When a Pooled Curve Joins Different Subjects
A concentration-time curve is often drawn by averaging all observed concentrations at each time. When most subjects contribute measurements across the full interval, those averages can be compared with within-subject patterns. But with sparse sampling, one group of subjects may supply the early-time average and another group the late-time average. The joined points then describe a sequence of observed group averages, not necessarily the trajectory followed by any individual.
This distinction matters because concentrations vary between subjects. If the subjects observed at later times tend to have higher concentrations for reasons unrelated to time, a pooled curve can flatten or rise even while individual subjects’ concentrations decline. Sparse sampling does not automatically make a pooled mean biased: under suitable representative sampling, each time-specific mean can estimate the population mean at that time. It does, however, make it harder to learn how concentrations change within the same subjects.
A simple way to expose the issue is to represent concentration as a subject-specific level plus a shared time trend:
Here \(a_i\) is subject \(i\)’s level at time zero and \(\beta\) is the change in concentration per unit time. If the observed subjects differ across times, the pooled mean depends on both the time and the average \(a_i\) among the subjects observed then:
Thus, comparing two pooled means combines the time trend with a possible change in the average subject level. The derivation is direct: averaging \(Y_i(t)=a_i+\beta t\) over the \(n_t\) subjects observed at time \(t\) gives \(\bar{Y}_{\text{pool}}(t)=\sum_{i\in S_t}a_i/n_t+\beta t\), where \(S_t\) is the set of subjects observed then. If the sets \(S_t\) differ, their average levels need not match.
Worked Example: A Rising Pooled Curve From Declining Subjects
Suppose four hypothetical subjects each decline at \(2\) mg/L per hour, but their starting concentrations differ. Only A and B are sampled at time zero; only C and D are sampled at hour four. The underlying profiles follow \(Y_i(t)=a_i-2t\).
| Subject | Starting concentration \(a_i\) (mg/L) | Observed time (h) | Observed concentration (mg/L) |
|---|---|---|---|
| A | 30 | 0 | 30 |
| B | 32 | 0 | 32 |
| C | 50 | 4 | 42 |
| D | 52 | 4 | 44 |
At time zero, the pooled mean is \((30+32)/2=31\) mg/L. At hour four, it is \((42+44)/2=43\) mg/L. The plotted pooled mean therefore rises by \(43-31=12\) mg/L over four hours.
Yet every subject’s underlying change over four hours is \(-2\times4=-8\) mg/L. The apparent rise occurs because the later observations come from subjects with higher starting concentrations. Since no subject was measured at both times, the observed data alone cannot reveal those individual declines. The example’s full profiles are known only to demonstrate how the misleading pooled pattern can arise.
Separating Time Change From Composition Change
Under the simple model above, for two times \(t_0\) and \(t_1\), the difference in pooled means is:
The first term is the shared time trend. The second is the composition difference: how much higher or lower the average starting level is in the later-time sample. This identity does not require that the observed pooled change equal the within-subject change. It shows exactly why the two can differ in this model.
Worked Example: Calculating the Composition Contribution
Consider four hypothetical subjects with starting concentrations \(a_A=10\), \(a_B=20\), \(a_C=30\), and \(a_D=40\) mg/L. Each declines at \(1\) mg/L per hour, so \(Y_i(t)=a_i-t\). A, B, and C are observed at time zero; B, C, and D are observed at hour four.
At time zero, the pooled mean is \((10+20+30)/3=20\) mg/L. At hour four, the observed concentrations are \(16\), \(26\), and \(36\) mg/L, so the pooled mean is \((16+26+36)/3=26\) mg/L. The pooled change is \(26-20=6\) mg/L.
The time contribution is \(\beta(t_1-t_0)=(-1)(4-0)=-4\) mg/L. The average starting level among the later-time subjects is \((20+30+40)/3=30\) mg/L; among the time-zero subjects it is \((10+20+30)/3=20\) mg/L. The composition contribution is therefore \(30-20=10\) mg/L. Adding the components gives \(-4+10=6\) mg/L, exactly the observed pooled change.
Subjects B and C were observed at both times. Their changes are \(16-20=-4\) and \(26-30=-4\) mg/L, averaging \(-4\) mg/L. This within-subject result matches the time contribution in the assumed model, while the pooled difference also includes the \(10\) mg/L composition shift.
In real data, the individual levels and time trend are not generally known, and the simple shared-slope model may not describe the profiles. The decomposition is still useful as a way to reason about what changed: the pooled mean can move because concentrations changed over time, because the observed subjects changed, or because both happened. Subject-specific slopes or more complex profiles add further ways for changing subject composition to affect a pooled curve.
Sparse Sampling Can Add Noise Without Systematic Bias
A crucial qualification is that different subjects at different times do not, by themselves, guarantee a misleading estimate of the population mean at each time. If subjects are sampled representatively from the same target population at each time, and sampling is independent of their concentrations and subject-specific characteristics, then the time-specific sample mean can estimate the population mean at that time. Sparse sampling still means that few or no individual changes are observed directly. In a particular finite sample, the subjects selected at different times can also differ by chance, producing a composition difference.
Worked Example: Chance Composition in Representative Samples
Suppose a hypothetical population has four subjects with concentrations at time zero of \(10\), \(20\), \(30\), and \(40\) mg/L. Each declines by \(1\) mg/L per hour. At hour four, their concentrations are therefore \(6\), \(16\), \(26\), and \(36\) mg/L. The population means are \(25\) mg/L at time zero and \(21\) mg/L at hour four, a population mean change of \(-4\) mg/L.
Imagine that, by chance, A and C are sampled at time zero, while B and D are sampled at hour four. The observed time-zero mean is \((10+30)/2=20\) mg/L. The observed hour-four mean is \((16+36)/2=26\) mg/L. This particular pooled comparison rises by \(26-20=6\) mg/L, even though the population mean falls by \(4\) mg/L.
If each time’s two subjects are selected by simple random sampling from the same four-subject population, the expected sample mean at each time equals that time’s population mean. For example, each subject has probability \(2/4\) of inclusion, so the expected time-zero sample mean is \((2/4)(10+20+30+40)/2=25\) mg/L. The same calculation at hour four gives \((2/4)(6+16+26+36)/2=21\) mg/L. Thus this realized reversal is sampling variability, not evidence of systematic bias under the stated sampling scheme. With sparse data, however, it may be difficult to distinguish such a chance composition shift from a genuine change in the population mean.
What the Data Can and Cannot Show
The pooled curve and the individual trajectories address different questions. A time-specific pooled mean describes the average among the subjects represented at that time. A within-subject comparison describes change among subjects observed at multiple times. If the scientific question concerns how the same subjects’ concentrations evolve, a pooled curve from non-overlapping samples cannot answer it on its own.
Several checks help identify when a plotted curve may be fragile:
- Count the number of distinct subjects contributing at each time, rather than counting concentration records alone.
- Check which subjects contribute at multiple times and whether the overlap changes substantially across the interval.
- Compare the observed subject characteristics or early concentrations across the groups contributing at different times, where such information is available.
- Plot subject-specific observations and connect them only within subjects; do not imply that a line joining pooled means is an individual profile.
- Report how each time-specific mean was calculated and how many subjects contributed. A mean supported by a different small group at each time may be uncertain even if the plotted line looks smooth.
A subject-level plot makes repeated observations visible, while time-specific means summarize the observations at each time. When there is adequate overlap, paired changes can show what happened within the same subjects; when overlap is limited, those changes describe only the subset with repeated measurements. A mixed-effects analysis can represent subject-level differences while using repeated measurements, but it does not create observed within-subject information where the design provides none. The choice of analysis should follow the question and the sampling design.
The main limitation is interpretive: a pooled curve can be a useful description of observed means at each time, yet it may not summarize a typical subject’s course. Sparse sampling also limits the ability to diagnose the shape of individual profiles and to separate between-subject variation from within-subject change without additional assumptions. The curve’s appearance alone cannot determine whether a rise or fall is a time effect, a composition shift, or a combination.
Check Your Understanding
Try each question, then compare your reasoning with the solutions below.
- At time zero, subjects with an average starting level of \(18\) mg/L are observed; at a later time, subjects with an average starting level of \(25\) mg/L are observed. Under a shared decline of \(1\) mg/L per hour over three hours, what pooled change is expected from the decomposition?
- Why does a time-specific pooled mean not necessarily describe the change experienced by a typical individual?
- Four subjects have baseline concentrations \(8\), \(12\), \(16\), and \(20\) mg/L and each declines by \(2\) mg/L over the interval. Subjects with baselines \(8\) and \(12\) are observed early; those with baselines \(16\) and \(20\) are observed late. What is the pooled change, and what is the within-subject change in the stated model?
- Under what sampling condition can sparse time-specific samples estimate the population mean at each time without systematic composition bias?
Solutions. 1. The time contribution is \((-1)(3)=-3\) mg/L and the composition contribution is \(25-18=7\) mg/L, so the pooled change is \(-3+7=4\) mg/L. 2. Different subjects may contribute at different times, so the means can differ both because time has passed and because the average characteristics of the observed subjects have changed. 3. The early pooled mean is \((8+12)/2=10\) mg/L. The late concentrations are \(14\) and \(18\) mg/L, giving a mean of \(16\) mg/L, so the pooled change is \(+6\) mg/L. Each subject’s modeled within-subject change is \(-2\) mg/L. 4. Subjects must represent the same target population at each time, with selection independent of their concentrations and relevant subject characteristics; the time-specific samples may still differ by chance.