Why Correction Cannot Recover Missing Comparisons

Why a correction cannot supply an unobserved comparison

Batch correction is often used to make measurements from different processing batches more comparable. But if one condition appears only in one batch and another condition appears only in a different batch, their observed difference has two possible sources: biological condition and batch. The data contain no within-batch comparison to separate those sources. A correction algorithm may still return adjusted values, but that output does not by itself identify which part of the original difference was biological.

The issue is not simply that an estimate is imprecise. Under perfect confounding, the separate condition and batch effects are not identifiable from the observed design: multiple combinations of those effects give exactly the same fitted values. More cells or samples in the same condition-batch arrangement can measure the combined difference more precisely, but cannot reveal its separate components.

A model for the missing comparison

Consider one gene measured on a normalized expression scale. For clarity, let \(Y_i\) denote the expression for observational unit \(i\), such as a sample-level summary for that gene. The units are expression-scale units. Suppose there are two conditions, A and B, and two batches. Use an additive model with no condition-by-batch interaction:

Here \(C_i=1\) for condition B and \(0\) for condition A; \(B_i=1\) for batch 2 and \(0\) for batch 1. The parameter \(\mu\) is the mean for condition A in batch 1, \(\tau\) is the condition effect at a fixed batch, and \(\gamma\) is the batch 2 effect at a fixed condition. The error \(\varepsilon_i\) has mean zero under the model. This model is used to display the identifiability problem; the problem occurs even if the additive model is exactly correct.

Suppose every A unit was processed in batch 1 and every B unit in batch 2. Then the only observed condition-batch combinations are \((A,\text{batch 1})\) and \((B,\text{batch 2})\). The model implies

Consequently, the difference between the observed group means is \(\tau+\gamma\), not \(\tau\) alone. There is no observed mean for condition B in batch 1 or condition A in batch 2 to anchor a within-batch comparison.

Why the coefficients cannot be separated

The same conclusion can be seen directly from the design matrix. Its columns correspond to the intercept, condition indicator, and batch indicator. With perfect confounding, each observed A row is \((1,0,0)\), and each observed B row is \((1,1,1)\). For any coefficient vector \((\mu,\tau,\gamma)\), the fitted values are therefore \(\mu\) for A and \(\mu+\tau+\gamma\) for B.

Now change the coefficients by adding \(t\) to the condition effect and subtracting \(t\) from the batch effect, where \(t\) is any real number. The new coefficients give the same fitted value for every observed row:

For A, both indicators are zero, so neither fitted value changes. For B, both indicators equal one, so the changes \(+t\) and \(-t\) cancel. Thus the data cannot distinguish the original coefficients from the altered ones. Yet the condition effect changes from \(\tau\) to \(\tau+t\). This proves that the condition effect is not identified by these observations.

In design-matrix language, the coefficient change \((0,t,-t)\) is in the null space: multiplying it by the observed design matrix gives zero. A contrast is identifiable only if it is unchanged by every coefficient change in that null space. The condition contrast selects \(\tau\), so it changes when \(t\) changes; the observed combined contrast \(\tau+\gamma\) does not.

What correction can and cannot change

A correction procedure can transform the recorded values, but a transformation cannot create the missing condition-batch observations. In a simple additive adjustment, suppose the batch 2 value is shifted by an amount \(\delta\), while the batch 1 value is left unchanged. For the hypothetical means above, the adjusted contrast is

Choosing \(\delta=0\) leaves a contrast of 4.0 units; choosing \(\delta=4.0\) yields 0 units; choosing \(\delta=6.0\) yields -2.0 units. Those results agree with different decompositions of the same observed difference. The adjustment value has determined how the difference is allocated, rather than the confounded observations having identified that allocation.

Real correction procedures may use many genes, cell-level distributions, or other structure, rather than one simple shift. Such information can be useful only through additional restrictions or assumptions about how batch effects behave and what biological signals should be preserved. When the design itself contains no overlap, an algorithm's output is not evidence that the missing comparison has been recovered. Its result depends on the information and constraints it uses.

Comparing responses to perfect confounding

Several tempting responses answer different questions, and it helps to distinguish them. The unadjusted group difference describes the observed difference between the two condition-batch groups. It is valid as that descriptive contrast, but it is not a condition-only effect. A correction that subtracts a batch offset reports a contrast after applying that offset; without identifying information, it cannot establish that the offset is the true batch effect. A model that includes both condition and batch terms cannot estimate their separate coefficients uniquely from this design.

The appropriate response depends on the scientific aim. If the aim is to describe the observed groups, report the combined contrast and state the confounding. If the aim is to estimate a condition effect separate from batch, the present data do not provide that comparison. A future design can include both conditions within batches; other sources of information may also be relevant, but their role must be made explicit rather than treated as if the design had contained overlap.

This is different from a large standard error due to near confounding. With near confounding, some within-batch information may exist, even if it is weak; with perfect confounding, the condition effect can vary without changing any fitted observed values. Increasing precision for the observed combined contrast does not resolve this structural ambiguity.

Diagnostics and common pitfalls

The first diagnostic is a table of counts for condition by batch. If every condition occurs only in its own batch and the other condition-batch cells are empty, the design has no within-batch condition comparison. A second diagnostic is the design matrix: duplicate or dependent columns reveal that distinct coefficient combinations give the same fitted values. These checks identify a property of the design, not a problem that can be solved by selecting a different correction algorithm.

• Mistake: “The groups differ by 4 units, so the condition effect is 4.” The calculation is the observed contrast, but under perfect confounding it equals the condition effect plus the batch difference. The correction is to name it as a combined condition-batch contrast unless additional identifying information is supplied.
• Mistake: “The correction algorithm returned adjusted values, so it removed the batch effect.” An output is not proof that a non-identifiable decomposition has been recovered. State what assumptions or external information determine the adjustment.
• Mistake: “More cells will solve the problem.” More observations in the same empty-cell pattern can improve precision for observed group means, but they do not add the missing comparison. Replication within an existing group is not the same as overlap across conditions and batches.
• Mistake: “A coefficient printed by software must be uniquely estimated.” When the design is rank deficient, software may drop a term, impose a constraint, or use a convention to return values. Those choices do not make the separate biological effect identifiable from the observed data.