Two Different Forms of Incomplete Data
A donor can be absent from an analysis, or present with only a small number of observed cells. These are not interchangeable situations. If an entire donor is missing, that donor contributes no outcome estimate at all. If a donor has few cells, the donor may still contribute an estimate, but that estimate can be less precise and may fail to represent the donor's relevant cell population.
Suppose the scientific target is the average of a donor-level quantity \(Y_d\), such as a donor-specific treatment contrast or a donor-level expression summary. Let \(R_d=1\) if donor \(d\) has an observed, usable outcome and \(R_d=0\) otherwise. Let \(m_d\) be the number of observed cells for that donor and cell type. A donor with \(R_d=0\) is missing from the donor-level analysis; a donor with \(R_d=1\) and small \(m_d\) is incompletely sampled within donor.
The distinction matters because the remedies differ. More cells from donors already represented can improve their within-donor estimates, but cannot restore information about donors who were never observed. Conversely, adding donors with a handful of cells may improve coverage of donor-to-donor variation while leaving their cell-level summaries noisy. Inference therefore requires stating both the donor-level target and how each donor's estimate was obtained.
When Missing Donors Change the Target Estimate
For a donor population with mean outcome \(\mu_Y=E(Y_d)\), the complete-case mean is the mean among donors with observed outcomes:
This estimates the donor-population mean under conditions such as donors being selected independently of their outcomes, or under a justified adjustment for variables that explain selection. If donors with particular outcomes are more likely to be missing, the observed donor set can have a different mean from the target population. Having many cells per observed donor does not fix that selection problem.
Three standard descriptions help clarify the assumptions. Missing completely at random (MCAR) means donor observation is independent of the outcome and relevant covariates. Missing at random (MAR), conditional on observed covariates \(X_d\), means \(R_d\) is independent of \(Y_d\) given \(X_d\). Missing not at random (MNAR) means that, even after conditioning on the observed covariates used, observation still depends on the unobserved outcome or other unobserved information. These labels describe assumptions about the missingness process; the observed data alone generally cannot establish which one is true.
A Short Derivation of Complete-Case Bias
For a randomly selected donor, the mean outcome among observed donors is \(E(Y_d\mid R_d=1)\). Since \(R_d\) is binary, this conditional mean is \(E(R_dY_d)/P(R_d=1)\). By the covariance identity \(E(R_dY_d)=E(R_d)E(Y_d)+\operatorname{Cov}(R_d,Y_d)\), and because \(E(R_d)=P(R_d=1)\), the difference from the target mean is:
Thus, if donors are observed independently of their outcomes, the covariance is zero and the observed-donor mean has no selection bias under this model. If observation tends to occur for higher-outcome donors, the covariance is positive and the observed-donor mean tends to be too high. This result concerns selection of donors; it says nothing by itself about the accuracy of cell sampling within an observed donor.
Worked Example: Outcome-Related Donor Missingness
Consider a hypothetical target set of five donors with donor-level outcomes \(1, 2, 3, 4,\) and \(8\) units. The outcome is observed for the first four donors, while the donor with outcome \(8\) is missing. Compare the target mean with the complete-case mean.
The target mean across all five donors is:
The complete-case mean is:
In this example, the observed donors' mean is \(1.1\) units below the target mean because the donor with the largest outcome is missing. The arithmetic illustrates the consequence of this particular missingness pattern; it does not show that the direction or size of bias can be inferred from observed outcomes alone. In real data, the unobserved donor's outcome is not available for this direct comparison.
Adjustment When Donor Observation Is Explained by Covariates
If donor observation is plausibly MAR conditional on measured covariates \(X_d\), adjustment can recover a target-population mean under additional conditions. One option is standardization: estimate the outcome mean within covariate groups using observed donors, then average those estimates over the covariate distribution of the target donor population. Another is inverse-probability weighting (IPW). Let \(\pi_d=P(R_d=1\mid X_d)\) be the donor's observation probability. If these probabilities are correctly specified, positive for every covariate pattern in the target population, and the outcome is conditionally independent of observation given \(X_d\), an IPW mean is:
The weights give more influence to observed donors who represent donors with similar covariates but lower observation probabilities. Weighting cannot recover outcome information for a group with no chance of being observed: positivity requires \(\pi_d>0\) throughout the target population. It also does not solve unmeasured outcome-dependent selection. Estimated weights can be unstable when observation probabilities are near zero, so their distribution and sensitivity to extreme values should be examined.
Worked Example: Weighting for Unequal Observation Probabilities
Suppose a hypothetical target population has eight donors: four in covariate group A and four in group B. Assume their mean outcomes are \(2\) units in A and \(8\) units in B. The target mean is therefore \(5\) units. In an observed sample, two A donors and all four B donors are observed. Assume observation probabilities are known to be \(\pi_A=0.5\) and \(\pi_B=1\), and that within each group observation does not depend on the outcome.
The unweighted observed mean is:
For the IPW mean, each observed A donor receives weight \(1/0.5=2\), and each observed B donor receives weight \(1/1=1\). The weighted numerator is \(2(2\cdot2)+4(1\cdot8)=40\), and the denominator is \(2(2)+4(1)=8\). Thus:
The weighting recovers the target mean in this constructed example because the group membership and observation probabilities are known, every target group has a positive observation probability, and selection is independent of outcome within groups. If those conditions fail, the agreement with the target is not guaranteed.
What Few Cells Do to a Donor Estimate
For an observed donor, let \(Z_{di}\) be a cell-level quantity for cell \(i\), and let the donor's target be the mean \(M_d\) in the relevant cell population. A simple cell-sample estimate is \(\bar{Z}_d=m_d^{-1}\sum_{i=1}^{m_d}Z_{di}\). If the observed cells are a representative random sample from that population, then \(E(\bar{Z}_d\mid d,m_d)=M_d\). Under independent cell sampling with within-donor variance \(\sigma_d^2\), its conditional variance is \(\sigma_d^2/m_d\). Fewer cells therefore increase sampling uncertainty, even when they do not systematically shift the donor estimate.
The independence calculation is an idealized reference, not a default guarantee. Cells from the same donor can be correlated, and cells may be preferentially recovered or retained according to their state, quality, or expression. In those cases, the uncertainty need not shrink at the independent-cell rate, and the observed-cell mean may target a selected subset rather than the intended cell population. The earlier material on cell sampling uncertainty within donors and intraclass correlation explains why cell count alone does not determine precision.
A further choice is whether the scientific target weights donors equally or weights cells equally. An equal-donor average gives each donor's estimate the same weight. A pooled-cell average gives more influence to donors with more observed cells. The second can estimate a cell-weighted quantity, but it is not automatically an estimate of the average donor. If cell recovery differs with donor biology or outcome, cell weighting can change the target as well as the precision.
Worked Example: Few Cells Versus Cell-Weighted Averaging
Suppose three hypothetical donors have estimated donor-level quantities of \(1\), \(3\), and \(5\) units. Their observed cell counts are \(2\), \(2\), and \(10\). If the target is the mean donor, each donor contributes equally:
Pooling the cell-level observations gives the donor estimates weights proportional to their cell counts:
The cell-weighted result is higher because the donor with an estimate of \(5\) units contributes ten cells, while each other donor contributes two. Neither calculation is universally correct: \(3\) units answers the equal-donor question, whereas \(4.143\) units answers a cell-weighted question under this aggregation. A small cell count for a donor is not, by itself, a reason to omit that donor from an equal-donor analysis.
Diagnostics, Alternatives, and Limits
A useful diagnostic begins with the donor flow: record how many donors were eligible, enrolled, processed, passed quality control, and contributed each cell type. Tabulate cell counts by donor and relevant covariates, and compare observed with unobserved donors on covariates available for both. For represented donors, inspect donor-level estimates against cell count and assess whether conclusions change when low-cell donors are excluded or retained with appropriate uncertainty. These checks can reveal problems, but they cannot prove that missingness is MCAR or MAR.
The main approaches answer different questions and depend on different assumptions:
| Approach | What it does | Key limitation |
|---|---|---|
| Complete-case donor analysis | Uses donors with observed outcomes. | Can be biased if inclusion is related to outcome or relevant covariates. |
| Covariate standardization or IPW | Adjusts for observed differences in donor observation, under a justified MAR model. | Needs measured predictors, correct modeling, and positivity; cannot remove unmeasured selection bias. |
| Equal-donor summary with uncertainty for cell sampling | Retains donor as the unit of the target while recognizing that some donor estimates are noisier. | Requires representative cell sampling or a model for the sampling process; few cells can leave estimates imprecise. |
| Cell-pooled summary | Lets donors with more observed cells contribute more. | Targets a cell-weighted quantity, which may differ from the mean donor. |
Imputation is another possible strategy when missing outcomes can be modeled using informative observed variables, but it does not make untestable assumptions disappear. Any adjustment needs to describe the target, the assumed missingness process, and the information used. If a cell type has no observed cells for a donor, a cell-type-specific expression estimate is unavailable for that donor; it should not silently be recorded as zero expression. A biological absence, a technical failure to capture cells, and an unobserved cell type are distinct possibilities that require scientific context.
Common Pitfall
Wrong reasoning: “One donor has very few cells, so I will drop that donor; another donor has many cells, so those cells can compensate for the missing information.” This treats cell count as a substitute for donor representation. It can change an equal-donor target and does nothing to recover an entirely missing donor. The correction is to keep the two issues separate: evaluate selection into the donor analysis, and evaluate the precision and representativeness of cells within each observed donor. State whether the analysis targets an average donor or an average cell before choosing weights.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A donor outcome is observed with probability \(0.7\), independently of the outcome. Under the stated model, is the complete-case donor mean biased for the population mean? Explain.
- In a target population, 30% of donors are in group A and 70% are in group B. Their mean outcomes are \(2\) and \(6\) units, respectively. Compute the population mean.
- A donor has four randomly sampled cells with within-donor standard deviation \(3\) units. Under independent cell sampling, what is the standard error of the sample mean?
- Why can a cell-pooled estimate differ from an equal-donor estimate even when every donor's estimate is measured correctly?
- A cell type is not observed for one donor after processing. Why is it generally incorrect to enter an expression value of zero for that donor without further evidence?
Solutions. (1) No. Independence implies \(\operatorname{Cov}(R_d,Y_d)=0\), so the derived selection-bias expression is zero, assuming the inclusion probability is positive. (2) The weighted mean is \(0.30(2)+0.70(6)=0.6+4.2=4.8\) units. (3) The standard error is \(\sigma_d/\sqrt{m_d}=3/\sqrt{4}=1.5\) units under the independent-sampling assumption. (4) Cell pooling weights donors according to their observed cell counts, whereas an equal-donor estimate assigns equal weight to each donor; the two estimates therefore target different averages when donor estimates or counts differ. (5) No observed cells can reflect capture or sampling failure rather than true biological absence, and an unobserved expression quantity is not established to be zero.