Batch-by-Condition Interactions

Why Allow the Condition Effect to Vary by Batch?

An additive model assumes that the condition difference is the same in every batch. That may be a reasonable simplification when batches shift expression by similar amounts in both conditions. But if the measured condition difference varies across batches, an additive model cannot represent that pattern: it forces one shared condition coefficient to summarize potentially different batch-specific effects.

A batch-by-condition interaction relaxes that restriction. It allows the modelled condition difference to depend on batch. This is a statement about the statistical mean model, not by itself a claim that batch causes the difference or that a biological mechanism has been identified. In single-cell data, the response could be a gene’s suitably transformed expression, with the model fitted separately for each gene. The interpretation of an interaction is always on the scale used by the model.

Model and Interpretation

Let \(Y_{ig}\) be the response for cell or observational unit \(i\) and gene \(g\). Let \(b(i)\) denote the unit’s batch, and let \(x_i=0\) for the reference condition and \(x_i=1\) for the other condition. For \(B\) batches, use batch 1 as the reference and write the linear mean model for one gene as follows:

Here \(I\{\cdot\}\) is an indicator that equals 1 when its condition holds and 0 otherwise. The reference-batch coefficients are \(\alpha_{1g}=0\) and \(\delta_{1g}=0\) by convention; they are not additional parameters. The term \(\alpha_{bg}\) describes the difference between batch \(b\) and batch 1 in the reference condition. The coefficient \(\beta_g\) is the condition difference in batch 1. For another batch \(b\), \(\delta_{bg}\) is the change in that condition difference relative to batch 1.

This parameterization requires care in interpretation: \(\beta_g\) is not an average condition effect across all batches. It is specifically the condition effect in the reference batch. Changing the reference batch changes the individual coefficient meanings, though fitted cell means and appropriately defined comparisons do not change when the model is identifiable.

Why the Interaction Is a Difference of Differences

The formula follows directly from the four expected means in a two-batch, two-condition model. With batch 1 as the reference, the expected responses are \(\mu_g\) for batch 1 in the reference condition, \(\mu_g+\beta_g\) for batch 1 in the other condition, \(\mu_g+\alpha_{2g}\) for batch 2 in the reference condition, and \(\mu_g+\alpha_{2g}+\beta_g+\delta_{2g}\) for batch 2 in the other condition. Subtracting the within-batch condition differences gives:

The shared baseline \(\mu_g\) and batch shift \(\alpha_{2g}\) cancel. What remains is exactly the amount by which the batch-2 condition effect differs from the batch-1 condition effect. In a model with several batches, each \(\delta_{bg}\) is a comparison to the reference batch. A joint test that all interactions are zero asks whether the model’s condition effects are equal across batches; it does not test whether the condition effect itself is zero.

Estimability Depends on the Observed Combinations

Including an interaction term in a formula does not guarantee that the data identify it. An effect is estimable when the observed design contains enough information to determine it uniquely, despite any alternative coefficient values that produce the same fitted means for the observed units. For a design matrix \(X\) and coefficient vector \(\theta\), a linear function \(c^\mathsf{T}\theta\) is estimable precisely when it has the same value for every coefficient vector that gives the same \(X\theta\). Equivalently, \(c\) must lie in the row space of \(X\).

With reference coding, a complete crossing—observations in every batch-condition combination—makes all cell means and their contrasts estimable, provided the model has no other exact redundancies. If a combination is absent, the pattern of missingness matters. Some effects may still be estimable, while others are not. In particular, a within-batch condition difference requires observations in both conditions in that batch when the model allows that batch its own condition effect.

Assumptions, Diagnostics, and Alternatives

The interaction model describes conditional means. For the linear model as written, it assumes that the expected response is represented by the stated batch, condition, and interaction terms, with errors having conditional mean zero. Standard least-squares standard errors additionally require an appropriate variance model; conventional independent, homoskedastic errors are often used for basic inference. Exact small-sample \(t\)- or \(F\)-test calibration additionally relies on normally distributed errors. These assumptions should not be transferred uncritically to raw single-cell counts: the response scale and error model must suit the data and the analysis.

In single-cell studies, cells from the same biological sample are not generally independent experimental replicates. Treating every cell as an independent replicate can understate uncertainty. The experimental unit and dependence structure must therefore be considered when making inferential claims; a cell-level interaction coefficient alone does not solve that issue.

Useful checks include tabulating the number of observations in each batch-condition combination, inspecting the design matrix rank, and examining fitted means and residual patterns by batch and condition. Empty combinations warn of non-estimable effects. Very small counts in some combinations may leave effects technically estimable but imprecise. Residual trends, unequal spread, or outlying combinations can indicate that the chosen mean or variance model is inadequate. A large fitted interaction can also be driven by a small or unusual group, so it should be interpreted alongside the data supporting each cell mean.

The main alternative is an additive model, which omits all \(\delta_{bg}\) terms and assumes one common condition effect across batches. It is simpler and may be more precise when that restriction is credible, but it cannot describe genuine variation in the condition effect. An interaction model is more flexible but uses more parameters and can be unstable when combinations are sparse or absent. A cell-means model estimates each observed batch-condition mean directly; with a complete design, it represents the same mean patterns as the fully interacted categorical model, though its parameterization differs. No parameterization can supply a missing comparison without an added assumption.