A Two-Axis Taxonomy of Integration Designs

Why Two Axes Are Needed

An integration design cannot be described adequately by saying only that a study combines two omics data blocks. One question is whether observations in the blocks come from the same identifiable subjects. A separate question is whether the blocks measure some of the same features. These answers determine how the data blocks relate, and they are not interchangeable: having the same subjects does not imply measuring the same features, and measuring the same features does not imply observing the same subjects.

The earlier material on horizontal integration and vertical integration introduced the broad ideas of combining samples across blocks and combining feature sets across blocks. Here, the two ideas are organized as independent axes. The classification is descriptive: it identifies the structure of the available measurements, not a particular statistical analysis or a guarantee that integration will be scientifically useful.

For two blocks, the relevant intersections are \(S_A\cap S_B\) and \(F_A\cap F_B\). The sets must be defined consistently. A feature counts as shared only when it represents a comparable measured feature in both blocks; a shared pathway annotation or a possible biological relationship between two different molecules is not, by itself, feature overlap. Likewise, sample correspondence requires evidence that observations refer to the same subject, not merely similar demographics or row positions.

A design may have complete, partial, or no usable correspondence. In a fully matched design, every relevant observation in one block has a valid counterpart in the other at the chosen unit. In a partially matched design, only some observations have counterparts. “Unmatched” is most useful for blocks with no usable cross-block subject links. If the identity or linkage information is simply unavailable, the design is not evidence that the underlying subject sets are truly disjoint; it is a design with unavailable correspondence.

Feature overlap also varies in degree. Exact equality of feature sets is not required for horizontal integration: blocks may share a common feature panel while one or both also contain additional features. When feature sets are distinct, they are vertically arranged in the broad sense. Partial overlap can produce a mixed structure rather than a clean horizontal or vertical case.

The Two-Axis Taxonomy

Matched-horizontal means that the blocks contain linked observations on the same subjects and share measured features. For example, two assay platforms might measure a common panel in the same participants. The shared subjects make cross-block comparisons at the subject level possible, while shared features give the blocks a common measurement dimension. Extra features in either block do not erase the shared portion.

Matched-vertical means that blocks are linked through the same subjects but measure different feature sets, often from different molecular layers. The subject link is what makes the blocks vertically complementary: one can consider distinct measurements for the same individuals. The feature sets need not have the same units or interpretation.

Unmatched-horizontal means the blocks share a feature space but do not have usable subject links. For example, two cohorts may measure a comparable assay panel in different participants. Their data can have a common feature schema, but a row in one block cannot be paired with a row in the other as the same person.

Unmatched-vertical means that neither subject correspondence nor common measured features links the blocks. This is a particularly weak basis for direct joint subject-level analysis: the blocks differ both in who was observed and in what was measured. It may still be possible to relate aggregate summaries under additional assumptions, but the design alone does not provide matched multi-omics observations.

These names describe the dominant structure, not an automatic instruction to concatenate matrices. Partial matching, partial feature overlap, repeated measurements, or multiple cohorts can yield several structures in one dataset. In those cases, state the pattern for each relevant subset rather than forcing one label on the entire study.

Worked Classifications

Why the Matrix Structure Follows the Classification

A simple matrix argument explains why the two axes matter. Let \(X_A\) be an \(n_A\)-by-\(p_A\) matrix and \(X_B\) an \(n_B\)-by-\(p_B\) matrix, with rows representing observations and columns representing features. Horizontal concatenation requires row \(i\) in each block to refer to the same analysis unit; otherwise the combined row would falsely pair different subjects. Vertical stacking requires columns to represent the same harmonized features in the same order; otherwise a column would change meaning between rows.

The justification is direct. In \([X_A\ X_B]\), each row combines measurements from two blocks, so the row has a coherent subject interpretation only if the observations align. In the stacked matrix, each column must have a single feature meaning for every row, so the two blocks need a common feature schema. These are structural conditions for those particular concatenations, not universal requirements for every possible integration method. Missing entries created by unequal feature panels or incomplete assay coverage are not automatically measured zeros.

Diagnostics, Alternatives, and Limitations

Before assigning a label, check the analysis unit, subject identifiers, and feature definitions. Ask whether the sample intersection is known, merely assumed, or unavailable; whether links are one-to-one at the stated unit; and whether feature names refer to genuinely comparable measurements. Report counts of intersections and unions when possible. Jaccard overlap, \(J(A,B)=|A\cap B|/|A\cup B|\), is a descriptive index between 0 and 1; it has no universal cutoff that decides whether a study is “matched” or “overlapping.”

Common signs that a classification is going wrong include apparent perfect matching based only on equal row counts, a shared feature label with different measurement definitions, and a matrix of missing values interpreted as biological zeros. Equal numbers of rows do not establish correspondence. Similar names do not guarantee equivalent features. An assay not run on a subject is not evidence that the subject's feature value was zero.

A frequent mistake is to call any two datasets with the same subjects “horizontal,” because they can be placed side by side. The correction is to classify both axes separately: if their feature sets are distinct, they are matched-vertical. Another mistake is to call cohorts matched because they measure the same feature panel. Shared columns establish feature overlap, not shared people; absent subject links, the design is unmatched-horizontal.

The taxonomy can be compared with simpler one-axis descriptions. Calling a design only “horizontal” emphasizes shared features but can conceal whether observations are paired. Calling it only “matched” emphasizes subject links but can conceal whether the feature measurements are common or complementary. The two-axis description retains both facts. For a complex study, a block-by-block table of sample and feature intersections is often more informative than one global label.

Finally, design classification does not resolve every analytical issue. It does not establish measurement comparability, remove batch effects, identify causal effects, or determine how to handle missing data. It also cannot recover correspondence that was never recorded. The taxonomy is a first structural audit: it clarifies which links are present before an analysis relies on them.