Sample Scores As Cross-Omics Profiles

What a Sample Score Represents

A loading describes how a latent factor relates to measured features; a sample score describes where one sample lies along that factor. Taken together, a sample’s scores form coordinates for summarizing its molecular profile. In multi-omics analysis, those coordinates may include factors shared across views and factors that describe variation specific to one view.

A score is not a directly measured molecular quantity. It is an estimated latent coordinate whose meaning comes from the model, the fitted loadings, and the scaling used for the data. A positive score, for example, means movement in the positive direction of the chosen factor coordinate. Which features increase or decrease in that direction is determined by their loadings.

The equation describes the factor contribution to the fitted profile. In a continuous observation model, residuals account for differences between observed and fitted values. For count or binary data, the factor terms may instead form a linear predictor that is mapped to the observation scale by the model’s link function. Thus, a score should be interpreted on the scale of the model component in which it appears, not automatically as a change in observed counts or probabilities.

Meaningful interpretation requires that samples are aligned across views, the fitted factor structure is appropriate, and the data preprocessing and model are adequate for the measurements. Scores and loadings must also be read together: the same score can correspond to different feature patterns in different views because each view has its own loading matrix. The numerical value depends on the fitted coordinate system and its scaling; it is not an absolute biological unit.

Reading Shared and View-Specific Coordinates

A shared score is a coordinate for a direction represented in multiple views. Its estimated value is the same for a sample wherever that shared factor is used, but the resulting feature contributions need not have the same sign or magnitude across views. For example, a positive shared score could contribute positively to some RNA features and negatively to some protein features if the corresponding loadings have opposite signs.

A view-specific score represents a direction included for one view only. A high value on a view-specific RNA score can help explain that sample’s RNA profile, but it is not itself evidence that the same direction is present in protein or metabolite measurements. It is also not a “less important” score by definition: it may capture substantial, reproducible variation in its own view.

From Coordinates to Fitted Feature Contributions

The fitted profile is useful because it connects compact sample coordinates to the features that were measured. Each feature’s fitted value is the sum of its contributions from shared and view-specific factors. The same decomposition can be used to compare two samples or to ask which latent directions account for a particular fitted feature value.

This expression also gives a direct way to interpret score differences. Holding the fitted loadings fixed, the difference between two samples’ fitted values for a feature is the loading-weighted difference in their scores. This follows by subtracting the fitted-value equation for sample \(a\) from the equation for sample \(b\):

Thus, a score difference alone does not tell how much a specific feature differs. Its contribution depends on that feature’s loading. This is a model-based decomposition of fitted profiles, not a causal explanation of why the samples differ.

Assumptions, Diagnostics, and Alternatives

A score interpretation is only as useful as the representation that produced it. At minimum, the factor model must match samples across views correctly, use an appropriate observation model and preprocessing, and provide an adequate low-dimensional description of the patterns being interpreted. For a linear reconstruction, the score-loading products must correspond to the fitted component on the stated scale. For non-Gaussian observations, interpretation must respect the model’s linear predictor and link rather than treating that predictor as an observed measurement.

Scores are estimated quantities. Their uncertainty may matter, especially for samples with missing measurements, sparse counts, or weak information for a factor. A point estimate should not be mistaken for a perfectly measured biological state. Nor should scores be compared numerically across separately fitted models unless their coordinates have been aligned and their scales made comparable.

Useful diagnostics include inspecting score distributions and outliers, checking whether a sample’s score is driven by a few unusual observations, and comparing fitted profiles with observed data on the appropriate scale. Assess whether samples with similar score patterns have coherent fitted feature contributions across views. If scores change substantially under reasonable changes in preprocessing, model fit, or sample inclusion, interpretations based on small score differences are fragile. A large score is not automatically an error; it is a reason to check the corresponding data and fitted contributions.

Scores also differ from other low-dimensional representations. PCA scores are coordinates in principal directions for a single analyzed matrix (or a specifically constructed joint matrix); they do not inherently separate shared from view-specific structure. NMF represents data through nonnegative components and sample weights, which can be useful for additive patterns but has a different constraint and interpretation. Separate factor models for each view produce coordinates that need not represent common directions. The shared-and-specific factor model makes the distinction explicit, but its score meanings still depend on its fitted assumptions and parameterization.

Common Pitfall: Treating a Score as a Measured Trait

A common mistake is to describe a positive shared score as “high expression across all omics,” or to interpret a view-specific score as if it were a shared biological signal. Both claims skip the loading map. A score indicates a coordinate; a feature’s loading determines whether moving along that coordinate contributes positively or negatively to that feature, and whether the coordinate is shared or view-specific determines where it enters the model.

Another mistake is to infer causation from a score-feature pattern. A factor model summarizes patterns under a statistical representation. Even if a shared factor is associated with features in several views, that alone does not establish a biological mechanism or causal direction. Use score interpretations as descriptions of fitted multivariate structure, and consider other evidence before making biological claims.