Why Wide Blocks Dominate Concatenated Analyses

Why block width matters

A concatenated multi-omics data set places the features from several measurement platforms side by side. A distance-based analysis of that combined data uses every coordinate in the concatenated vector. Consequently, a block with many features can contribute substantially to distances even when each feature has only moderate variance. A smaller block can also contribute substantially if its features vary more.

The calculation is simple but important: squared Euclidean distance adds the squared differences across coordinates. For a centered block, its expected squared norm is the sum of its feature variances. This makes both the number of features and their variances relevant. It also clarifies what the calculation does not establish: a large contribution is not, by itself, evidence that a block contains more biological signal.

Expected squared norm of one block

Suppose a block has \(p_b\) features, and let \(X_b=(X_{b1},\ldots,X_{bp_b})\) denote the feature vector for a randomly selected specimen. Assume the features have finite second moments. For now, suppose each feature has been centered so that \(E(X_{bj})=0\). Write \(\sigma_{bj}^2=\operatorname{Var}(X_{bj})\) for the variance of feature \(j\) in block \(b\).

The result follows by taking expectations of the squared coordinates one at a time. No assumption that features within the block are independent is needed: the squared norm contains a sum of individual squares, not cross-products. Correlations can affect how variable the squared norm is from specimen to specimen, but they do not change its expectation once the marginal variances are fixed.

If features have not been centered, the corresponding identity is \(E(\|X_b\|^2)=\sum_j(\sigma_{bj}^2+\mu_{bj}^2)\), where \(\mu_{bj}=E(X_{bj})\). Thus, a large uncentered mean can increase a block’s expected squared norm even if its within-feature variance is small. For comparisons intended to reflect variation around feature means, center the features first; this is the setting assumed in the calculations below.

What concatenation adds together

Let \(X=(X_1,\ldots,X_B)\) be the vector formed by concatenating \(B\) blocks. Concatenation preserves each feature as a separate coordinate. Therefore the squared norm of the combined vector is exactly the sum of the blocks’ squared norms. The blocks need not be statistically independent for this identity or its expectation to hold.

Under the simplifying assumption that every feature in block \(b\) has variance \(\sigma_b^2\), its expected contribution is \(p_b\sigma_b^2\). The ratio between two blocks’ expected contributions is then determined by both their widths and their per-feature variances. If their variances are equal, the wider block has the larger expected contribution in proportion to its feature count. If their widths are equal, the block with greater feature variance contributes more.

For a centered block, the root mean square (RMS) norm is \(\sqrt{E(\|X_b\|^2)}\). It is useful to distinguish this from the expected norm \(E(\|X_b\|)\): in general, \(E(\|X_b\|)\) is not equal to \(\sqrt{E(\|X_b\|^2)}\). The squared-norm calculation concerns the quantity directly added by squared Euclidean distance.

Unequal width and unequal variance

When per-feature variances differ, counting features alone is not enough. With a common variance within each block, compare the products \(p_b\sigma_b^2\). With heterogeneous variances, add the feature-specific variances. The following example shows why a narrow, variable block can still contribute strongly.

A useful extension is to allow different variances among features within a block. For example, if a centered block has feature variances \(0.5\), \(1.5\), and \(3\) units squared, its expected squared norm is \(0.5+1.5+3=5\) units squared. Replacing these variances by their average and multiplying by three gives the same result, but the sum makes clear that equal variances are not required.

Connection to distances and observed data

For two independently drawn specimens from the same centered distribution, the expected squared distance in a block is twice the expected squared norm. For each feature, independence of the two draws gives \(E[(X_{bj}-X'_{bj})^2]=E(X_{bj}^2)+E((X'_{bj})^2)=2\sigma_{bj}^2\). Summing over features yields the block result below. Independence is required between the two draws for this particular form; it is not required among features within either draw.

In observed data, let \(x_{ij}\) be the centered value for specimen \(i\) and feature \(j\), with \(n\) specimens. If the sample variance \(s_j^2\) uses the denominator \(n-1\), then \(\sum_{i=1}^{n}x_{ij}^2=(n-1)s_j^2\). Summing over features shows that the average squared row norm is \((n-1)/n\) times the sum of the feature sample variances. This supplies a direct empirical check of the variance-based calculation for a centered data matrix.

The factor \((n-1)/n\) appears because the sample variances use denominator \(n-1\), while the left side averages over \(n\) specimens. It does not alter the relative contributions of blocks calculated on the same specimens.

Assumptions, diagnostics, and alternatives

The expected squared-norm identity requires finite second moments. The interpretation as a variance sum also requires centering each feature at its relevant mean. In a practical comparison, check that blocks refer to the same specimens, that their feature values have been processed as intended, and that missingness has not changed the set of observations differently across blocks. The simple \(p_b\sigma_b^2\) expression additionally assumes a common variance within each block; otherwise use the sum of feature-specific variances.

Useful diagnostics include tabulating the number of retained features in each block, inspecting the distribution of feature variances, and calculating each block’s sum of variances or empirical average squared row norm. A block whose total is dominated by a few extreme variances deserves closer inspection. Outliers, remaining technical effects, or poorly measured features may inflate the sum without representing meaningful biological variation. Large differences in these summaries should prompt investigation, not automatic removal or rescaling.

Feature-wise scaling changes the calculation by changing the variances that enter it. If every feature is scaled to variance one, then each block’s expected squared-norm contribution is proportional to its feature count. This equalizes per-feature variance but does not equalize total contribution across blocks of different widths. Other feature-wise scaling choices change the variance pattern in different ways. By contrast, analyzing blocks separately avoids summing their coordinates into one distance, but does not directly provide a single combined distance. Decisions about block-level weighting are a separate choice; the calculation here identifies the contribution before such a choice is made.

The central limitation is that expected squared norm is a measure of variation in the coordinates, not a measure of predictive value, biological relevance, or measurement reliability. It also describes an average: individual specimens or pairs can have contributions above or below that average. Correlation between features does not change the expected squared-norm sum, but can change how concentrated or variable contributions are across specimens. Thus, report the assumptions and inspect the data rather than interpreting the expectation as a complete description of block behavior.