Diagnosing Time-Structured Residuals

Why Residual Patterns Matter

An observed-predicted plot can show whether predictions tend to lie above or below measurements, but it may hide when those differences occur. A model can match an overall summary while predicting the peak too early, declining too quickly at late times, or describing the early distribution phase poorly. For held-out concentration-time data, plotting residuals against time helps expose these structured discrepancies without refitting the model to the validation observations.

A residual is a signed difference between an observation and its corresponding fixed prediction. The sign convention must be stated because reversing it reverses every visual interpretation. Residual plots are diagnostic displays, not automatic identification of a unique mechanism: several model or data features can produce similar patterns.

Raw residuals express absolute concentration differences: a residual of \(1\ \mathrm{mg/L}\) has the same vertical distance wherever it occurs. Log residuals express relative differences: \(r_i=0\) means exact agreement, \(r_i>0\) means the observation exceeds the prediction, and \(r_i<0\) means it is lower. For example, \(r_i=\ln(2)\) means the observation is twice the prediction. Log residuals require positive concentrations; they are not defined for zero or negative values.

The basic diagnostic uses residual on the vertical axis and time on the horizontal axis, retaining the recorded units and sampling times. A second useful display places residual against predicted concentration, or shows residuals across concentration ranges. That view can reveal whether discrepancy changes with concentration magnitude. Neither plot should be interpreted in isolation: time and concentration often change together over a profile.

Assumptions and Reading the Displays

These diagnostics are most interpretable when observations are correctly matched to the prediction for the same participant, dose, study, sampling time, and concentration units. Predictions must remain those generated for independent validation; adjusting them after viewing residuals turns a diagnostic exercise into calibration on the held-out data. The comparison also presumes that recorded times and concentrations are sufficiently reliable for the question being asked.

A residual plot does not require residuals to be statistically independent in order to display their pattern. But serial observations from one participant are not automatically independent replications. A run of residuals with the same sign may reflect a coherent within-profile discrepancy; counting every sample as an independent confirmation would exaggerate the evidence. Overlaying participant-specific profiles, where available, helps show whether a pattern recurs across participants or is driven by one profile.

Use time relative to a clearly defined event, such as dose administration, and examine the phases relevant to the prediction question. Sparse sampling can miss the true peak or the transition into a terminal phase. The apparent time of the largest measured concentration is therefore not necessarily the continuous-time peak. Interpret visual patterns as evidence about where and how predictions diverge, not as proof of a specific parameter error.

Terminal Slope and Distribution-Phase Patterns

For a terminal phase that is approximately log-linear, a change in the slope of log concentration over time appears as a systematic trend in log residuals. The connection follows directly from the definition. Since \(r(t)=\ln C_{\mathrm{obs}}(t)-\ln C_{\mathrm{pred}}(t)\), differentiating over a time interval where both profiles are positive and smooth gives:

Thus, if the observed log concentration declines less steeply than the predicted log concentration, the observed log-slope is less negative, \(r(t)\) tends to increase with time, and late observations tend to lie progressively above predictions. The reverse pattern is consistent with predictions declining less steeply than observations. This interpretation concerns the slopes of the compared profiles; it does not by itself identify an elimination parameter or prove that the sampled interval represents a single terminal process.

A distribution-phase mismatch often appears as a transient pattern: residuals change systematically during an early post-peak interval and then move closer to zero later. For example, predictions may be too high during an early decline but align at later times, or the reverse. Such a localized pattern differs visually from a persistent late residual trend, but its timing alone cannot separate distribution from absorption, sampling, or other sources of model-data mismatch.

Using Concentration Plots and Avoiding Misdiagnosis

Residuals plotted against predicted concentration can help distinguish a roughly constant absolute discrepancy from one that changes across concentration magnitudes. If raw residuals spread or shift with predicted concentration, the mismatch may depend on concentration level. Log residuals instead emphasize proportional differences. For instance, the same ratio of observed to predicted concentration produces the same log residual at low and high concentrations, even though the corresponding raw differences are different.

Concentration and time are often linked: concentrations commonly rise and then fall as time advances. A pattern against concentration may therefore be another view of a time-dependent pattern, not independent evidence of a concentration-dependent mechanism. Compare both displays and, when useful, inspect early, peak, and late portions separately. If participants have different concentration ranges or sampling schedules, a pooled display can also conceal opposing individual patterns.

Common failure modes include treating every residual as independent, interpreting a few noisy points as a smooth slope, and inferring a unique physiological cause from a visual signature. A sign change near the peak is not enough to locate peak-time error unless sampling is adequate on both sides. Likewise, an increasing late log-residual trend is informative only if the interval being examined plausibly represents the phase whose slope is being compared. Check units, time alignment, participant-level profiles, and the actual sampling schedule before assigning meaning to a pattern.

Observed-predicted plots and summary endpoints remain useful complements: they answer different questions from time-residual plots. A summary can agree while a phase-specific pattern remains, and a time plot can reveal that pattern without reducing the entire profile to one number. These diagnostics do not establish a pass-or-fail threshold, quantify predictive error, or replace the prespecified validation question. Their value is in locating and describing model-data disagreement while preserving the independence of the held-out predictions.