Why Pairing Changes the Question
A multimodal data set may contain RNA and protein measurements from the same cells, or it may combine RNA measurements from one collection of cells with protein measurements from another. Both designs can support analyses involving multiple modalities, but they do not provide the same information. The key distinction is whether the measurements can be linked to the same observational unit.
When RNA and protein are measured in the same cell, a researcher can examine whether cells with higher values in one modality also tend to have higher or lower values in the other. When the modalities were measured in separate cells, their separate distributions can still be summarized and compared, but the cell-by-cell relationship is not observed. An integration algorithm may place the separate data sets in a shared representation; that operation does not turn unobserved cell-level matches into measurements.
The earlier material on paired measurement structures distinguishes the observation unit from the sample or donor. That distinction matters here: sharing a donor, condition, or tissue does not by itself establish that an RNA observation and a protein observation came from the same cell. Pairing is a property of the measurements and their identifiers, not a conclusion drawn from similar cell types or nearby positions in an embedding.
What Is Identifiable From Paired Measurements?
Suppose each of \(n\) paired cells has an RNA-derived value \(X_i\) and a protein-derived value \(Y_i\). These could be transformed measurements or scores, provided the definitions and scales are appropriate to the question. The observed pairs are \((X_i,Y_i)\), and the association between modalities can be summarized, for example, by sample covariance or sample correlation.
The association is identifiable from the observed data only if the matching is meaningful and the paired measurements are available. For a population interpretation, the paired units must also represent the target population under an appropriate sampling design. If cells are clustered within donors, treating every cell as an independent replicate for inferential uncertainty may be inappropriate; a cell-level descriptive association and a donor-level inferential claim are not interchangeable.
Worked Example: Computing Association From Four Paired Cells
Suppose a hypothetical experiment measures an RNA score and a protein score in each of four cells. The paired values, in the recorded cell order, are RNA \(X=(1,2,3,4)\) and protein \(Y=(1,2,3,4)\). Treat these values as equally scaled scores for this illustrative calculation.
Both means are \(\bar{x}=\bar{y}=(1+2+3+4)/4=2.5\). The centered values in each modality are \((-1.5,-0.5,0.5,1.5)\). Thus the sum of paired centered products is
The sample covariance is \(s_{XY}=5/(4-1)=1.667\). For each modality, the sum of squared centered values is also 5, so \(s_X^2=s_Y^2=5/3\), and \(s_X=s_Y=\sqrt{5/3}\). Therefore,
In this deliberately simple example, the four observed RNA and protein scores increase together perfectly. This describes these paired values; by itself, it does not establish a causal effect or guarantee the same relationship in a broader population.
Why Separate Modality Data Cannot Recover Cell-Level Association
Unpaired data provide the values observed in each modality, but not which value in one modality belongs with which value in the other. The marginal distributions, means, and variances do not specify the joint distribution. In particular, they do not determine the centered products in the covariance formula.
A short derivation makes the missing information explicit. The covariance numerator contains \(\sum_i(X_i-\bar{x})(Y_i-\bar{y})\). Reordering the \(Y_i\) values leaves \(\bar{y}\) and \(s_Y\) unchanged, but changes which centered \(Y\) value multiplies each centered \(X\) value. Consequently, the covariance numerator can change even though all modality-specific summaries stay fixed.
Worked Example: Same Marginals, Different Pairings
Consider the same hypothetical RNA values \(X=(1,2,3,4)\), but now suppose the protein values are observed in a separate, unpaired collection and are \(1,2,3,4\). The marginal means and sample variances are 2.5 and \(5/3\) for both modalities. The separate lists do not reveal their cell-by-cell matching.
If one were to pair the lists in the same order, the calculation from the preceding example gives covariance \(5/3\) and correlation \(1.000\). If instead one paired the same RNA list with the protein list in reverse order, \(Y=(4,3,2,1)\), its mean and variance would remain 2.5 and \(5/3\). The centered protein values would be \((1.5,0.5,-0.5,-1.5)\), so the centered products sum to
That hypothetical matching gives covariance \(-5/3\) and correlation \(-1.000\). The two possible matchings have exactly the same modality-specific values and summaries but opposite associations. Since the unpaired experiment observed neither assignment, neither correlation is supported by its data. Choosing a matching after the fact would add an assumption, not reveal a measured relationship.
What Unpaired Integration Can Still Support
Unpaired data are not useless. Each modality can be analyzed on its own, and summaries can be compared across samples or conditions when the design and measurement scales make that comparison meaningful. For example, researchers may compare the distribution of an RNA score in an RNA sample with the distribution of a protein score in a protein sample. Such comparisons concern modality-specific distributions; they do not establish that high-scoring RNA cells correspond to high-scoring protein cells.
An unpaired integration method may use shared cell-type labels, sample metadata, or distributional structure to align data sets. Any resulting correspondence is an inferred alignment under the method's assumptions. It should be described as such, rather than as a directly measured cell-level relationship. If cell types, conditions, or batches differ between the modality-specific samples, an apparent alignment may reflect those differences rather than a common biological structure.
Worked Example: Separate Samples From the Same Donors
In a hypothetical study, RNA is measured in three cells from donor A with scores \(2,4,6\), and protein is measured in three different cells from donor A with scores \(1,3,8\). The data share a donor, but no cell identifiers link an RNA measurement to a protein measurement.
The RNA sample mean is \(\bar{x}=(2+4+6)/3=4\), and the protein sample mean is \(\bar{y}=(1+3+8)/3=4\). These modality-specific means are computable. The RNA sample variance is \(\{(2-4)^2+(4-4)^2+(6-4)^2\}/(3-1)=8/2=4\). The protein sample variance is \(\{(1-4)^2+(3-4)^2+(8-4)^2\}/2=(9+1+16)/2=13\).
The equal means do not imply that RNA and protein are equal within cells, and the different variances do not describe within-cell coupling. No paired centered products can be calculated because the three RNA cells are not matched to the three protein cells. Donor identity can support donor-level grouping or comparisons if the wider design permits them, but it does not supply the missing cell-level correspondence.
Design Checks, Failure Modes, and Alternatives
Before interpreting a multimodal result, verify the observational unit and the data linkage. In a genuinely paired design, each row used for a cross-modal calculation must represent the same cell in both modalities. In unpaired designs, rows should remain separate and any alignment should be labeled as inferred. Useful checks include:
- Confirm that cell identifiers are unique and refer to the same cell across modalities; matching row order alone is not evidence of pairing.
- Check how many cells have measurements in each modality and whether exclusions or filtering changed the set of linked cells.
- Inspect whether modality and donor, condition, or batch are confounded. If, for example, all RNA cells come from one condition and all protein cells from another, a cross-modal comparison cannot separate modality from condition.
- Check that the chosen measurements and scales are suitable for the intended comparison. Two numerically similar values need not have the same meaning or units.
- Keep the level of interpretation aligned with the design: cells, donors, and samples are different observational units.
The main alternatives differ in what they can claim. Paired analysis uses measured correspondences and can describe cross-modal relationships among the linked cells. Separate modality-specific analysis preserves the observed data without inventing matches, but cannot estimate cell-level association. Unpaired integration can construct a shared representation from separate data sets, but its alignments depend on modeling assumptions and do not substitute for measured pairing.
| Data structure | Supported directly by observed data | Not supported without extra assumptions |
|---|---|---|
| Same cells measured in both modalities | Modality-specific summaries and cell-level paired summaries | Automatic causal interpretation or population generalization without suitable design |
| Different cells, same donors | Modality-specific summaries and donor-level grouping | Cell-level cross-modal covariance or correlation |
| Different cells and donors | Separate summaries for the samples represented | Cell-level matching and clean modality comparisons if donor or condition is confounded |
| Unpaired data computationally aligned | Results conditional on the alignment method and its assumptions | Claim that the inferred matches were directly measured |
A frequent mistake is to reason, “The RNA and protein embeddings overlap, so the same RNA-high cells must be protein-high.” The overlap may indicate that the method placed similar distributions or cell populations near one another; it does not reveal the missing cell-level pairing. The correction is to state what was observed, report the method-dependent nature of any inferred alignment, and reserve cell-level association claims for genuinely paired measurements.
Check Your Understanding
Answer each question using the distinction between observed pairing and inferred alignment. Brief solutions follow.
- RNA and protein are measured on different cells from the same donor. Are they cell-paired? What can donor identity support?
- Two modality-specific data sets have the same values in each modality, but no linked cell identifiers. Can their cell-level correlation be determined?
- A paired study has four cells with RNA scores \(1,2,3,4\) and protein scores \(4,3,2,1\) in the same cell order. What is the sample correlation?
- An integration method aligns unpaired RNA and ATAC data into a shared embedding. What must be said before interpreting an apparent cross-modal correspondence?
- All RNA cells come from condition A and all protein cells come from condition B. Why is a difference between the modality-specific samples difficult to interpret?
Solutions. (1) No. They are donor-matched but not cell-paired; donor identity may support grouping by donor, but does not identify within-cell relationships. (2) No. Different assignments of one modality to the other can leave the marginal values unchanged while changing the association. (3) The centered products sum to \(-5\), and each sample standard deviation is \(\sqrt{5/3}\), so \(r=(-5/3)/(5/3)=-1.000\). (4) State that the correspondence is inferred by the method and depends on its assumptions; it is not a directly measured match. (5) Modality is confounded with condition, so observed differences cannot be attributed to modality rather than condition using that design alone.