Why Technical Structure Matters
Single-cell samples pass through a chain of people, places, and equipment. A site may have several operators, a processing center may serve multiple sites, and a platform may be used at several centers—or be available at only one. These relationships determine which samples share technical influences. They matter when designing a comparison because a condition difference can be difficult or impossible to distinguish from a technical difference if the two are allocated together.
The key is to describe the structure actually used, rather than infer it from labels. Two operators are nested within a site when each operator works at only one site in the study. Operators are crossed with sites when the same operators work at multiple sites. A platform is crossed with sites if the same platform level occurs at multiple sites; it is nested if each platform level belongs to just one site. The same distinction applies to processing centers and the other factors.
Crossing need not mean that every possible combination is observed. A factor can be partially crossed: for example, some operators may work at more than one site, while others work at only one. Record the observed combinations explicitly. Treating a partially crossed structure as fully nested or fully crossed can imply sharing relationships that the data do not have.
Indexing and a Working Model for Shared Technical Variation
For a clear design map, index each biological sample by \(i\), and record its site \(s(i)\), operator \(o(i)\), platform \(p(i)\), and processing center \(c(i)\). Use actual level identities consistently: if an operator works at two sites, the operator should have the same identifier at both. A sample-level response \(Y_i\) might be a prespecified summary for a gene or cell population. It is not an individual cell treated as an independent biological sample.
The following model illustrates how technical levels can induce dependence. It is a working random-effects model, not a claim that all experiments follow it exactly:
Here \(g(i)\) denotes the condition of sample \(i\), and \(\tau_{g(i)}\) is its condition effect. The terms \(b_s\), \(u_{o:s}\), \(v_p\), and \(w_c\) represent site, operator-within-site, platform, and processing-center effects. In this illustration, operator is nested within site, while platform and center are modeled as factors whose levels may be shared across sites. The sample-specific residual is \(\varepsilon_i\).
For the covariance implications below, assume the random effects have mean zero, are mutually independent across terms, and have constant variances \(\sigma^2_{\text{site}}\), \(\sigma^2_{\text{operator}}\), \(\sigma^2_{\text{platform}}\), and \(\sigma^2_{\text{center}}\); assume residuals are independent across distinct samples and have variance \(\sigma^2_{\text{residual}}\). These assumptions are a simplifying model. If platform or center is nested rather than shared, its term must be indexed by its parent as well. If operator works across sites, the operator term should not be written as nested within site.
The justification is direct: in the covariance of \(Y_i\) and \(Y_j\), independent terms from different levels have covariance zero, and independent residuals contribute zero for distinct samples. A random-effect term contributes its variance only when both samples contain the same random effect. Thus sharing site and platform, but not operator or center, contributes \(\sigma^2_{\text{site}}+\sigma^2_{\text{platform}}\). Sharing platform alone contributes \(\sigma^2_{\text{platform}}\). This level-by-level reasoning is safer than memorizing a short list of special cases.
Worked Example: Mapping a Nested Operator Structure
Worked Example: Mapping a Nested Operator Structure
Suppose a hypothetical study recruits samples at Sites A and B. Two operators work at each site, and each operator processes samples only at that site. At each site, both control and treated samples are assigned to each operator. Is operator nested or crossed with site, and what does this allocation achieve?
Label the operators A1, A2, B1, and B2. A1 and A2 occur only at Site A, while B1 and B2 occur only at Site B. Therefore operator is nested within site: the observed combinations are A with A1/A2 and B with B1/B2, not each site with the same operator levels.
For a concrete allocation, place two control and two treated samples with each operator. Each site then has \(2 \times 2 \times 2=8\) samples: two conditions, two operators per site, and two samples per condition-operator combination. Across the two sites there are 16 samples, with 8 per condition. Both conditions occur at each operator, so condition is not confined to one operator within either site. The allocation also represents both conditions at both sites, avoiding complete condition-site confounding.
The design does not make operator effects disappear. It makes within-operator condition comparisons possible, while replication across operators and sites allows technical variation to be represented rather than equated with condition. The example specifies allocation, not a guarantee of a particular precision or a substitute for checking the full design matrix.
Worked Example: Covariance From All Shared Levels
Worked Example: Covariance From All Shared Levels
Consider a hypothetical study with the model above but no center term. Let the site, operator, platform, and residual variance components be 4, 1.5, 2.5, and 9, respectively, in squared units of the sample-level response. Operators are nested within site, and platform identifiers are shared across sites. Find the covariance and correlation for three distinct sample pairs.
For samples at the same site and platform but handled by different operators, the site and platform indicators equal 1, and the operator indicator equals 0. Therefore $$ \operatorname{Cov}(Y_i,Y_j)=4+2.5=6.5. $$
For samples sharing the same platform but neither site nor operator, only the platform indicator equals 1, so their covariance is \(2.5\). For samples sharing site and operator but using different platforms, the site and operator indicators equal 1, giving covariance \(4+1.5=5.5\).
Every sample has variance \(4+1.5+2.5+9=17\), because it contains one level of each random effect and its residual. The corresponding correlations are covariance divided by 17: \(6.5/17=0.3824\), \(2.5/17=0.1471\), and \(5.5/17=0.3235\), respectively. These calculations use the assumed independent variance components; they do not establish that those values are appropriate for an actual study.
Allocation for Interpretable Condition Comparisons
A useful design question is whether each condition is represented across the technical levels that could otherwise explain its difference. When feasible, distribute each condition across sites and across operators within sites. If a platform or processing center is shared across sites, use it for more than one condition and avoid assigning every sample in one condition to one level and every sample in another condition to a different level. In a nested structure, balance condition within each parent level where practical: for example, include both conditions under each site-specific operator.
This is not a demand for every possible combination or equal sample counts everywhere. Feasibility, sample availability, and the scientific comparison matter. The design should nevertheless make clear which comparisons are within a site, operator, platform, or center and which comparisons rely on between-level contrasts. Earlier material on rank and alias checks provides the formal way to assess whether parameters in a proposed design matrix can be distinguished; a balance table is a practical first diagnostic, not a replacement for that check.
Worked Example: Condition Crossed With Site Versus Confounded With Site
A hypothetical study compares control and treated samples at two sites. In a crossed allocation, the sample-level means for a chosen summary are 10 and 13 units for control and treated at Site A, and 14 and 16 units at Site B. Calculate the within-site differences and their equally weighted average.
At Site A, the treated-minus-control difference is \(13-10=3\) units. At Site B, it is \(16-14=2\) units. The equally weighted average of these site-specific differences is \((3+2)/2=2.5\) units. Both conditions occur at each site, so this descriptive comparison does not simply compare one site's samples with the other's. It is a design-based illustration, not evidence that treatment caused a change.
Now suppose instead that all control samples were collected at Site A and all treated samples at Site B. The difference between the same two site means, \(14-10=4\) units, would combine condition and site. No arithmetic on those means can separate the two effects because condition and site are perfectly aligned in this allocation. Replicating more samples within those same condition-site assignments would not create the missing within-site comparison.
Practical Design Checks and Alternatives
Before processing begins, make a table that records, for every planned sample, condition, site, operator, platform, and center. Then check the following:
- Confirm the hierarchy. Verify whether each operator, platform, and center level is genuinely nested within a parent or reused across parents. Names that restart at each site should not be mistaken for globally shared identities.
- Check condition coverage. Identify technical levels containing only one condition. If condition is confined to one site, operator, platform, or center, that factor may be confounded with condition for the intended comparison.
- Inspect sparse combinations. A factor may be technically crossed but have only one sample in some combinations. Record such gaps; nominal crossing alone does not ensure a robust comparison.
- Check the model against the workflow. If each platform is used at one center only, model platform within center rather than as a globally shared platform effect. If operators work at multiple sites, do not assign them separate nested identities unless they truly are different people or units.
- Recheck after deviations. Reassignments, failed preparations, and replacements can turn a balanced plan into an incomplete or confounded realized design.
The main alternatives are not simply “crossed is good” and “nested is bad.” A fully crossed design can support comparisons across shared levels, but may be expensive or operationally impossible. A nested design may reflect reality and can still be useful when each condition is represented within the relevant parent levels. A partially crossed design lies between them; it should be described using its actual incidence pattern. Allocation decisions should be made around the target comparison, not around an abstract preference for a particular design label.
A common mistake is to assume that including “site” or “operator” as a model term repairs any allocation. Correction: a model term can represent variation only when the design contains information to distinguish it from the condition contrast. If every control is processed at one site and every treated sample at another, adding both labels does not recover a separate condition effect from the data. Another mistake is to treat cells from the same sample as independent replicates for the site or operator allocation. The technical structure is assigned at the sample-processing level; many cells from one sample do not create additional independently assigned sites or operators.
These plans also have limits. A balanced table cannot prevent unmeasured technical differences, guarantee that the random-effects assumptions hold, or establish that a chosen summary captures all relevant biology. Some variance components may be difficult to estimate when there are few levels of a factor. When the workflow forces nesting, explicitly state that structure and ensure condition comparisons are represented within the levels that matter whenever possible.
List the parent-child relationships among sites, operators, platforms, and centers using actual level identities.
Allocate each condition across relevant levels, especially within nested parent levels, where feasible.
Distinguish unavoidable sparse combinations from assignments that confound condition with a technical level.
Specify which shared levels are assumed to induce dependence, and revise the model if the workflow does not match it.
Check Your Understanding
Answer each question, then compare with the solutions.
- Two operators work at both of two sites, and each operator processes samples at each site. Is operator nested or crossed with site?
- Under the model with site variance 3, operator-within-site variance 2, platform variance 1, and independent residuals, what is the covariance for two distinct samples sharing site and platform but not operator?
- All treated samples are assigned to Platform P1 and all control samples to Platform P2. What design problem does this create?
- Why does processing many cells from one sample not provide many independent replicates for a site-level allocation?
Solutions. (1) Operator is crossed with site because the same operator levels occur at both sites. (2) The pair shares site and platform but not operator, so the covariance is \(3+1=4\) squared response units; the operator term contributes zero. (3) Condition is confounded with platform in the planned allocation, so a difference between conditions cannot be separated from a platform difference using that allocation alone. (4) The cells share the sample's site assignment and processing history, so they do not constitute independent assignments to sites; they are subsamples within that sample.