Separating Shared and Modality-Specific Cell Variation

Why Separate Shared From Modality-Specific Variation?

A cell can vary from other cells in ways that appear across several measurements and in ways that are visible mainly in one modality. A biological state may affect RNA abundance, surface protein abundance, and chromatin accessibility together. At the same time, an RNA program may vary without a corresponding protein change, an antibody-derived protein signal may vary independently of RNA, or accessibility at a set of regions may distinguish cells even when the measured RNA scores do not.

These distinctions matter when interpreting a joint representation. A pattern that contributes strongly to one modality may be diluted when combined with other modalities; conversely, a shared pattern may dominate a joint display and make a distinct modality-specific pattern harder to see. Paired measurement means the modalities are observed on the same cell identities, so their cell-to-cell covariation can be examined. Pairing alone, however, does not prove that covariation is shared biological signal.

The decomposition below applies to centered, comparable cell-level scores, such as scores for a feature set or a component within a modality. It is not a model of raw RNA, protein-tag, or accessibility counts. The notation $R_i$, $P_i$, and $A_i$ denotes the respective score for cell $i$; a score may be derived from a processed representation. The same logic applies to any pair of modalities.

Definition: A shared component is cell-to-cell variation represented in more than one modality. A modality-specific component is variation in a modality not represented by the shared component in the other modalities. Measurement error is variation in an observed score not assigned to either biological component in the model. These are model-based categories: a specific component can include real biology, technical effects, or both unless further evidence distinguishes them.

A Paired-Cell Decomposition and Its Assumptions

For a cell sampled from the population of interest, consider the model

\begin{aligned} R &= \lambda_R S + U_R + \varepsilon_R,\\ P &= \lambda_P S + U_P + \varepsilon_P,\\ A &= \lambda_A S + U_A + \varepsilon_A. \end{aligned}

Here $S$ is a centered shared cell-level component; $U_R$, $U_P$, and $U_A$ are centered RNA-, protein-, and accessibility-specific components; and the $\varepsilon$ terms are centered measurement errors. The $\lambda$ values are loadings that describe how strongly the shared component appears in each score. For a simple variance partition, assume finite variances and that all components are mutually uncorrelated: $S$ is uncorrelated with every $U$ and every $\varepsilon$; each $U$ is uncorrelated with every other $U$ and every error; and errors from different modalities are uncorrelated. Within each modality, the specific component and its error must also be uncorrelated. Cells must be paired correctly, and scores must describe the same cell population under a compatible preprocessing and sampling design.

These are substantive assumptions, not consequences of having paired data. In particular, the requirement that $S$ be uncorrelated with each error is essential. Without it, covariance between two observed scores can include terms involving shared signal and error, so it cannot be attributed just to the shared component.

Theorem: Under the stated zero-mean and mutual-uncorrelation assumptions, $$ \operatorname{Var}(R)=\lambda_R^2\operatorname{Var}(S)+\operatorname{Var}(U_R)+\operatorname{Var}(\varepsilon_R), $$ and likewise for $P$ and $A$. For two distinct modalities, such as RNA and protein, $$ \operatorname{Cov}(R,P)=\lambda_R\lambda_P\operatorname{Var}(S). $$ The covariance identity is justified only when all cross-component covariance terms outside the shared component vanish.

The variance result follows by expanding the variance of a sum: all cross-products have zero covariance under the assumptions, leaving only the three component variances. For the covariance result, expand both sums before applying those assumptions:

\begin{aligned} \operatorname{Cov}(R,P) &=\operatorname{Cov}(\lambda_R S+U_R+\varepsilon_R,\, \lambda_P S+U_P+\varepsilon_P)\\ &=\lambda_R\lambda_P\operatorname{Var}(S) +\lambda_R\operatorname{Cov}(S,U_P) +\lambda_R\operatorname{Cov}(S,\varepsilon_P)\\ &\quad+\lambda_P\operatorname{Cov}(U_R,S) +\operatorname{Cov}(U_R,U_P) +\operatorname{Cov}(U_R,\varepsilon_P)\\ &\quad+\lambda_P\operatorname{Cov}(\varepsilon_R,S) +\operatorname{Cov}(\varepsilon_R,U_P) +\operatorname{Cov}(\varepsilon_R,\varepsilon_P). \end{aligned}

Every term after the first is zero by the mutual-uncorrelation assumptions, including $\operatorname{Cov}(S,\varepsilon_R)$ and $\operatorname{Cov}(S,\varepsilon_P)$. Thus only $\lambda_R\lambda_P\operatorname{Var}(S)$ remains. If, for example, errors track cell quality in both modalities, or a supposedly specific component is correlated across modalities, the extra terms need not vanish; the covariance identity then does not hold.

Worked Example: Partitioning Variance Under the Model

Suppose hypothetical centered scores follow the stated assumptions, with $\operatorname{Var}(S)=4$, $\lambda_R=1.5$, $\lambda_P=0.5$, $\operatorname{Var}(U_R)=4$, $\operatorname{Var}(U_P)=8$, $\operatorname{Var}(\varepsilon_R)=1$, and $\operatorname{Var}(\varepsilon_P)=3$. Find the two total variances and their covariance.

For RNA, the shared contribution is $1.5^2(4)=9$, so $\operatorname{Var}(R)=9+4+1=14$ score-units squared. Its shared fraction of total variance is $9/14=0.6429$, or about 64.3%.

For protein, the shared contribution is $0.5^2(4)=1$, so $\operatorname{Var}(P)=1+8+3=12$ score-units squared. Its shared fraction is $1/12=0.08333$, or about 8.33%.

The cross-modality covariance is $\operatorname{Cov}(R,P)=1.5(0.5)(4)=3$ score-units squared. The shared component is the same latent source in both scores, but it accounts for very different fractions of their total variance because the loadings and specific and error variances differ. These values are properties of the hypothetical model, not estimates from a real dataset.

Examples of Shared Structure and Specific Signals

The covariance identity gives a precise interpretation only under the assumptions. In practice, paired data provide empirical covariances, while the decomposition explains what those covariances would represent in an idealized model. Neither an observed covariance nor a high correlation by itself establishes which biological program produced it. The examples below separate the arithmetic from the assumptions needed to interpret it.

Worked Example: A Protein Signal With a Strong Specific Component

Consider hypothetical paired cells with $\operatorname{Var}(S)=1$. Let $R=S+U_R$ and $P=S+U_P$, with $\operatorname{Var}(U_R)=0.25$, $\operatorname{Var}(U_P)=4$, and zero measurement error. Assume $S,U_R,U_P$ are mutually uncorrelated. Calculate each total variance, their covariance, and their correlation.

The within-modality variances are $\operatorname{Var}(R)=1+0.25=1.25$ and $\operatorname{Var}(P)=1+4=5$. By the covariance identity, $\operatorname{Cov}(R,P)=1(1)(1)=1$. Therefore

\operatorname{Corr}(R,P) =\frac{1}{\sqrt{1.25\cdot 5}} =\frac{1}{2.5} =0.4000.

The shared component contributes 80.0% of RNA variance ($1/1.25$) but 20.0% of protein variance ($1/5$). The moderate correlation is compatible with a strong shared component in absolute terms alongside substantial protein-specific variation. It does not mean that 40% of cells are shared, nor does it identify the source of the protein-specific variance.

Worked Example: Averaging Can Attenuate an RNA-Specific Pattern

Suppose four hypothetical paired cells have centered shared scores $S=(-1,-1,1,1)$. RNA has an additional specific pattern $U_R=(1,-1,-1,1)$, and protein has no corresponding specific pattern. Thus $R=S+U_R=(0,-2,0,2)$ and $P=S=(-1,-1,1,1)$. The cellwise average is $(R+P)/2=(-0.5,-1.5,0.5,1.5)$.

Using the population variance convention for these four listed values, $\operatorname{Var}(S)=1$ and $\operatorname{Var}(U_R)=1$; their covariance is zero because the products sum to zero. RNA therefore has variance $1+1=2$. The averaged score is $S+U_R/2$, whose variance is $1+(1/2)^2(1)=1.25$. Its RNA-specific variance contribution has fallen from 1 to 0.25, while the shared contribution remains 1.

This simple average has attenuated, not eliminated, the specific pattern. Other joint representations can attenuate a modality-specific signal differently, depending on their scaling, feature selection, and construction. The example is not evidence that a particular integration method necessarily loses signal; it shows why a joint coordinate should not automatically be treated as a complete summary of each input modality.

Worked Example: Calculating Covariance From Paired Cells

For four hypothetical paired cells, suppose centered RNA scores are $R=(-2,-1,1,2)$, and protein scores are $P=(-1,0,0,1)$. Compute the sample covariance and correlation. Use the usual sample covariance denominator $n-1$.

Both sample means are zero. The sum of cross-products is $(-2)(-1)+(-1)(0)+(1)(0)+(2)(1)=4$, giving $s_{RP}=4/(4-1)=1.333$ score-units squared. The sums of squared scores are 10 for RNA and 2 for protein, so $s_R^2=10/3=3.333$ and $s_P^2=2/3=0.6667$. The sample correlation is

r_{RP}=\frac{s_{RP}}{\sqrt{s_R^2s_P^2}} =\frac{4/3}{\sqrt{(10/3)(2/3)}} =\frac{4}{\sqrt{20}} =0.8944.

The paired scores show a strong positive linear association in this small hypothetical set. This calculation alone cannot tell whether the association is a shared biological component: with only four cells, it is also sensitive to individual observations, and correlated errors or correlated modality-specific biology could contribute. It is a descriptive estimate, not proof of the decomposition assumptions.

Diagnostics, Alternatives, and Limitations

A useful analysis asks whether the decomposition is plausible for the scores and cells at hand, rather than treating the equations as guaranteed. Inspect paired cell identities and modality-specific score distributions first. Check for batch or cell-quality patterns that could induce cross-modality covariance, and examine whether a few extreme cells drive the observed relationship. Repeat descriptive calculations within relevant cell groups or batches when appropriate: pooling distinct groups can create covariance even if scores are unrelated within each group. Conversely, centering or scaling choices can change the relative contribution of modalities to a joint representation.

A practical warning sign is a strong association between modality-specific residual patterns, or a common technical covariate associated with both modalities. Such patterns challenge the assumption that specific components and errors are uncorrelated across modalities. A paired-cell permutation diagnostic can also be informative: randomly permuting the cell pairing in one modality breaks cell-level correspondence, so a marked reduction in association indicates that the observed pairing contributes to the relationship. This is a diagnostic, not a proof of biological sharing; confounding or technical effects tied to cell identity can also produce paired association.

The decomposition is deliberately simple. Alternatives answer different questions. Separate modality analyses preserve the ability to inspect each modality on its own, but do not summarize shared structure in one representation. A joint representation combines information across modalities, but its coordinates may mix shared and specific patterns and depend on how the modalities are scaled and weighted. Explicit latent-factor models can represent several shared and modality-specific factors, but require additional modeling choices and identification assumptions. No approach makes modality-specific biology disappear from the data; it may simply become less visible in a chosen summary.

The simple model also has limits. A single shared factor may be inadequate when several biological processes are shared in different ways. Real modality-specific components can be correlated across modalities, and measurement errors can share technical causes. Nonlinear relationships and cell-state-dependent loadings are not represented by the linear equations. Finally, the variance fractions depend on the chosen score definition and scale: they are not universal properties of a cell or a modality.

Common pitfall: “The RNA–protein covariance equals the shared biological variance, so a positive covariance proves shared biology.” This is incorrect: the covariance equals $\lambda_R\lambda_P\operatorname{Var}(S)$ only under the stated orthogonality assumptions, and even then it is the shared variance multiplied by two loadings. Correlated modality-specific components, shared errors, or correlation between $S$ and an error add terms to the covariance. Treat the identity as a conditional model result, not as an automatic interpretation of an empirical correlation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In the decomposition, what assumption involving $S$ and the measurement errors is needed for the cross-modality covariance identity?
Suppose the shared component has variance 2, and the RNA and protein loadings are 2 and 0.5. Under the assumptions, what is their covariance?
Why might a joint average reduce the visibility of an RNA-specific pattern even when RNA and protein were measured on the same cells?
Give one reason that a positive paired-cell covariance might not be evidence of shared biological variation.

Solutions. (1) $S$ must be uncorrelated with the error in each modality, so $\operatorname{Cov}(S,\varepsilon_R)=\operatorname{Cov}(S,\varepsilon_P)=0$, in addition to the other required cross-component covariances being zero. (2) The covariance is $2(0.5)(2)=2$ score-units squared. (3) Averaging combines the RNA score with another modality that lacks the RNA-specific pattern, attenuating that pattern in the average; in the worked example its variance contribution fell from 1 to 0.25. (4) For example, a technical factor affecting both modalities, or correlated modality-specific biological components, can create positive covariance without satisfying the simple shared-only covariance model.