Why Separate the Two Sources of Variation?
A concentration dataset often contains several observations from each participant. A pooled display can show that concentrations differ, but it may not show how they differ: some variation is among subjects, some occurs across observations for the same subject, and some reflects the expected change in concentration over time. These patterns matter for exploratory interpretation. A large difference between two participants at a comparable sampling time is not the same visual pattern as a large change across times for one participant.
The useful starting point is to keep subject identity visible. Individual profiles show the sequence of observed concentrations for each subject. Subject-level summaries compress each person's observations into a smaller set of values. The two views complement one another: profiles preserve timing and shape, while summaries make differences in typical level or observed spread easier to compare.
These are descriptive distinctions, not estimates of population-model variance components. A plot can suggest where variation occurs, but it cannot by itself identify its biological, design, or measurement sources. Interpretation depends on what was measured, when it was measured, and whether the observations being compared represent comparable conditions.
Pair Profiles With Subject-Level Summaries
An individual concentration-time plot places time on the horizontal axis and concentration on the vertical axis, using a separate trace for each subject. Points should remain visible; connecting points in time order helps show each observed trajectory but does not assert that the concentration was measured continuously between them. When subjects overlap, transparency, small multiples, or restrained highlighting can help preserve visibility.
A second display can summarize each subject. For a subject with \(n_i\) observations \(C_{ij}\), the arithmetic mean is \(\bar{C}_i = \frac{1}{n_i}\sum_{j=1}^{n_i} C_{ij}\). A dot plot of these means gives one point per subject. A range or other spread display alongside each mean can reveal whether similar subject means conceal very different observed within-subject spreads. The summary and its spread should be interpreted in the context of the sampling schedule.
A subject mean is only a useful comparison of typical levels when the observations contributing to it are meaningfully comparable. If one subject is sampled mainly early after a dose and another mainly late, their raw means can differ simply because concentrations change with time. Similarly, combining samples taken under different dosing or study conditions may make a subject summary hard to interpret. A profile plot helps diagnose these problems because it retains the time and sequence of each observation.
Worked Example: Separating Spread at a Matched Sampling Condition
Suppose three hypothetical subjects each have concentration measured on three separate occasions at the same nominal time after dosing, under the same recorded study condition. The concentrations, in mg/L, are A: 8, 10, 12; B: 14, 16, 18; and C: 20, 22, 24. The matched conditions make a descriptive comparison of subject levels and within-subject spread reasonable.
The subject means are \(\bar{C}_A=(8+10+12)/3=10\) mg/L, \(\bar{C}_B=(14+16+18)/3=16\) mg/L, and \(\bar{C}_C=(20+22+24)/3=22\) mg/L. The grand mean across all nine observations is \(\bar{C}=(8+10+12+14+16+18+20+22+24)/9=16\) mg/L.
A profile or subject-by-observation display would show that each subject's observations span 4 mg/L, while the subject means differ by 6 mg/L between adjacent subjects. For a numerical check, the within-subject sum of squared deviations is \( [(8-10)^2+(10-10)^2+(12-10)^2] + [(14-16)^2+(16-16)^2+(18-16)^2] + [(20-22)^2+(22-22)^2+(24-22)^2] = 8+8+8=24\) \((\mathrm{mg/L})^2\). This describes scatter around each subject's own mean. It does not explain why those observations vary.
The between-subject sum of squares of the means, weighted by three observations per subject, is \(3[(10-16)^2+(16-16)^2+(22-16)^2]=3(36+0+36)=216\) \((\mathrm{mg/L})^2\). Here the visible differences among subject levels are substantial relative to the within-subject scatter. This is a descriptive statement about these hypothetical observations, not a conclusion that subjects differ by a particular biological mechanism.
A Simple Decomposition and Its Conditions
For a balanced set of observations—\(m\) observations for each of \(N\) subjects—let \(C_{ij}\) be observation \(j\) from subject \(i\), \(\bar{C}_i\) be subject \(i\)'s mean, and \(\bar{C}\) be the grand mean. The total squared deviation from the grand mean can be split into within-subject and between-subject parts:
The identity follows by writing each deviation as \(C_{ij}-\bar{C}=(C_{ij}-\bar{C}_i)+(\bar{C}_i-\bar{C})\), squaring, and summing. The cross-product is \(2\sum_i(\bar{C}_i-\bar{C})\sum_j(C_{ij}-\bar{C}_i)\), which is zero because for each subject \(\sum_j(C_{ij}-\bar{C}_i)=0\). The remaining between-subject term repeats each subject-mean deviation \(m\) times, giving the factor \(m\).
This algebraic split is always an identity for the stated balanced data. Its interpretation as separating the two sources of variation is most useful when observations are comparable: for example, measurements under the same nominal time and condition. When observations occur at different points on a concentration-time curve, the within-subject term also contains the time trend. The identity does not remove that trend or turn the terms into model-based variance components.
With unequal numbers of observations \(n_i\), a corresponding descriptive identity uses \(n_i\) as the weight on each subject mean:
The pooled grand mean in this expression weights subjects according to their number of observations. Consequently, subjects with denser sampling contribute more to the pooled description. A plot showing each subject equally, alongside the number and timing of that subject's observations, makes this feature easier to notice.
Worked Example: Why a Subject Mean Can Mix Level and Time
Consider three hypothetical subjects measured at 1, 4, and 8 hours after a dose. Their concentrations, in mg/L, are A: 12, 6, 2; B: 24, 12, 4; and C: 36, 18, 6. Assume for this illustration that each subject has the same sampling times and that these values are the recorded concentrations.
The raw means are \(\bar{C}_A=(12+6+2)/3=20/3=6.67\) mg/L, \(\bar{C}_B=(24+12+4)/3=40/3=13.3\) mg/L, and \(\bar{C}_C=(36+18+6)/3=60/3=20.0\) mg/L. A subject-mean dot plot therefore shows increasing average observed concentrations from A to C.
But the individual profiles reveal an important feature that the means alone omit: every subject's concentration decreases over the observed times. At 1 hour the values are 12, 24, and 36 mg/L; at 4 hours they are 6, 12, and 18 mg/L; and at 8 hours they are 2, 4, and 6 mg/L. Thus, the within-subject changes include a clear time pattern. The raw means summarize the three sampled times together; they are not a time-independent measure of each subject's concentration level.
Here the common sampling schedule makes the means more comparable than they would be under different schedules, but it does not make the temporal pattern disappear. A profile display paired with the subject means supports both observations: concentrations decline over time within each subject, and the observed levels differ across subjects at each of the matched times. The display alone does not establish the reason for either pattern.
Reading Within-Subject Spread Without Hiding the Data
A compact subject-level display can show a mean or median per subject together with a range, interquartile range, or the individual observations. For small numbers of observations, showing the points is often most informative: a range based on two points, for example, is not a stable description of a distribution. With more observations, a box or interval can be useful, but should not obscure how many observations support it or when they occurred.
A useful visual pairing is a profile panel and a subject-summary panel. The profile panel preserves the time course; the summary panel allows a quick comparison of subject-level means, medians, or observed spreads. If summaries are calculated within a prespecified common time window or at matched nominal times, state that restriction. If different summaries use different numbers of observations, report or display those counts.
Worked Example: Equal Means, Different Observed Spread
Suppose two hypothetical subjects each have three measurements at the same nominal time and under the same condition. Subject A has 8, 10, and 12 mg/L; subject B has 4, 10, and 16 mg/L. Both means equal 10 mg/L: \((8+10+12)/3=10\) and \((4+10+16)/3=10\).
For A, the sample standard deviation is \(s_A=\sqrt{[(8-10)^2+(10-10)^2+(12-10)^2]/(3-1)}=\sqrt{8/2}=2.00\) mg/L. For B, it is \(s_B=\sqrt{[(4-10)^2+(10-10)^2+(16-10)^2]/(3-1)}=\sqrt{72/2}=6.00\) mg/L.
A plot showing only one mean per subject would make the subjects appear identical. Showing their observations or their mean with a spread measure reveals that B's observed values are more dispersed. With only three observations per subject, these sample standard deviations are descriptive and imprecise; they do not establish stable long-run within-subject variability.
Diagnostics, Alternatives, and Limitations
Before interpreting a display, check whether it preserves the design information needed for the comparison. Verify subject identifiers, time units, nominal sampling times, dose or visit context, and the number of observations per subject. Look for missing or duplicated records, unusual gaps, and apparent jumps that may reflect a changed condition or data issue rather than a typical trajectory. These checks are especially important when lines connect observations: an incorrect order or an unmarked visit boundary can make a misleading profile.
Several patterns should prompt caution:
- Unequal timing: subject means may differ because subjects contribute observations at different points along a changing concentration-time curve.
- Unequal observation counts: pooled summaries give more weight to subjects with more records, while an unweighted display of subject means gives each subject one point.
- Time trends inside a summary: a broad within-subject range may primarily reflect expected change over time, not random scatter around a stable level.
- Overplotting: many similar profiles can conceal individual trajectories. Use small multiples or carefully selected highlighting rather than interpreting a dense band as identical behavior.
- Small numbers of observations: a mean, range, or standard deviation from very few points may be highly sensitive to a single measurement.
An overlay of individual profiles is the main choice when timing and trajectory shape are central. Faceted profiles trade compactness for easier inspection of each subject. Subject-level dot plots or spread displays are the alternative when the question is about differences in summaries, but they sacrifice the ordering and timing of observations. Time-binned medians and quantiles describe the distribution across observations within time bins; because they do not retain each observation's subject identity, they cannot by themselves separate within-subject from between-subject patterns.
No single display answers every question. Subject summaries can hide changes over time, while profile overlays can become unreadable in a large dataset. Unequal sampling complicates comparisons, and observed differences can reflect design, measurement, or recording features as well as underlying subject differences. Exploratory plots support description and diagnosis; they do not establish causation or replace an appropriately specified population model.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- What does a subject-level mean conceal that an individual profile can reveal?
- Two subjects have different raw means, but one was sampled mostly early and the other mostly late after dosing. What is the main interpretive concern?
- In the balanced decomposition, why does the between-subject sum of squares contain the factor \(m\)?
- Why can a large within-subject range across a concentration-time profile not automatically be interpreted as large random residual variability?
Solutions. (1) A mean conceals the timing, ordering, shape, and spread of the observations contributing to it. (2) Their means may differ because the sampling times differ along a changing concentration-time curve; the comparison does not isolate subject level. (3) Each subject-mean deviation occurs once for each of that subject's \(m\) observations, so its squared contribution is repeated \(m\) times. (4) The range may largely reflect systematic concentration change over time, rather than scatter around a time pattern; the plot and sampling schedule must be examined before interpreting it.