Why Connect Measurements by Subject?
A concentration-time data set contains observations made at particular times and measurements belonging to particular subjects. A pooled mean at each time can summarize the observed concentrations, but it does not show how one subject’s measurements relate to that same subject’s other measurements. A spaghetti plot makes this identity visible by plotting each subject’s concentration-time observations and connecting that subject’s points in time order.
The purpose is descriptive: to inspect patterns of repeated measurements, not to establish a cause for differences or to prove that a particular model is correct. Look for overall direction, subject-to-subject separation, similarities in profile shape, and trajectories that differ from one another. Those patterns help explain why a pooled curve may not adequately describe the data.
A plot is easiest to interpret when time and concentration use clearly labelled axes with units, and when each subject’s line can be distinguished without overwhelming the display. If observations are taken at different times for different subjects, the plot should show the actual observation times rather than imply that all subjects were measured at an identical schedule.
What the Patterns Can Show
Parallel profiles have broadly similar changes over time, even if subjects have different concentrations throughout. They suggest that subjects may share a similar time course while differing in level. A plot alone does not tell us why these levels differ; possible explanations could include biological variability, design features, or measurement variation.
Divergent trajectories differ in their shape or rate of change. For example, one subject’s concentration may decline quickly while another’s changes more slowly. Lines may also converge or cross. Such patterns indicate that a single pooled trajectory may conceal important variation in how concentrations change across subjects. The plot does not, by itself, determine whether the differences are systematic or just reflect a small and variable sample.
An unconnected scatterplot retains the locations of the observed points but removes the visible link between measurements from the same subject. As a result, the overall cloud may look similar whether individuals have nearly parallel profiles or markedly different paths. The information lost is not another concentration value; it is the identity-based pairing of measurements.
Worked Example: Recognizing Parallel Profiles
Hypothetical data: Three subjects have concentrations measured at 0, 2, and 4 hours.
| Subject | 0 h | 2 h | 4 h |
|---|---|---|---|
| A | 24 mg/L | 16 mg/L | 8 mg/L |
| B | 32 mg/L | 24 mg/L | 16 mg/L |
| C | 40 mg/L | 32 mg/L | 24 mg/L |
Plot each row as a connected profile. For A, the changes are \(16-24=-8\) mg/L over 2 hours and \(8-16=-8\) mg/L over the next 2 hours. For B, they are \(24-32=-8\) mg/L and \(16-24=-8\) mg/L. For C, they are \(32-40=-8\) mg/L and \(24-32=-8\) mg/L. Each subject’s observed concentration therefore decreases at the same average rate over each interval: \(-8/2=-4\) mg/L per hour.
The three lines have similar shape and remain separated. Their concentrations differ by 8 mg/L at each measured time, while their observed interval changes match. The pooled means are \((24+32+40)/3=32\) mg/L at 0 hours, \((16+24+32)/3=24\) mg/L at 2 hours, and \((8+16+24)/3=16\) mg/L at 4 hours. That mean curve shows the shared decrease, but only the connected profiles show that each subject has a similar decrease while maintaining a different level.
This small example illustrates a visual pattern; it does not establish that real subjects follow straight lines between measurements or share an identical underlying rate. A plot connects observed points as a guide to their sequence, not as proof about unobserved concentrations between them.
Why Pairing Matters
At any single time, a pooled summary depends on the concentrations observed at that time. It does not depend on which subject had which value at another time. The subject connections therefore add information about the relationship between repeated measurements, beyond the time-specific distributions and their means.
For two measurements per subject, write the first and second concentrations as \(C_{i1}\) and \(C_{i2}\). The change for subject \(i\) is \(D_i=C_{i2}-C_{i1}\). Its variance depends on the association between the two measurements:
This identity follows by expanding the variance of a difference: variance adds for the two separate terms, and the covariance terms enter with a minus sign because the second measurement is subtracted. Connecting points shows which observations form each pair, so it helps us see whether subjects with relatively high concentrations at one time tend also to be high at another. A plot does not calculate the covariance or prove a particular dependence structure, but it preserves the pairing needed to examine one.
Worked Example: Divergent Profiles Behind a Pooled Curve
Hypothetical data: Three subjects are measured at 0, 1, and 2 hours.
| Subject | 0 h | 1 h | 2 h |
|---|---|---|---|
| A | 30 mg/L | 18 mg/L | 9 mg/L |
| B | 24 mg/L | 20 mg/L | 17 mg/L |
| C | 18 mg/L | 16 mg/L | 14 mg/L |
The pooled means are \( (30+24+18)/3=24\) mg/L at 0 hours, \( (18+20+16)/3=18\) mg/L at 1 hour, and \( (9+17+14)/3=13.33\) mg/L at 2 hours. They suggest a decreasing average concentration.
The subject connections reveal more. A decreases by 12 mg/L in the first hour and 9 mg/L in the second. B decreases by 4 mg/L and then 3 mg/L. C decreases by 2 mg/L in each interval. The profiles differ in both level and amount of decline: A drops much more over the observed period than B or C. The pooled curve represents the average at each time, not the trajectory of a particular subject.
All three profiles decrease in this example, so the data do not show crossing lines or an increase. But their visibly different changes are enough to show why it would be misleading to assume that the pooled curve describes every subject equally well. The plot prompts questions about variation; it does not decide which explanation or model is appropriate.
Worked Example: Same Pooled Means, Different Connections
Hypothetical data: Consider two possible ways to pair measurements from three subjects at 0 and 2 hours. Both versions have the same set of concentrations at each time.
| Version | Subject | 0 h | 2 h | Change |
|---|---|---|---|---|
| Similar changes | A | 10 mg/L | 0 mg/L | -10 mg/L |
| Similar changes | B | 20 mg/L | 10 mg/L | -10 mg/L |
| Similar changes | C | 30 mg/L | 20 mg/L | -10 mg/L |
| Different changes | A | 10 mg/L | 20 mg/L | 10 mg/L |
| Different changes | B | 20 mg/L | 10 mg/L | -10 mg/L |
| Different changes | C | 30 mg/L | 0 mg/L | -30 mg/L |
In both versions, the mean at 0 hours is \((10+20+30)/3=20\) mg/L and the mean at 2 hours is \((0+10+20)/3=10\) mg/L. Thus the pooled means alone cannot distinguish the two patterns. In the first version, every subject’s change is \(-10\) mg/L. In the second, changes are \(10\), \(-10\), and \(-30\) mg/L, which differ substantially. The second version includes an increase for A, a pattern that could occur during an absorption phase; the numbers are illustrative, not claims about a particular dosing situation.
The arithmetic mean change is \(-10\) mg/L in both versions, but the subject-specific changes are not alike in the second. The spaghetti plot makes that difference visible because it connects each subject’s two measurements. The pooled mean curve retains the average change but loses the distribution of individual changes.
Reading Carefully: Checks and Limitations
A spaghetti plot is a diagnostic and exploratory display. Before interpreting its apparent patterns, check whether the axes and units are clear, whether subjects have actually been connected by identity, and whether the horizontal position represents the actual sampling time. If there are many subjects, overlapping lines can make the plot unreadable; transparency, selected subject labels, or separate panels may help, provided the display still communicates the data honestly.
Also check the observation schedule. If one subject has many more measurements than another, that subject may dominate a pooled point or a line-based visual impression. If measurements are missing or subjects drop out, later-time profiles may represent a different subset of the original group. A plot can reveal that the set of observed subjects changes, but it cannot recover unobserved concentrations or establish why measurements are missing.
Connected lines should be interpreted cautiously when measurement times differ. A straight segment between two observations is a visual aid, not evidence that concentration changed linearly between those times. When observations are sparse, a short line may conceal a complex course; when the schedule differs across subjects, apparent differences may partly reflect when each person was measured. The plot also does not quantify uncertainty or separate true subject differences from measurement error.
Compared with an unconnected scatterplot, the spaghetti plot preserves subject identity and sequence. Compared with a pooled mean curve, it reveals the range of profiles and how those profiles vary. A pooled curve is still useful for summarizing the average observed concentration at each time, and an unconnected scatterplot can show the distribution of points without emphasizing individual paths. These displays answer different descriptive questions; none should be treated as a substitute for all the others.
The central limitation is that a spaghetti plot describes the observed data, not the population beyond them. With few subjects, a striking pattern may not be representative. With many subjects, overplotting may hide it. Differences in apparent profiles are not, by themselves, evidence of a mechanism or a treatment effect. Their value is to direct attention to the repeated, subject-specific structure that pooled summaries cannot display.
Check Your Understanding
Answer each question using the definitions and examples above. Brief solutions follow the questions.
- What does connecting a subject’s points add that an unconnected scatterplot does not?
- Three profiles decline by 6 mg/L over the same 2-hour interval, but one subject’s concentrations remain 10 mg/L higher than the others. What pattern does this suggest?
- Two hypothetical data sets have identical pooled means at 0 and 2 hours. In one, every subject declines by 5 mg/L; in the other, subject changes are \(-15\), \(-5\), and \(5\) mg/L. What can be concluded about the pooled means and the individual changes?
- Why should a line joining observations not be taken as proof of the concentration between those observations?
Solutions. 1. It shows which measurements belong to the same subject and their observed order, making individual changes and profile shapes visible. 2. It suggests parallel profiles with different levels over that interval; it does not identify the cause of the level difference. 3. The pooled means can be identical even though subject-specific changes differ. The second set includes an increase for one subject, which the pooled mean change alone does not reveal. 4. The segment is a visual connection between observed values; it does not establish the unobserved concentration path, which could be nonlinear or otherwise different.