Population Predictions, Individual Predictions, And Residuals

Why One Observation Can Have More Than One Residual

A concentration measurement does not have a unique residual until the prediction used as its reference is specified. A prediction based on the typical subject in the population model answers a different question from one that incorporates an estimate of the subject’s individual characteristics. Both can be compared with the same observed concentration, so they generally produce different residuals.

This distinction matters when assessing a population PK model. A population prediction shows how the model performs without using a subject-specific random-effect estimate. An individual prediction incorporates information attributed to that subject and therefore describes how closely the model can represent the subject after that information is used. The comparisons are related, but they are not interchangeable measures of performance.

What the Two Predictions Represent

In a population PK model, \(\theta\) describes the population-level structural parameters, while \(\eta_i\) represents subject-level deviations under the model’s random-effects specification. The population prediction sets those deviations to zero while retaining the subject’s known covariates, dose, and sampling time. It is therefore a prediction for a typical subject with those covariates and dosing conditions—not necessarily for a subject who is typical in every respect.

The individual prediction substitutes an estimate \(\hat{\eta}_i\), commonly obtained as an empirical Bayes estimate, into the same structural model. That estimate reflects the model and the subject’s observed data. Some estimation approaches report a conditional mean and others a conditional mode; the precise estimate depends on the model-fitting method. In either case, \(IPRED\) is a subject-specific, data-informed prediction.

A population prediction with \(\eta_i=0\) is also not generally the same as the population-marginal mean prediction. The marginal mean averages predictions over the distribution of random effects. For a nonlinear function \(f\), evaluating at the mean random effect need not equal averaging over that distribution: \(f(\theta,0)\) is not generally \(E_{\eta}[f(\theta,\eta)]\). Thus, “population prediction” here means the conventional typical-subject prediction, not an automatic integration over all subjects.

These definitions assume the model, dosing history, covariates, time, and observation being compared are aligned. They also assume that each prediction and its residual are expressed on the same scale. If the model describes log concentrations, a residual on the log scale is \(\log(y_{ij})-\log(\widehat{y}_{ij})\), not \(y_{ij}-\log(\widehat{y}_{ij})\). The latter subtracts quantities with incompatible scales.

Residual Sign and the Difference Between Residuals

A raw residual is a signed difference in the measurement’s units. A positive observed-minus-predicted residual means the observation exceeds the prediction, so the model prediction is below the observed concentration for that record. A negative residual means the prediction exceeds the observation. A residual of zero means the prediction matches the observation exactly on the scale used.

The two residuals for one observation are connected by a simple identity. Subtracting the individual residual from the population residual gives:

Equivalently, the individual residual equals the population residual minus the change in prediction from population to individual:

This is an algebraic identity, not an assumption about how well either prediction performs. If the individual prediction moves toward the observed value, the magnitude of its residual decreases. If it moves away, the magnitude increases. The identity also shows why an individual residual cannot be interpreted without knowing which prediction was used.

Worked Calculations Across Records

For each record, calculate the residual using the observation and prediction that belong to the same subject, time, and scale. The following examples use hypothetical concentration data. They illustrate arithmetic and interpretation, not results from a real study.

Assumptions, Diagnostics, and Common Pitfalls

Calculating observed-minus-predicted residuals requires little beyond correctly matching the record to its prediction. Interpreting a pattern of residuals, however, depends on the model and the diagnostic purpose. Before interpreting them, check that the prediction uses the intended dose history, covariates, sampling time, and parameter estimates. Confirm that the observed and predicted values are on the same scale and that units match. A mismatch in any of these can produce apparent discrepancies that are calculation or data-alignment errors rather than evidence about model fit.

Raw residuals retain the original measurement units and may have different variability at different concentration levels. A residual of \(2\) mg/L has a different relative size when the prediction is \(3\) mg/L than when it is \(100\) mg/L. If the model is defined on a transformed scale, the corresponding residual should be calculated on that scale for analyses that require it. Standardized or weighted residuals are different quantities: they account for an error-scale model and should not be confused with the raw differences defined here.

Population and individual residuals also answer different diagnostic questions. Population residuals compare observations with predictions made without subject-specific random-effect estimates. Individual residuals show fit after incorporating estimates informed by a subject’s data. Because those estimates use the observations, individual residuals may appear smaller even when population-level predictions are not correspondingly improved. With limited information per subject, individual estimates may also be pulled toward the population value, so a small or large difference between prediction types should not be read as a direct measure of the amount of information in the data.

A frequent mistake is to call every observed-minus-predicted difference “the residual” without naming its prediction source. That hides whether the comparison is to \(PRED\) or \(IPRED\), which can change the magnitude and even the sign. Another common error is reversing the subtraction: predicted minus observed has the opposite sign. State the definition explicitly and retain the units.