Batch Effects and Confounding in Cross-Layer Models

Why Cross-Layer Associations Need a Technical Check

Suppose expression and protein abundance are measured in the same set of samples, and the two measurements are strongly associated. The association may reflect shared biology, but it may also arise because both assays were processed in the same laboratory batch, because the samples differ in cell composition, or because a processing feature affects both measurements. The observed correlation alone cannot distinguish these explanations.

This tutorial focuses on diagnosing those sources of cross-layer association and on using design-aware adjustment and negative-control comparisons to assess robustness. A covariate is not automatically a nuisance to remove: its role depends on the scientific question. In particular, cell composition can be a biological feature of interest, a source of confounding, or part of a pathway through which a biological exposure operates.

A Model for Cross-Layer Adjustment

Let \(X_i\) denote a molecular measurement in one omics layer and \(Y_i\) a measurement in another layer for sample \(i\). A basic linear model for their conditional association is

Here, \(\mathbf{C}_i\) contains measured covariates such as batch indicators, estimated cell proportions, and sample-processing variables. The coefficient \(\beta_X\) describes the mean difference in \(Y\) associated with a one-unit difference in \(X\), conditional on the included covariates, under the specified linear model. It is an adjusted association, not automatically a causal effect.

For the usual least-squares interpretation, the model assumes a correctly specified conditional mean, independent samples (or an analysis that accounts for dependence), and errors with conditional mean zero given the predictors and covariates. Standard errors and tests require additional variance assumptions or a suitable robust procedure. A categorical batch variable should generally be represented by indicator columns with a reference level, rather than treated as a numeric scale whose values imply a linear trend.

The adjustment is informative only when the data contain enough overlap to distinguish the predictor from the covariates. If every sample with one value of \(X\) is in one batch and every sample with another value is in another batch, the data may not identify separate effects of \(X\) and batch. No regression option can recover information absent from the design.

Confounding and Identifiability

A simple derivation shows how an omitted batch factor can distort a cross-layer slope. Suppose the true model is \(Y_i=\beta_X X_i+\gamma B_i+\varepsilon_i\), where \(B_i\) is a numeric indicator for a batch contrast. Assume \(E(\varepsilon_i\mid X_i,B_i)=0\). If \(Y\) is instead regressed on \(X\) alone, the population slope is

To obtain this expression, substitute the true model into the covariance: \(\operatorname{Cov}(X,Y)=\beta_X\operatorname{Var}(X)+\gamma\operatorname{Cov}(X,B)+\operatorname{Cov}(X,\varepsilon)\). The final term is zero under the stated conditional-mean assumption. Dividing by \(\operatorname{Var}(X)\) gives the result. Thus, when batch affects \(Y\) and is associated with \(X\), the unadjusted slope generally combines the cross-layer association and a batch-related component.

Adjustment does not always solve the problem. If \(X\) and batch are perfectly aligned, their separate coefficients cannot be estimated. For example, if the batch indicator vector equals the predictor vector for every sample, the two corresponding columns of the design matrix are identical. Adding both to a regression does not create two independent sources of information; only a combined contribution is estimable. With partial overlap, adjustment may be possible but imprecise, and conclusions can depend heavily on a small number of samples in the overlapping groups.

Covariate columns can also be nonorthogonal without being identical. The following cross-product calculation illustrates why checking arithmetic and the design itself matters.

Choose Adjustments With the Design and Scientific Question in View

A robust analysis begins before fitting a model. If samples from distinct biological groups were processed in separate batches, batch and biology may be inseparable. Whenever possible, randomize samples across plates and runs, balance important biological groups within each batch, and record processing variables at collection and assay. These choices create overlap that makes adjustment more credible.

When adjusting, include covariates that plausibly affect the measured outcomes and are related to the predictor or outcome of interest. For batch, use an appropriate categorical representation. For cell composition, use suitable measured or estimated proportions and consider whether the target is a within-cell-type association or an association that includes changes in cell mixture. If composition lies on a biological pathway relevant to the question, adjusting for it changes the estimand and may remove part of the relationship being studied. Processing variables can likewise be consequences of sample quality or biological state; automatic adjustment can introduce bias if a variable is affected by both the exposure and unmeasured causes of the outcome.

Inspect group counts and cross-tabulations, examine whether covariate values overlap across biological groups, and assess the rank and conditioning of the design matrix. A fitted coefficient can be returned by software even when it is unstable or its interpretation depends on near-confounding. Report which covariates were adjusted for and why, and distinguish the adjusted association from the unadjusted one.

What Robustness Checks Can and Cannot Establish

Cross-layer results are more persuasive when they are not driven by one obvious technical contrast. Useful checks include comparing estimates before and after adjustment, examining within-batch associations when there is adequate within-batch variation, and repeating the analysis under reasonable alternative representations of batch or composition. The purpose is not to search for a preferred coefficient, but to determine whether the conclusion changes materially under plausible specifications.

Negative-control outcomes or exposures can help identify residual structure. A negative-control outcome should be affected by relevant technical processes but not by the hypothesized biological link. A negative-control exposure should share important confounding structure but not have the proposed causal connection. Neither is a generic substitute for design or a universal test of confounding; the scientific assumptions that make a control negative must be stated.

Common Mistakes

• Wrong: “The adjusted coefficient is biological because batch was included.” Correction: Adjustment controls only modeled covariates under the model assumptions; it does not rule out unmeasured or misspecified technical effects.
• Wrong: “A large batch effect means the molecular layers are unrelated.” Correction: Batch may explain some observed association, while a biological association may still remain. Estimate and interpret both with uncertainty and design limitations in view.
• Wrong: “Adding batch to the model separates batch and biology even when they never overlap.” Correction: Perfect confounding prevents separate estimation; additional samples with overlap or a redesigned experiment are needed.
• Wrong: “Cell composition is always a nuisance covariate.” Correction: Its role depends on the target question. Adjustment may be appropriate for a within-composition association but may remove part of a biologically meaningful pathway.
• Wrong: “A null negative-control result proves there is no residual technical confounding.” Correction: A control can be insensitive to some artifacts, and its interpretation depends on whether it truly shares the relevant technical structure.