Why Reporting Is Part of the Analysis
A batch-corrected multi-omics data set is not self-explanatory. A reader needs to know what the analyst called a batch, which cohorts and platforms contributed samples to each batch, which biological variables were included in the model, and what the correction changed. Without those details, a small batch association after correction might reflect successful adjustment, removal of biological signal, or simply a design in which the relevant effects cannot be separated.
The goal of reporting is not to claim that corrected values are free of technical effects. It is to make the model and evidence inspectable, and to mark the boundary between effects supported by the observed design and effects that remain ambiguous. The earlier material on uncertainty in estimated batch corrections concerns uncertainty in the estimated adjustment itself; here, the focus is on how to describe the model, its design, its diagnostics, and its limitations.
Define the Data Structure and the Model
For a feature \(g\) measured on sample \(i\), a common additive model is
Here \(Y_{ig}\) is the measured value of feature \(g\) for sample \(i\), \(\alpha_g\) is an intercept, \(\mathbf{z}_i\) contains the biological or design covariates to be represented in the model, \(\boldsymbol{\beta}_g\) contains their feature-specific coefficients, \(b_i\) is sample \(i\)'s batch, and \(\gamma_{g,b_i}\) is the corresponding batch effect. The error is \(\varepsilon_{ig}\). A reference batch with effect zero, or a sum-to-zero constraint on batch effects, is needed to make the intercept and batch effects uniquely parameterized.
This model states that covariate effects and batch effects add on the modeled measurement scale; it does not include batch-by-covariate interactions unless those are specified. For ordinary feature-wise regression inference, the conditional mean must be represented adequately, and the error assumptions used for the reported standard errors must be stated or justified. For example, conventional standard errors generally rely on independent errors with a suitable variance model. Empirical-Bayes procedures such as ComBat add distributional and sharing assumptions for stabilizing feature-wise batch estimates. Report the method actually used rather than describing all correction methods as if they fit the same model.
Report Overlap Before Interpreting Effects
A reader should see how biological groups are distributed across the batches. A cohort-by-batch table is useful when cohort membership is important; a batch-by-disease table is essential when disease effects are being estimated. Include zero cells, not just the nonzero combinations. Those absences show whether a comparison is supported within batches or must depend on the model's additive assumptions.
In a model with categorical batch and categorical disease, the design matrix must have sufficient rank for the contrasts being reported. A practical way to inspect the simplest batch-disease structure is to form a bipartite graph: one set of nodes represents batches, the other represents disease groups, and an edge joins a batch to a disease group when that combination has observations. If this graph is disconnected, batch and disease contrasts across disconnected components cannot be separated by the additive model. If it is connected, the two-factor additive contrasts can be estimable, subject to the model and the actual design matrix. With additional covariates, rank of the full model matrix—not this graph alone—is the general criterion.
Worked Example: A Mixed Batch Can Connect Exclusive Batches
Suppose a hypothetical study measures one feature in three batches. Batch A has 8 controls and no cases, batch B has no controls and 8 cases, and batch C has 5 controls and 5 cases. The outcome is modeled with an intercept, disease status, and batch indicators.
| Batch | Controls | Cases |
|---|---|---|
| A | 8 | 0 |
| B | 0 | 8 |
| C | 5 | 5 |
The graph has edges A-control, C-control, C-case, and B-case. It is connected through batch C, which contains both disease groups. In particular, the within-batch contrast in C informs the disease effect; the control observations in A and C inform an A-versus-C batch contrast, and the case observations in B and C inform a B-versus-C contrast. Under the additive model, the disease effect and batch contrasts can therefore be separated. The design is not equivalent to having only two batches, one containing controls and the other cases.
A suitable report would give the table, identify batch C as the only within-batch disease comparison, and state that inference relies on the additive model. It should not imply that every batch independently contains both groups. If the disease effect varies by batch, this sparse pattern gives little direct information for assessing that variation.
Worked Example: Two Exclusive Batches Do Not Identify Both Contrasts
Now suppose the only two batches are A, containing 6 controls with a hypothetical feature mean of 10 units, and B, containing 6 cases with a mean of 17 units. Under a two-group additive description, the observed difference is
This is one equation in two unknown contrasts. For example, a disease contrast of 7 units and a batch contrast of 0 units fit the difference; so do a disease contrast of 0 units and a batch contrast of 7 units. More generally, every pair \((d,b)\) satisfying \(d+b=7\) gives the same two observed group means after choosing the intercept. The design matrix is rank deficient for estimating both contrasts separately.
The appropriate report says that the observed 7-unit difference cannot be attributed separately to disease and batch from these data under this model. It is not correct to report a uniquely estimated disease effect or claim that a correction has isolated disease biology. This non-identifiability statement applies when the design lacks overlap connecting batch and disease groups; it does not follow merely from the fact that one batch is exclusive to one group if other batches provide connecting overlap.
Describe the Adjustment Covariates and Target
List every covariate included in the batch model and state why it is there. Distinguish variables whose effects the model is intended to preserve—such as disease status, age, or sex—from variables used to describe technical structure. Give coding and reference levels for categorical covariates, units and centering for continuous covariates, and the sample count used in the fitted model. If a covariate was omitted because of missingness, collinearity, or a design decision, say so.
For a correction intended to remove estimated batch effects while retaining modeled covariate effects, a simple additive adjustment can be written as
This expression describes the correction target: subtract the estimated batch contribution while retaining the observed covariate-associated component. It does not guarantee that the retained component is purely biological or that the removed component is purely technical. Those interpretations depend on the design, model specification, and assumptions. State whether values were adjusted on a transformed scale, whether batch effects were estimated feature by feature or with a sharing procedure, and how missing values or excluded samples were handled.
Worked Example: Reporting a Covariate-Preserving Adjustment
Consider hypothetical feature measurements in two batches, A and B, with 10 controls and 10 cases in each batch. At age 50, the fitted group means are 10 and 13 units in batch A for controls and cases, and 14 and 17 units in batch B. A fitted additive model uses disease status, batch, and age centered at 50 years:
where \(D_i=1\) for a case and \(0\) for a control, and \(I(b_i=B)=1\) for batch B and \(0\) for batch A. The fitted disease contrast is 3 units at a fixed batch and age; the fitted B-versus-A batch contrast is 4 units at a fixed disease group and age. The age coefficient is 0.2 units per year.
For a 54-year-old case in batch B, the fitted value is \(10+3+4+0.2(54-50)=17.8\) units. The adjustment subtracts the batch B estimate of 4 units, giving \(17.8-4=13.8\) adjusted units. For a 54-year-old case in batch A, the fitted value is \(10+3+0+0.8=13.8\) units, and no batch estimate is subtracted. Thus the adjustment aligns these fitted values at the same disease status and age under this additive model. A report should state that age and disease were included and preserved by the specified adjustment, and that the numerical illustration is model-based—not evidence that the correction has removed every technical difference.
Diagnostics and Limits of the Evidence
Diagnostics should be tied to the model and presented as evidence, not as a pass/fail certificate. Useful reporting includes batch and cohort counts, missingness by group, distributions before and after adjustment, and residual summaries or plots grouped by batch and by protected biological covariates. For feature-wide analyses, summarize the distribution of diagnostic quantities across features and explain any filtering or aggregation. Report influential batches and whether conclusions change under reasonable alternative specifications; the earlier material on sensitivity to batch model specification and influential cohorts provides the relevant analytical background.
A post-adjustment reduction in batch separation is not sufficient by itself. It can occur because a real cohort or disease difference was also removed, particularly when biology and batch are strongly associated. Conversely, a remaining batch pattern may indicate residual technical signal, a misspecified model, or an unmodeled interaction. Report diagnostics for preserved biological variables as well as batch variables, and compare their patterns before and after adjustment. If a diagnostic test is used, give its statistic, reference procedure, and the blocking or exchangeability assumptions relevant to its interpretation; a non-significant result does not establish absence of batch effects.
A transparent report also acknowledges what the data cannot resolve. Perfect or near-perfect confounding limits attribution; sparse overlap can make estimates imprecise even when the design matrix technically has full rank. Additive models may be inadequate if batch effects vary with disease status, and platform differences may not be captured by a single offset or scale term. Correction can depend on the measurement transformation, selected features, reference materials, and the samples included. State these constraints in context rather than presenting adjusted values as universally comparable measurements.
Comparing What the Report Should Distinguish
| Analysis or claim | What to report | What it does not establish |
|---|---|---|
| Batch included as a model covariate | Model formula, coding, covariates, and the contrast being estimated | That a separate corrected data set is free of batch signal |
| Batch-adjusted measurements | Adjustment target, method, scale, retained covariates, and diagnostics | That all removed variation was technical or all retained variation biological |
| Analysis within batches or cohorts | Which groups are compared within each stratum and how results are combined, if applicable | That differences between non-overlapping strata are identified without assumptions |
| Batch and biology not separately identifiable | The confounded groups, rank or overlap limitation, and unsupported contrasts | A uniquely estimated biological effect from the confounded comparison |
These approaches answer different analytical questions. Including batch as a covariate estimates specified contrasts conditional on the modeled batch terms. Adjustment produces a transformed data set for subsequent use. Within-batch analysis limits comparisons to observed overlap. When effects are not identifiable, reporting that limitation is more informative than presenting a model-dependent decomposition as data-determined.
Common Reporting Mistakes
- Calling every cohort a batch. Cohort and technical batch may coincide, overlap, or cross; define each from the recorded study structure.
- Reporting only the correction algorithm. Name the model, covariates, coding, samples, scale, and target of adjustment as well.
- Claiming that exclusive batches always make effects unidentifiable. The key issue is whether the design has overlap connecting the groups and whether the full model matrix has sufficient rank. A mixed batch can connect otherwise exclusive batches.
- Equating reduced batch separation with successful correction. Check whether biological patterns intended for preservation remain, and describe the limits of each diagnostic.
- Interpreting an unidentifiable contrast as zero. Lack of identification means the data and model do not uniquely determine the contrast; it is not evidence that the contrast is absent.
Check Your Understanding
Answer each question, then compare with the solutions below.
- What information should accompany the label “batch” in a report?
- Two batches each contain both controls and cases. What does that overlap contribute to an additive batch-and-disease model?
- One controls-only batch and one cases-only batch are connected by a third mixed batch. Is separate estimation necessarily impossible? Explain.
- Why does a non-significant post-adjustment test of batch association not prove that batch effects are absent?
- In the adjustment formula, what is retained when the estimated batch contribution is subtracted, and what assumption does that interpretation rely on?
Solutions. (1) State the operational batch definition, assignment rule, observational unit, and relevant cohort/platform composition. (2) Within-batch disease comparisons are available, helping distinguish disease from batch effects under the additive model; the full model matrix still needs adequate rank. (3) No. The mixed batch can connect the disease groups and batch levels, so the additive contrasts may be identifiable; verify the full design matrix and disclose the sparse pattern. (4) A non-significant result may reflect limited power or the test's assumptions, and it does not demonstrate zero residual signal. (5) The adjustment is designed to retain modeled covariate contributions while subtracting estimated batch contributions. That interpretation depends on the specified model and its assumptions; it is not guaranteed by the arithmetic alone.