Why Look at Diagnostics One Subject at a Time?
A pooled diagnostic plot can summarize a model’s overall behavior, but it can also hide a pattern that repeats within particular subjects. In a repeated-dose study, each subject contributes observations at multiple times and often across multiple dose occasions. Keeping those observations in sequence helps answer a distinct question: does the model miss in a consistent way for an individual, and does that pattern recur after dosing?
A subject-level diagnostic panel is a set of aligned plots for one subject, or a small group of subjects, showing observed concentrations, model predictions, and residuals against an appropriate time axis. These panels are descriptive tools. A pattern can flag a model or data issue for investigation, but a plot alone does not establish its cause.
The chosen prediction must be stated. Residuals from population predictions and residuals from individual predictions answer different diagnostic questions; they should not be mixed in one series or treated as interchangeable. Weighted residuals may also be displayed when their definition and scale are appropriate for the fitted model. The panels do not replace the earlier material on residual trends across time, residuals versus predictions, or the choice of time since dose versus clock time. They apply those diagnostic ideas while retaining subject identity.
Building a Panel That Preserves the Study Sequence
A useful panel makes it easy to follow what happened to one subject over time. For a repeated-dose study, mark dose occasions and display observations in their recorded chronological order. If a concentration profile is expected to repeat at comparable times after each dose, time since dose can support comparisons across occasions; if the question concerns accumulation or change over calendar time, clock time or dose-occasion order may be more informative. The axis should represent the actual sampling and dosing record, not merely the planned schedule.
A typical panel has a concentration plot and a residual plot. The concentration plot shows observed concentrations and the selected prediction, with a clear legend. The residual plot shows the corresponding residuals against the same time coordinate and a horizontal zero reference. Use consistent axes when comparing subjects, and state when different axis limits are necessary. A shared scale makes the size of differences comparable; a very wide shared scale, however, can make small but recurrent deviations hard to see.
Before interpreting any pattern, check that the panel’s records are aligned correctly. Confirm subject identifiers, concentration units, dose times, actual sample times, prediction records, and the definition of each residual. Missing observations should remain missing rather than being drawn as if they were measured. Connecting points across long gaps can suggest a continuous trajectory that was not observed, so use connecting lines only when they clarify the sequence without implying data between samples.
Retain actual concentration measurements, corresponding predictions, dose times, and actual sample times. Verify units and identifiers before plotting.
Use a coordinate suited to the question, such as time since dose for comparing within-interval profiles or clock time for viewing change across the study.
Plot observed concentrations with the selected prediction, then show residuals from that same prediction against the corresponding time points.
Look for a similar direction or shape across occasions, then check whether other subjects, dose records, or sampling features support that interpretation.
Worked Examples
Worked Example: A Late-Interval Pattern Repeats Across Dose Occasions
The following hypothetical data are from one subject in a repeated-dose study. Concentrations are in mg/L, and samples were collected at the listed times after each dose. The table shows individual predictions and raw residuals defined as observed concentration minus individual prediction.
| Dose occasion | Time after dose (h) | Observed (mg/L) | Individual prediction (mg/L) | Residual (mg/L) |
|---|---|---|---|---|
| 1 | 1 | 9.8 | 10.0 | -0.2 |
| 1 | 4 | 6.6 | 7.0 | -0.4 |
| 1 | 8 | 3.2 | 4.0 | -0.8 |
| 1 | 12 | 1.0 | 2.0 | -1.0 |
| 2 | 1 | 9.9 | 10.0 | -0.1 |
| 2 | 4 | 6.5 | 7.0 | -0.5 |
| 2 | 8 | 3.3 | 4.0 | -0.7 |
| 2 | 12 | 0.9 | 2.0 | -1.1 |
For example, at 8 h after the first dose, \(e_{ij}=3.2-4.0=-0.8\) mg/L. At 12 h after the second dose, the residual is \(0.9-2.0=-1.1\) mg/L. Each occasion shows increasingly negative residuals later in the interval: the observed concentrations are below the individual predictions, with the largest discrepancies at 12 h.
A panel makes this recurring shape visible in both the prediction overlay and the residual plot. It suggests investigating whether the model’s within-interval profile, the recorded dose or sample times, or another relevant feature is contributing to the late-interval mismatch. It does not by itself identify which explanation is correct. This interpretation assumes that dose and sampling times are recorded accurately, the observations and predictions are correctly matched, and the displayed residuals use the same prediction definition throughout.
Worked Example: Opposing Subject Patterns Cancel in a Pooled Plot
Suppose four hypothetical subjects each have two residuals at the same two post-dose sampling times. The residuals, in mg/L, are:
| Subject | First time | Second time | Subject mean |
|---|---|---|---|
| A | 1.0 | 1.0 | 1.0 |
| B | 1.0 | 1.0 | 1.0 |
| C | -1.0 | -1.0 | -1.0 |
| D | -1.0 | -1.0 | -1.0 |
At the first time, the pooled mean residual is \((1.0+1.0-1.0-1.0)/4=0\) mg/L. The same calculation at the second time also gives \(0\) mg/L. Yet subjects A and B have positive residuals at both times, while C and D have negative residuals at both times. A pooled mean near zero therefore does not show that every subject is well described.
More generally, if \(e_{ij}\) is the residual for subject \(i\) at a matched time \(j\), then the pooled mean at that time is
Here \(N_j\) is the number of subjects with a residual at that time. Positive and negative subject residuals can cancel in this sum even when each subject has a consistent pattern. Subject panels retain the index \(i\), so that cancellation no longer hides which individuals tend to lie above or below their predictions. This reasoning assumes the compared residuals refer to a meaningful common time or time stratum; pooling unmatched times can obscure patterns for additional reasons.
Worked Example: An Apparent Outlier Caused by the Wrong Time Coordinate
A hypothetical subject was expected to have a concentration of 4.0 mg/L at 8 h after a dose and was observed at 2.6 mg/L. If the record is incorrectly plotted as an 8 h sample, the raw residual is \(2.6-4.0=-1.4\) mg/L, which looks like a substantial discrepancy.
Review of the hypothetical dosing record shows that the sample was actually collected 10 h after the dose. At 10 h, the corresponding model prediction is 2.5 mg/L. Using the actual sample time gives \(2.6-2.5=0.1\) mg/L. The residual is small and positive, rather than large and negative. The concentration measurement did not change; the time used to select and display its prediction did.
A panel that marks actual dose and sample times can reveal this mismatch, particularly when a point appears isolated from the subject’s other dose occasions. Before treating an apparent outlier as evidence of model failure, check whether the time coordinate reflects the actual record and whether the prediction was evaluated at that same time. This example assumes the stated times and predictions are accurate; in practice, the record should be verified rather than corrected by visual guesswork.
Interpreting Patterns and Checking Failure Modes
A repeated pattern is more informative than a single unusual point, but recurrence is not proof of a particular mechanism. Similar residual shapes across dose occasions can motivate examination of time dependence, dose history, covariates, or the observation record. A pattern limited to one subject can also reflect a subject-specific feature, a data issue, or an unstable estimate. Use the panel to define a question for further checking rather than to make a causal claim.
Several practical checks help distinguish a meaningful pattern from a plotting artifact:
- Confirm the time alignment. A dose-time error or a plot against planned rather than actual sample times can create apparent residual trends.
- Check residual definition and units. A raw residual in mg/L should not be interpreted as if it were a standardized residual. Keep prediction type and residual type explicit.
- Look across occasions and subjects. A shape that recurs within a subject is different from a single large residual; a pattern shared across subjects may point to a broader model feature.
- Review missingness and sparse sampling. Few samples may not establish the shape of a profile. A line between two distant observations does not supply the unobserved concentrations between them.
- Compare scales carefully. Shared axes support comparisons across panels, but excessive range can hide small deviations. If axes differ, label that clearly and avoid comparing visual magnitudes directly.
Panels are most useful alongside, not instead of, other diagnostic views. Pooled plots provide an efficient overview and can show broad population-level trends; subject panels reveal how those trends are distributed across individuals and dose occasions. Numerical summaries, such as a subject’s mean residual, are compact but can hide a change in direction over time. Conversely, panels can be visually busy when there are many subjects or observations. Selecting a manageable set of subjects for review should not mean showing only the most striking cases; the selection rule should be transparent.
Individual predictions are informed by subject-level data, so a close overlay of observations and individual predictions should not be treated as independent confirmation that the population model is adequate. Population predictions and individual predictions serve different diagnostic purposes. A complete assessment uses the prediction type appropriate to the question and checks the same records from more than one useful view.
Check Your Understanding
Answer each question using the definitions and reasoning in this tutorial. The solutions follow the questions.
- A subject has residuals of \(-0.3\), \(-0.6\), and \(-0.9\) mg/L at successively later times after a dose. What pattern does this suggest, and what can the panel not establish by itself?
- At one matched time, four subjects have residuals \(2\), \(1\), \(-1\), and \(-2\) mg/L. Calculate the pooled mean and explain why it is not a complete description.
- A residual is defined as observed concentration minus prediction. If the observed concentration is 5.4 mg/L and the selected prediction is 6.0 mg/L, calculate the residual and interpret its sign.
- Why should actual sample times be used when matching observations to predictions in a repeated-dose panel?
- Name one strength of subject-level panels and one limitation relative to pooled diagnostic plots.
Solutions. 1. The residuals become progressively more negative, suggesting that observations fall increasingly below predictions later in the interval. The panel alone cannot determine the cause. 2. The mean is \((2+1-1-2)/4=0\) mg/L. Opposing residuals cancel, even though individual subjects have nonzero residuals. 3. The residual is \(5.4-6.0=-0.6\) mg/L; the observation is 0.6 mg/L below the prediction. 4. The prediction must correspond to the time the sample was actually collected; otherwise a timing mismatch can create an apparent residual pattern. 5. A panel preserves subject identity and repeated patterns; unlike a pooled plot, it can become visually unwieldy when many subjects and observations are shown.