Crossed, Nested, and Partially Crossed Batch Structures

Why the Batch Structure Matters

A batch label is not enough to tell whether its effect can be separated from other sources of variation. The crucial information is which combinations of factor levels actually occur in the samples. For example, a platform used at every site creates different comparison opportunities from a platform used at only one site. Likewise, processing runs that are unique to each site do not directly link samples across sites.

This tutorial focuses on the structure of the design, not on choosing a particular batch-adjustment procedure. The same labels can support some comparisons but not others. Classifying the structure before adjustment helps prevent a numerical method from being asked to estimate a contrast that the data cannot identify.

For instance, “platform” and “site” are fully crossed if every platform is used at every site. “Run” is nested within “site” if each run belongs to a single site and run labels identify different runs at different sites. Partial crossing is not the same as nesting: in a partially crossed design, at least some levels are shared across combinations, even though the grid is incomplete.

The word crossed describes the allocation pattern, not whether the factors are statistically independent or whether their effects are equal. Nor does crossing alone guarantee that a biological contrast is estimable: the biological groups of interest must also be represented in a way that separates them from the batch factors.

Reading the Incidence Pattern

A practical first step is to make a table whose rows and columns are factor levels and whose entries give the number of samples in each combination. A positive entry is an observed combination; zero means that cell is absent. The same pattern can be represented as a bipartite graph: one set of nodes represents the levels of factor A, another represents levels of factor B, and an edge joins two levels when at least one sample has that combination.

In this graph, complete crossing means every possible A node is linked to every B node. A nested factor has each of its nodes linked to just one parent node. Partial crossing has some missing edges, but shared nodes can still connect multiple levels. If the graph breaks into disconnected components, there is no observed path linking levels across components.

The justification follows from the additive mean model. For an observation in cell \((a,b)\), write its expected value as \(\mu+\alpha_a+\gamma_b\), where \(\alpha_a\) and \(\gamma_b\) are the effects of the two factor levels. Adding a constant \(c\) to all \(\alpha_a\) values and subtracting \(c\) from all \(\gamma_b\) values leaves every fitted cell mean unchanged. This is a harmless location ambiguity and is resolved by a reference level or a sum-to-zero constraint.

If the graph has two disconnected components, there is a further ambiguity. Add \(c\) to the A effects in one component and subtract \(c\) from the B effects in that component. Every observed cell mean remains unchanged, because every observed edge lies wholly within one component. Yet comparisons of A effects across components change. The data therefore cannot determine those cross-component comparisons under the additive model. This is the graph version of the rank and connectivity checks from the earlier material on design identifiability.

This result assumes an additive model for the factor effects, a common response scale, and observations that provide information in the cells used for a comparison. If an A-by-B interaction is important, the effect of A can differ across levels of B, and a single additive contrast may not describe the data. A connected graph does not supply information for an unobserved cell or prove that an interaction is absent.

Worked Design Examples

Diagnostics, Alternatives, and Limits

Before adjustment, list the levels of every relevant factor and tabulate their observed combinations. Then inspect both the empty cells and the links among levels. The earlier material on rank and connectivity gives the corresponding design-matrix perspective; the incidence graph makes the source of a missing comparison easier to see.

• Check factor labels. Confirm whether run identifiers are truly shared across sites or merely reused labels. Incorrectly treating local run names as global levels can invent comparisons.
• Check the biological allocation too. A platform may be crossed with site but perfectly confounded with disease group. Batch-factor crossing does not resolve confounding with an unexamined biological variable.
• Look for empty cells and disconnected components. A zero cell can limit a particular within-level comparison; a disconnected component prevents cross-component comparisons under an additive model.
• Check for heterogeneity. Large differences among within-site platform contrasts may indicate interaction, instability, or sampling noise. A single average contrast can conceal this variation.
• Check replication and precision. An observed edge based on one sample technically links levels, but provides little protection against outliers and may yield highly uncertain estimates.

A complete factorial design is the clearest alternative when planning is possible: it observes every combination and facilitates comparisons across factors. A connected partial crossing can be more economical and still support selected comparisons, but conclusions rely more heavily on which cells exist and on assumptions such as additivity. A nested design is appropriate when lower-level units genuinely belong to only one higher-level unit; it supports within-parent comparisons, not direct comparisons of a shared lower-level effect across parents.

These structures do not prescribe a fixed-effect or random-effect treatment; that is a separate modeling decision. Nor does connectivity establish that a batch correction is scientifically appropriate. The target biological contrast, the feature scale, and the possibility of interactions remain relevant. If biology and batch are inseparable in the observed allocation, software cannot recover the missing comparison merely by applying an adjustment.