Why the Time Axis Changes the Diagnostic Question
A residual plotted against time answers a question only after the horizontal axis has been defined. In a population pharmacokinetic analysis, two common choices are time since the most recent dose and study clock time. The first aligns observations by their position within a dosing interval; the second preserves their position in the study timeline. Neither is universally preferable. A pattern may be visible on one axis and blurred on the other because each preserves different information.
Use the same residual definition throughout a comparison. For a signed residual based on observed minus predicted concentration, a positive value indicates that the observation is above its prediction and a negative value indicates that it is below. Weighted residuals also reflect the model’s error scale. A trend in either kind of residual is a diagnostic signal to investigate, not by itself a diagnosis of why the model is wrong.
A predose observation before the first administered dose therefore has no postdose time since dose. It can still be plotted against clock time, or it can be identified separately as predose. If an analysis assigns a special value or category to such observations, that convention must be stated; it is not the same as a mathematically defined time since the most recent dose.
What Each Axis Preserves
Time since dose is useful for asking whether residuals vary with postdose phase. For example, residuals that are systematically positive shortly after dosing and negative later may indicate a mismatch in the model’s description of the concentration-time profile. This axis is most directly interpretable when dose times are known, dose amounts and intervals are comparable, and prior dosing histories do not make observations at the same postdose time fundamentally different.
Clock time preserves the order and timing of observations across the study. It can make patterns associated with accumulation, changes in regimen, study occasion, or other time-linked processes easier to notice. But observations at the same clock time can be at very different positions in their dosing intervals, especially when subjects have different dose schedules or dose times. A clock-time pattern can therefore combine observations with different expected postdose profiles.
For a regular regimen with interval \(\Delta\), a first dose at time \(a\), and observations after that dose, the dose times are \(a, a+\Delta, a+2\Delta,\ldots\). For an observation at time \(t\), the interval number and time since the latest dose are
Thus clock time determines both the dosing-interval number \(n\) and the within-interval time \(\tau\) in this synchronized, regular example. Plotting against \(\tau\) alone discards \(n\). That matters because the concentration at a given \(\tau\) can differ between early doses and later doses as drug accumulates. The following derivation makes the loss of information explicit: two observations can have the same \(\tau\) while their \(n\) values differ, so the same horizontal coordinate on a postdose plot does not imply the same dosing history.
A postdose plot is most straightforward to interpret when the regimen is stable and observations are at comparable accumulation states, such as after a common steady-state regimen. A clock-time plot is especially useful for examining whether the residual pattern changes over the study timeline. In practice, viewing both plots can help distinguish a within-interval pattern from a pattern that develops across intervals.
Worked Example: A Postdose Plot Can Hide Accumulation
Suppose subjects receive hypothetical doses every 12 hours, beginning at elapsed study time 0 hours. For one subject, observations are collected at 10, 58, and 106 hours. The latest preceding doses occurred at 0, 48, and 96 hours, respectively. The same signed residual measure gives values \(+0.1\), \(+0.8\), and \(+1.5\).
For the first observation, \(\tau=10-0=10\) hours. For the second, \(\tau=58-48=10\) hours. For the third, \(\tau=106-96=10\) hours. All three observations therefore appear at 10 hours on a time-since-dose plot, even though they occur in different dosing intervals.
On a clock-time plot, the residuals increase from \(+0.1\) to \(+0.8\) to \(+1.5\) over 96 hours. Because positive residuals indicate observations above predictions under the stated convention, this hypothetical pattern is consistent with increasing underprediction over the study timeline. If the model’s dosing history or accumulation description were incomplete, that could be one explanation to investigate. The residuals alone do not establish that explanation: other changes over time or data problems could also produce a pattern.
On the postdose plot, the three points share the same horizontal coordinate. Their residual differences show that matching the latest-dose phase has not made their histories equivalent. A useful diagnostic display would retain both views, with the clock-time plot revealing change across intervals and the postdose plot showing that the observations occurred at the same phase after their latest dose.
Recording Conventions and Dose Histories
The calculation is only as reliable as the event times used in it. Use the recorded administered dose times that correspond to the analysis data, rather than silently substituting scheduled times. A missed, delayed, or extra dose changes the most recent administered dose and can change \(\tau\). Likewise, a dosing record entered at a nominal time may not represent the actual administration time. When the distinction is material, investigate it and state which timestamp convention is used.
For an infusion, a single “time since dose” can be ambiguous unless the reference point is specified. Depending on the diagnostic question and the model’s event-time convention, a plot may use time since infusion start or time since infusion end; during infusion, it may be more informative to mark the infusion phase explicitly. Do not switch conventions between observations without explanation. A postdose time axis also does not encode dose amount, infusion duration, or earlier doses. Those parts of the event history may be essential to interpreting the model predictions.
Worked Example: Actual and Scheduled Dose Times Give Different Axes
Consider a hypothetical observation recorded at 14:00. The dose was scheduled for 08:00 but was actually administered at 09:30. Assume the subject had no later dose before the observation, and use elapsed hours for the arithmetic.
Using the scheduled time gives \(14{:}00-08{:}00=6.0\) hours since dose. Using the actual administration time gives \(14{:}00-09{:}30=4.5\) hours. The difference is \(6.0-4.5=1.5\) hours. A residual plotted at 6.0 hours using the schedule would be placed at the wrong postdose phase if the actual administration time is the relevant event time.
The clock-time coordinate remains 14:00, or the corresponding elapsed study time, under either calculation. Thus, a clock-time plot can preserve the observation’s study position while the postdose coordinate changes with the dose-recording convention. If the model itself used scheduled rather than actual dose time, the discrepancy also raises a model-input question; changing the plotting coordinate alone does not correct the model or the data.
If an observation occurred before this subject’s first administered dose, there would be no dose at or before that observation, so \(\tau\) would be undefined. It could be shown at its clock time and labelled predose, but it must not be assigned a postdose time by subtracting a nonexistent prior dose.
Reading the Two Plots Together
The choice of axis should follow the question, and comparing both can clarify what a pattern means. A consistent residual pattern across \(\tau\) values suggests an association with postdose phase, but the interpretation is stronger when dose amounts, prior histories, and study conditions are comparable. A trend over clock time suggests that residual behavior changes across the study timeline, but may reflect differing postdose phases if dosing is not synchronized. Neither plot, on its own, separates all possible causes.
| Diagnostic question | Useful first view | Important qualification |
|---|---|---|
| Do residuals vary within the interval after a dose? | Residuals versus time since the most recent dose | Check whether dose histories and accumulation states are comparable. |
| Do residuals change as the study progresses? | Residuals versus elapsed study clock time | Check whether observations at a given clock time occupy different dose phases. |
| Could a pattern reflect both dose phase and study period? | Compare both axes, retaining the event history | A visual pattern is a clue, not evidence of a unique cause. |
A practical diagnostic sequence is to verify event timestamps first, then identify what each axis represents, and only then interpret the residual pattern. If residuals differ at the same \(\tau\) across intervals, examine whether accumulation, dose changes, or other history-dependent factors differ. If residuals differ at the same clock time across subjects, examine whether their dose phases differ. These checks prevent an axis choice from concealing the very feature the diagnostic is intended to reveal.
Confirm the study origin, observation times, dose times, and whether timestamps represent scheduled or administered events.
Use time since dose to inspect postdose phase; use clock time to inspect change over the study timeline.
Time since dose omits interval number and earlier dose history; clock time does not directly align observations by dose phase.
Compare both views where relevant and investigate plausible explanations against the dosing and study records.
Common Pitfalls and Limitations
- Treating the latest dose as the whole history. Two observations can have identical time since the latest dose but different prior doses and accumulation. The correction is to retain the full dose history when interpreting the pattern.
- Using clock time as though it were postdose time. A shared clock-time coordinate does not mean observations have the same position in their dosing intervals. Check each observation’s dose phase.
- Computing a time since dose for predose observations. If no administered dose occurred at or before the observation, the maximum in the definition does not exist. Mark time since dose undefined or use an explicitly stated predose category; do not invent a value.
- Assuming a visible trend identifies its cause. Dose timing, accumulation, protocol changes, data-recording issues, or other time-linked factors may all be relevant. A residual plot indicates where to investigate, not which explanation is correct.
- Changing the timestamp convention without notice. Scheduled and actual times, or infusion start and end times, can produce different coordinates. State the convention and use it consistently.
These plots are descriptive diagnostics. Uneven sampling can make parts of a timeline sparse, and a broad range of histories can be collapsed onto one postdose coordinate. Conversely, a clock-time display can mix dose phases. For that reason, neither axis should be treated as a replacement for checking the dosing records and the intended model inputs.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A subject receives doses at elapsed times 0, 12, and 24 hours. An observation occurs at hour 29. What is the time since the most recent dose?
- An observation occurs before a subject’s first administered dose. Is time since the most recent dose defined by the formula in this tutorial? What can be plotted instead?
- Two residuals occur 3 hours after a dose, one after the first dose and one after the sixth dose. Why might their postdose times match while their expected concentrations differ?
- Why can residuals at the same clock time have different interpretations for two subjects with different dose schedules?
Solutions. (1) The latest dose at or before hour 29 occurred at hour 24, so \(\tau=29-24=5\) hours. (2) No: the set of administered doses at or before the observation is empty, so the maximum and \(\tau\) are undefined. Plot against clock time or label the observation as predose. (3) Both have \(\tau=3\) hours, but the first and sixth doses occur at different points in the dosing history; accumulation or other history-dependent effects can make the expected concentrations differ. (4) The subjects may be at different postdose phases at that clock time, so a clock-time comparison can combine different positions in their dosing intervals.