Choosing Residuals for Standard Diagnostics

Why Residual Choice Matters

A residual is not a single universal quantity. It is an observation compared with a prediction, possibly scaled to account for variability, and the prediction itself may represent the population or a particular individual. A plot can therefore appear to show poor fit when it actually uses a residual intended for a different question—or appear reassuring because the residual has been normalized in a way that hides the distinction.

For standard diagnostics, the key choice is whether the display is assessing the population model or the fit after using individual-specific information. That choice determines which prediction and residual are appropriate. The display should also identify its residual definition, scale, and source; labels such as “weighted residual” alone may not be sufficient because implementations can differ.

Predictions and Raw Residuals

Let \(y_{ij}\) be the observed dependent-variable value for individual \(i\) at observation \(j\). Write the model prediction as \(f_{ij}(\theta,\eta_i)\), where \(\theta\) denotes population parameters and \(\eta_i\) denotes individual random effects. A population prediction, often called PRED, sets the random effects to their typical value, conventionally zero. An individual prediction, often called IPRED, uses an estimate \(\hat{\eta}_i\) for that individual.

Population raw residuals include discrepancies due to both individual deviations from the typical prediction and unexplained observation-level error. Individual raw residuals compare the observation with a prediction informed by that individual’s data; they are not an independent assessment of population fit. In particular, estimating \(\hat{\eta}_i\) using the same observations being plotted can make individual predictions track those observations more closely.

A common observation model is \(y_{ij}=f_{ij}(\theta,\eta_i)+\epsilon_{ij}\), with \(E(\epsilon_{ij}\mid\eta_i)=0\) and conditional variance \(R_{ij}\). The variance may depend on the predicted response—for example, under a proportional error model—or differ across observations. Thus, raw residuals with the same magnitude need not represent the same degree of discrepancy.

Weighted Residuals and Their Assumptions

A weighted residual expresses a discrepancy relative to its expected variability. In the simplest case, if a prediction is treated as fixed and the residual has mean zero and known variance \(V_{ij}\), its standardized form is the raw residual divided by \(\sqrt{V_{ij}}\). The result is dimensionless. If the model and variance are correct, this standardized quantity has mean zero and variance one under those conditions.

The denominator must match the residual being standardized. For an individual weighted residual (IWRES), the numerator is typically \(y_{ij}-\mathrm{IPRED}_{ij}\), and the denominator is based on the observation-level residual variance at the individual prediction. Under an additive error model with standard deviation \(\sigma\), this denominator is \(\sigma\). Under a proportional error model with standard deviation \(\sigma f_{ij}\), it depends on the predicted response. The exact variance formula should follow the fitted observation model.

A population weighted residual (often abbreviated WRES) is intended to scale a discrepancy associated with population predictions. However, “WRES” is not a guarantee of one universal formula: implementations can differ in how they calculate the variance used for weighting, including their treatment of random effects and model linearization. Do not silently substitute the observation-level residual variance for a marginal predictive variance, or assume two packages’ WRES columns are interchangeable. Record the exact software-defined quantity and its calculation.

Conditional weighted residuals (CWRES) are designed to assess population-model discrepancies while accounting for the conditional structure induced by individual random effects. In nonlinear mixed-effects models their calculation generally uses a conditional approximation or linearization; it is not simply the individual raw residual divided by a residual standard deviation. Under a correctly specified model and the method’s approximation conditions, CWRES are intended to be approximately centered at zero and scaled to unit variance. That reference behavior is approximate, not a guarantee that every finite-sample set of CWRES will look normally distributed.

Scaling a Residual: A Short Justification

Suppose \(e=y-\hat{y}\), with \(E(e)=0\) and \(\operatorname{Var}(e)=V>0\), where the prediction is treated as fixed for this calculation. Define \(z=e/\sqrt{V}\). Then

This explains why standardization helps compare discrepancies across observations with different variances. It also shows the assumptions behind the simple result: the variance must be correctly specified, positive, and appropriate for the error being scaled; the residual must be centered under the model; and the prediction’s estimation must not invalidate the calculation. In a fitted mixed-effects model, predictions and individual effects are estimated, and observations within an individual may be conditionally related. The simple calculation therefore motivates weighting but does not establish that every software residual is exactly standard normal.

Which Residual Belongs in Which Display?

Choose the residual according to the question the display is designed to address, and label the prediction axis separately from the residual. A reliable diagnostic specification states the target (population or individual), residual definition, scaling rule, and any transformation or approximation.

These are choices of quantity, not claims that a particular plot has a universal interpretation. For example, plotting IWRES answers a different question from plotting CWRES even if the axes and units look identical. For an observed-versus-prediction display, show the observed values and the appropriate prediction directly; do not replace that display with a residual plot and call it equivalent.

Diagnostics, Failure Modes, and Alternatives

Before generating a standardized residual display, check that observations, predictions, and residuals refer to the same record and response scale. Confirm that the residual denominator is finite and positive, that the residual uses the intended prediction, and that missing or failed calculations have not been silently converted to zero. A reported residual with a missing prediction, a nonpositive variance, or an incompatible transformation is not interpretable as a valid standardized residual.

When population residuals are unexpectedly much smaller than individual residuals, or vice versa, first verify the definitions and joins rather than concluding that one model component has failed. Check whether one column is raw and another standardized, whether the denominator includes the intended error components, and whether one quantity was calculated on a transformed response scale. Record whether the residual is conditional or population-oriented, and whether its calculation depends on an approximation.

• Raw residuals retain the response units and are easy to interpret in magnitude, but are not directly comparable across observations with different residual variability.
• IWRES scale individual deviations using observation-level residual error and are useful for individual-fit assessment, but use predictions informed by the individual’s data.
• CWRES target population-model assessment with conditional structure accounted for approximately; their construction is more involved and implementation-dependent.
• WRES may be convenient when produced by the chosen software, but the name alone does not fully specify the calculation.

No standardized residual repairs a misspecified model, sparse information, or poor prediction support. Standardization can also make a plot look comparable while concealing that residuals were computed under different definitions or on different scales. For automated reports, store the residual type, prediction type, variance convention, software or calculation method, and response transformation with the diagnostic data. This makes subsequent comparisons reproducible and helps distinguish a change in model behavior from a change in residual calculation.

Common Mistake

Wrong reasoning: “IWRES and CWRES are both weighted residuals, so either can be used in any residual panel.” This ignores which prediction each quantity uses and what variability it represents.

Correction: Match the quantity to the diagnostic target. Use population predictions for population prediction displays, individual predictions for individual fit displays, and name the standardized residual explicitly. If the software’s WRES definition is not known, consult and record that definition rather than treating the abbreviation as self-explanatory.