Two Ways to Represent the Same Experiment
Single-cell data contain many measured cells, but cells from the same donor share biological and technical features. A differential-expression analysis must account for that dependence. Two broad strategies are to aggregate measurements within each donor and condition before modeling, or to retain cell-level observations and represent donor dependence in a mixed model. These strategies are not simply two computational routes to an identical answer: their weighting, assumptions, and sometimes their estimands differ.
The experimental unit is the unit independently assigned to, or sampled under, the biological condition of interest. In many studies it is the donor. Cells are observations within that unit, not automatically independent biological replicates. The distinction matters especially when one donor contributes many more cells than another.
For pseudobulk counts, let \(Y_{ig}\) be the aggregate count for gene \(g\) in biological sample \(i\), with a sample-specific library-size adjustment in the model. In a paired design, the sample-level model must also represent the pairing, for example through donor fixed effects or a suitable random effect. For a cell-level analysis, let \(y_{ijg}\) denote the measurement for cell \(j\) from donor \(i\). A schematic model for a continuous or transformed response is
Here \(x_{ij}\) indicates condition, \(b_{ig}\) is a donor-specific random intercept, and \(\varepsilon_{ijg}\) is cell-level residual variation. This equation illustrates the dependence structure; it is not a specification of a count distribution. Count-scale cell-level models require an appropriate count response and link, while pseudobulk count models also need an appropriate sample-level mean and variance model.
What Is Being Estimated?
An estimand is the quantity a model is intended to estimate. A pseudobulk analysis usually targets a condition contrast in aggregate gene abundance or normalized expression across biological samples. A cell-level model may target a condition contrast in the cell-level response, conditional on its modeled donor effects and covariates. How that contrast relates to a population-average contrast depends on the response scale, link function, random-effects assumptions, and how donors and cells are weighted.
For example, a pooled cell-level mean gives more weight to donors contributing more cells. An equal-donor summary instead gives each donor the same weight. A model can use yet another weighting implied by its likelihood and design. Thus, “the condition effect” does not identify a unique estimand unless the target population, response scale, and weighting are specified.
Aggregation also changes what information is retained. A sum of counts preserves total observed counts within the aggregation group, but discards within-donor cell-to-cell variation and cell-level covariates. If cell numbers differ, raw sums are not directly comparable as expression levels without accounting for library size and the intended scale. Restricting analysis to a defined cell type helps clarify interpretation, but does not remove dependence among cells from the same donor.
Worked Example: Cell-Weighted and Donor-Weighted Contrasts
Suppose a hypothetical study measures normalized expression for one gene in two donors under control and treatment. In control, donor A contributes 10 cells with total expression 100, and donor B contributes 100 cells with total expression 200. Under treatment, donor A contributes 10 cells with total expression 100, and donor B contributes 100 cells with total expression 500. Assume these normalized values are comparable across the four donor-condition samples.
The donor-specific control means are \(100/10=10\) and \(200/100=2\); the treatment means are \(100/10=10\) and \(500/100=5\). The equal-donor condition means are therefore
The corresponding equal-donor contrast is \(7.5-6=1.5\) expression units. In contrast, pooling all cells within each condition gives control mean \(300/110\) and treatment mean \(600/110\). The pooled-cell contrast, calculated from the unrounded totals, is
The difference arises because donor B contributes most of the cells and has the larger treatment-control change. The pooled contrast weights cells; the equal-donor contrast weights donors. Neither calculation alone establishes a population-level treatment effect or its uncertainty. The example shows why an analysis must specify its target and account for donor replication.
Dependence and the Source of Uncertainty
Pseudobulk analysis treats each aggregated biological sample as a model observation. Its central independence assumption is that samples are independent after accounting for the design and any modeled repeated structure. If a donor contributes both conditions, those observations are not independent; pairing must be represented. Pseudobulk does not make dependence disappear—it moves the analysis to the sample level, where the dependence structure can be specified directly.
A cell-level mixed model represents correlation among cells from the same donor. Under the random-intercept model above, two distinct cells from the same donor share the same \(b_{ig}\), so their responses have covariance \(\tau_g^2\), assuming independent residuals. Cells from different donors have covariance zero under the model, conditional on the fixed predictors. The random-effects distribution and residual assumptions must be plausible enough for the intended inference, and the model must distinguish donor-level variation from cell-level residual variation.
A short derivation shows why counting cells as independent can badly understate uncertainty. Suppose each donor contributes \(m\) cells and
with independent donor effects and independent residuals. The mean across \(D\) donors and \(m\) cells per donor is \(\bar y = (Dm)^{-1}\sum_i\sum_j y_{ij}\). Substituting the model gives
The two sums are independent. Their variances are \(\tau^2/D\) and \(\sigma^2/(Dm)\), respectively, so
Increasing cells per donor reduces the residual contribution, but it does not reduce the donor contribution as if each cell were a new donor. An analysis that treats all \(Dm\) cells as independent omits the shared donor term and can report overly small standard errors.
Worked Example: Why More Cells Do Not Replace More Donors
For a hypothetical gene, suppose the donor-level variance component is \(\tau^2=4\) expression-units squared and the cell-level residual variance is \(\sigma^2=9\) expression-units squared. There are \(D=4\) donors, each contributing \(m=25\) cells. Under the random-intercept assumptions just stated, the variance of the overall mean is
If the 100 cells were incorrectly treated as independent with total variance \(\tau^2+\sigma^2=13\), the resulting standard error would be \(\sqrt{13/100}=0.3606\) expression units. If the donor component were omitted entirely, it would be \(\sqrt{9/100}=0.3000\). Both calculations are much smaller than 1.044 because they fail to account adequately for shared donor variation. The values are illustrative; real analyses estimate variance components and must also represent the design and response distribution.
With more cells from the same four donors, the residual term can shrink, but the donor term remains \(\tau^2/D=1\). Adding independent donors, rather than only cells, is what increases biological replication and reduces that component of uncertainty.
Comparing the Approaches in Practice
| Feature | Donor-level pseudobulk | Cell-level mixed model |
|---|---|---|
| Analysis unit | One aggregated observation per sample and cell type | Cells, with donor dependence represented in the model |
| Dependence | Represented through sample-level design, such as pairing or repeated measures | Represented through donor random effects or another suitable covariance structure |
| Information retained | Aggregate expression; cell-level variation and covariates are lost | Cell-level measurements and covariates are retained |
| Weighting | Typically based on sample-level likelihood and normalization; donors are not automatically weighted equally in every model | Likelihood and design can give more influence to donors with more observed cells |
| Main inferential concern | Few or unrepresentative biological samples, incorrect sample-level model, or poorly handled repeated observations | Misspecified within-donor dependence, weak donor replication, or unsuitable response and random-effects assumptions |
Pseudobulk is often attractive when the scientific target is a sample-level condition contrast and the study has a modest number of cells per donor but adequate biological replication. It provides a direct route to using donor-level replication and established sample-level count models. A cell-level mixed model can be useful when cell-level covariates, cell-specific responses, or within-donor heterogeneity are scientifically important. It requires a dependence model that is credible for those cells and a clear definition of its target contrast.
These approaches should not be compared by asking which produces the smaller p-value. Compare the estimated effects on compatible scales, the implicit weighting, the design terms, and the uncertainty. A discrepancy can arise because the methods target different weighted quantities, because one model is misspecified, or because a small number of donors makes estimates unstable. Agreement is reassuring only when the estimands and assumptions are also aligned.
Worked Example: A Paired Donor Contrast
Consider three hypothetical donors measured under both conditions. Their normalized pseudobulk summaries for a gene are: donor 1, control 4 and treatment 6; donor 2, control 5 and treatment 4; donor 3, control 7 and treatment 10 expression units. The paired differences are \(2,-1,3\) expression units. The mean paired difference is
The sample standard deviation of the differences is
Thus the standard error of the mean paired difference is \(s_d/\sqrt{3}=1.202\) expression units. Under independent, approximately normal donor-level differences, the one-sample \(t\) statistic for \(H_0:\mu_d=0\) is \(1.333/1.202=1.109\), with \(df=2\). This small hypothetical study provides very little precision; the estimated positive contrast is not, by itself, strong evidence of a nonzero population mean difference. A cell-level mixed model could use the paired structure and cells, but it must still reflect that only three donors supply independent biological replication.
Assumptions, Diagnostics, and Limitations
For either approach, the design must support the contrast being estimated. Condition should not be completely confounded with donor, batch, or another covariate. Samples should be assigned to the correct donor and cell type, and the model must reflect paired or repeated observations. In pseudobulk, aggregation and normalization must be appropriate for the intended expression scale. In a mixed model, the response distribution, link, random-effects structure, and conditional independence assumptions must be suitable. Neither method can recover information about between-donor variability when there are too few independent donors.
Useful checks include verifying donor and sample counts by condition, cell counts per donor-condition group, the presence of missing or highly unbalanced samples, and whether any condition is represented by only a small subset of donors. Check that the estimated contrast corresponds to the intended comparison and is not driven by one donor or a large difference in cell representation. For mixed models, examine whether the model converges and whether its estimated donor variability is plausible; for pseudobulk, inspect sample-level fit and whether the normalization and design reflect the data. These checks identify potential problems, but cannot prove that the assumptions hold.
A common mistake is to argue that thousands of cells provide thousands of independent replicates. The correction is to distinguish measurement volume from biological replication: cells within one donor share donor-level influences, and the effective information about population-level condition differences depends strongly on the number and diversity of donors. More cells can improve the characterization of a donor, but they do not substitute for additional independently sampled donors.
Another limitation is that aggregation can hide heterogeneity: two donors with the same pseudobulk mean may have very different cell-level distributions. Conversely, cell-level modeling is not automatically more informative for a population-level effect; its extra observations can make estimates appear precise if the donor dependence or estimand is mishandled. The choice is therefore scientific and statistical, not just computational.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A study has 8 donors, each contributing 500 cells in one condition. What is the number of independent biological replicates, and why are there not 4,000?
- In one condition, donor A contributes 20 cells and donor B contributes 200. What weighting difference can arise between a pooled-cell mean and an equal-donor mean?
- Under a random-intercept model, what covariance is induced between two different cells from the same donor?
- If the number of cells per donor doubles while the number of donors stays fixed, which component of the mean's variance decreases in the derivation above, and which does not?
Solutions. (1) There are 8 independent biological replicates, assuming donors were independently sampled. Cells from the same donor share donor-level influences, so 4,000 measurements do not represent 4,000 independent donors. (2) The pooled-cell mean gives donor B ten times as much weight as donor A because B contributes ten times as many cells. An equal-donor mean assigns one-half weight to each donor; the two summaries can differ when their expression levels differ. (3) With independent residuals and a shared random intercept, the covariance is \(\tau^2\), the variance of the donor random effect. (4) The residual component \(\sigma^2/(Dm)\) decreases as \(m\) increases. The donor component \(\tau^2/D\) does not decrease, so adding cells alone cannot eliminate uncertainty from between-donor variation.