Donor-Level Covariates and Information Limits

Why the Number of Cells Is Not the Number of Covariate Replicates

A donor’s age, sex, or exposure status is usually recorded once and then attached to every cell from that donor. If a study profiles 500 cells from each of 12 donors, it has 6,000 cell records but only 12 independently observed ages, sex values, or exposure statuses. Treating those 6,000 records as independent replications of a donor-level covariate can make uncertainty appear far smaller than the design supports.

Cells are still useful: additional cells can improve measurement of a donor’s expression pattern. But once the within-donor measurement is sufficiently precise, the remaining uncertainty in a covariate effect is largely about differences across donors. The distinction follows from the data-generating structure, not from a rule that cells never add information.

Definition: A donor-level covariate is a variable whose value is shared by all analyzed cells from a donor, such as that donor’s age or exposure group. The donor is the independent unit at which this covariate varies. The number of cells can affect how precisely each donor’s outcome is measured, but it does not create additional independent covariate values.

The earlier material on intraclass correlation and random slopes highlights why cells from the same donor cannot automatically be treated as independent. To make the information limit precise, consider a simple random-intercept model for a continuous cell-level outcome:

Y_{ij} = \beta_0 + \beta_1 x_i + b_i + \varepsilon_{ij}, \qquad i=1,\ldots,D,\quad j=1,\ldots,m_i.

Here $Y_{ij}$ is the outcome for cell $j$ from donor $i$, $x_i$ is a donor-level covariate, $D$ is the number of donors, $m_i$ is the number of sampled cells from donor $i$, $\beta_1$ is the covariate association of interest, and $b_i$ is a donor-specific random intercept. For the calculations below, assume $x_i$ is fixed at its observed value; the donor effects are independent across donors and independent of all $x_i$, with $E(b_i)=0$ and $\operatorname{Var}(b_i)=\tau^2$. Assume also that the cell residuals are independent across cells and donors, have mean zero and variance $\sigma^2$, and are independent of the donor effects and covariates. These assumptions imply a common conditional variance for donor means only when the cell counts are balanced.

What Averaging Within Donors Can—and Cannot—Do

For donor $i$, let $\bar{Y}_i$ be the mean outcome across that donor’s $m_i$ sampled cells. Averaging reduces the contribution of cell-level residual noise, but it does not average away the donor effect:

\bar{Y}_i = \beta_0 + \beta_1x_i + b_i + \bar{\varepsilon}_i, \qquad \operatorname{Var}(\bar{Y}_i\mid x_i) = \tau^2 + \frac{\sigma^2}{m_i}.

The conditional variance follows because the assumptions give $\operatorname{Var}(b_i\mid x_i)=\tau^2$, $\operatorname{Var}(\bar{\varepsilon}_i\mid x_i)=\sigma^2/m_i$, and zero covariance between these terms. Independence of $b_i$ and $x_i$ is important here: knowing only that $E(b_i\mid x_i)=0$ and that the marginal variance of $b_i$ is $\tau^2$ would not justify the conditional variance used in this calculation.

Key result: With $m_i=m$ cells for every donor, the donor means have common conditional variance $v=\tau^2+\sigma^2/m$. Under the stated model, ordinary least squares of the $D$ donor means on $x_i$, with an intercept, has conditional slope variance $$ \operatorname{Var}(\hat{\beta}_1\mid x_1,\ldots,x_D) = \frac{\tau^2+\sigma^2/m}{\sum_{i=1}^{D}(x_i-\bar{x})^2}. $$ The denominator measures how much the covariate varies across donors; the numerator measures the remaining outcome variation per donor mean.

This result separates two sources of limited precision. More cells reduce $\sigma^2/m$, the cell-sampling component, but they do not reduce $\tau^2$, the between-donor component. More donors can add independent covariate values and increase the denominator, provided the added donors contribute useful covariate variation. If cell counts differ, donor means have variances $\tau^2+\sigma^2/m_i$, which are generally unequal; the balanced-design formula should not be applied as though those variances were common.

Worked Example: Estimating an Exposure Difference Across Donors

Suppose six hypothetical donors contribute 100 cells each. Three are unexposed, with donor mean outcomes 1.0, 1.6, and 0.8 units; three are exposed, with means 2.1, 2.7, and 1.8 units. Assume the random-intercept model above, with independent normally distributed donor effects and cell residuals, and suppose for illustration that $\tau^2=0.64$ and $\sigma^2=4$ are known. Exposure is coded $x_i=0$ or $1$. These assumptions make the donor means independent and normally distributed with a common variance.

The unexposed mean is $(1.0+1.6+0.8)/3=1.1333$ units. The exposed mean is $(2.1+2.7+1.8)/3=2.200$ units. Thus the estimated exposure difference is $2.200-1.1333=1.0667$ units.

Each donor mean has variance $0.64+4/100=0.6800$ units squared. The two group means use disjoint sets of three independent donors, so the estimated difference has variance $0.6800/3+0.6800/3=0.4533$ units squared and standard error $\sqrt{0.4533}=0.6733$ units. Under the stated normal model and known variances, a 95% interval is $1.0667\pm1.96(0.6733)=(-0.2530,\ 2.386)$ units. It describes uncertainty in the model-based exposure association; it does not establish a causal effect.

The calculation uses six independent covariate-bearing donors, not 600 independent exposure assignments. In practice, variance components are usually estimated rather than known, and six donors provide little information for estimating them. This example’s interval is therefore an illustration of the variance calculation, not a recommendation to rely on a known-variance interval in a small study.

The Cell-Number Limit and the Value of More Donors

The balanced-design variance makes the limit explicit. If $\tau^2>0$, then as the number of cells per donor grows, $\sigma^2/m$ approaches zero, but the slope variance approaches $\tau^2/S_{xx}$, where $S_{xx}=\sum_i(x_i-\bar{x})^2$. This is a limit for the stated model and fixed donor covariate values, not a universal claim that additional cells never help. If between-donor variation is negligible, cell-level noise can remain an important source of uncertainty. Also, more donors can change $S_{xx}$, while more cells per donor do not change the observed covariate values.

Worked Example: Comparing More Cells With the Limiting Precision

Consider eight hypothetical donors, four in each of two exposure groups. For a binary covariate coded 0 and 1, the mean is 0.5 and $S_{xx}=8(0.5)^2=2$. Assume the model’s independence conditions, balanced cell counts, and known variance components $\tau^2=1$ and $\sigma^2=4$, in outcome-units squared.

With $m=10$ cells per donor, the slope variance is $(1+4/10)/2=0.7000$, giving $SE=\sqrt{0.7000}=0.8367$ outcome units. With $m=1{,}000$, it is $(1+4/1{,}000)/2=0.5020$, giving $SE=\sqrt{0.5020}=0.7085$ outcome units. As $m$ grows without bound, the variance approaches $1/2=0.5000$ and the standard error approaches $\sqrt{0.5000}=0.7071$ outcome units.

In this illustration, increasing the cell count from 10 to 1,000 per donor reduces the standard error, but it cannot remove the between-donor component. The numerical limit relies on the assumed positive $\tau^2$ and the fixed set of eight donors; it should not be reported as a property of every study.

The number of donors and the spread of their covariate values both matter. A large donor count with almost no variation in age or exposure can still provide little information about a slope. Conversely, for a binary covariate, a highly unbalanced design can leave one group represented by very few independent donors, even if every donor contributes many cells.

Worked Example: Imbalance in a Donor-Level Exposure

Suppose ten hypothetical donors each contribute 100 cells, but only two donors are exposed and eight are unexposed. Assume the same random-intercept model and known variance components as in the preceding example: $\tau^2=1$ and $\sigma^2=4$. Assume independent donor effects and residuals as specified above. Each donor mean then has variance $1+4/100=1.04$ units squared.

The variance of the difference between the exposed and unexposed donor means is $1.04(1/2+1/8)=1.04(0.625)=0.6500$ units squared. Its standard error is $\sqrt{0.6500}=0.8062$ units. With five donors in each group and the same 100 cells per donor, the corresponding standard error would be $\sqrt{1.04(1/5+1/5)}=\sqrt{0.4160}=0.6450$ units.

The comparison shows how group balance affects the precision of a donor-level contrast. The 100 cells per donor reduce the cell-noise contribution, but they do not turn the two exposed donors into 200 independent exposed donors. Adding independent donors, especially to the sparse group, can be more informative for this contrast than adding cells to donors already represented.

Assumptions, Alternatives, and Failure Modes

A cell-level mixed model and an analysis of donor summaries can both respect the donor-level structure, but their implementations and assumptions differ. A correctly specified mixed model can use cell-level outcomes and account for within-donor dependence. A donor-summary analysis makes the replication unit explicit, but may need methods that account for unequal precision when donors contribute different numbers of cells. In a balanced design under the simple random-intercept model, the donor-mean slope has the variance derived above. Neither approach makes a donor-level covariate vary independently from cell to cell.

By contrast, a regression that treats all cells as independent observations typically uses the cell count as though it were the number of independent covariate observations. When cells from one donor share a random effect, that independence assumption fails. The resulting standard errors can be too small, and the apparent evidence can be overstated. Correcting standard errors for clustering can address dependence in uncertainty calculations, but it does not create additional donors or fix a lack of covariate overlap.

Model-based justification: Under the stated random-intercept assumptions with balanced cell counts, the donor-mean slope is a linear combination of independent donor means. Its numerator is $\sum_i(x_i-\bar{x})(\bar{Y}_i-E[\bar{Y}_i\mid x_i])$, and its denominator is $S_{xx}$. Independence and common variance $v$ give numerator variance $$ \sum_i(x_i-\bar{x})^2v = vS_{xx}. $$ Dividing by $S_{xx}^2$ yields $\operatorname{Var}(\hat{\beta}_1\mid x)=v/S_{xx}$, as stated. If donor means have unequal variances, this equal-variance derivation no longer applies; a weighted analysis or a model that handles the unequal variances is needed.

A further limitation concerns the independence assumption between $b_i$ and $x_i$. If donor effects are associated with age or exposure—for example, if unmeasured donor characteristics related to the outcome also differ systematically by exposure—the simple random-intercept model may not isolate the covariate association as intended. More cells do not resolve that problem. The covariate coefficient is an association under the model and design; a causal interpretation requires additional design and identification assumptions.

A donor fixed-effects model provides a useful contrast. With an unrestricted indicator for every donor, any covariate that is constant within donor is also represented by the donor indicators. Its separate coefficient is therefore not identifiable: the covariate does not vary within a donor to distinguish its effect from that donor’s intercept. A random-intercept model can estimate a donor-level coefficient by comparing donors, but this depends on its assumptions about the donor effects and covariates. The approaches answer different modeling problems; neither can manufacture within-donor variation in a fixed covariate.

Diagnostics and Practical Interpretation

Before interpreting a donor-level coefficient, check how many independent donors contribute each covariate value, how much continuous covariates vary across donors, and whether a small number of donors dominate the comparison. Examine cell counts by donor as well: highly unequal counts affect the precision of donor measurements, and the equal-variance formula does not apply when $m_i$ differs. Sensitivity to influential donors is especially important when the donor count is small.

The random-intercept variance and intraclass correlation can help describe the relative amount of within- and between-donor variation, but they do not replace inspection of the covariate design. The main warning signs are a covariate group represented by very few donors, little covariate spread, severe cell-count imbalance, or conclusions that change substantially when one donor is removed. In those cases, report the limitation directly rather than presenting a large cell count as evidence of strong donor-level replication.

Common mistake: “There are thousands of cells, so the age effect is estimated from thousands of independent observations.” This is wrong when age is recorded once per donor and cells within a donor are dependent. The correction is to identify the donor as the unit at which age varies, model the within-donor dependence, and assess precision using the number and distribution of donors as well as the cell-sampling contribution.

Check Your Understanding

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

A study has 20 donors and 300 cells per donor. How many independent age values are observed, and what can the additional cells contribute?
Under the balanced random-intercept model, write the variance of a donor mean and explain which component remains as the cell count grows.
For a binary exposure with four donors in each group, $S_{xx}=2$. If $\tau^2=0.5$, $\sigma^2=2$, and there are 50 cells per donor, calculate the model-based slope standard error, treating the variance components as known.
Why can a donor fixed-effects model not separately estimate the coefficient of a covariate that is constant within donor?

Solutions. (1) There are 20 independent age values, one per donor. The additional cells can improve the precision of each donor’s outcome measurement by reducing cell-sampling noise, but they do not add independent ages. (2) The variance is $\tau^2+\sigma^2/m$; as $m$ grows, the cell-noise term approaches zero and the between-donor component $\tau^2$ remains. (3) The donor-mean variance is $0.5+2/50=0.5400$. The slope variance is $0.5400/2=0.2700$, so the standard error is $\sqrt{0.2700}=0.5196$ outcome units. (4) A donor indicator already captures every variable constant within that donor. The covariate is therefore a linear combination of donor indicators and the intercept, so its separate coefficient is not identifiable in that model.