Residual Stratification Diagnostics for Covariate Effects

Why Stratify Residuals by Covariates?

A fitted NLME model can describe the overall concentration-time profile reasonably well while still leaving systematic differences across subjects. For example, residuals might tend to be positive in one category of a categorical covariate and negative in another. That pattern suggests the model is, on average, underpredicting one group and overpredicting the other. Stratifying residuals by covariate values provides a direct way to look for such remaining structure.

The purpose is diagnostic: to ask whether residual behavior changes with a covariate after fitting the current model. A pattern is a reason to investigate the model, data, and covariate representation. It is not, by itself, a formal test, proof of a covariate effect, or a basis for a causal claim.

In an NLME model, let \(y_{ij}\) be the response for subject \(i\) at time \(t_{ij}\), and let the individual parameter vector be \(\theta_i\). A basic representation is

Here \(x_i\) denotes subject-level covariates, \(\beta\) denotes fixed-effect parameters, \(\eta_i\) denotes between-subject random effects, and \(\epsilon_{ij}\) denotes residual error. For a population-prediction residual, use the fitted prediction with random effects set to their population mean, commonly zero: \(\hat{y}_{ij,\mathrm{pop}}=f(t_{ij},g(x_i,\hat{\beta}))\). A conditional prediction instead uses an estimate of that subject’s random effects. These predictions answer different questions, so the residual type must be stated when interpreting a plot.

For comparability, a diagnostic may use \(z_{ij}=r_{ij}/\sqrt{\widehat{V}_{ij}}\), where \(\widehat{V}_{ij}\) is the model-based predictive variance appropriate to the chosen residual definition. The variance used for standardization should be documented: it may include residual error alone or additional modeled uncertainty, depending on the diagnostic. A standardized residual is not automatically a standard normal observation, particularly when parameters are estimated, residuals are correlated, or the variance approximation is imperfect.

What a Group Pattern Means

If the fitted model is adequate for the diagnostic being considered, residuals should not show a systematic mean pattern across a covariate after accounting for the model’s structure. This expectation can be stated simply. Let \(m_0(x)\) be the fitted conditional mean under the current model, and suppose the true mean is \(m_0(x)+\delta(x)\), where \(\delta(x)\) is a discrepancy not represented by the fitted model. Then

Thus, if \(\delta(x)\) changes across covariate values or categories, the expected residual changes with the covariate. This is the justification for looking at residual summaries by covariate. The argument concerns a mean pattern; residual spread or shape can also change, indicating possible variance misspecification rather than a mean covariate effect.

The interpretation depends on the residual definition. Population-prediction residuals retain variation associated with unexplained subject effects and can reveal between-subject patterns. Conditional residuals use subject-specific estimates and may appear more centered because those estimates have already adapted to each subject’s observations. Conditional residuals are useful for some checks, but they should not be the only basis for judging whether a subject-level covariate is missing.

Practical Stratification

For a categorical covariate, show the residual distribution separately for each category. Useful summaries include the group mean or median, spread, and the number of subjects represented. For a continuous covariate, plot residuals against the covariate and, when helpful, add summaries over prespecified ranges. Binning can make broad patterns easier to see, but the choice of cut points can hide a trend or create an appearance that depends on arbitrary boundaries.

When the model has repeated observations per subject, show the observation-level pattern while also checking subject-level summaries. A subject with many recorded times contributes more rows than a subject with few times. Consequently, a simple average over all rows weights subjects in proportion to their number of observations. That may be appropriate for describing the observation-level data, but it is not the same as giving each subject equal weight. Within-subject correlation also means rows should not be treated as independent evidence.

The diagnostic should match the covariate’s role. For a subject-level covariate, examine whether subject-level residual behavior differs across subjects’ covariate values. For a time-varying covariate, consider whether the residual pattern changes with its value within the relevant observations, while accounting for the model’s time and repeated-measures structure. A residual-versus-time pattern can also be mistaken for a covariate pattern if covariate values change systematically with time.

Diagnostics, Alternatives, and Limitations

A useful review checks more than whether group means differ. Look for whether the pattern is consistent across the covariate range, whether residual spread changes, whether the result is driven by a small number of subjects, and whether it depends on the residual definition. Confirm that the plotted covariate values and categories are correct, that missing values have not silently changed the groups, and that the residuals are on a comparable scale. For repeated data, inspect both observation-level plots and subject-level summaries where relevant.

Stratified residuals are complementary to formal covariate-effect procedures. Wald statistics and likelihood-ratio tests evaluate specified model parameters or comparisons under their assumptions. Residual plots instead ask whether the fitted model leaves a visible pattern. They can flag a poorly represented functional relationship, an omitted covariate, an incorrect variance structure, or another model-data mismatch; they do not identify which explanation is correct. Conversely, a formal test can provide evidence for a specified effect but does not guarantee that the model’s residual behavior is satisfactory.

Residual stratification is also different from screening covariates with empirical Bayes estimates. Empirical Bayes estimates are subject-specific estimates of random effects; population-prediction residuals are observation-level discrepancies from population predictions. Each has different sources of variability and can reveal different patterns. Neither should be interpreted as a substitute for assessing the fitted model and its assumptions.

A residual pattern can be obscured by a poor choice of scale, arbitrary bins, few subjects in some ranges, or shrinkage of conditional predictions toward the population mean. A pattern can also arise from misspecification unrelated to the covariate, such as a time-course error or a variance model that does not capture changing variability. These diagnostics are exploratory and model-dependent; visual evidence should be followed by a scientifically justified model comparison and appropriate checks, not an automatic covariate addition.