Fixed-Effect and Random-Effect Batch Adjustment

Why the Choice of Batch Model Matters

A batch term represents systematic differences associated with processing groups, such as assay runs, laboratories, or platforms. In an additive model, including this term can help estimate biological contrasts without attributing batch-associated shifts to biology. The modeling choice matters because a fixed-effect model estimates a separate effect for each observed batch, whereas a random-effect model treats batch deviations as draws from a shared distribution. Those models answer related but distinct questions: what are the effects of these particular batches, and what is the distribution of effects across a population of batches?

The distinction is not a device for repairing a poor design. As established in the earlier material on rank, connectivity, and confounding, a biological effect cannot be separated from a batch effect when the observed design does not identify that contrast. Treating batch as random does not create the missing comparison. The choice also does not determine whether an adjustment is safe: a model can remove real biological signal if biology and batch are confounded or if relevant biology is omitted.

Two Models for a Batch Term

For one omics feature, let \(y_{ib}\) be the measured value for sample \(i\) in batch \(b\). Let \(x_{ib}\) denote the measured biological and other design covariates, and let \(\beta\) be their coefficient vector. A fixed-effect model can be written as

Here \(\alpha_b\) is the fixed effect for batch \(b\). To identify the intercept and batch terms, use a reference-batch constraint such as \(\alpha_1=0\), or a stated sum-to-zero constraint. Either parameterization gives the same fitted values and estimable contrasts. The effects need not follow a distribution across batches.

A random-intercept model instead writes

The usual model assumes the batch deviations are independent across batches, the residual errors are independent across samples conditional on the batch effects, and the two sets of random terms are independent of one another. It also assumes the random batch effects are independent of the included covariates. For standard small-sample likelihood-based inference, normality and a correctly specified mean and variance structure are commonly assumed; the distributional assumptions should not be confused with the design requirement that the biological contrast be identifiable.

A fixed model is often natural when inference concerns a small, particular set of observed batches. A random model is often useful when the batches can reasonably be regarded as exchangeable representatives of a broader population, especially when there are many batches or their sample sizes vary. In either case, covariates and batch structure must be represented so the desired biological contrast is estimable.

Random Effects and Partial Pooling

A random-effect model does not estimate each batch independently of the others. Instead, it combines the batch’s own data with the assumed distribution of batch effects. This is called shrinkage: an estimated batch deviation is pulled toward the population mean of zero. The pull is stronger when the batch has little information and weaker when it has more information.

The mechanism can be derived for a simple case. Suppose covariate contributions are known or removed, leaving residual values \(r_{ib}=y_{ib}-x_{ib}^{T}\beta\) in batch \(b\). Assume \(r_{ib}=u_b+\varepsilon_{ib}\), with \(u_b\sim N(0,\tau^2)\) and independent \(\varepsilon_{ib}\sim N(0,\sigma^2)\). Let \(\bar{r}_b\) be the mean residual for that batch. The likelihood for \(u_b\), up to a factor not depending on \(u_b\), is proportional to \(\exp[-n_b(\bar{r}_b-u_b)^2/(2\sigma^2)]\). Multiplying by the normal prior density for \(u_b\) adds the term \(u_b^2/(2\tau^2)\) in the exponent. Completing the square gives the conditional distribution of \(u_b\) given the residual data:

The conditional mean is the best linear predictor in this simple normal model. Its multiplier on the unpooled batch mean is between zero and one:

For a small batch, \(n_b\) is small and \(\lambda_b\) is closer to zero, so the estimate is pulled more toward zero. For a large batch, \(\lambda_b\) is closer to one, so the batch’s observed mean has more influence. In practice, \(\tau^2\), \(\sigma^2\), and \(\beta\) are usually estimated from the full data; treating the variance components as known is an explanation of the shrinkage mechanism, not a claim that they are known.

Choosing Between the Models

The choice depends on the scientific target, the number and sampling of batches, and the design. A fixed-effect model has fewer distributional assumptions about batch effects and directly represents the observed batch levels. Its cost is that it estimates many batch parameters as the number of batches grows. A random-effect model uses a shared variance distribution and partial pooling, which can stabilize estimates, but its conclusions depend on the exchangeability and distribution assumptions.

These are tendencies, not automatic selection rules. A few batches do not guarantee that fixed effects will be adequate: a biological contrast may still be unidentifiable, or residual assumptions may be poor. Likewise, many batches do not guarantee that a random-effect model is appropriate: the observed batches may be the specific targets of inference, or their effects may not be plausibly exchangeable. A random-effect distribution is especially questionable when batch labels represent systematically different settings rather than levels drawn from a common population.

Diagnostics, Failure Modes, and Limits

Batch-model diagnostics should address both the design and the model assumptions. Before interpreting adjusted estimates, check that the biological contrasts of interest are estimable and that the design contains the comparisons needed to separate those contrasts from batch. Then examine whether residual patterns, outliers, or markedly different within-batch variability make the assumed error model implausible. Residual checks can reveal problems but cannot prove that batch levels are exchangeable or that confounding is absent.

• Too few random-effect levels: estimates of \(\tau^2\) can be unstable or near a boundary. Report that limitation and consider whether inference about the observed batches is more appropriate than inference about a batch population.
• Very small or highly uneven batches: fixed estimates for small batches can be noisy; random effects can shrink them strongly. Check how sensitive the biological estimate is to the batch specification rather than treating shrinkage as automatic correction.
• Batch associated with biology: the random-effect assumption that batch deviations are independent of included covariates may fail. A fixed-effect formulation avoids that particular distributional independence assumption, but it cannot recover a biological contrast that the design does not identify.
• Unmodeled variation in biological effects: a single additive treatment coefficient assumes the treatment contrast is common across batches. If that assumption is scientifically implausible, a common-effect estimate may conceal heterogeneity.
• Uncertainty after adjustment: subtracting estimated batch effects and then treating the adjusted values as known can understate uncertainty. Prefer inference from the fitted model when uncertainty in batch estimates matters.

A frequent mistake is to claim that a random-effect model is always superior because it “corrects” small batches by borrowing information. The correction is conditional on the assumed common distribution and independence structure; if those assumptions are wrong, shrinkage can pull estimates in an inappropriate direction. The converse mistake is to insist that fixed effects are assumption-free. They avoid a probability distribution for the batch coefficients, but still require an adequate outcome model, identifiable contrasts, and defensible error assumptions.