Uncertainty in Estimated Batch Corrections

Why an Estimated Correction Is Not Exact

A batch correction is calculated from data, so its estimated parameters are uncertain. If a feature is shifted upward in one batch, subtracting the estimated shift produces adjusted values; it does not make the shift known without error. This distinction matters when comparing biological groups: uncertainty in the correction can make an adjusted contrast much less precise than its point estimate suggests.

The introductory material on location-scale batch effects and ComBat established common ways to represent batch differences and adjust feature values. Here the focus is narrower: how to quantify uncertainty in the fitted adjustment and carry it through to an adjusted feature comparison. An interval describes uncertainty under a specified model and design. It cannot recover information absent from the data, such as a biological contrast that is inseparable from batch.

Model-Based Uncertainty

For one feature, a simple location model is \(y_i=\mu+\gamma_{b(i)}+\epsilon_i\), where \(y_i\) is the measured feature value for sample \(i\), \(\mu\) is a reference level, \(\gamma_{b(i)}\) is the shift for that sample’s batch relative to the reference batch, and \(\epsilon_i\) is residual variation. A model may also include biological covariates. The correction for a non-reference batch subtracts its estimated shift, \(\hat{\gamma}_b\), from that batch’s values.

In a linear model \(y=X\beta+\epsilon\), suppose the design matrix \(X\) has full column rank and the errors are independent with \(\epsilon\sim N(0,\sigma^2I)\). The least-squares estimate is \(\hat{\beta}=(X^\mathsf{T}X)^{-1}X^\mathsf{T}y\), and its estimated covariance matrix is \(\widehat{\mathrm{Var}}(\hat{\beta})=s^2(X^\mathsf{T}X)^{-1}\), where \(s^2\) is the residual mean square. If the batch shift is a coefficient or a linear combination of coefficients, its standard error follows from this covariance matrix. For a contrast \(a^\mathsf{T}\beta\), the estimated standard error is \(SE=s\sqrt{a^\mathsf{T}(X^\mathsf{T}X)^{-1}a}\); under the stated Gaussian model, a two-sided \(100(1-\alpha)\%\) interval is \(\hat{\theta}\pm t_{1-\alpha/2,df}SE\), with \(\hat{\theta}=a^\mathsf{T}\hat{\beta}\) and residual degrees of freedom \(df\).

These formulas rely on a correctly specified mean structure, independent errors with a common variance, full-rank design, and Gaussian errors for the exact \(t\)-interval. If observations are paired, clustered, heteroscedastic, or otherwise dependent, the corresponding covariance model or resampling scheme must reflect that structure. A narrow interval obtained from a model that treats dependent samples as independent can be misleading.

How Correction Uncertainty Affects a Feature Comparison

An adjusted biological contrast is often a function of both the observed feature contrast and one or more estimated batch parameters. Its uncertainty must account for all estimated quantities in that function. For a simple case, let \(\hat{\delta}_{raw}\) be an unadjusted group contrast and let \(\hat{\gamma}\) be the estimated batch difference that must be removed from it. The adjusted estimate is \(\hat{\theta}=\hat{\delta}_{raw}-\hat{\gamma}\). Its variance is

If the two estimates come from independent data, the covariance term is zero. If they use overlapping samples or are fitted together, it generally need not be zero. In a joint linear model, the direct approach is to define the adjusted comparison as a contrast \(a^\mathsf{T}\beta\) and use the full estimated covariance matrix of \(\hat{\beta}\). This automatically retains covariance among the biological and batch coefficients.

Resampling the Correction Procedure

When a reliable analytic standard error is unavailable, resampling can approximate sampling variability. A bootstrap replicate is made by sampling the independent observational units with replacement, refitting the batch model, and recalculating the correction or adjusted contrast. The spread of replicate estimates approximates sampling uncertainty, while their empirical quantiles can form a percentile interval.

The resampling unit must match the sampling design. For paired bridge measurements, resample specimen pairs together, not each batch measurement separately. For repeated observations from one person, resample people or clusters rather than individual rows. If the target is generalization to new cohorts, resampling only samples within the observed cohorts does not represent between-cohort variation. With very few cohorts, resampling cohorts may itself be unreliable.

Diagnostics, Alternatives, and Limits

Useful diagnostics focus on whether the uncertainty calculation reflects the information in the design and whether the fitted model behaves plausibly. Examine standard errors and interval widths across features, the distribution of bootstrap estimates, sensitivity to individual bridge specimens, and whether resamples frequently produce singular or unstable fits. An interval that changes sharply when one specimen is removed suggests that the correction is influential and weakly supported. Residual patterns or unequal variability may indicate that the model-based covariance assumptions need reconsideration.

Model-based intervals are efficient and interpretable when the mean and covariance assumptions are defensible. Bootstrap intervals can better accommodate skewness or complicated fitting procedures, but they require enough independent sampling units and a resampling design that mirrors the intended sampling process. Neither approach automatically accounts for uncertainty in a chosen model specification. A sensitivity analysis across plausible specifications can reveal that instability, but it is not a substitute for a correctly defined sampling model.

Empirical-Bayes adjustments can stabilize feature-specific estimates by sharing information across features. An interval conditional on estimated prior or hyperparameters may understate uncertainty if those quantities are treated as fixed. A bootstrap that refits the empirical-Bayes procedure can capture more of this estimation variability, provided the resampling and fitting steps are appropriate. This does not remove the need to examine model assumptions.

A common mistake is to correct every value using \(\hat{\gamma}\), then calculate a downstream standard error as though \(\hat{\gamma}\) were fixed. That calculation omits the correction’s estimation uncertainty and can yield intervals that are too narrow. The correction is not a constant supplied by nature: include its variance and any relevant covariance, or refit the whole procedure within each valid resample. When overlap is inadequate, the honest result may be a wide interval or an explicitly non-identifiable contrast, not a more elaborate correction.