Why the Independent Unit Matters
A single-cell experiment can contain thousands of cells but only a small number of donors. Those cells are not automatically thousands of independent biological replicates. Cells from one donor share genetics, environment, disease history, sample handling, and other donor-level characteristics. Consequently, their abundance measurements can be correlated even when the cells are measured separately.
The independent unit for a condition comparison is generally the unit independently sampled or assigned to that condition. In an observational comparison of donors, that unit is the donor. The number of cells helps describe each donor’s sample, but does not replace the number of independently sampled donors. Treating correlated cells as if they were independent replicates is called pseudoreplication.
The sampling models for cell-type counts describe how counts or proportions can vary as cells are sampled from a donor. That within-donor sampling uncertainty is real, but it is distinct from variation between donors. For an inference about a condition’s donor population, the latter determines how much independent replication supports the comparison.
How Correlation Changes the Standard Error
A simple exchangeable-correlation model makes the issue explicit. Suppose there are \(D\) independent donors, each contributing \(m\) cells, and consider a cell-level measurement \(Z_{di}\), where \(d\) indexes donors and \(i\) indexes cells. Assume all cells have variance \(\sigma^2\), and that two different cells from the same donor have correlation \(\rho\). For this derivation, assume cells from different donors are independent and the within-donor correlation is the same for every pair.
The overall cell mean is \(\bar{Z}=(Dm)^{-1}\sum_d\sum_i Z_{di}\). Its variance includes both the \(Dm\) individual variances and the covariance for every within-donor cell pair:
If the cells were independent, \(\rho=0\), and the variance would be \(\sigma^2/(Dm)\). With positive within-donor correlation, the variance is larger by the design effect, \(1+(m-1)\rho\). The corresponding standard error is multiplied by the square root of that factor. This derivation assumes equal cluster sizes and exchangeable within-donor correlation; real data may be more complex, but the central point remains: correlated cells contain less independent information than the same number of unrelated cells.
Worked Example: Quantifying the Design Effect
Suppose 8 donors each contribute 1,000 cells, and a cell-level measurement has within-donor correlation \(\rho=0.02\). Under the exchangeable model above, calculate the design effect and the standard-error inflation relative to an independent-cell calculation.
The design effect is \(1+(m-1)\rho=1+(1{,}000-1)(0.02)=1+19.98=20.98\). Thus the variance is 20.98 times the variance an independent-cell calculation would use, and the standard error is inflated by \(\sqrt{20.98}\approx4.58\).
The independent-cell calculation would treat all \(8{,}000\) cells as unrelated. The correlation-adjusted variance is instead equivalent, for this particular mean and model, to dividing the independent-cell variance by an effective cell count of approximately \(8{,}000/20.98=381.3\). That effective count is a way to express lost precision; it is not 381 independent donors. There are still only 8 independent donor units for assessing between-donor replication.
What Pseudoreplication Does to Inference
A cell-level test that assumes independence uses the wrong sampling variance when cells within donors are correlated. Its standard error is usually too small, its test statistic too large in magnitude, and its p-value too small. The nominal significance level no longer guarantees the advertised probability of a false positive under the null hypothesis. The problem is not simply that the test has “too much data”; it is that the calculation counts dependent observations as if they supplied independent evidence.
The point estimate can also target the wrong quantity. If donors contribute very different numbers of cells, a pooled cell-level proportion gives greater weight to donors with more captured cells. An equally weighted mean of donor-level abundance summaries instead describes the average donor, provided that is the prespecified estimand. These summaries need not be equal. Choosing the target quantity is a scientific decision; the next step is to use an analysis whose weighting and uncertainty match that target.
Worked Example: Many Cells, Few Independent Donors
Consider a hypothetical two-group study with 4 donors per group and 10,000 cells per donor. A prespecified abundance summary is 0.10, 0.12, 0.08, and 0.14 for the four donors in Group A, and 0.20, 0.22, 0.18, and 0.24 for the four donors in Group B. For illustration, suppose each donor’s displayed value corresponds to the observed fraction among exactly 10,000 cells.
Pooling cells gives Group A a fraction of 0.11 and Group B a fraction of 0.21. An independence-based two-proportion calculation would use \(SE=\sqrt{0.11(0.89)/40{,}000+0.21(0.79)/40{,}000}\approx0.002568\). The difference, 0.10, divided by this standard error is about 38.94. That enormous statistic is a nominal cell-level result: its independence assumption is not justified because the cells are clustered within only four donors per group. It must not be reported as valid donor-level evidence.
The relevant replication for a comparison across donors is four independent donors in each group. A donor-level analysis retains the variation among their eight abundance summaries, rather than allowing 80,000 cells to masquerade as 80,000 independent biological units. This example does not show that cell counts are useless: they help estimate each donor’s summary. It shows that they cannot, by themselves, establish how reproducible a group difference is across donors.
Formulating a Donor-Level Comparison
For two independent groups, let \(x_{1j}\) and \(x_{2j}\) be the prespecified abundance summaries from donor \(j\) in Groups 1 and 2. A basic estimand is the difference in average donor-level abundance, \(\mu_1-\mu_2\). An independent-groups analysis treats each donor summary as one observation. A Welch two-sample \(t\) procedure is one option when donor summaries are approximately normally distributed within groups; it does not require equal group variances.
The assumptions are about donors and their summaries: donors are independent across groups; the group assignment or sampling supports the intended comparison; the summary is defined consistently across donors; and the within-group distributions are sufficiently well behaved for the small-sample \(t\) approximation. With few donors, distributional assumptions are difficult to diagnose, and results should be interpreted with that uncertainty in mind. No test can compensate for unrepresentative donor sampling or confounding in an observational study.
Worked Example: Welch Comparison of Donor Summaries
Use the eight hypothetical donor summaries above. Group A has values 0.10, 0.12, 0.08, and 0.14; Group B has 0.20, 0.22, 0.18, and 0.24. Assume the donors are independent across groups and the summaries are approximately normal within each group. These assumptions are illustrative; four donors per group provide little ability to assess normality.
The sample means are \(\bar{x}_A=(0.10+0.12+0.08+0.14)/4=0.11\) and \(\bar{x}_B=0.21\). For either group, the squared deviations from its mean sum to \(0.002\), so the sample variance is \(s^2=0.002/(4-1)=0.0006667\). The estimated difference in means, Group B minus Group A, is \(0.21-0.11=0.10\), or 10 percentage points.
The Welch standard error is \(SE=\sqrt{s_A^2/4+s_B^2/4}=\sqrt{0.0006667/4+0.0006667/4}\approx0.01826\). Therefore \(t=0.10/0.01826\approx5.477\). Because the estimated variances and sample sizes are equal, Welch’s degrees of freedom are \(df=6\). The two-sided \(p\)-value is approximately 0.0015. Under the stated model, this is evidence against equal group means; it is not the probability that the null hypothesis is true.
Using \(t_{0.975,6}\approx2.447\), a 95% confidence interval for the difference is \(0.10\pm2.447(0.01826)=0.10\pm0.0447\), or approximately \([0.0553,0.1447]\). In this hypothetical example, the interval is about 5.53 to 14.47 percentage points higher in Group B. The interval procedure has 95% coverage over repeated samples under its assumptions; it does not assign a 95% probability to this realized interval containing a fixed parameter.
Designs, Diagnostics, and Alternatives
The independent-donor comparison above is not appropriate for every study design. If the same donor contributes samples under multiple conditions, those outcomes are related: the donor remains the independent unit, while the within-donor pairing must be represented in the analysis. Repeated time points and other clustered designs likewise require their dependence structure to be respected. The appropriate model depends on the design and the estimand; simply pooling all cells across conditions does not remove donor dependence.
Three broad approaches are useful to distinguish:
| Approach | What it treats as independent | Main consideration |
|---|---|---|
| Cell-level test assuming independence | Every cell | Generally invalid for donor-level inference when within-donor correlation is present. |
| Donor-level analysis | Each donor’s prespecified summary | Directly reflects between-donor replication; may be imprecise with few donors. |
| Cluster-aware model | Donors are clusters | Can model within-donor dependence, but depends on correct specification and enough independent clusters for reliable inference. |
Diagnostics should focus on the donor-level evidence and the design. Check how many independent donors occur in each group, whether any donor dominates a pooled cell count, and whether donor summaries show outliers, skewness, or markedly different variability. Examine whether sample processing or other donor characteristics align with condition. A large number of cells cannot resolve a confounding pattern that occurs between donors.
Cluster-aware mixed models or variance estimators can be alternatives when their assumptions suit the design. They do not make the number of donors irrelevant: with few clusters, variance estimates and reference distributions can be unreliable. A donor-level analysis can be more transparent, but may discard some information about the precision of each donor’s summary. The right choice depends on the target and model, not on which method produces the smallest p-value.
Common Mistake and Practical Limits
Wrong reasoning: “There are 50,000 cells, so the study has 50,000 replicates and extremely high power.” This confuses measurement volume with independent replication. If those cells came from six donors, the study has six independent donor units for a donor-level condition comparison.
Correction: Report both the number of donors and the number of cells, then base uncertainty on the independent-unit structure. More cells can sharpen each donor’s estimate, but generalization across people depends primarily on the number and representativeness of donors. Donor-level inference also has limits: a small donor sample can yield wide uncertainty, and a well-calibrated test cannot make an observational association causal.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A study has 12 donors, each contributing 2,000 cells. How many independent donor units support a between-donor condition comparison?
- Under exchangeable correlation with \(m=200\) cells per donor and \(\rho=0.01\), calculate the design effect and standard-error inflation.
- Why can a cell-level test yield an extremely small p-value while donor-level evidence remains uncertain?
- When the same donors contribute observations in two conditions, are the two sets of donor observations independent? What must the analysis reflect?
Solutions. (1) There are 12 independent donor units, not 24,000. (2) The design effect is \(1+(200-1)(0.01)=2.99\); the standard-error inflation is \(\sqrt{2.99}\approx1.73\). (3) The cell-level test may count correlated cells as independent and underestimate uncertainty; the donor-level analysis reflects variation across the relatively few donors. (4) No. The observations from the same donor are paired or otherwise dependent, so the analysis must account for that within-donor structure.