Why Stratify Concentration-Time Profiles?
A categorical covariate divides observations into distinct levels, such as categories of sex or treatment group. Displaying concentration-time data separately for those levels can reveal whether the observed profiles appear similar, shifted, more variable, or differently covered over time. This is a descriptive comparison: a plot can suggest patterns worth examining, but by itself it does not establish a statistical difference or explain its cause.
A useful comparison makes the groups visually comparable. Use the same time axis, concentration axis, units, and plotting scale for every category. Also show, or otherwise report, how many observations contribute at relevant times. Without those choices, differences in plotting range or data coverage can look like differences in concentration profiles.
The stratifying variable must be defined at the level where it is measured. For a subject-level covariate, each concentration record for that subject belongs to the same category. For a treatment group that can change by period or visit, records must be assigned according to the treatment actually relevant to that record. A category label is not a substitute for checking the analysis data and its assignments.
Constructing a Fair Comparison
Begin by checking that time and concentration are defined consistently across categories. Then choose common axes and a time window that covers the observations you intend to compare. If the profiles are shown on a linear concentration scale, the vertical distances represent absolute concentration differences. A logarithmic concentration axis can make multiplicative differences and lower concentrations easier to inspect, but it requires positive values and should be labelled clearly. Do not use different vertical scales for different categories when the purpose is direct visual comparison.
When observations are dense, a group summary can complement individual points or subject profiles. For each group and time bin, report the number of observations and a center, such as the median, with a spread summary such as the interquartile range. The median is the middle ordered value for an odd number of observations; for an even number, it is the average of the two middle values. Quartile conventions can differ, especially with small samples, so state the convention if exact quartiles matter. A summary line should not imply observations exist at times where the group has no data.
Check category labels, units, record assignments, and whether each category represents the intended covariate level.
Compare the time ranges and observation counts across categories. Identify empty or sparsely sampled parts of the window.
Use shared axes, a stated concentration scale, and either individual observations or summaries on comparable time intervals.
Report overlap, apparent shifts, spread, and coverage. Keep the description distinct from claims about statistical significance or causation.
Worked Example: Comparing Two Categories at Matched Times
Suppose a hypothetical dataset contains concentrations in mg/L from two categories, A and B, at 1, 2, and 4 hours after dosing. Three observations are available for each category at each time. For this small example, calculate the arithmetic mean at each time to show one possible summary; the values are not presented as a real study.
| Time (h) | Category A concentrations (mg/L) | Category B concentrations (mg/L) |
|---|---|---|
| 1 | 8, 10, 12 | 6, 8, 10 |
| 2 | 5, 6, 7 | 4, 5, 6 |
| 4 | 1, 2, 3 | 0.5, 1, 1.5 |
At 1 hour, the category A mean is \((8+10+12)/3=10\) mg/L, while the category B mean is \((6+8+10)/3=8\) mg/L. At 2 hours the means are \((5+6+7)/3=6\) mg/L and \((4+5+6)/3=5\) mg/L. At 4 hours they are \((1+2+3)/3=2\) mg/L and \((0.5+1+1.5)/3=1\) mg/L.
A display with shared axes could show category A's mean above category B's mean at all three observed times. The ranges nevertheless overlap at each time: 8–10 mg/L at 1 hour, 5–6 mg/L at 2 hours, and 1–1.5 mg/L at 4 hours. A careful description is that the means differ at these sampled times while the observed ranges overlap. These data alone do not establish whether a population-level difference exists.
Why Pooled Averages Can Mislead
A single average per category can hide how observations are distributed across time. If one category is sampled mostly early and another mostly late, their pooled means combine different parts of the concentration-time course. This can reverse the apparent ordering seen at matched times.
The arithmetic explains the problem. If group \(g\) has \(n_{gj}\) observations in time bin \(j\), with mean concentration \(\bar{x}_{gj}\), its pooled mean is a weighted average of the bin means:
Thus, the weights are determined by the number of records in each bin. If the groups have different time distributions, they use different weights. A pooled comparison is then not a like-for-like comparison at a common time.
Worked Example: Unequal Sampling Reverses the Pooled Ordering
Consider hypothetical observations in two time bins: early and late. Category A has one early observation of 10 mg/L and nine late observations, each 2 mg/L. Category B has nine early observations, each 8 mg/L, and one late observation of 1 mg/L. At both times, A's concentration is higher than B's: 10 versus 8 mg/L early, and 2 versus 1 mg/L late.
The pooled mean for A is \((1\times10+9\times2)/10=28/10=2.8\) mg/L. The pooled mean for B is \((9\times8+1\times1)/10=73/10=7.3\) mg/L. The pooled means make B appear higher, even though A is higher in each time bin.
The calculation is correct for the observations collected, but it answers a different question: the average among each category's observed records, with the observed time distribution acting as weights. To compare profiles, show time-specific summaries on common bins and make the counts visible. If a category has little or no coverage in a bin, do not imply a well-supported comparison there.
This is a reason to avoid treating a pooled average as a profile comparison, not a reason to discard all summaries. State what a summary averages over, and use matched times or bins when the descriptive question concerns concentrations at comparable points in the sampling window.
Summaries, Overlap, and Their Limits
A center alone does not describe a category's observations. Two categories can have similar medians but different spreads, or different medians with substantial overlap. Showing individual observations, ranges, or quantiles helps distinguish these patterns. When a bin contains only a few observations, however, its median and quantiles may change substantially if a single record changes. Report the count and avoid presenting a jagged summary line as if it were a precise continuous trajectory.
Worked Example: Reporting a Small-Bin Median and Spread
In a hypothetical 2-hour time bin, five concentrations for category C are 4, 5, 6, 8, and 12 mg/L. Five for category D are 3, 4, 5, 7, and 9 mg/L. With five ordered values, the median is the third value. Using the median-of-halves convention for quartiles, exclude the overall median and find the median of the lower and upper pairs.
For C, the median is 6 mg/L; \(Q_1=(4+5)/2=4.5\) mg/L and \(Q_3=(8+12)/2=10\) mg/L. For D, the median is 5 mg/L; \(Q_1=(3+4)/2=3.5\) mg/L and \(Q_3=(7+9)/2=8\) mg/L. The interquartile ranges are therefore 5.5 mg/L for C and 4.5 mg/L for D.
A reasonable description is that C has a median 1 mg/L higher than D in this bin, while the interquartile ranges overlap from 4.5 to 8 mg/L. Each estimate is based on only five observations, so the displayed summaries are exploratory and should be read with that limited count in mind.
Overlap is a visual description, not a formal test. Overlapping ranges or quantiles do not prove that groups are equal; non-overlap does not, by itself, establish a statistically significant difference. Likewise, repeated observations from a subject are not independent people. A plot can show their measured records, but a formal analysis of uncertainty must account for the study design and within-subject dependence.
Diagnostics and Common Pitfalls
Before interpreting a stratified display, check whether the plot is being driven by the way the data were collected or displayed. Useful checks include:
- Unequal coverage: compare counts and time ranges by category. Missing observations at a time are not evidence of zero concentration.
- Unequal bin counts: show counts, since a summary based on two observations is less stable than one based on many.
- Scale mismatch: verify that categories share the same axes and that linear or logarithmic scaling is clearly identified.
- Unequal dosing or design conditions: consider whether apparent category differences could reflect differences in dose, visit, or other study features rather than the category alone. The plot does not separate those explanations.
- Overplotting: if many points coincide, use a readable display that preserves the data and makes group membership clear; do not let hidden points appear to be absent.
A common mistake is to say, “Category A has a higher profile because its pooled mean is higher,” without checking whether both categories were sampled at comparable times. The correction is to compare time-specific summaries with their counts, inspect the underlying observations, and describe any remaining imbalance. Another mistake is to interpret a visible separation as a causal effect of the category. A descriptive plot establishes neither causality nor statistical significance; those conclusions require an appropriate analysis and design context.
Compared with an overlay of every subject profile, a time-binned group summary is easier to read when there are many observations, but it can hide individual variation and depends on bin choices. An individual-level display preserves variation but may become crowded. A pooled category mean is compact, yet can obscure time patterns and unequal sampling. These displays answer different descriptive questions; none removes the need to assess coverage and interpret the plotted data in context.
Check Your Understanding
Answer each question, then check the solutions below.
- Category A has 12 observations and category B has 3 observations in a time bin. What information should accompany the bin summaries?
- At a matched time, the mean is 7 mg/L in one category and 5 mg/L in another, but their observed ranges overlap. State one appropriate descriptive conclusion.
- Two categories have different pooled means but were sampled at very different times. What comparison is more relevant to describing their profiles?
- Why should a plot use the same concentration axis for all categories?
Solutions. (1) Report both counts alongside the summaries and interpret the summary based on three observations cautiously; the bin counts reveal unequal support. (2) State that the first category's observed mean is 2 mg/L higher at that time and that the observed ranges overlap; do not infer significance from the plot. (3) Compare time-specific summaries using common time points or bins, show the counts, and inspect the coverage. (4) Shared axes ensure that visual distances and positions are comparable; separate scales can exaggerate or conceal apparent differences.