Partial Confounding and Recoverable Comparisons

What Partial Confounding Does to a Comparison

A study may distribute conditions across batches unevenly without assigning every condition to every batch. In that setting, a condition contrast can have different levels of support. It may be observed directly among samples processed in the same batch, or it may be recoverable only by combining comparisons across batches and assuming that batch effects act additively. Those are not equivalent kinds of evidence.

For example, if conditions A and B occur together in one batch, their difference can be estimated without comparing that batch to another. If B and C occur together in a second batch, the A-to-C difference can be pieced together through B. But no batch directly compares A with C. That final comparison therefore depends on a model connecting the batches, and especially on the assumption that the condition effect does not change by batch.

The introductory material on diagnosing complete condition-batch confounding establishes why a condition that occurs in only one batch can be inseparable from that batch under an additive analysis. Partial overlap can provide more information: shared conditions may link otherwise different batches. The practical task is to say exactly which comparisons those links support, and what assumptions are needed to interpret them.

Model and Assumptions

The terms \(\tau_c\) and \(\gamma_b\) need a reference convention, such as setting one condition effect and one batch effect to zero, or imposing sum-to-zero constraints. The convention sets the origin of the effects; it does not change a contrast such as \(\tau_B-\tau_A\). The model concerns the outcome at the experimental-unit level. Multiple cells from one donor do not become independent donors merely because they produce many measurements.

The additive model does not assert that batches are harmless or that condition assignment was randomized. It says that batch shifts the outcome by an amount that is common across conditions, while condition shifts it by an amount that is common across batches. For a causal interpretation, the study’s assignment and sampling design must also justify that interpretation; adjustment alone does not establish causation.

Represent the allocation as a set of condition-batch pairs that have observations. A condition pair is directly compared if both conditions occur in at least one common batch. A chain of such shared-batch comparisons can connect conditions that never occur together. A comparison obtained only by following a chain is model-dependent: it is supported under the additive model, but it is not a direct within-batch comparison.

Why a Shared Condition Can Bridge Batches

Suppose conditions A and B are observed in batch 1, while B and C are observed in batch 2. Under the additive model, the expected difference between B and A in batch 1 is

The batch effect cancels. Similarly, the B-to-C contrast in batch 2 estimates \(\tau_C-\tau_B\). Adding the two contrasts gives

This is the basis for recovering a comparison along a path of overlapping conditions. The cancellation is valid only if the B effect is the same in both batches and condition effects do not change by batch. If B behaves differently across batches, the two contrasts do not share a common B effect to cancel.

A short derivation also shows why a direct comparison is less dependent on cross-batch assumptions. In any batch containing both A and B, subtracting their expected outcomes eliminates both \(\mu\) and that batch’s \(\gamma_b\). When A and C are never together, no such subtraction is available; an estimate of their contrast must instead connect observations from multiple batches through the model.

Direct Evidence, Model-Based Recovery, and Extrapolation

It is useful to label each planned contrast before interpreting an adjusted estimate. A direct contrast has observations for both conditions in the same batch. A path-based contrast can be assembled from direct contrasts that connect the conditions through shared batches. A comparison between disconnected groups of condition-batch observations cannot be recovered from condition and batch effects alone: the data provide no shared condition or batch link to separate those effects.

Even a connected path does not make every conclusion equally secure. A direct contrast does not require comparing batch levels to estimate the condition difference. A path-based contrast requires the condition effect to remain stable across the batches used in the path. Longer paths can also accumulate uncertainty, since each estimated contrast contributes to the final estimate. The resulting standard error depends on the data and the covariance structure; it is not generally correct to treat the component contrasts as independent if they reuse observations.

The term extrapolation is apt when a model supplies a condition comparison in a batch where that comparison was not observed. It does not necessarily mean predicting beyond the numerical range of the outcome. It means extending an assumed pattern of condition and batch effects to an unobserved condition-batch combination. Report this distinction rather than presenting every adjusted contrast as if it came from a direct comparison.

Checking Whether the Additive Interpretation Is Plausible

When a condition pair is present in multiple batches, compare its within-batch contrasts. Materially different estimates can indicate a condition-by-batch interaction, although sampling variability may also explain differences. For instance, if the B-minus-A contrast is 2 units in one batch and -1 unit in another, a single common contrast is not a good description of both batches unless the discrepancy is consistent with their uncertainty.

Partial overlap offers some diagnostics, but it cannot diagnose every failure. In the A-B, B-C arrangement, the A-versus-B contrast is observed only in the first batch and the B-versus-C contrast only in the second. There is no repeated condition contrast across batches with which to check stability. The additive assumption may then be scientifically reasonable, but the allocation itself cannot verify it.

• Inspect the allocation. List which conditions occur in each batch and identify the shared conditions linking batches.
• Compare repeated overlaps. Where the same condition pair occurs in multiple batches, examine the separate within-batch estimates and their uncertainty.
• Look for sparse or influential links. A bridge supported by very few donors can make a path-based contrast unstable.
• Check the experimental unit. Treating cells from the same donor as independent units can understate uncertainty and does not repair a weak allocation.
• Check the target estimand. A contrast averaged over a specified set of batches is not automatically the same as a universal condition effect.

Choosing an Analysis and Avoiding Overclaiming

There are three common strategies, each answering a somewhat different question. A within-batch analysis uses only batches containing both conditions. It is transparent and avoids estimating a contrast by bridging batches, but may discard useful data and may target only the subset of batches with overlap. An additive model uses all connected observations to estimate condition contrasts while adjusting for batch; it can improve precision or recover path-based comparisons, but its interpretation depends on the additive assumption. A model allowing condition-by-batch interaction permits different contrasts by batch, but needs adequate overlap and replication to estimate those differences; it cannot create information in unobserved condition-batch cells.

Choose between these approaches based on the scientific target and the allocation, not on which produces the most favorable result. If the target is a condition contrast among batches where both conditions were processed, a direct within-batch analysis may align naturally with that target. If the target is a common effect across batches, additive adjustment may be useful when the common-effect assumption is credible. If meaningful effects may differ by batch, report batch-specific contrasts or a clearly defined average rather than relying on one common effect.

A common mistake is to say, “The adjusted model includes batch, so the A-versus-C comparison is directly controlled for batch.” That reasoning confuses statistical adjustment with observed overlap. In the A-B, B-C allocation, the model can estimate A-versus-C only by combining the two links and assuming that the condition effects transport across batches. The correction is to state both the estimate and its support: direct within-batch evidence for the component contrasts, and model-dependent recovery for the end-to-end comparison.