Why Repeated Measurements Add Information
A concentration measured once describes exposure at one time for one participant. It cannot, by itself, show whether that participant’s concentration was rising, falling, or near a peak. Measurements at several times for the same participant can show that trajectory. They also allow comparisons of change within a participant, which can be more informative than comparing two concentrations from different participants.
This distinction matters because participants can have persistently different concentration levels. For example, one participant may have concentrations about twice those of another at several times, while both show a similar proportional decline. A comparison at one time mixes that persistent difference with the time-related change. A paired comparison within each participant can remove a stable level difference and focus on the change.
The earlier material on peak and trough samples concerns the information available at selected points in the dosing interval. Here the focus is on what is gained when observations are linked within participants: a visible time-course and information about within-participant variation. Repeated observations are not equivalent to observations from the same number of additional independent participants.
A Simple Model for Separating Levels and Changes
Concentrations are positive and participant differences are often naturally expressed as proportional differences. A simple way to describe these data is to work with log concentration. Let \(C_{ij}\) be the concentration for participant \(i\) at time \(t_{ij}\), and define \(y_{ij}=\ln(C_{ij})\). One illustrative model is
Here, \(\mu\) is a reference log concentration, \(f(t)\) is the common time-related pattern, \(b_i\) is participant \(i\)’s persistent deviation from that reference, and \(\epsilon_{ij}\) is the residual at that observation. This is a simplified model, not a full pharmacokinetic model: it does not prescribe the shape of \(f(t)\), and it assumes participants share that shape apart from a stable offset.
For the variance interpretation below, suppose \(b_i\) has mean zero and variance \(\tau^2\); residuals have mean zero and variance \(\sigma^2\); and participant effects and residuals are independent. In the simplest version, participant effects for different participants are independent, and residuals from different observations are also independent. Under these assumptions, two observations from the same participant share \(b_i\), whereas observations from different participants do not share that effect.
For two observations from the same participant, the model implies
The participant effect \(b_i\) cancels. This is the key reason paired measurements can clarify time-related change despite persistent differences in level. If the two residuals are independent with variance \(\sigma^2\), then the variance of their difference is
If residuals at the two times are correlated, the more general expression is \(2\sigma^2-2\operatorname{Cov}(\epsilon_{i1},\epsilon_{i2})\). Thus, pairing removes a stable participant offset, but it does not remove residual variation or all forms of within-participant dependence.
In this simple model, two measurements from the same participant at distinct times have covariance \(\tau^2\), because they share \(b_i\) and their residuals are independent. Measurements from different participants have covariance zero under the stated assumptions. This provides a precise meaning for persistent between-participant differences. It does not mean that every observed within-participant deviation is assay noise: true individual time-course differences, occasion effects, and timing or dose deviations can also contribute.
Worked Example: Paired Changes Remove a Stable Level Difference
Suppose three hypothetical participants have concentrations measured 2 hours and 8 hours after a dose. The values are in mg/L: participant A has 16 and 4, participant B has 32 and 8, and participant C has 8 and 2. Calculate each participant’s log-concentration change and compare it with the change in concentration.
For each participant, the log change is the log of the later concentration divided by the earlier concentration. For participant A,
The same calculation for B and C gives
All three participants have the same observed proportional change: each later concentration is one quarter of that participant’s earlier concentration. Their absolute concentrations differ twofold between A and B at both times, and twofold between C and A. The paired log changes remove those stable level differences and show the common proportional decline in these hypothetical data.
The elapsed time is \(8-2=6\) hours. The average log change per hour is \(-1.386/6\approx-0.2310\) per hour. This is a descriptive summary of the two observed times, not evidence by itself that the same rate applies at every intervening time or that a specific pharmacokinetic process has been established.
Repeated Measurements Show Time-Course Shape
A pair of measurements shows a net change between two times, but multiple times can reveal whether the concentration first rises and then falls, whether the decline appears steeper in one interval than another, or whether observations are inconsistent with a simple smooth pattern. The value of the series depends on the recorded times: a set of repeated measurements can only describe the portions of the time course it samples.
Worked Example: Detecting a Rise Followed by a Fall
Suppose a hypothetical participant has concentrations of 3, 8, 6, and 2 mg/L at 0.5, 1, 2, and 4 hours after a dose. Calculate the successive ratios and describe what the measurements show.
From 0.5 to 1 hour, the ratio is \(8/3\approx2.667\), so the observed concentration rises. From 1 to 2 hours, it is \(6/8=0.7500\), and from 2 to 4 hours it is \(2/6\approx0.3333\), so the observed concentration falls across both later intervals. Among the recorded observations, the highest concentration is 8 mg/L at 1 hour.
These repeated measurements show a rise followed by a fall in this participant’s observed time course. A single measurement at 2 hours would have shown 6 mg/L but would not have revealed that an earlier observed concentration was higher. The series does not establish the exact time or value of the true maximum between visits, because no measurement was taken at every possible time.
Repeated measurements can also help separate a common pattern from a persistent participant-level offset when several participants are measured over time. If concentrations are recorded at comparable times and on the same scale, the time pattern can be viewed alongside each participant’s trajectory. Participants who remain consistently higher or lower may differ in persistent level; participants whose trajectories change differently may show departures from the shared pattern. Those departures should not automatically be called measurement error.
Worked Example: Comparing a Paired Difference With a Cross-Participant Difference
Use the hypothetical 2-hour and 8-hour concentrations from the first example. Compare A’s change between 2 and 8 hours with the difference between A and B at 2 hours.
A’s paired log change is \(\ln(4/16)=\ln(1/4)\approx-1.386\). It compares A with A and therefore cancels A’s stable log-level effect under the simple model.
At 2 hours, the log difference between A and B, expressed as A minus B, is
This cross-participant contrast says A’s concentration is half B’s concentration at that time. It does not isolate a time-related change: it reflects the difference between two participants at one time. In these particular data, the 8-hour contrast is also \(\ln(4/8)=\ln(1/2)\), consistent with a persistent level difference and the same proportional change in both participants. With only one time point, that persistence could not be checked.
What Repeated Samples Can and Cannot Separate
Repeated observations are useful because their structure carries information. A paired difference removes a stable participant offset under the model above. Multiple participants with repeated observations can show whether a pattern is shared or whether participants differ in level. However, a series does not by itself identify every source of variation. In particular, a residual at one time can combine assay error, transient biological variation, imperfectly recorded timing, dose deviations, and a mismatch between the assumed pattern and the actual trajectory.
Separating these sources requires assumptions and appropriate replication. For example, the simple covariance result relies on a stable participant effect and independent residuals. If residuals are serially correlated, the observed similarity between two samples may reflect both the participant effect and residual correlation. If participants have different time-course shapes, a single stable offset is not enough to describe them. And if every participant is measured at different times, participant differences may be difficult to distinguish from differences associated with sampling time.
The main alternatives answer different questions:
| Approach | What it can show | Main limitation |
|---|---|---|
| One sample per participant | Between-participant concentration variation at the sampled times | Cannot show each participant’s change over time |
| Paired samples per participant | Within-participant change between two times; stable offsets cancel in a paired contrast | Shows net change only between those times, not the intervening shape |
| Several samples per participant | More detail about the observed trajectory and repeated within-participant departures | Does not automatically distinguish residual noise from individual time-course differences |
The comparison is not a claim that one approach is always preferable. Repeated samples provide within-participant information, but the interpretation depends on the times, the consistency of dosing and observation, and the assumptions used to describe dependence. The design question of how to distribute samples across participants is separate from the question of what repeated measurements can reveal.
Diagnostics and Common Pitfalls
Before interpreting a repeated series, check that each concentration is linked to the correct participant and that actual sample times and dose times are available. A nominal time can be misleading if a sample was collected substantially early or late. Also inspect whether dose amounts, dosing conditions, or relevant occasions differ across the observations being compared. An apparent trajectory change may reflect a changed dose or a timing discrepancy rather than time alone.
Plotting each participant’s concentrations against actual time can reveal patterns that a single pooled summary hides. Look for consistently higher or lower trajectories, unexpected reversals, isolated extreme observations, and missing time points. If a log scale is used, concentrations must be positive; zero or below-quantification observations need a treatment appropriate to the analysis and cannot simply be logged. A smooth-looking plot does not prove that the assumed model is correct, but visible irregularities can prompt closer review.
- Mistake: treating every sample as an independent participant. Two measurements from one person share participant characteristics, so counting them as independent evidence can overstate how many independent people were observed. Keep participant identity attached to every sample.
- Mistake: calling all within-participant variation assay noise. Residual variation may include biological changes, timing errors, dose differences, or model mismatch. Use “residual” unless the source has been isolated.
- Mistake: assuming two points reveal the full curve. Two measurements establish a difference between those times, not the path between them. More time points can reveal more shape, but only over the sampled interval.
- Mistake: interpreting a paired contrast as a causal effect. Pairing controls stable participant-level differences in the comparison, but it does not control time-varying factors or establish that time alone caused the observed change.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A participant’s concentrations are 20 mg/L at 3 hours and 5 mg/L at 9 hours. What is the log-concentration change, and what does it say about the proportional change?
- In the model \(y_{ij}=\mu+f(t_{ij})+b_i+\epsilon_{ij}\), which term cancels in a within-participant difference, and what does not necessarily cancel?
- Two participants have concentrations of 10 and 20 mg/L at one time. What can the comparison establish, and what can it not establish?
- Why might repeated observations from one participant be more similar than observations from different participants under the simple model?
Solutions. (1) The change is \(\ln(5/20)=\ln(1/4)\approx-1.386\). The later concentration is one quarter of the earlier concentration, a 75% decrease. (2) The stable participant effect \(b_i\) cancels; the time-related difference \(f(t_{i2})-f(t_{i1})\) and residual difference \(\epsilon_{i2}-\epsilon_{i1}\) remain. (3) The observed concentrations differ twofold at that time, but one cross-participant comparison cannot show either participant’s time course or whether the difference persists. (4) The observations share the participant effect \(b_i\), which creates covariance \(\tau^2\) under the stated assumptions; independent residuals do not remove that shared component.