Outlier Counts and Robust Gene-Level Inference

Why a Large Count Needs Context

In pseudobulk differential expression, a gene’s count is recorded for each biological sample, such as a donor-condition combination. A count can look large in isolation yet be entirely plausible given the sample’s library size, the gene’s expected expression, and the gene’s variability across samples. Conversely, a count that is not the largest in the table can be surprising under the fitted model and substantially affect a gene-level comparison.

The relevant question is therefore not simply whether a count is numerically high. It is whether it is unusually high relative to a suitable model, and whether it has enough influence to alter an estimate or its uncertainty. A flagged count is a diagnostic signal to investigate, not evidence by itself that the observation is erroneous or should be removed.

A Model-Based Reference for Counts

A common starting point for pseudobulk counts is a negative binomial generalized linear model, fitted separately for each gene. Let \(Y_{gj}\) be the count for gene \(g\) in biological sample \(j\), \(s_j\) a sample-specific size factor, and \(\mathbf{x}_j\) the row of the design matrix. One parameterization is

Here \(\mu_{gj}\) is the fitted mean, \(\alpha_g\) is the gene-specific dispersion, and \(\log(s_j)\) is an offset. The design encodes the comparison and any appropriate covariates. The model assumes that the conditional mean structure and dispersion adequately describe the counts, that the offset is appropriate, and that biological samples are independent conditional on the model (or that their dependence is handled by the specified model). A misspecified design, an inadequate mean-variance relationship, or unmodeled dependence can make outlier diagnostics misleading.

A simple diagnostic is the Pearson residual:

This scales the difference between observed and fitted count by the model’s estimated standard deviation. A large positive residual indicates a count above the fitted mean relative to the expected variability. It is a diagnostic, not a formal outlier test with a universal cutoff: parameters were estimated from the data, and leverage, small sample size, and dispersion estimation affect the residual’s behavior. Examine residuals alongside fitted values, sample quality, design, and influence measures rather than applying one threshold mechanically.

From a Flag to an Inference Decision

A useful investigation proceeds in stages. First verify the observation: check count generation, sample identity, library size, cell-type assignment, and relevant quality-control information. Then ask whether the fitted mean and dispersion are credible for that gene and design. A condition-specific pattern may be real and should not be “corrected” just because it is large. Finally, assess influence by comparing the gene-level estimate and test with and without the observation, or by using a fitting method that limits the influence of extreme residuals.

The main handling choices have different meanings:

• Retain the count and use the ordinary model. This is appropriate when the count is plausible and the model is adequate. It preserves the observation but may leave inference sensitive to an extreme value.
• Exclude a sample. Exclusion may be justified by a documented sample-level failure, not merely because removing the sample improves a p-value. It reduces information and can change the target population or design balance.
• Downweight or robustly fit the gene model. A robust coefficient procedure limits how strongly large residuals affect estimation. Its weighting rule and assumptions matter; robust fitting is not a license to ignore sample quality or model misspecification.
• Use an outlier replacement procedure, if appropriate and pre-specified. Replacement changes the data entering the test and requires a method that accounts for the replacement in its inferential procedure. Ad hoc substitution with a convenient value generally does not provide valid uncertainty or testing.
• Report a sensitivity analysis. Show whether the estimated effect and conclusion are stable under defensible alternatives. A sensitivity result complements, rather than replaces, the primary analysis rationale.

Robust standard errors are not the same as robust coefficient estimation. A sandwich-type standard error can protect uncertainty estimates against some forms of variance misspecification, but it does not necessarily stop an extreme count from pulling the fitted coefficient. Conversely, robust coefficient estimation changes the way observations contribute to the estimate; its standard error and test must be calculated for that procedure. State which quantity is robust.

Effects on Tests, Diagnostics, and Interpretation

For a gene-level test of a coefficient \(\beta_g\), a Wald statistic is often formed as \(\widehat{\beta}_g/SE(\widehat{\beta}_g)\), with a reference distribution appropriate to the model and inferential method. An influential count can change the coefficient, its standard error, or both. Consequently, the test statistic and p-value may change even if the underlying biological pattern has not. A small p-value from one fit does not validate the model or prove that a flagged count is a measurement error.

Diagnostics should be interpreted together. A large positive residual may arise from a bad offset, a missing covariate, an incorrect condition label, an influential sample, or a genuinely unusual biological response. A dispersion that is too small can make many counts appear extreme; an incorrect mean structure can make residuals systematically positive in one group. Examine residual patterns across samples and genes, compare fitted and observed counts, and assess whether the flagged sample has broader quality concerns. The earlier material on residual diagnostics, dispersion fit, and donor-sample influence provides relevant complementary checks.

No single outlier rule is universal. A residual cutoff does not account in a simple way for all genes and samples being examined, and a gene selected because it has the most extreme residual is subject to selection. With few biological replicates, one observation can dominate an estimate, while the data provide limited ability to distinguish a true sample-specific response from an anomaly. Robust procedures can reduce sensitivity, but may sacrifice efficiency when the ordinary model is correct and may not protect against design errors or systematic bias.

Common Pitfall: Treating “Large” as “Wrong”

Incorrect reasoning: “This is the largest count in the table, so it must be an error; I will remove it and report the smaller p-value.” This confuses raw magnitude with model-based surprise, ignores the possibility of a real effect, and makes inference depend on a post hoc decision.

Correction: Compare the count with its fitted mean and dispersion, verify sample and design information, and evaluate how the estimate and test change under defensible procedures. If there is independent evidence of a technical failure, document the exclusion rule and its consequences. Otherwise, retain the observation in the primary analysis or use a justified robust procedure, and report sensitivity when the result depends on the choice.