Comparing Factor Scores Across Views

Why Compare Scores Across Views?

A multi-omics factor model can describe each subject with a score for a pattern shared across data views and, depending on the model, additional scores specific to each view. Comparing those scores helps answer two distinct questions: do subjects show a similar pattern in different views, and does a particular view contribute a subject-level deviation beyond the shared pattern? A plot can reveal the shape of these relationships; a correlation summarizes linear association. Neither, by itself, establishes biological meaning or causation.

A valid comparison requires the same subjects on both axes and a consistent factor interpretation. The factor's identity, sign, and scale must be comparable across the scores being plotted. The earlier material on rotational invariance and scale and sign constraints explains why these conditions matter: without a common orientation and scale, an apparent disagreement can be a parameterization artifact rather than a substantive difference.

Definition: Let $i=1,\ldots,n$ index subjects and $m$ index an omics view. Write $\hat h_i$ for the estimated score of subject $i$ on a shared factor, and $\hat u_{im}$ for that subject's estimated score on a view-specific factor for view $m$. A matched score comparison compares scores for the same subjects under a common factor interpretation and compatible orientation and scale. A view-dependent contribution is a subject's estimated component $\hat u_{im}$, interpreted relative to the shared score and the fitted model's factorization.

In a plot of $\hat h_i$ against $\hat u_{im}$, each point represents one subject. The horizontal coordinate describes that subject's shared score; the vertical coordinate describes the view-specific score. If both scores are centered at zero, positive and negative values indicate positions above and below the fitted score center, respectively. They do not automatically represent clinical categories or meaningful thresholds.

Reading Shared and View-Specific Score Plots

A shared-versus-specific plot is useful for examining whether subjects with larger shared scores also tend to have larger view-specific scores, whether the scores vary largely independently, or whether a few subjects have unusual combinations. Begin with axis labels, score units, and a description of the factor represented. If scores have been standardized, state that the units are standard deviations on the chosen score scale; standardization does not make the scores error-free.

The quadrants are a simple descriptive aid. A subject in the upper-right quadrant has positive shared and specific scores; one in the lower-left has negative scores on both. A subject may instead have scores with opposite signs. These sign combinations describe estimated coordinates, not whether the factor is present or absent. Their interpretation also depends on the chosen factor orientation.

Worked Example: Interpreting Shared and Specific Coordinates

Suppose a fitted model gives the following hypothetical standardized scores for four subjects on one shared factor and one view-specific factor. The scores are already oriented consistently, and zero is the center of each score scale.

Subject	Shared score $\hat h_i$	View-specific score $\hat u_{iA}$
1	-1.5	-1.0
2	0.8	-0.6
3	-0.7	-0.4
4	1.4	1.2

On a plot with $\hat h_i$ on the horizontal axis and $\hat u_{iA}$ on the vertical axis, subject 4 is high on both scores, while subject 1 is low on both. Subjects 2 and 3 have opposite shared-score signs but both have relatively low specific scores. Thus, the points do not support a description in which subjects 1 and 4 are both high on both axes.

The Pearson correlation can summarize linear association. The mean shared score is $( -1.5+0.8-0.7+1.4 )/4=0$, and the mean specific score is $( -1.0-0.6-0.4+1.2 )/4=-0.2$. The cross-product sum of centered scores is $1.2-0.32+0.14+1.96=2.98$. The centered sums of squares are $5.34$ for the shared scores and $2.80$ for the specific scores. Therefore,

r=\frac{\sum_i(\hat h_i-\bar h)(\hat u_{iA}-\bar u_A)} {\sqrt{\sum_i(\hat h_i-\bar h)^2\sum_i(\hat u_{iA}-\bar u_A)^2}} =\frac{2.98}{\sqrt{5.34(2.80)}}\approx 0.771.

This positive sample correlation describes a moderately strong positive linear association in these four hypothetical subjects. With only four observations, it should be treated as a descriptive summary, not as precise evidence about a population relationship. The scatterplot still matters: a high correlation does not mean that the two scores are equal or that each subject follows the same pattern.

Comparing View-Specific Scores Across Views

When the model estimates view-specific scores for two views, a direct comparison plots $\hat u_{iA}$ against $\hat u_{iB}$ for every subject with both scores. This is a paired comparison: the two coordinates in each point belong to the same subject. A positive correlation indicates that subjects with larger scores in one view tend also to have larger scores in the other, in a linear sense. A correlation near zero indicates little linear association, but can coexist with a nonlinear pattern.

Before calculating a correlation, verify that the scores refer to comparable factors and that their orientations and units are interpretable together. If one score has been multiplied by $-1$, its correlation with the other score changes sign even though the underlying fitted direction may be unchanged. If scales differ, correlation is unchanged by positive rescaling, but the plotted distance from the diagonal is not directly comparable in the original units.

Definition: For paired scores $x_i$ and $y_i$, Pearson's sample correlation is $$ r=\frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)} {\sqrt{\sum_{i=1}^{n}(x_i-\bar x)^2\sum_{i=1}^{n}(y_i-\bar y)^2}}. $$ It is defined when both scores have nonzero variation. Its value lies between $-1$ and $1$, and it summarizes the direction and strength of linear association in the observed pairs. It is not a measure of equality or agreement.

The numerator measures whether centered values tend to deviate in the same direction; the denominator rescales that sum by the two observed spreads. Consequently, multiplying either score by a positive constant leaves $r$ unchanged, while multiplying one score by $-1$ reverses its sign. This gives a useful check on interpretation: correlation cannot tell whether two views have the same score magnitude or whether their units match.

Worked Example: Correlating Scores From Two Views

Consider hypothetical view-specific scores for five matched subjects. Assume both views use the same factor orientation and standardized score units.

Subject	$\hat u_{iA}$	$\hat u_{iB}$
1	1.2	1.0
2	0.5	0.2
3	-0.4	-0.6
4	-1.0	-0.8
5	-0.3	0.2

The two means are both zero. The sum of paired products is $1.2(1.0)+0.5(0.2)+(-0.4)(-0.6)+(-1.0)(-0.8)+(-0.3)(0.2)=2.28$. The sums of squares are $2.94$ for view A and $2.08$ for view B. Substitution gives

r=\frac{2.28}{\sqrt{2.94(2.08)}}=\frac{2.28}{2.473}\approx 0.922.

The scores show a strong positive linear association across these five subjects. However, subject 5 has a negative score in view A and a positive score in view B. The correlation does not erase that view-dependent difference, and it does not show that the paired scores agree in magnitude. A plot of the points, preferably with a reference line $y=x$, makes such departures visible.

Paired Differences and View-Dependent Contributions

Correlation is not the only useful summary. If scores from two views have the same scale and orientation, define the paired difference $d_i=\hat u_{iA}-\hat u_{iB}$. A positive value means subject $i$'s estimated view-A score exceeds the view-B score on that common scale; a negative value means the reverse. The mean difference summarizes an average offset, while the individual differences identify subjects with larger departures. These comparisons require commensurate score scales: subtracting scores with incompatible units has no clear interpretation.

Some model parameterizations define a view-level coordinate as the sum of a shared and a view-specific contribution. In that case, and only when the fitted model uses this additive definition, one may calculate $\hat z_{im}=\hat h_i+\hat u_{im}$. The decomposition separates a common component from a view-dependent offset. It is not a universal rule that every model's view-level score is obtained by adding two estimates; consult the model's stated parameterization.

Worked Example: Separating Shared and View-Specific Contributions

Suppose a hypothetical model explicitly defines the coordinate for view $m$ as $\hat z_{im}=\hat h_i+\hat u_{im}$. For one subject, let the shared score be $0.5$, the view-A-specific score be $0.4$, and the view-B-specific score be $-0.3$, all in the same standardized score units.

For view A, $\hat z_{iA}=0.5+0.4=0.9$. For view B, $\hat z_{iB}=0.5+(-0.3)=0.2$. Their difference is $0.9-0.2=0.7$, which also equals the difference in specific contributions, $0.4-(-0.3)=0.7$, because the shared contribution cancels:

\hat z_{iA}-\hat z_{iB} =(\hat h_i+\hat u_{iA})-(\hat h_i+\hat u_{iB}) =\hat u_{iA}-\hat u_{iB}.

This arithmetic shows how a common component can coexist with a view-dependent difference. It does not establish why the difference occurred, and it should not be used for a model whose definition of view-level coordinates is not additive.

Assumptions, Diagnostics, and Alternatives

For descriptive plots and correlations, the essential conditions are correctly matched subjects, meaningful score estimates, and a consistent factor interpretation. Pearson correlation additionally summarizes a linear relationship; it can be strongly influenced by an outlying subject. A population-level confidence interval or test for correlation would require additional inferential assumptions, including an appropriate sampling model and independent subject pairs. A descriptive $r$ alone does not justify a population claim.

Inspect the plots before interpreting summaries. Check whether points form a roughly linear cloud, whether one or two subjects dominate the pattern, whether there is a curved relationship, and whether a few extreme scores determine the apparent association. Check the ranges and score centers on both axes. If uncertainty estimates for scores are available, remember that plotting point estimates alone hides that uncertainty; a subject with a poorly determined score may appear more distinctive than the evidence warrants.

Question	Useful display or summary	Interpretive limit
Do scores tend to increase together?	Scatterplot and Pearson correlation	Correlation summarizes linear association, not equality.
Are paired scores similar in magnitude?	Scatterplot with $y=x$, plus paired differences	Requires compatible scales and orientation.
Is a relationship monotone but not linear?	Scatterplot; Spearman rank correlation as a supplementary summary	Rank association does not describe score-unit differences.
Do some subjects have distinctive view-dependent offsets?	Subject-level paired differences or labeled points	Offsets describe estimates, not their cause.

Spearman rank correlation is a useful alternative when the association is monotone but not well summarized by a straight-line trend, or when ranks are the intended comparison. It replaces the observed values with their ranks before calculating correlation. It does not answer whether score values agree in magnitude. For that question, paired differences and a common-scale plot are more direct.

Verify comparability.
Confirm that both scores refer to the same subjects and factor, with compatible orientation and scale.

Plot matched scores.
Use one point per subject, clear axis labels, and a reference line when comparing agreement on a common scale.

Summarize the relevant feature.
Use Pearson correlation for linear association, or paired differences for subject-level offsets and agreement.

Check sensitivity and uncertainty.
Look for outliers, nonlinear patterns, restricted ranges, and imprecisely estimated scores before making a substantive interpretation.

Common Pitfalls and Limitations

Confusing correlation with agreement. A high correlation means the scores tend to vary together; it does not mean they are equal. Compare paired differences and the reference line $y=x$ when agreement is the question.
Ignoring sign orientation. A reversed score axis reverses the sign of the correlation. Confirm the factor orientation before calling a relationship positive or negative.
Misreading the quadrants. A subject with negative shared and negative specific scores is low on both axes relative to zero, not high on both. Read the numerical coordinates, not an impression of the point cloud.
Overinterpreting a small sample correlation. A large $r$ from a few subjects can be unstable. Report the number of matched subjects and inspect individual points.
Treating estimates as known without error. Factor scores are estimated quantities. Their apparent separation does not, by itself, demonstrate a reliable subject-level difference.
Assuming a score difference has a biological cause. A view-specific score is a model component. Its interpretation depends on the fitted factorization and cannot alone identify the source of a difference.

These comparisons are descriptive tools for interpreting a fitted factor model. They do not establish that a view-specific contribution is biologically meaningful, that a shared score is free of technical influence, or that score differences are stable to modeling choices. State the population and views represented, the score scale, the number of matched subjects, and whether the reported quantity is association, agreement, or a paired difference.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In a shared-versus-specific plot, a subject has $\hat h_i=-1.5$ and $\hat u_{iA}=-1.0$. Is the subject high or low on each axis relative to zero?
Two matched view-specific score vectors have Pearson correlation $r=0.90$. Does this show that their scores agree in magnitude? What additional display or summary would help?
Why can reversing the sign of one factor's scores change the sign of a cross-view correlation without changing the underlying fitted direction?
If $\hat z_{im}=\hat h_i+\hat u_{im}$ is the stated model definition, show how the shared component cancels when comparing views A and B.
What is a suitable alternative to Pearson correlation when the relationship is monotone but clearly nonlinear, and what does that alternative still fail to show?

Solutions. (1) The subject is low on both axes relative to zero. (2) No; correlation measures linear association, not equality. A scatterplot with $y=x$ and paired differences can assess departures in magnitude. (3) Multiplying one score vector by $-1$ changes every centered cross-product's sign, so $r$ changes sign; the model direction may simply have been represented with the opposite orientation. (4) Subtracting gives $(\hat h_i+\hat u_{iA})-(\hat h_i+\hat u_{iB})=\hat u_{iA}-\hat u_{iB}$. (5) Spearman rank correlation can summarize monotone association, but it does not show agreement in the original score units.