Readable Overlays of Subject Concentration Profiles

Why Profile Overlays Become Hard to Read

An individual concentration-time plot can reveal a subject’s observed concentrations and their sequence over time. When many subjects share one set of axes, however, curves can cross, overlap, and hide one another. The display may become a dark tangle in which neither individual trajectories nor the spread of observed profiles is easy to inspect. The goal of a readable overlay is not to make every line equally prominent; it is to show how profiles relate while retaining enough detail to detect unusual trajectories, gaps, and differences in sampling.

Let \(C_{ij}\) denote the observed concentration for subject \(i\) at time \(t_{ij}\), where \(j\) indexes that subject’s observations. An individual profile is a plot of the paired values \((t_{ij}, C_{ij})\) for one subject. A line connecting successive observations is a visual guide between recorded points; it does not establish that concentrations were measured continuously or that the connecting path is the true concentration curve.

Before overlaying profiles, confirm that the time variable has a coherent meaning across subjects, such as time since dose, and that concentration and time use consistent units. If subjects have different relevant dosing or sampling contexts, make those contexts explicit rather than treating all curves as directly comparable. Use a linear or logarithmic concentration axis that suits the display; the earlier material on Log-Scale Individual Concentration-Time Plots covers the interpretation of that choice.

Three Strategies for Dense Overlays

The main options are to draw all profiles with transparency, show a deliberately selected subset, or give subjects separate panels in a small-multiple display. These approaches solve different problems. Transparency retains all subjects on one set of axes, selection reduces visual load at the cost of omitting profiles from view, and small multiples preserve individual separation at the cost of space.

With transparency, each line is drawn with opacity \(\alpha\), where \(0<\alpha\leq 1\). On a white background, repeated strokes of the same color darken where they overlap. Under ordinary alpha compositing, the combined opacity of \(m\) exactly overlapping strokes is:

This is a display calculation, not a statistical estimate. It assumes the same opacity and color for every stroke and exact overlap at the location considered. In a real plot, lines often only partly overlap, have different positions, or cross at different points. Therefore, darkness can suggest where strokes accumulate, but it is not a calibrated count of subjects and should not be read as a concentration distribution.

A selected-subject display is useful when the purpose is to inspect individual trajectories and a full overlay is too crowded. The selection rule should be stated and reproducible—for example, a random sample generated with a recorded seed, or a prespecified selection of subjects with particular sampling patterns. Do not select subjects because their profiles make a preferred story look convincing. A subset can illustrate individual shapes, but it cannot stand in for all subjects unless its representativeness is justified for the intended purpose.

Small multiples use a separate panel for each subject, usually with common axis limits. They reduce direct line overlap and make it easier to associate observations with a subject. Their cost is a larger figure, smaller panels, and the possibility that many panels will overwhelm the reader. Shared scales are especially important when the purpose is to compare plotted magnitudes across panels; independently scaled axes can make profiles look more similar or more different than their concentrations warrant.

Worked Examples

How to Build and Check the Display

A reliable overlay begins with the intended visual question. If the aim is to inspect all observed trajectories together, start with all subjects and consider transparency. If the aim is to show examples of individual profile shapes, select subjects according to a transparent rule. If the aim is to inspect subjects one by one while retaining direct visual comparability, use small multiples. These are display choices, not methods for estimating a typical profile.

Connecting observations requires care. Connect points within a subject only, and in time order. Do not draw a line across records that represent different dosing occasions or a substantial interruption if that connection would imply a continuous observed trajectory. Avoid inventing values to fill gaps. Where an observation is censored or otherwise not a measured numerical concentration, do not plot it as an exact zero merely to complete a line.

Inspect the plot at the size at which it will be read, not only while zoomed in on a large screen. Look for dark saturated regions, points hidden beneath lines, indistinguishable colors, and panels whose labels or tick marks cannot be read. If a sparse set of subjects appears unusually tidy, verify that the selection procedure did not inadvertently favor complete profiles or a narrow range of observed behavior. A clear caption should describe the plotted population of records, any subset selection, axis transformations, and relevant units.

Comparison, Limitations, and Common Pitfalls

No strategy eliminates the need to interpret the display in context. Transparency can make dense regions appear like a surface even though the data are discrete subject records. A selected overlay cannot show subjects who were omitted. Small multiples can encourage rapid scanning but make subtle comparisons difficult when the figure is too large or the panels too small. If there are many subjects, a single figure may not be able to show every record legibly; use complementary displays rather than forcing all purposes into one plot.

A common mistake is to interpret a dark patch in a transparent overlay as proof that many subjects had nearly the same concentration. The reasoning is that darker color must directly encode the number of subjects. The correction is that darkness depends on line opacity, color, stroke width, exact overlap, and rendering; the compositing formula describes only an idealized point of exact overlap. Treat dark regions as a prompt to inspect the profiles, not as a numerical frequency summary.