Scaling Binary and Ordinal Omics Features

Why Binary and Ordinal Features Need a Separate Decision

Omics blocks may contain more than continuous measurements. A feature can indicate whether a molecule was detected, record a binary assay call, or encode an ordered category such as an ordinal severity grade. These values can be included in a mixed-type block, but scaling them is not automatic: the numerical operation may implicitly assign weights or meanings that the encoding did not originally have.

The earlier material on within-block centering and feature scaling and the comparison of standard, Pareto, and robust scaling establish the general role of scale parameters. Here the key question is narrower: when is it reasonable to treat a binary or ordinal feature as numeric, and when is it better to retain an indicator representation? Standardization itself does not require normality. The important assumptions concern what the codes mean, whether the feature varies, and whether the resulting influence is appropriate for the analysis.

If the feature is observed for all \(n\) samples, standardization gives it sample mean zero and sample variance one. This is a numerical rescaling, not a claim that the feature is continuous, normally distributed, or measured on an interval scale. In particular, coding ordered categories as 1, 2, 3, and so on treats the numerical gaps between adjacent categories as equal. That assumption needs subject-matter justification.

Binary Features: Prevalence Determines the Standardized Values

For a binary feature \(x_i\in\{0,1\}\), let \(\hat{p}\) be the observed proportion of 1s. Its mean is \(\bar{x}=\hat{p}\). The sample standard deviation depends directly on prevalence, so standardizing the feature does not give the same numerical transformation across different prevalence levels.

The formula follows because \(x_i^2=x_i\) for binary values. Thus \(\sum_i x_i^2=n\hat{p}\), and the sum of squared deviations is \(n\hat{p}-n\hat{p}^2=n\hat{p}(1-\hat{p})\). Dividing by \(n-1\) gives the stated sample variance. When the 1 category is rare, its standardized value can be large in magnitude; this may give a rare call substantial influence in analyses sensitive to feature scale. At \(\hat{p}=0\) or \(1\), \(s=0\), so standardization is undefined.

It can be reasonable to standardize a binary feature when the analysis expects numerical inputs on comparable sample-variance scales and the resulting weighting is intentional. Retaining 0/1 values is often preferable when those codes are already interpretable, when the feature should have a chosen fixed contribution, or when standardizing a rare indicator would make its positive value disproportionately large. Centering without dividing by \(s\) changes the feature’s origin but not the difference between its two values; whether centering is needed depends on the downstream method.

Ordinal Features: Order Does Not Guarantee Equal Spacing

Ordinal categories have a meaningful ordering, but their category labels do not by themselves define distances. Standardizing successive integer codes is defensible when treating adjacent category changes as comparable is a reasonable working approximation for the analysis. It is not justified merely because the categories can be numbered. If equal spacing is not credible, retain category indicators or use a representation designed around the category structure rather than treating the codes as interval measurements.

Retaining Indicators and Choosing an Encoding

For a binary feature, the 0/1 encoding is already an indicator representation. Standardizing it applies a feature-specific shift and multiplier; it does not create new information. The practical choice is whether the analysis should use that raw binary scale or the sample-standardized scale. For a categorical feature with more than two levels, one-hot encoding creates a separate 0/1 indicator for each category. This represents category membership without assigning numerical gaps between category labels.

One-hot encoding has trade-offs. If all category indicators are included, they sum to one for every complete observation, so they are linearly dependent with an intercept in a linear model. Omitting a reference category avoids that particular redundancy in such models, though the coding then expresses categories relative to the reference. Independently standardizing indicators can also magnify rare categories because each has its own prevalence-dependent standard deviation. Retaining indicators in their 0/1 form avoids that magnification, but their influence still depends on the analysis and the other features.

The choice is therefore not a universal rule that every feature must have sample variance one. A mixed-type block may intentionally retain binary indicators while standardizing continuous measurements, provided the encoding and relative influence are documented and suitable for the downstream method. Conversely, when using a method whose behavior depends strongly on numeric scale, retaining raw indicators without considering their scale is not automatically neutral.

Assumptions, Diagnostics, and Failure Modes

Before scaling, verify that the coding is consistent across samples, that categories are correctly ordered where appropriate, and that missing values have not been silently treated as a category or as zero. Compute means and standard deviations from the observed values under a clearly stated missing-data rule. If a feature is constant in the fitting data, \(s=0\) and standardization cannot be performed; remove or separately handle it rather than dividing by zero.

For ordinal variables, examine category frequencies and the meaning of adjacent levels. Sparse or empty categories can make an indicator representation unstable, while collapsing categories changes the scientific definition and should not be done solely to obtain more convenient scaling. For binary features, report prevalence as well as the chosen encoding so that unusually large standardized positive values are interpretable.

When an analysis has training and test samples, estimate \(\bar{x}\), \(s\), and any category decisions using the training data, then apply those same values to the test data. Recomputing the parameters on the test set changes the transformation and can leak information into model evaluation. If a test value falls outside the training categories, define its handling in advance; do not silently assign it a numerical score that was never specified.

A common mistake is to standardize every numeric-looking column and conclude that all features now have equivalent meaning because each has variance one. The correction is that standardization equalizes a sample variance, not the validity of the coding or the scientific importance of a feature. A code of 1 may mean “detected,” while an ordinal code of 1 may mean the lowest category; equal numerical variance does not make those measurements interchangeable. Likewise, leaving all indicators unscaled is a choice, not a guarantee that they will have equal influence.