Cells Nested Within Donors

Why the Sampling Structure Matters

A single-cell data set may contain thousands of measured cells, but those cells are often collected from a much smaller number of people or animals. Cells from one donor are not simply a large set of unrelated biological sources: they come from an organism with shared characteristics, such as genotype, age, disease history and treatment exposure. The tissue from that organism also has a shared biological and collection context.

A sampling diagram makes those relationships visible. It records which observations came from which donors, and whether samples such as tissue regions or visits sit between donors and cells. This is a description of how observations are organized and generated; it is not, by itself, a choice of statistical method or a conclusion about which observations should be used for inference.

Definition: In a nested sampling structure, lower-level observations belong to higher-level units. For example, cells are nested within a tissue specimen, and specimens may be nested within donors. Cells that share a donor also share donor-level biological sources of variation. The number of cells and the number of donors therefore describe different aspects of the data.

Drawing the Hierarchy

Begin with the population or collection of donors relevant to the study. Each donor contributes one or more specimens; each specimen may contain one or more sampled regions; and each sampled region contributes captured cells. The actual structure depends on the protocol. If one specimen is taken per donor, it is reasonable to draw donors directly above cells while noting the specimen collection. If several specimens or regions are recorded separately, include those levels rather than hiding them.

For a simple design with \(D\) donors and \(n_i\) cells measured from donor \(i\), let \(i=1,\ldots,D\) index donors and \(j=1,\ldots,n_i\) index cells within donor \(i\). If expression is recorded for gene \(g\), write the observed value as \(Y_{ijg}\). The index \(i\) identifies the donor shared by all cells in that group; \(j\) distinguishes cells within that donor.

\text{Donor population} \longrightarrow \text{Donor } i \longrightarrow \begin{cases} \text{Cell } (i,1)\\ \text{Cell } (i,2)\\ \vdots\\ \text{Cell } (i,n_i) \end{cases} \longrightarrow Y_{ijg}

The arrows indicate a sampling or measurement path, not a claim that a donor biologically produces identical cells. The cells can differ in cell type, state, gene expression and other features. The point is that their observations have a common parent in the sampling structure. A diagram should also show the actual number of donors and, where relevant, the differing number of cells captured from each donor.

Worked Example: A Basic Donor-to-Cell Diagram

Suppose a hypothetical study collects tissue from three donors. After quality control, the data contain 4 cells from donor A, 3 from donor B and 5 from donor C. Show the hierarchy and identify its levels.

The diagram has a donor level with three donors. Each donor has its own set of captured cells:

\begin{aligned} \text{Donor A} &\longrightarrow \{A1,A2,A3,A4\}\\ \text{Donor B} &\longrightarrow \{B1,B2,B3\}\\ \text{Donor C} &\longrightarrow \{C1,C2,C3,C4,C5\} \end{aligned}

There are \(D=3\) donors and \(4+3+5=12\) measured cells. The 12 cells are observations at the cell level, grouped into three donor clusters. The unequal cell counts are part of the observed sampling structure; they do not change the fact that there are three donors. For example, a cell labelled C2 shares donor C with C1, C3, C4 and C5, but not with cells from donors A or B.

What Cells From One Donor Share

Cells from the same donor share more than a label. They arise from the same organism, and therefore can share biological influences that affect many cells. These may include inherited genetic background, age, sex, systemic exposures, disease history, medication, and physiological state at collection. Cells from the same specimen may additionally share tissue environment, anatomical location, collection time and handling history.

These influences do not act identically on every cell. A donor's genotype may affect expression differently across cell types; disease state may affect only particular pathways or cell populations; and local tissue conditions may vary across a specimen. Thus, “shared source” means a common biological context, not identical expression values or identical effects in every cell.

A useful conceptual representation for a gene is

Y_{ijg}=\mu_g+B_{ig}+E_{ijg},

where \(\mu_g\) is a reference level for gene \(g\), \(B_{ig}\) represents donor-level influences on that gene, and \(E_{ijg}\) represents cell-specific departures and other variation not represented by the donor term. This is a schematic decomposition, not a complete count model for single-cell data. It does not assert that expression values are normally distributed, that all cells within a donor are otherwise independent, or that the donor term is the only shared influence.

Interpretation: The donor term \(B_{ig}\) is common to cells with the same donor index \(i\). Consequently, observations from one donor have a shared component in this representation. Cells from different donors have different donor terms. Further structure, such as tissue regions or local cell neighborhoods, may create additional shared components.

A Short Justification for the Shared-Source Representation

The role of a shared donor influence can be seen without choosing a specific distribution for expression. Let \(B_i\) denote the donor's biological context, and let \(Y_{i1}\) and \(Y_{i2}\) be measurements from two cells of that donor. In a simplified model, suppose that, conditional on \(B_i\), the cells' remaining variation is independent. Then the conditional joint probability factors, but the unconditional joint probability averages over the same donor context for both cells:

P(Y_{i1}=y_1,Y_{i2}=y_2) = \int P(Y_{i1}=y_1\mid B_i=b) P(Y_{i2}=y_2\mid B_i=b) \,dP_{B_i}(b).

The same value \(b\) appears in both conditional terms because both cells come from donor \(i\). In general, this integral is not equal to the product of the two marginal probabilities, which would average each cell over donor contexts separately. Thus, conditional independence given a donor context does not generally imply unconditional independence across cells from that donor.

This justification depends on a simplifying assumption: after conditioning on the donor context, the two cells have independent remaining variation. In real data, cells can also share specimen-, region- or processing-level influences, so that assumption may fail. The diagram helps expose such possibilities; it does not prove conditional independence.

Worked Example: Shared Donor Context in a Gene-Level Model

Consider hypothetical expression values for one gene, on an illustrative normalized scale, from two donors. Let the reference level be \(\mu_g=10\). Suppose donor A has a shared donor contribution \(B_{Ag}=2\), while donor B has \(B_{Bg}=-1\). The cell-specific contributions are shown below.

Donor	Cell	\(B_{ig}\)	\(E_{ijg}\)
A	A1	2	1
A	A2	2	-2
B	B1	-1	0
B	B2	-1	3

Using \(Y_{ijg}=\mu_g+B_{ig}+E_{ijg}\), the four values are:

\begin{aligned} Y_{A1g}&=10+2+1=13, & Y_{A2g}&=10+2-2=10,\\ Y_{B1g}&=10-1+0=9, & Y_{B2g}&=10-1+3=12. \end{aligned}

The cells within donor A share the contribution \(+2\), and the cells within donor B share \(-1\), while their cell-specific contributions differ. The example illustrates the bookkeeping of shared and cell-specific sources; the values are hypothetical and do not estimate actual donor effects or establish a particular probability distribution.

Adding Specimens, Regions and Processing

A donor-to-cell diagram is useful only when it matches how material was collected. If two tissue regions are sampled from each donor, the structure could be represented as donor \(\rightarrow\) region within donor \(\rightarrow\) cells within region. Cells from a region share both their donor and their region-specific context. If multiple visits are sampled, visits may be nested within donors, with cells nested within visits. In contrast, if the same processing batch contains cells from several donors, batch is not nested within donor: it is a separate grouping that crosses the donor structure.

Worked Example: Two Regions and a Shared Processing Batch

In a hypothetical study, tissue is collected from two regions of each of two donors. Two cells are recorded per region. All four region samples are processed in the same laboratory batch. A suitable sampling diagram is:

\begin{aligned} \text{Donor A} &\longrightarrow \begin{cases} \text{Region A1}\longrightarrow \{A11,A12\}\\ \text{Region A2}\longrightarrow \{A21,A22\} \end{cases}\\ \text{Donor B} &\longrightarrow \begin{cases} \text{Region B1}\longrightarrow \{B11,B12\}\\ \text{Region B2}\longrightarrow \{B21,B22\} \end{cases}\\ \text{Processing batch 1} &\longrightarrow \text{cells from both donors and all four regions.} \end{aligned}

There are 2 donors, 4 regions nested within donors and 8 cells nested within regions. The batch includes samples from both donors, so it is a shared processing context across donor groups rather than a level below one donor. Drawing it separately prevents the mistaken diagram in which batch 1 is shown as belonging only to donor A or only to donor B. The design may also have further technical details, but this diagram captures the stated grouping structure.

Checking the Diagram and Its Assumptions

Before using a hierarchy to describe a data set, check what each row represents and which identifiers connect observations. A cell barcode identifies a cell, but a donor identifier is needed to recover which cells share a donor. If donor identifiers are missing or ambiguous, the intended grouping cannot be reliably reconstructed from cell counts alone.

Check the biological sampling path. Verify whether cells were collected from one specimen, multiple regions, or repeated visits, and draw the levels that were actually sampled.
Keep biological and technical sources distinct. Donor, tissue region and processing batch may all matter, but they refer to different sources and can have different nesting or crossing relationships.
Do not infer identical cells from shared membership. A donor grouping indicates shared context, not uniform expression, cell type or state.
Do not infer independence from different cell labels. Two distinct cells can still share donor- or specimen-level influences.
Check what is absent from the diagram. If local neighborhoods, repeated time points or multiple batches exist but are omitted, the displayed structure may be incomplete.

The main alternative to a hierarchical description is to list every cell as if it were an unrelated observation. That flat representation can be convenient for storing expression values, but it hides which cells share donors and specimens. A donor-level diagram preserves the biological grouping. A crossed-factor diagram is more appropriate when a source such as batch connects observations from multiple donors, rather than sitting strictly inside a donor. These are complementary descriptions of the sampling design, not competing biological explanations.

Common Mistakes and Limitations

Wrong: “There are 1,000 cells, so the study has 1,000 donors.” Correction: Count donors from donor identifiers. The cells are grouped observations, and many may come from each donor.
Wrong: “Cells from the same donor must have the same expression.” Correction: They share some biological sources, but cell-specific and local variation can still produce different measurements.
Wrong: “Batch is nested within donor because every donor's sample was processed.” Correction: Determine membership from the actual design. A batch containing samples from several donors crosses donor groups.
Wrong: “A simple donor-and-cell diagram describes every dependence in the data.” Correction: It describes the stated nesting, but tissue regions, repeated visits, local environments and technical groupings may require additional levels or crossed structure.

A diagram cannot recover unrecorded biological history, prove that a source affected expression, or determine a suitable inferential procedure on its own. It is a disciplined account of where observations came from and which contexts they share. Those distinctions are essential for interpreting cell-level data in a study with biological replication.

Check Your Understanding

Answer each question from the sampling structure described. Solutions follow the questions.

A hypothetical data set has 6 donors, with 20 cells from each. How many donors and how many cell observations are represented?
Two cells from one specimen have the same donor and region identifiers but different expression values. What does the shared grouping imply, and what does it not imply?
Four donors contribute samples to one processing batch. Is that batch nested within one donor? Explain.
Write a simple expression decomposition for cell \(j\), donor \(i\), and gene \(g\), identifying the donor-shared and cell-specific terms.

Solutions. (1) There are 6 donors and \(6\times20=120\) cell observations. (2) The cells share donor- and region-level biological context; this does not imply equal expression or identical cell states. (3) No. The batch contains samples from multiple donors, so it is a shared grouping across donor groups, not a level nested within one donor. (4) One schematic decomposition is \(Y_{ijg}=\mu_g+B_{ig}+E_{ijg}\), where \(B_{ig}\) is shared by cells from donor \(i\) and \(E_{ijg}\) is the cell-specific contribution. This notation alone does not specify a full probability model.

What You'll Learn