What Exploratory Covariate Plots Can Show
Before fitting or changing a covariate model, plots can reveal whether a candidate covariate has a plausible descriptive relationship with observed concentrations or with an individual-level summary. They can expose curvature, outliers, sparse regions, and differences between categories that a single numerical summary would conceal. Their role is to help formulate and scrutinize hypotheses, not to decide whether a covariate has an independent effect.
A parameter-independent plot is constructed from recorded data and summaries calculated from those data, without using parameter estimates from a fitted NLME model. For example, a plot can show observed concentrations against body weight, or each participant’s dose-normalized AUC against a categorical renal-function group. “Parameter-independent” does not mean free of assumptions: the meaning of a summary still depends on dose, timing, and how it was calculated.
Let \(Y_{ij}\) denote the observed concentration for individual \(i\) at observation \(j\), and let \(x_i\) denote a continuous covariate such as body weight. A plot of \(Y_{ij}\) against \(x_i\) displays all measured concentrations; a plot of a summary \(S_i\), such as an individual’s dose-normalized AUC, against \(x_i\) displays one value per individual. For a categorical covariate, the corresponding displays compare concentrations or summaries across its levels.
Choose the Display to Match the Data
For a continuous covariate, a scatterplot is usually the starting point. Plot individual-level summaries against the covariate when the question concerns between-individual differences in those summaries. Show the individual observations, label the axes with units, and inspect the actual data density: a smooth curve can be misleading where few individuals are observed. A reference line or a descriptive smoother may help reveal a broad pattern, but it is not an estimate of an adjusted covariate effect.
If raw concentrations are plotted against a covariate, repeated measurements from one individual share the same covariate value. Points will therefore form vertical stacks or bands. Use transparency, jitter where appropriate, or a small-multiple display to reveal overlap. When concentrations are strongly right-skewed or vary multiplicatively, a logarithmic vertical axis may make relative differences easier to see. State the scale clearly; visual distances on a log scale represent ratios rather than equal absolute concentration differences.
For a categorical covariate, plot one individual-level summary per individual using, for example, a dot plot, boxplot, or violin plot with the individual observations visible where possible. Show group sizes. A boxplot summarizes the distribution within each group, but can hide small sample sizes and clusters of tied values. When the levels are ordered, retain that order on the horizontal axis; for unordered categories, choose a clear and consistent display order.
Concentration-time data need special care. Concentration depends on time since dose, dose amount, dose history, and sampling schedule as well as on individual characteristics. A plot that puts every concentration on the vertical axis and a covariate on the horizontal axis can mix measurements taken at very different times. Consider plotting by selected sampling times, using concentration-time panels by covariate group, or first examining a clearly defined individual-level summary. Do not compare apparent group differences as if sampling times were interchangeable.
Worked Example: A Dose-Normalized Summary Versus Body Weight
Suppose four hypothetical participants each receive the same dose. Concentrations are measured at times 0, 2, and 6 hours. For participant A, the hypothetical concentrations are 0, 8, and 4 mg/L. Calculate the trapezoidal AUC from 0 to 6 hours, divide by dose, and describe how this value could be plotted against weight.
For adjacent observations, the trapezoidal area is the time interval multiplied by the average of the two endpoint concentrations. Thus, for participant A:
If the hypothetical dose is 100 mg, the dose-normalized summary is \(32/100=0.320\ \mathrm{h/L}\). Suppose the four participants have weights 50, 60, 70, and 80 kg and dose-normalized AUC values 0.320, 0.350, 0.370, and 0.410 h/L, respectively. A scatterplot places weight in kilograms on the horizontal axis and dose-normalized AUC in h/L on the vertical axis. The points rise across these four observations, a descriptive pattern that may motivate further investigation.
This calculation is an approximation based on the stated sampling times, and the plot does not show an adjusted effect of weight. It does not account for other covariates, differences in dose history, or uncertainty in the individual summaries. The four-point pattern is not evidence by itself that increasing weight causes AUC to increase.
Compare Individuals, Not Just Measurement Counts
A raw-concentration plot and an individual-summary plot answer different descriptive questions. The raw plot shows the observed measurements and their distribution across sampling times. The summary plot gives each individual one contribution, which can make between-individual patterns easier to inspect. Neither plot automatically controls for differences in sampling or other covariates.
The importance of the unit of display can be seen directly from the arithmetic. If individual \(i\) contributes \(m_i\) observations with mean concentration \(\bar{Y}_i\), then the mean across all observations is
This is a measurement-weighted mean: individuals with more observations receive more weight. The unweighted mean of individual means, \(\sum_i\bar{Y}_i/n\), gives each individual equal weight, but it is not necessarily a meaningful overall concentration summary either, especially when sampling times differ. The two quantities answer different questions; neither should be substituted for the other without explanation.
Worked Example: Unequal Numbers of Concentration Measurements
At one nominal sampling time, suppose three hypothetical participants have observed concentrations as follows: participant A has 8, 10, 9, and 11 mg/L; participant B has 15 mg/L; participant C has 6 mg/L. Calculate the pooled measurement mean and the mean of the three participant means.
Participant A’s mean is \((8+10+9+11)/4=9.5\) mg/L. Participants B and C have means of 15 and 6 mg/L. The pooled mean is
The unweighted mean of the three participant means is
These differ because participant A contributes four measurements while each other participant contributes one. A plot with all six points makes the measurement distribution visible; a plot with one mean per participant gives each participant one displayed value. Before interpreting either display, verify that the observations really are comparable in sampling time and context. If they are not, these arithmetic means do not represent a common-time concentration comparison.
Comparisons Across Categories
For a categorical covariate, displaying individual summaries helps distinguish a shift in the group center from differences in spread or isolated observations. Always show the number of individuals in each category, since a visually striking pattern based on only a few people is unstable. If group sizes differ greatly, avoid relying on overlapping boxplots alone; add the individual points or use another display that makes sample sizes apparent.
Worked Example: Dose-Normalized AUC Across Two Groups
Consider a hypothetical study with four participants in each of two renal-function categories. Their dose-normalized AUC values, in h/L, are:
Category 1: 0.24, 0.28, 0.32, 0.36.
Category 2: 0.30, 0.34, 0.38, 0.42.
The median for Category 1 is \((0.28+0.32)/2=0.300\) h/L. The median for Category 2 is \((0.34+0.38)/2=0.360\) h/L. The difference between these sample medians is \(0.360-0.300=0.060\) h/L. A dot plot would show the eight individual values and their category; a boxplot could summarize the distributions.
The display suggests higher dose-normalized AUC values in Category 2 in this small hypothetical sample. It does not establish that renal-function category has an adjusted association with AUC: the groups could differ in other covariates, dose conditions, or sampling. The median difference describes these observed summaries only.
Why a Visible Pattern Is Not an Adjusted Effect
A plot compares what was observed, often one covariate at a time. An adjusted effect asks a different question: how the model’s description of the response changes with a covariate while accounting for other modeled factors. The visible pattern can be distorted or even reversed when covariates are related to one another, or when the design allocates different doses, sampling times, or study sites to different covariate groups. This is why the earlier material on separating covariate effects from study-design confounding matters when interpreting exploratory displays.
Plots also have limits as evidence. The apparent slope can depend on a few influential individuals, the choice of scale, or sparse data at the extremes. A smoother can conceal gaps in covariate support, and a group-level display can conceal substantial overlap between individuals. Measurement error in covariates, missing observations, and uncertainty in calculated individual summaries are not resolved simply by drawing a graph.
Use plots alongside knowledge of the data collection and the covariate support, not instead of them. A visual pattern can help prioritize a candidate for evaluation, flag data-quality concerns, or suggest that a linear relationship may be inadequate. It cannot supply a formal test, establish causation, or guarantee that a covariate should be retained in a model.
Diagnostics and Common Pitfalls
Before drawing a conclusion from a plot, check that its construction matches its interpretation. Confirm the covariate and outcome units, identify whether points represent measurements or individuals, and verify that the summary’s dose and time window are stated. For raw concentrations, inspect sampling times and dose histories; for summaries, check how many observations contributed to each value and whether the calculation is comparable across individuals.
- Unequal sampling treated as equal evidence. A cloud with many points from one participant may look persuasive, but those points are not many independent individuals. Show individual identity or use an individual-level display for between-person patterns.
- Different sampling times compared as one outcome. Concentrations at early and late times are not interchangeable. Stratify or otherwise display timing rather than attributing the difference to the covariate.
- A visual trend called an adjusted effect. A one-covariate plot does not control for other covariates or design features. Describe the pattern as unadjusted and exploratory.
- A smooth curve mistaken for data support. Check the number and distribution of individuals across the covariate range. A smooth line cannot create information where observations are sparse.
- Different scales interpreted alike. On a logarithmic concentration axis, equal vertical distances correspond to equal ratios, not equal concentration differences. State the scale and describe the pattern accordingly.
The main alternatives are complementary rather than interchangeable. Raw concentration-time displays retain measurement-level detail; individual-summary plots emphasize between-person summaries; fitted-model plots can present conditional or adjusted model-based relationships but depend on a model and its estimates. Exploratory plots are useful before fitting or revising that model precisely because they do not depend on its parameter estimates, but that independence limits what they can establish.
Decide whether a point is one observed concentration or one individual-level summary, and label the plot accordingly.
Review dose, sampling time, summary calculation, group size, and covariate support before comparing patterns.
Report the observed direction, shape, overlap, and limitations as exploratory evidence, not as an adjusted effect.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why can a raw concentration-versus-weight plot give disproportionate visual influence to some individuals?
- A plot of individual dose-normalized AUC against body weight rises across the observed range. What can and cannot be concluded from this pattern?
- Two covariate categories have different concentration distributions, but one category was sampled mostly early after dosing and the other mostly late. What is the main interpretive problem?
- On a logarithmic concentration axis, what does an equal vertical distance represent?
- Why should a categorical comparison show individual points or group sizes in addition to boxplots?
Solutions. (1) Individuals may contribute unequal numbers of measurements, so the raw display contains more points for some people; repeated points do not represent additional independent individuals. (2) The plot describes an unadjusted pattern in the observed summaries and may motivate further evaluation, but it does not establish an adjusted effect or causation. (3) Sampling time is entangled with category, and concentrations at different times are not directly comparable as if they measured the same condition. (4) Equal vertical distances represent equal concentration ratios, not equal absolute differences. (5) Boxplots can hide small group sizes, ties, and individual overlap; points and counts make the underlying data and support more visible.