Separating Biological Factors From Technical Structure

Why Compare Biological and Technical Annotations?

A latent factor can summarize coordinated variation across omics measurements, but its statistical prominence does not say what generated that variation. A factor may align with a biological characteristic such as disease status, or with a technical feature such as assay batch, collection site, or sequencing depth. It may align with both. Comparing annotations is therefore a useful interpretation step: it helps describe what varies with the factor and identify technical structure that needs attention.

The comparison is evidence about association, not a verdict about cause. A factor associated with disease is not thereby established as a disease mechanism; likewise, association with batch does not prove that every feature loading on the factor is an artifact. Biological and technical metadata can track one another, and a factor’s interpretation also depends on how the factor was estimated and which measurements were included.

Write \(Z_{ik}\) for the score of sample \(i\) on factor \(k\), and \(L_{jk}^{(v)}\) for the loading of feature \(j\) in omics view \(v\) on that factor. A factor model represents the measured data, approximately, through combinations of sample scores and feature loadings. Earlier material on rotational invariance and scale and sign constraints explains why factor labels, signs, and scales must be treated consistently. The comparisons below concern a specified fitted factor and a clearly defined set of annotations.

Set the Comparison Up Before Testing

First state the factor, the annotation, and the unit of analysis. For a score comparison, the units are usually samples; for a loading comparison, they are features within a specified omics view. Do not treat thousands of feature loadings as independent biological replicates when the claim concerns variation among subjects. Conversely, a sample-level score test does not establish which features drive that factor.

For a categorical sample annotation with two groups, a useful descriptive quantity is the difference in mean factor scores. Welch’s two-sample \(t\)-test assesses whether the population means differ without assuming equal group variances. Its usual small-sample inference assumes independent observations within and between groups and approximately normal score distributions in each group; with sufficiently large samples it is often less sensitive to nonnormality, but outliers and dependence still matter. For more than two groups, an omnibus comparison can test whether at least one group mean differs, followed by carefully specified comparisons if needed.

For a continuous annotation, Pearson correlation measures the strength and direction of a linear association. Its conventional test and confidence interval rely on independent sample pairs and, for exact small-sample inference, a bivariate normal model. A scatterplot should support an approximately linear pattern without influential outliers. A nonlinear relationship can be strong but have a small Pearson correlation, so a single coefficient is not a substitute for looking at the data.

Loading comparisons require a definition of the selected features and the null comparison. For example, one can compare the observed overlap of a high-loading feature set with a pathway against overlaps from random feature sets of the same size. A simple random-set null assumes eligible features are exchangeable. That assumption can fail if high-loading features are systematically more abundant, variable, or well-measured than other features. A more credible permutation procedure preserves relevant feature properties when constructing null sets.

Worked Score Comparisons

Comparing Loading Patterns With Feature Annotations

A loading’s sign indicates the direction of association under the chosen factor orientation, while its magnitude indicates how strongly the feature contributes on that scale. Since a factor’s sign can be reversed without changing the represented fit, do not interpret “positive” as inherently biological and “negative” as inherently technical. For a question about which features contribute most strongly regardless of direction, rank by \(|L_{jk}^{(v)}|\). For a question about a coordinated direction, retain the signed loadings and state the orientation convention.

A simple enrichment analysis asks whether a selected loading set contains more annotated features than expected under a specified null. Let \(N\) be the number of eligible measured features, \(K\) the number annotated to a set, \(n\) the number selected as high-loading, and \(x\) the observed overlap. Under uniform random selection without replacement, the overlap \(X\) follows a hypergeometric distribution:

This upper-tail probability asks how surprising an overlap at least as large as the observed one would be if the selected set were a uniform random subset of the eligible features. It does not account automatically for feature-selection bias, dependence among annotations, or differences in measurement quality. A permutation null that matches the selection process and preserves important feature properties can be preferable.

Justification and Practical Interpretation

The enrichment calculation has a direct probability justification. Under the simple null, each subset of \(n\) features is equally likely among the \(\binom{N}{n}\) possible subsets. A subset with exactly \(a\) pathway members can be formed by choosing \(a\) from the \(K\) annotated features and \(n-a\) from the \(N-K\) others. Therefore,

The assumptions are doing real work: the denominator counts all equally likely subsets. If high-loading features are more likely to be highly expressed or variable, uniform selection is not a credible reference distribution. Matching or permuting on relevant properties changes the null comparison to better reflect how the observed set could have arisen.

For score analyses, report the estimated difference or correlation alongside its uncertainty and p-value. For loading analyses, report the selected-set rule, overlap, enrichment ratio or another effect size, eligible background, and null construction. If many factors, annotations, views, or pathways are examined, some small p-values can occur by chance. Define the family of tests and use a multiple-testing procedure such as the Benjamini–Hochberg false discovery rate adjustment when appropriate. Adjusted significance still does not establish biological validity.

A useful comparison is not just “which p-value is smaller?” Different tests can have different sample sizes and different null hypotheses. A biological annotation may be strongly associated with scores while a technical annotation is enriched among loadings; these are complementary observations, not quantities that can be ranked on one universal scale. Replication in a distinct data set, coherent feature-level evidence, and knowledge of data collection can strengthen an interpretation, but none converts an observational association alone into proof of mechanism.

Diagnostics, Alternatives, and Limitations

Before interpreting a result, check the score distributions and score-versus-annotation plots; inspect group sizes, missingness, extreme points, and whether samples are dependent (for example, repeated measurements from the same subject). For loading analyses, check how the high-loading threshold was chosen, whether the factor orientation was handled consistently, how many eligible features have annotation data, and whether the feature background matches the fitted view. A result that changes sharply under a modest change in threshold or background should be described as sensitive rather than definitive.

The main alternatives answer somewhat different questions. Rank-based association can be less sensitive to extreme values and captures monotone rather than specifically linear trends, though its assumptions and interpretation differ from Pearson correlation. An omnibus group test is appropriate for more than two categories, while pairwise tests need multiplicity control. Enrichment testing is compact and interpretable for sets, whereas examining signed loading distributions can preserve direction and avoid an arbitrary high-loading cutoff. No alternative removes the need to specify a valid null and appropriate unit of analysis.

A major limitation is that biological and technical annotations may be associated with each other in the observed samples. Consequently, separate marginal tests cannot determine which annotation accounts for a factor’s pattern when both are related. This is a warning about interpretation, not a reason to ignore either result. Factor scores and loadings are estimated quantities, too; uncertainty in their estimation is not represented by a simple downstream test unless the analysis explicitly accounts for it. Finally, annotations may be incomplete or inaccurate, and a factor may represent biology not captured by the available labels.