Why the Unit of Analysis Matters
In CITE-seq, RNA and antibody-derived protein measurements are paired for each measured cell. A positive association across all cells might seem to indicate that cells with more RNA for a gene also have more of its corresponding protein. But a pooled pattern can instead reflect differences between cell types: one type may tend to have both higher RNA and higher protein than another, even if RNA and protein vary little together within either type. The reverse can also occur: a pooled relationship can obscure a clear within-type association.
The question here is therefore not simply whether RNA and protein are correlated. It is which association is being estimated: a relationship among cells of the same type, a relationship across a mixture of types, or a relationship adjusted for type while allowing types to have different slopes. These are different statistical targets.
The RNA and protein values need not share units. For example, \(R_i\) might be a normalized or transformed RNA expression value and \(P_i\) a transformed, background-adjusted antibody measurement. A slope then has units of protein-scale units per RNA-scale unit. The scale and preprocessing must be stated because the numerical association can change under transformations. A log transformation, for instance, changes the meaning of a one-unit increase.
Estimating a Within-Type Association
A useful starting model, fitted separately within cell type \(g\), is the simple linear regression \(P_i=\beta_{0g}+\beta_{1g}R_i+\varepsilon_i\) for cells with \(C_i=g\). The slope \(\beta_{1g}\) is the expected difference in transformed protein measurement associated with a one-unit difference in transformed RNA measurement within that type, under the model. It is an association, not a causal effect.
The fitted slope is the centered cross-product divided by the centered RNA sum of squares. For \(n_g\) cells in type \(g\), write \(\bar{R}_g\) and \(\bar{P}_g\) for the type-specific means. Then \(\hat{\beta}_{1g}=S_{RP,g}/S_{RR,g}\), where \(S_{RP,g}=\sum_{i:C_i=g}(R_i-\bar{R}_g)(P_i-\bar{P}_g)\) and \(S_{RR,g}=\sum_{i:C_i=g}(R_i-\bar{R}_g)^2\). The intercept is \(\hat{\beta}_{0g}=\bar{P}_g-\hat{\beta}_{1g}\bar{R}_g\). When the linear-model assumptions are appropriate, the slope standard error is \(SE(\hat{\beta}_{1g})=s_g/\sqrt{S_{RR,g}}\), where \(s_g^2\) is the residual sum of squares divided by \(n_g-2\). A test of \(H_0:\beta_{1g}=0\) uses \(t=\hat{\beta}_{1g}/SE(\hat{\beta}_{1g})\) with \(df=n_g-2\), under the stated assumptions.
Worked Example: Fitting a Slope Within One Cell Type
Suppose six hypothetical cells assigned to one cell type have the following transformed measurements. RNA and protein are in their respective transformed-expression units.
| Cell | RNA \(R_i\) | Protein \(P_i\) |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 2 |
| 3 | 3 | 3 |
| 4 | 4 | 4 |
| 5 | 5 | 4 |
| 6 | 6 | 5 |
The means are \(\bar{R}=3.5\) and \(\bar{P}=20/6=3.3333\). The centered RNA sum of squares is \(S_{RR}=17.5\). The centered cross-product is \(S_{RP}=11\), obtained by summing the six products of centered values. Therefore,
The slope means that, in this fitted line, a one-unit higher RNA value is associated with an estimated \(0.6286\)-unit higher protein value within this type. For inference, the centered protein sum of squares is \(S_{PP}=22/3\), so the residual sum of squares is \(SSE=S_{PP}-S_{RP}^2/S_{RR}=22/3-121/17.5=44/105\approx0.4190\). With \(df=6-2=4\), \(s^2=(44/105)/4=11/105\approx0.1048\), and \(SE(\hat{\beta}_1)=\sqrt{(11/105)/17.5}\approx0.07737\). Thus \(t=(11/17.5)/\sqrt{(11/105)/17.5}=\sqrt{66}\approx8.1240\), with 4 degrees of freedom; the two-sided \(p\)-value is approximately \(0.001249\) under the model assumptions.
This hypothetical calculation supports evidence against a zero linear slope under the stated model. It does not establish a causal effect, and the very small sample makes the normal-error assumption consequential. In real data, six cells from one donor would not provide six independent biological replicates.
Why Pooling Can Reverse the Pattern
The pooled cross-product has a within-type part and a between-type part. This exact decomposition explains why pooled and within-type associations can differ. Let \(\bar{R}\) and \(\bar{P}\) be the overall means, and let \(n_g\) be the number of cells in type \(g\). Then
To see why, write \(R_i-\bar{R}=(R_i-\bar{R}_g)+(\bar{R}_g-\bar{R})\), and write the corresponding decomposition for \(P_i-\bar{P}\). On multiplying and summing within group \(g\), each cross-term involving just one group-centered value vanishes because \(\sum_{i:C_i=g}(R_i-\bar{R}_g)=0\) and \(\sum_{i:C_i=g}(P_i-\bar{P}_g)=0\). The remaining terms are the within-type cross-product and the between-type contribution shown above. The same decomposition applies to the RNA sum of squares. Consequently, the pooled slope combines within-type covariation with shifts in the type-specific means; it is not generally an average of the type-specific slopes.
Worked Example: A Pooled Slope With the Opposite Sign
Consider four hypothetical cells from each of two types. Their transformed RNA and protein values are:
| Type | RNA values | Protein values | Within-type slope |
|---|---|---|---|
| A | 1, 2 | 8, 9 | 1 |
| B | 7, 8 | 2, 3 | 1 |
Within each type, the two values rise together: the slope is \(1\) protein-scale unit per RNA-scale unit. Across all four cells, however, \(\bar{R}=4.5\) and \(\bar{P}=5.5\). The within-type cross-products sum to \(0.5+0.5=1\). The between-type contribution is \(2(1.5-4.5)(8.5-5.5)+2(7.5-4.5)(2.5-5.5)=-36\). Therefore the pooled cross-product is \(1-36=-35\). The pooled RNA sum of squares is \(1+36=37\), giving
The pooled line slopes downward even though both within-type slopes are positive. Here, the between-type differences dominate: type A has lower RNA and higher protein measurements than type B. The negative pooled slope describes this particular mixture of cells, not the within-type pattern.
Comparing Cell Types With an Interaction Model
Separate regressions make each type’s slope visible, but a regression with a cell-type indicator and an RNA-by-type interaction can compare slopes directly. For two types, let \(G_i=1\) for type B and \(G_i=0\) for type A. The model \(P_i=\beta_0+\beta_1R_i+\gamma G_i+\delta(R_iG_i)+\varepsilon_i\) has slope \(\beta_1\) in type A and slope \(\beta_1+\delta\) in type B. The interaction coefficient \(\delta\) is the difference between those slopes. A test of \(H_0:\delta=0\) asks whether the fitted slopes differ, subject to the model’s assumptions. It does not test whether either type-specific slope is zero.
Worked Example: Comparing Two Type-Specific Slopes
In a separate hypothetical data set, four cells of type A have RNA values \(1,2,3,4\) and protein values \(2,4,5,8\). Four cells of type B have RNA values \(1,2,3,4\) and protein values \(5,6,7,7.5\). Both types have \(\bar{R}_g=2.5\) and \(S_{RR,g}=5\).
For type A, \(\bar{P}_A=4.75\). The centered cross-product is \(({-1.5})({-2.75})+({-0.5})({-0.75})+(0.5)(0.25)+(1.5)(3.25)=9.5\), so its slope is \(9.5/5=1.90\). For type B, \(\bar{P}_B=6.375\), and its centered cross-product is \(({-1.5})({-1.375})+({-0.5})({-0.375})+(0.5)(0.625)+(1.5)(1.125)=4.25\), so its slope is \(4.25/5=0.8500\). The estimated difference, type B minus type A, is \(0.8500-1.90=-1.05\) protein-scale units per RNA-scale unit.
In the interaction model with A as the reference type, this corresponds to \(\hat{\beta}_1=1.90\) and \(\hat{\delta}=-1.05\); the B slope is \(1.90-1.05=0.8500\). These data show different fitted slopes, but the arithmetic alone does not establish that the difference is statistically distinguishable from zero. That requires an uncertainty estimate and an appropriate sampling model.
Diagnostics, Design, and Alternatives
Before interpreting a fitted slope, inspect a scatterplot of paired RNA and protein values within each cell type. A pooled plot alone can hide the pattern of interest. Look for curvature, outliers, changing vertical spread, clusters or bands, and restricted RNA range. A fitted straight line can be misleading if the relationship is nonlinear; a slope can also be unstable when RNA varies little within a type. Residual-versus-fitted plots help reveal curvature and unequal variance, while a normal quantile plot can flag severe departures from normality when small-sample \(t\) inference is planned.
The model’s assumptions concern the measurement and sampling structure as well as the plotted shape. RNA dropout, antibody background, saturation, and normalization choices can distort measured values. Misclassified cell types can mix unlike cells within a stratum. Cells from one donor or sample may share biological and technical influences, making their errors dependent. If the target is association across donors, treating thousands of cells from a few donors as independent evidence can produce overly narrow uncertainty estimates. The experimental unit and intended population-level claim should determine the analysis; methods accounting for donor-level clustering or donor-specific slopes may be needed.
The main choices answer different questions:
| Approach | What it describes | Main caution |
|---|---|---|
| Pooled regression | Association across the observed mixture of cell types | Can be driven or reversed by differences in type means |
| Separate regressions | Association within each type, with type-specific slopes | May be imprecise in small or narrow-range groups |
| Type indicators without interaction | A common slope after allowing type-specific intercepts | Assumes the slope is the same in every type |
| Type indicators with interaction | Type-specific intercepts and slopes, and their differences | Uses more parameters and requires adequate data per type |
A common-slope model can be useful when the scientific question concerns a shared within-type trend and the data support that restriction. It should not be chosen merely because its estimate is more precise: if slopes truly differ, the shared estimate can conceal meaningful heterogeneity. Conversely, fitting a separate slope to every small cell population may yield unstable estimates. Report the chosen target, type-specific sample sizes, slopes with uncertainty where justified, and the measurement scale. Correlations can complement slopes, but a correlation is scale-free and does not express the change in protein units associated with a unit change in RNA.
Common Mistakes and Key Takeaway
- Wrong reasoning: “The pooled correlation is positive, so RNA and protein are positively associated within every cell type.” Correction: The pooled association includes between-type mean differences. Inspect and estimate within-type relationships.
- Wrong reasoning: “A significant within-type slope proves that RNA causes protein abundance to change.” Correction: The regression estimates an association; confounding, measurement processes, and direction of influence are not resolved by the slope test.
- Wrong reasoning: “A large cell count guarantees reliable population-level inference.” Correction: Cells from the same donor may not be independent. The number and structure of biological replicates matter.
- Wrong reasoning: “If the slope is not significant, the association is zero.” Correction: A non-significant test does not prove a zero slope; limited within-type variation or imprecision may leave substantial uncertainty.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- What does a within-type slope of \(0.4\) mean if RNA and protein are both transformed measurements?
- Can two positive within-type slopes produce a negative pooled slope? What data feature can cause this?
- In the two-type interaction model, what does \(\delta\) represent when type A is the reference?
- Name two reasons why treating every cell as an independent replicate can understate uncertainty.
Solutions. (1) Within that type, a one-unit higher transformed RNA value is associated with an estimated \(0.4\)-unit higher transformed protein value under the fitted linear model. (2) Yes. Opposing between-type shifts in RNA and protein means can dominate the within-type cross-products, as in the worked example. (3) \(\delta\) is the type B slope minus the type A slope. (4) Cells from the same donor or sample can share biological or technical influences, so their measurements are dependent; the number of independent donors may also be much smaller than the number of cells.