Residuals Against Individual Predictions

Why Plot Residuals Against Individual Predictions?

A residual-versus-prediction plot asks whether errors vary systematically with the level of a model's predictions. Against an individual's prediction, it can help reveal prediction-dependent bias or an inadequate residual-error model. But individual predictions are not independent reference values: they are calculated using empirical Bayes estimates informed by that individual's observations. The plotted horizontal and vertical coordinates are therefore linked, even when the model is correct.

This dependence changes how the plot should be read. A visible trend is a prompt to investigate, not by itself proof of misspecification; a flat-looking plot is not proof that the model is adequate. Interpretation should consider how the estimates were obtained and whether the pattern is larger than expected under the fitted model.

Definition and Model Setup

For individual \(i\) at observation time \(t_{ij}\), write the model prediction as \(f(t_{ij},\phi_i)\), where the individual parameter \(\phi_i\) depends on population parameters and the individual's random effects \(\eta_i\). An empirical Bayes estimate, denoted \(\hat{\eta}_i\), is estimated using that individual's observations along with the fitted population model. The individual prediction, often called IPRED, is the model prediction evaluated at that estimate.

For example, if the observation model has conditional variance \(R_{ij}\), a simple standardized residual is \(e^I_{ij}/\sqrt{R_{ij}}\), with the variance evaluated according to the fitted error model. Some models require a more general calculation to account for correlated observations or transformations. The standardization convention must therefore match the model and software being used. Raw residuals retain the response units; standardized residuals are dimensionless.

The plot places IPRED on the horizontal axis and either \(e^I\) or IWRES on the vertical axis. Before interpreting it, check that the structural and residual-error models are plausible, that the estimation has converged, and that predictions and residuals were calculated consistently. A plot can flag a problem, but it cannot identify its cause on its own.

Why the Coordinates Are Dependent

A short Gaussian example makes the key issue precise. Suppose one individual has one observation and the model is \(Y=\mu+\eta+\varepsilon\), where \(\eta\sim N(0,\omega^2)\), \(\varepsilon\sim N(0,\sigma^2)\), and \(\eta\) and \(\varepsilon\) are independent. Assume \(\mu\), \(\omega^2\), and \(\sigma^2\) are known. Under this model, the conditional mean of \(\eta\) given \(Y=y\), which is also its conditional mode, is \(\lambda(y-\mu)\), where \(\lambda=\omega^2/(\omega^2+\sigma^2)\). This is the empirical Bayes estimate in this simple case.

Eliminating \(y-\mu\) gives a direct relationship between the two plotted quantities:

Thus, even when the data come exactly from the assumed model, the raw residual tends to increase with IPRED in this one-observation example. The pattern is not evidence of a faulty model: the same observation moves both the estimated individual prediction and the residual. In more complex NLME models, the relationship need not be a straight line, but the underlying data reuse remains. The derivation assumes known population parameters and a Gaussian random effect and error; estimating population parameters adds further dependencies.

Reading the Plot Carefully

A useful plot displays residuals against IPRED with a horizontal reference at zero. Look for sustained departures from zero, curvature, or a change in spread over the prediction range. A smooth increase or decrease may be compatible with prediction-dependent bias, but the empirical Bayes dependence just derived can also produce a trend. Increasing spread may suggest an error variance that depends on response magnitude, but it can also reflect differences in observation precision or how residuals were standardized.

The plot is most informative when interpreted alongside the model's assumptions and a reference for patterns expected under those assumptions. A practical option is a simulation-based check: simulate datasets from the fitted model, refit the model using the same procedure, recalculate empirical Bayes estimates and residuals, and compare the resulting plots with the observed one. Refitting matters because it reproduces some of the estimation and data-reuse effects. A pattern outside the simulated range is more concerning than one commonly produced by the fitted model, though simulation checks still depend on the model used to generate the reference.

What This Diagnostic Can and Cannot Establish

Residuals against individual predictions focus on whether conditional discrepancies depend on the fitted level for each individual. They differ from residuals against population predictions, whose horizontal coordinate does not use that individual's empirical Bayes estimate in the same direct way. The population-prediction plot can be a useful complement, while the individual-prediction plot can show how well the model fits after individual-level adaptation. Neither plot is a stand-alone test of model adequacy.

A recurring pattern may motivate checking the structural model or the residual-error specification, but this plot alone cannot distinguish those explanations. Nor does a small residual necessarily imply strong evidence that the individual's parameters are well determined: an empirical Bayes estimate can adapt toward the observations, especially when individual information is limited. Accordingly, do not interpret residuals as independent observations or apply a simple regression p-value as if the horizontal coordinates were fixed independently of the data.

The main limitation is that individual predictions are partly fitted to the observations being assessed. Sparse data, strong shrinkage, nonlinear estimation, and misspecified error variance can all complicate interpretation. A sensible reading combines the plot with the model assumptions, other residual displays, and checks that account for the full estimation procedure.

Common Mistake

• Wrong reasoning: “The residuals rise with IPRED, so the model must systematically underpredict at high concentrations.”
• Correction: The same observations help determine both IPRED and the residual. In a correctly specified one-observation Gaussian example, this alone creates an increasing relationship. Treat the pattern as a diagnostic signal to investigate, and compare it with what the fitted model and estimation procedure would produce under simulation.
• Wrong reasoning: “The residuals are close to zero, so the model is adequate.”
• Correction: Empirical Bayes estimates are informed by the observations and can pull individual predictions toward them. Small residuals do not replace checks of the structural model, error assumptions, or residual patterns.