfall: “The global point lies between the partial points, so the blocks agree.” In a barycentric display, the global point is constructed as a weighted compromise; being between partial points is expected even when the blocks disagree substantially. Compare partial-to-global displacements, inspect the size and direction of those displacements, and check whether the pattern recurs across individuals.

Check Your Understanding

Answer the questions, then compare your reasoning with the brief solutions below.

What does a large distance between an individual's partial and global coordinates indicate, and what does it not establish?
Three equally weighted individuals have coordinates $1,-2,1$ on an axis. Calculate their percentage contributions.
Why can a high axis contribution not be interpreted as high representation quality?
What does a large JIVE residual RMS indicate, and name one reason it may occur other than model failure.
Why should analysts inspect more than the first two axes when diagnosing block agreement?

Solutions. (1) It indicates that the block's position differs from the compromise on the displayed axes; it does not establish error or biological cause. (2) The squared coordinates are $1,4,1$, with total $6$. Contributions are $100/6=16.67\%$, $400/6=66.67\%$, and $100/6=16.67\%$. (3) Contribution measures influence on one axis, whereas representation quality measures how much of the individual's information is captured by the displayed dimensions. (4) It indicates poorer reconstruction in that block on average; measurement noise can also produce residuals. (5) Disagreement may occur in dimensions not shown in the first two-axis display.