Why an Exploratory PK Plot Can Mislead
Exploratory plots help reveal concentration patterns, variability, and data coverage, but a visible pattern is not automatically a biological or treatment difference. A concentration-time display can look persuasive while hiding repeated observations at the same coordinates, unequal numbers of subjects, uneven sampling, or differences in the times at which groups were measured. The first task is therefore not to interpret the apparent shape; it is to check what observations, subjects, and sampling times produced it.
This tutorial focuses on five recurring sources of misreading: overplotting, unequal group sizes, axis choices, sparse regions, and time confounding. The relevant checks build on individual and overlaid profiles, time-binned summaries, covariate coverage displays, and plots stratified by formulation or route. These are descriptive tools. They can show what was observed, but by themselves they do not establish why a difference occurred or whether it is statistically reliable.
A point in a population PK plot usually represents a concentration observation from a subject at a sampling time. The number of points is not necessarily the number of subjects: one subject may contribute many samples, and multiple subjects may have samples at similar times and concentrations. Before interpreting a plot, establish the unit represented by each mark and the number of subjects and observations contributing to each part of it.
Overplotting: Visible Marks Are Not Observation Counts
In a dense overlay, many subject profiles may cross or lie close together. A region that appears to contain only a few marks may represent many observations, while a single isolated mark may correspond to one subject. This makes visual density an unreliable substitute for counting. Overplotting can conceal both the frequency of repeated values and the extent of between-subject variability.
Worked Example: Counting Hidden Observations
Suppose a hypothetical overlay contains 60 concentration observations from 12 subjects. After plotting, only 42 distinct marker locations are visible because some observations share the same time and concentration coordinates. How many observations are hidden by exact overlap, and what fraction of the observations does that represent?
The number hidden by exact overlap is \(60-42=18\) observations. The fraction of all observations hidden is \(18/60=0.30\), or 30%. Thus the visible locations do not represent 42 observations in a one-to-one way; they represent 60 observations, with 18 sharing locations with others.
This calculation addresses exact overlap only. Markers that are merely close together can still be hard to distinguish, so the true readability problem may be greater. A transparent-point display, a small amount of visual separation where appropriate, or a plot of subject-level summaries can help reveal structure. Any such display should preserve the original time and concentration values for interpretation; visual adjustments must not be mistaken for measured values.
Useful checks include comparing the number of plotted records with the number of visible marks, inspecting individual profiles, and displaying counts by time bin. Transparency can indicate dense regions, but it does not identify subjects or guarantee that overlapping marks have been counted correctly. If the scientific question concerns subjects rather than individual samples, show subject-level information explicitly instead of treating every sample as a separate subject.
Unequal Group Sizes: Density Is Not a Group Effect
Groups may differ in the number of enrolled subjects, samples per subject, or observations available at a particular time. A larger group can dominate an overlay simply because it contributes more marks. That visual dominance does not mean its typical concentration is higher. Conversely, a small group may look unusually distinct because a few observations are easy to pick out or because its profile is supported by very few subjects.
Unequal group size also affects pooled summaries. The overall mean of all observations gives more weight to groups that contribute more observations. That is a valid description of the pooled observations, but it is not the same as giving each group equal weight. Which summary is useful depends on the question being asked.
Worked Example: A Pooled Mean Gives More Weight to the Larger Group
Consider hypothetical concentrations at one sampling time. Group A has four observations: 2, 4, 6, and 8 mg/L. Group B has six observations: 8, 10, 12, 14, 16, and 18 mg/L. Compare the pooled mean with the mean of the two group means when the groups are weighted equally.
The Group A mean is \((2+4+6+8)/4=20/4=5\) mg/L. The Group B mean is \((8+10+12+14+16+18)/6=78/6=13\) mg/L. The pooled mean is \((20+78)/(4+6)=98/10=9.8\) mg/L. The equally weighted average of the group means is \((5+13)/2=9.0\) mg/L.
The difference, \(9.8-9.0=0.8\) mg/L, arises because the pooled mean gives Group B six-tenths of the weight and Group A four-tenths. In general, for group means \(\bar{x}_g\) and observation counts \(n_g\), the pooled mean is \(\sum_g n_g\bar{x}_g/\sum_g n_g\). The equally weighted group mean is \(\sum_g\bar{x}_g/G\), where \(G\) is the number of groups. These answer different descriptive questions; neither should be presented as the other.
When comparing formulations, routes, or covariate strata, report or display the number of subjects and observations contributing to each group and time region. If sampling frequency differs between subjects, counting raw samples alone can overrepresent subjects with more measurements. A helpful check is to compare both subject counts and observation counts, then inspect profiles or summaries that make the intended unit of comparison clear.
Axis Choices: Read the Scale Before the Pattern
The axis determines how distances in a plot correspond to concentration differences. On a linear concentration axis, equal vertical distances represent equal absolute differences in concentration. On a logarithmic axis, equal vertical distances represent equal multiplicative ratios. Neither scale is inherently misleading, but a reader can misread a pattern if the scale is not noticed or if a restricted range is mistaken for the full range.
For example, on a logarithmic axis, the same vertical distance from 1 to 2 mg/L as from 10 to 20 mg/L represents a doubling in both cases, even though the absolute differences are 1 and 10 mg/L. A log axis cannot display zero or negative concentrations. If zero values or values below a quantification limit are present, the display must make clear how those observations were handled; silently omitting them can give a distorted impression of the data.
Worked Example: Interpreting a Restricted Linear Axis
Suppose a hypothetical linear plot displays concentrations from 8 to 14 mg/L rather than starting at zero. Group A has a mean of 10 mg/L and Group B a mean of 12 mg/L. What is their absolute difference, and how can the displayed axis affect the visual impression?
The absolute difference is \(12-10=2\) mg/L. Relative to Group A's mean, this is \(2/10=0.20\), or 20%. On the displayed axis, the difference spans \(2/(14-8)=2/6=0.333\), or about one-third of the plotted vertical range. That can make the separation look large. If the same values were displayed on a linear axis from 0 to 14 mg/L, the difference would span \(2/14\approx0.143\), or about 14.3% of the range.
Both displays preserve the 2 mg/L difference, but the restricted axis gives it more visual space. It is therefore important to read the tick labels and state the actual concentration difference rather than describing the groups as “far apart” from appearance alone. A restricted axis may be useful for showing detail, provided its limits are clear and the visual emphasis is not confused with the magnitude of the effect.
Check whether an axis starts at zero, whether it is linear or logarithmic, whether limits differ between panels, and whether any values are clipped or excluded. When panels use different scales, direct visual comparisons of slopes or spread may be invalid. State units, and, when the scale is logarithmic, explain differences in terms of ratios as well as concentrations where helpful.
Sparse Regions and Unbalanced Sampling Times
A smooth-looking summary can conceal that some time regions contain few observations or subjects. A median or quantile based on many subjects is generally more descriptive of the sampled group than the same summary based on only one or two subjects. A sparse region is not automatically wrong or unusable, but it supports less stable visual conclusions and may fail to represent the intended population.
Worked Example: A Late-Time Median Based on Two Subjects
At 1 hour, a hypothetical study has concentrations from 30 subjects. At 12 hours, only two subjects have measured concentrations, 0.4 and 1.6 mg/L. Calculate the 12-hour median and explain what the sample size allows the plot to show.
For two ordered observations, the median is their average: \((0.4+1.6)/2=1.0\) mg/L. The plotted median is therefore 1.0 mg/L, but it is based on only two subjects. It does not show that most of the 30 subjects had a concentration near 1.0 mg/L at 12 hours, because 28 subjects contributed no measurement at that time.
The appropriate reading is that the two observed concentrations at 12 hours straddle a median of 1.0 mg/L. The plot should show the time-specific count, retain the individual observations where possible, and avoid implying the same coverage at 1 and 12 hours. Interpolation between these sparse observations would add an appearance of information that was not collected.
Time bins can help reveal uneven coverage, but bin width and boundaries also matter: changing them can change which observations are summarized together. Inspect the records within a bin and the number of distinct subjects, not only the summary line. A wide interval may mix concentrations from meaningfully different times, while a narrow interval may leave too few observations for a stable summary.
Time Confounding: Groups May Be Observed at Different Times
Concentration changes over time, so comparing groups without accounting for when their observations were collected can mix a time pattern with a group pattern. This issue is especially easy to miss when a plot places all observations from each group on the same panel or when groups have little overlap in their sampling schedules. A visible separation does not isolate a formulation, route, or covariate difference if one group is mostly observed earlier than the other.
Worked Example: Group and Time Change Together
In a hypothetical display, Group A is observed only at 1 hour, with a mean concentration of 12 mg/L. Group B is observed only at 8 hours, with a mean concentration of 3 mg/L. The apparent difference is \(12-3=9\) mg/L. Can that difference be attributed to group membership?
No. The calculation compares a Group A mean at 1 hour with a Group B mean at 8 hours; group and sampling time change together. Because concentration can change between 1 and 8 hours, these data do not separate a group difference from a time difference. The 9 mg/L contrast is a descriptive difference between the displayed sets of observations, not an isolated group effect.
A useful next display is to compare groups within time regions where both have observations and to show the number of subjects in each region. If the groups have no overlapping sampling times, a plot cannot resolve the confounding by itself. The study design and data collection determine what comparison is supported.
The key diagnostic is coverage: do the groups contain observations at comparable times, and are those observations contributed by comparable numbers of subjects? Stratifying by time can expose a mismatch that an overall overlay hides. It does not guarantee that all other group differences are absent, but it makes this particular ambiguity visible.
A Practical Diagnostic Sequence
Before describing an apparent pattern, use a consistent sequence of checks. These steps are useful across concentration-time plots, covariate displays, and comparisons across formulations or routes.
Determine whether marks are individual concentration records, subject summaries, or another quantity. Check subject and observation counts.
Look for shared coordinates and compare group counts, including how many distinct subjects contribute to each region.
Confirm units, linear or logarithmic scaling, panel-specific limits, and how zero, below-quantification, or clipped values are represented.
Find sparse regions, uneven sampling, and whether groups overlap in sampling time before comparing their concentrations.
Describe what is visible and its data support. Do not convert a descriptive pattern into a causal or inferential claim without appropriate evidence.
A useful justification for this sequence is that a plotted summary is computed from the observations that enter it. If those observations are obscured, unequally weighted, on a misunderstood scale, sparse, or collected at different times, the displayed feature may not answer the intended question. Making the contributing data visible is therefore part of interpreting the plot, not merely a presentation preference.
Common Mistakes and Alternatives
- Counting visible dots as subjects. Overlapping observations can hide records, and a subject can contribute several records. Check both counts rather than inferring either from marker density.
- Calling a denser group “higher.” More marks can reflect more subjects or more samples, not higher concentrations. Compare concentration values and group coverage separately.
- Ignoring axis transformations or limits. Vertical distances have different meanings on linear and logarithmic scales. Read labels and report differences in context and units.
- Treating a sparse summary as representative. A summary based on two subjects does not establish what most of a larger group experienced. Display its count and the underlying observations.
- Attributing a time-mismatched contrast to group membership. If groups were sampled at different times, the plot does not distinguish group from time patterns. Compare overlapping time regions where available, and acknowledge when they are not available.
There is no single alternative display that solves every problem. Individual profiles make subject contributions visible but can become crowded. Transparency and point separation can reveal overlap, but may not communicate exact counts. Time-binned summaries organize a dense sampling window, but can hide within-bin variation and depend on bin choices. Tables of counts clarify coverage but do not show concentration patterns. A practical presentation often pairs a readable summary with individual observations or counts, chosen to address the specific ambiguity.
The limitation is fundamental: an exploratory plot displays the data collected, not the unobserved concentrations or the cause of a pattern. Better plotting can uncover weaknesses in coverage and comparison, but cannot create missing time overlap, increase the number of subjects, or remove differences in study design. Interpretations should remain descriptive when those limits matter.
Check Your Understanding
Answer each question using the checks described above. Then compare your reasoning with the solutions.
- A plot contains 80 observations but 56 distinct visible marker locations. How many observations overlap exactly, and what proportion of all observations is that?
- Two groups have means of 4 mg/L and 10 mg/L, based on 3 and 9 observations, respectively. Calculate the pooled mean and the equally weighted average of the group means. Explain why they differ.
- A group median is displayed at 16 hours, but only one subject has an observation there. What can the plotted median establish about the group at that time?
- One route is sampled mainly at 1 hour and another mainly at 6 hours, with no shared sampling times. What is the main interpretive limitation of comparing their plotted concentrations?
Solutions. (1) \(80-56=24\) observations overlap; \(24/80=0.30\), or 30%. (2) The pooled mean is \((3\times4+9\times10)/(3+9)=(12+90)/12=8.5\) mg/L. The equally weighted average is \((4+10)/2=7.0\) mg/L. They differ because the pooled mean gives three-quarters of its weight to the group with nine observations, whereas the second calculation gives each group half the weight. (3) The median equals that subject's observed concentration; it describes the single observed subject, not the distribution or typical concentration of the group. (4) Route and sampling time are entangled, so the visual contrast cannot distinguish a route pattern from a time pattern.