Residual Diagnostics for Donor-Aware Models

Why Diagnose Residuals at the Donor Level?

A donor-aware model is intended to represent both the scientific relationship of interest and the dependence among observations from the same donor. Its diagnostics should therefore ask whether the fitted relationship is plausible across donors, not merely whether many cell-level residuals look small. A donor with many sampled cells can produce many residuals, but those cells do not constitute many independent biological replicates.

Residual diagnostics are descriptive checks on a specified model, not a guarantee that its assumptions are true. They can reveal patterns that merit investigation, such as a curved mean relationship, a donor with an unusual response, or variability that grows with the fitted mean. They cannot establish that a model is correct, and they should not be used as an automatic rule for deleting a donor.

Which Fitted Values and Residuals?

In a linear model, fitted values are generated by the model's estimated coefficients. In a mixed model, the choice of fitted value matters. A marginal fitted value uses the population-level fixed effects and averages over the random effects. A conditional fitted value also includes the estimated donor-specific random effect. The corresponding residuals answer different questions: marginal residuals show how observations depart from the population-level relationship, while conditional residuals show departures after accounting for the fitted donor-specific effect.

Conditional residuals can look relatively small because donor-specific effects were estimated using the same observations. In particular, estimated random effects are partially pooled toward the population mean, and the conditional residual pattern should not be read as if donor effects were known in advance. State which fitted values were used when presenting a diagnostic.

For models with one row per donor, residual diagnostics are directly examined at the donor level. For models with multiple observations per donor, inspect within-donor residual patterns as well as donor summaries. A large collection of cell-level residuals is not a substitute for assessing how the model performs across independent donors.

A Justification for Leverage-Adjusted Residuals

Even under a correctly specified homoskedastic linear model, raw residuals do not all have the same variance. This matters when comparing their magnitudes. To see why, consider a donor-level linear model in matrix form, \(y=X\beta+\varepsilon\), with independent errors of mean zero and common variance \(\sigma^2\). The ordinary least-squares fitted values are \(Hy\), where \(H=X(X^\mathsf{T}X)^{-1}X^\mathsf{T}\), so the residual vector is \(e=(I-H)y\). Since \((I-H)X=0\), this becomes \(e=(I-H)\varepsilon\).

The last equality uses the symmetry and idempotence of \(H\). Thus the variance of residual \(e_i\) is \(\sigma^2(1-h_{ii})\), where \(h_{ii}\) is the \(i\)th diagonal entry of \(H\), called leverage. A residual from a high-leverage donor can be smaller partly because the fit was influenced by that donor. Standardized or studentized residuals account for estimated scale and, in common linear-model diagnostics, leverage. Their exact form depends on the model and fitting method; a simple residual divided by an estimated standard deviation is not automatically a fully studentized residual.

This derivation also explains a basic caution: residuals are constrained by the fitting process. In a model with an intercept, their sum is zero, and in linear regression they are orthogonal to the included predictors. A zero average residual, by itself, is therefore not evidence that the model fits well.

Plots and Patterns to Examine

A useful diagnostic set begins with residuals plotted against fitted values and against important predictors. Under a well-specified mean relationship, residuals should fluctuate around zero without a systematic shape. A smooth curve, repeated sign changes, or a trend suggests that the fitted mean may be missing structure. A fan-shaped cloud, in which residual spread grows or shrinks across fitted values, suggests nonconstant variability.

Also inspect donor identity or donor-level summaries. A plot with donors labelled can reveal whether unusual residuals cluster in one donor, occur in donors with a particular covariate, or appear across many donors. A normal quantile plot can help identify strong departures from a normal error model when that assumption supports the chosen inference. With few donors, such a plot has limited resolution; it cannot certify normality.

Diagnostics should match the model. If the model allows donor-specific slopes, examine whether residual patterns remain across the predictor after those slopes are included. If repeated observations within donors have a meaningful order, inspect residuals against that order for systematic structure. Do not interpret a plot that treats cell-level residuals as independent evidence about between-donor variability.

Donor-Level Influence and Model Fit

An unusual residual and an influential donor are not the same thing. A residual measures a donor's discrepancy from the fitted outcome; influence asks how much the fitted result changes when that donor's information is removed or downweighted. A donor may have a large residual but little influence if it has low leverage, or a moderate residual but substantial influence if its covariates are unusual. Examine influence at the donor-cluster level: deleting individual cells while leaving their donor in the analysis does not answer whether the result depends on that donor.

Leave-one-donor-out coefficient comparisons are one practical diagnostic. They are sensitivity checks, not a formal test for whether a donor is valid. Interpret a change relative to the original estimate, its uncertainty, and the scientific scale of the effect. A donor can be biologically unusual and still belong to the target population.

Common Failure Modes and Alternatives

A useful diagnostic review considers what each finding does and does not imply:

• Curved residual pattern: the mean model may be missing nonlinear structure or an interaction. A variance adjustment alone will not fix a wrong conditional mean.
• Increasing or decreasing spread: the constant-variance assumption may be implausible. Consider whether the outcome scale, model variance structure, or an inference method appropriate to the design needs review.
• Donor-specific cluster of residuals: check data processing, donor-level covariates, and the assumed donor structure. The pattern may reflect genuine heterogeneity rather than a mistake.
• One large residual: investigate measurement, sample handling, and model fit, but do not delete a donor solely because it changes a result.
• Apparent normality across thousands of cells: this can be misleading when cells from a donor are dependent. For population-level claims, the number of independent donors remains central.

Alternatives are complementary rather than interchangeable. Model-based residual plots check the fit and variance structure implied by a model. Cluster-robust standard errors can protect coefficient uncertainty against some forms of within-donor dependence when there are enough independent clusters and the mean model is appropriate; they do not diagnose or cure poor fit. Design-based inference relies on the assignment or sampling design rather than a full outcome distribution model, but still requires diagnostics relevant to its specified estimand and design. Leave-one-donor-out analysis measures sensitivity to particular clusters, while residual plots show patterns across fitted values or predictors.

These checks have important limits. With few donors, patterns are noisy and formal distributional tests have little power; a lack of a visible pattern is not proof of an assumption. Residuals are estimated quantities, affected by model fitting, and mixed-model residuals depend on whether predictions are conditional or marginal. Diagnostics also cannot verify that donors represent the population to which results are generalized. They should inform model review and transparent reporting, not become a search for a preferred result.

A Practical Diagnostic Sequence