Why Compare Profiles by Formulation or Route?
A formulation or administration route can be an important grouping variable when exploring concentration-time data. A plot can show whether observed concentrations appear higher or lower at particular times, whether the sampling schedules cover the same time range, and how much profiles vary among individuals within each group. These are descriptive observations: a plot does not by itself establish why profiles differ, nor does it identify an absorption model.
The earlier material on Individual Concentration-Time Plots on Linear Scales, Log-Scale Individual Concentration-Time Plots, and Readable Overlays of Subject Concentration Profiles establishes the basic display choices. Here the central task is to make a comparison between groups readable and fair. That requires more than assigning colors: time origins, concentration units, axes, doses, sampling coverage, and group membership all affect what a visual comparison can support.
Design the Comparison Before Drawing It
Start by identifying what distinguishes the groups and what is held comparable. Record the formulation and route for each observation, and check whether groups differ in dose, dosing occasion, study visit, or subject composition. For every concentration, the relevant context includes its time relative to the dose or other prespecified time origin, its units, and whether it is a measured value or a value recorded as below the quantification limit. A label such as “Formulation B” is not enough if the dose or dosing occasion also changed.
Use the same time origin in every group. When time is relative to dose, zero should refer to the same dosing event for all displayed observations. Use the same concentration units and, when panels are intended for direct comparison, the same axis scales. Independently rescaling each panel can make groups look more alike or more different than they are. If a shared scale makes a low-concentration group hard to inspect, show a clearly labelled additional view rather than silently changing the scale.
An overlay is useful when the groups have comparable time coverage and a manageable number of profiles. Distinct colors or line types can distinguish groups, while thin, transparent individual trajectories help reveal within-group spread. If many profiles overlap, use aligned side-by-side panels or a summary with individual observations visible. Panels reduce clutter, but their scales and time axes should remain comparable. A legend should identify groups precisely, including route and formulation where both vary.
A group summary can help when individual profiles are numerous. At a given time, a median or quantile summarizes the observations available there, but it does not necessarily summarize the same subjects at every time. State the number of observations contributing to each summary, or display that number nearby. The earlier material on Time-Binned Concentration Medians and Quantiles covers summary construction; for this comparison, the key is to make the sampling support visible rather than implying a complete, equally sampled trajectory.
Worked Example: Comparing Two Formulations at Matched Times
Hypothetical data. At the same dose and under the same dosing conditions, three subjects in each formulation group have concentrations recorded at 1, 2, and 4 hours. Concentrations are in mg/L. The values are:
| Time (h) | Formulation A (mg/L) | Formulation B (mg/L) |
|---|---|---|
| 1 | 8, 10, 12 | 12, 14, 16 |
| 2 | 15, 18, 21 | 20, 24, 28 |
| 4 | 9, 12, 15 | 12, 15, 18 |
Plot each subject’s observations against time, using one color per formulation and the same axes. To make the group pattern easier to see, calculate the median at each matched time. With three ordered values, the median is the middle value. For example, at 2 hours the medians are \(18\) mg/L for A and \(24\) mg/L for B. Their descriptive difference at that time is
The corresponding medians at 1 hour are \(10\) mg/L for A and \(14\) mg/L for B; at 4 hours they are \(12\) mg/L and \(15\) mg/L. The plotted summaries therefore show higher observed medians for B at all three sampled times. The individual points remain important: with only three subjects per group, the medians do not show the full variation or establish that the formulations caused a difference. The comparison is interpretable as a formulation contrast only to the extent that dose, timing, and other study conditions are comparable.
Matched Times, Unequal Schedules, and Display Choices
A direct visual contrast at a particular time requires observations at that time, or a clearly stated method for obtaining a value there. If one group is sampled at 0.5 hours and another at 0.75 hours, those are not matched-time observations. Connecting nearby points or reading a smooth curve by eye can suggest a value between measurements, but it does not turn that value into an observation. Keep the actual sampling locations visible, especially when schedules differ.
A useful justification for matching time is simple. Let \(C_A(t)\) and \(C_B(t)\) denote observed group summaries at the same time \(t\), in the same concentration units. A descriptive difference at that time can be written as
Both terms must refer to the same time and compatible summaries for \(\Delta(t)\) to describe a time-specific contrast. If the available values instead refer to \(t_A\) and \(t_B\), subtracting them gives \(C_B(t_B)-C_A(t_A)\), which combines a group difference with a time difference. It is not the same-time contrast above. This is why unequal schedules should be shown honestly, rather than hidden by connecting or aligning unmatched observations.
Worked Example: Showing Different Sampling Schedules
Hypothetical data. An intravenous group has observations at 0.1, 0.5, and 1 hour; an oral group has observations at 0.5, 1, and 2 hours. Suppose their group medians are as follows, in mg/L:
| Time (h) | Intravenous median (mg/L) | Oral median (mg/L) |
|---|---|---|
| 0.1 | 30 | — |
| 0.5 | 12 | 8 |
| 1 | 6 | 14 |
| 2 | — | 10 |
Use a shared time axis with points only where data exist, and mark the missing group-time combinations as unsampled rather than as zero. At the matched time of 0.5 hours, the oral-minus-intravenous median difference is \(8-12=-4\) mg/L. At 1 hour it is \(14-6=8\) mg/L. At 0.1 and 2 hours there is no same-time comparison because only one route has observations.
The values do not justify filling in the missing medians or drawing a route-specific conclusion over the full interval. In particular, a line drawn between observed points is a visual connection, not a set of additional measurements. A display can show both schedules while making the unmatched time regions apparent.
Comparing Formulation and Route Without Confounding the Labels
When formulation and route both vary, distinguish their combinations rather than collapsing them into a single label. For example, if one formulation is given orally and another intravenously, a plot comparing those two groups cannot visually separate a formulation difference from a route difference. The group label represents both attributes at once. If the study contains multiple combinations, use panels or other clear grouping so each observed combination can be inspected.
Likewise, the people contributing data matter. In a crossover design, the same subjects may receive multiple conditions; in a parallel-group design, different subjects may contribute to each group. Showing subject identity can reveal whether apparent separation is consistent across individuals or driven by a few profiles. A group median alone can conceal those patterns. State which design applies and show individual profiles when feasible.
Worked Example: Recognizing a Design Limitation
Hypothetical data. A study compares two oral formulations, but every subject receiving A is observed at 1, 2, and 4 hours, while every subject receiving B is observed at 0.5, 1, and 2 hours. At 2 hours, the A observations are \(10, 12, 14\) mg/L and the B observations are \(16, 18, 20\) mg/L. The medians are \(12\) mg/L and \(18\) mg/L, so the same-time descriptive difference is
At this sampled time, B has the higher observed median. However, suppose all A subjects were enrolled in an earlier visit and all B subjects in a later visit. Formulation and visit are then aligned in the data: the display cannot show whether any difference is associated with formulation, visit, or both. The appropriate plot labels and a written caption should disclose this limitation. The arithmetic describes the observations; it does not resolve the design ambiguity.
Scale, Diagnostics, and Limits of the Display
Linear and logarithmic concentration scales answer different visual questions. A linear scale preserves absolute vertical differences, so equal vertical distances represent equal concentration differences. A logarithmic scale gives more room to inspect lower positive concentrations across a wide range, but equal distances then represent equal ratios rather than equal differences. Use an explicitly labelled scale and apply it consistently across the groups being compared. A logarithmic axis cannot display zero as an ordinary finite value; do not replace zero or below-quantification records with arbitrary positive numbers just to make them appear on that axis.
Before describing a pattern, check the data coverage and plot itself. Are time units consistent? Are all times relative to the intended dose? Are group labels correct? Do axes start and end at comparable limits? Are points hidden by overplotting? Are sample counts sharply different over time or between groups? Are any extreme values or below-quantification observations treated in a way that obscures their status? If a summary curve changes direction where its contributing subjects change, that may reflect changing coverage rather than a within-subject change. These checks support a more careful description; they do not replace a formal analysis.
Exploratory plots have important limits. They do not adjust for differences in dose, study conditions, or subject characteristics, and a visual separation does not quantify uncertainty or establish a causal effect. A summary curve can conceal individual variability and unequal numbers of observations. Sparse sampling limits what can be seen between recorded times. Comparisons across different doses need dose context, and dose normalization should not be applied automatically: it depends on whether that transformation is appropriate to the question and data. Do not use these displays to choose or fit an absorption model; the purpose here is to inspect and describe the observations.
A Practical Comparison Workflow
Identify formulation, route, dose, dosing occasion, and study design. Note whether the groups represent different subjects or repeated conditions in the same subjects.
Confirm time origin, units, and sampling coverage. Identify differences that prevent a straightforward comparison, such as unmatched times or a formulation that always coincides with one route.
Use an overlay when profiles remain distinguishable; otherwise use aligned panels. Keep scales consistent for direct comparison, and retain individual observations when possible.
State which observed groups appear higher or lower, at what sampled times, and how much individual variation or unequal coverage is apparent. Separate that description from explanations the plot cannot establish.
The main alternative to an overlay is a faceted or panel display: it can reduce clutter while preserving a common reference scale. A compact time-specific summary can make a pattern easier to scan, but should be paired with sample counts and individual data where feasible. Neither choice is universally best. Select the display that exposes both the group comparison and its limitations without implying measurements or comparability that the study does not provide.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two formulation groups have median concentrations of 9 mg/L and 13 mg/L at the same sampled time. What descriptive difference can be reported, and what can it not establish by itself?
- One route is sampled at 0.5 hours and another only at 0.75 hours. Why is subtracting those values not a same-time contrast?
- Every subject receiving formulation A is in one study visit, and every subject receiving B is in another. What limitation should accompany a visual difference?
- Why should a group summary be accompanied by information about the number of observations contributing at each time?
Solutions. (1) The second-minus-first median difference is \(13-9=4\) mg/L at that time. It describes the observed summaries, not a causal formulation effect or a parameter estimate. (2) The subtraction combines a group difference with a time difference: the values refer to different times, so it does not equal a contrast at one common time. (3) Formulation and visit are confounded in the displayed groups; the plot cannot distinguish which is associated with the observed difference. (4) The number can vary across times, affecting how much data supports each summary; without it, changing coverage may be mistaken for a consistent group profile.