Why Plot Concentration Against a Continuous Covariate?
A concentration-versus-covariate scatterplot places each recorded concentration on one axis and a continuous covariate on the other. It can reveal a trend, curvature, changing spread, clusters, or unusual observations that may be obscured by looking at the concentration-time profiles alone. But concentration depends strongly on sampling time. A pooled cloud can therefore show an association even when concentrations do not vary with the covariate among observations collected at comparable times.
For observation \(j\) from subject \(i\), let \(C_{ij}\) denote concentration, \(z_{ij}\) the continuous covariate, and \(t_{ij}\) the sampling time. A plot of \(C_{ij}\) against \(z_{ij}\) is a descriptive display of the observed pairs. It is not, by itself, an estimate of a causal effect or proof that the covariate explains concentration differences. Sampling design, repeated measurements, dose, visit, and other covariates can all contribute to the visible pattern.
Constructing a Useful Display
First identify what each point represents. If the covariate is measured once per subject, such as baseline body weight, the same \(z_i\) may appear beside several concentrations from that subject. Those points are repeated observations, not independent subjects. If the covariate changes over time, use the value measured at the relevant occasion and make that choice explicit. Subject identifiers can help show repeated records, but connecting every point with a line may create a confusing display when observations occur at different times or after different doses.
Put the covariate on the horizontal axis and concentration on the vertical axis unless a particular scientific question motivates another orientation. Label both axes with units. Use a linear concentration scale when absolute differences are the focus; a linear scale preserves absolute differences and can include zero, depending on the chosen axis limits. A logarithmic concentration scale can make multiplicative differences or a wide positive range easier to inspect. A logarithmic scale cannot display zero or negative values, so values below quantification require an explicit display convention rather than an unexamined logarithm.
Sampling time should be represented in a way that leaves the concentration-covariate relationship visible. Color or symbol can distinguish a small number of prespecified time windows in one panel. Separate panels can be clearer when points overlap heavily or time groups differ substantially in concentration range. Exact sampling times may be more informative than broad bins when there are enough observations; bins can conceal patterns within a window and their boundaries should not be chosen after searching for the strongest apparent trend. When sample sizes are small, report the number of observations in each time window and avoid interpreting an empty or sparsely populated part of a panel.
Repeated observations from the same subject can make a point cloud look denser than the number of independent subjects warrants. Use transparency, subject-specific symbols where practical, or a companion display that makes the repeated structure clear. A smooth curve may guide the eye, but it does not resolve dependence, confounding, or sparse coverage. Treat it as a visual aid, not as a fitted population pharmacokinetic relationship.
Why a Pooled Pattern Can Be Misleading
A useful justification comes from decomposing covariance by sampling time. Suppose \(C\) and \(Z\) represent concentration and a covariate for a randomly selected observation, and \(T\) is that observation's sampling time or time stratum. Assume the variables have finite means and variances. The overall covariance consists of the average covariance within time strata plus the covariance between the time-specific means:
The first term summarizes within-time association. The second is present when time strata with different mean concentrations also have different mean covariate values. Thus the pooled covariance can be nonzero even if the within-time covariance is zero.
To verify the identity, write the conditional covariance as \(E[CZ\mid T]-E[C\mid T]E[Z\mid T]\). Taking expectations gives \(E[CZ]=E[\operatorname{Cov}(C,Z\mid T)]+E[E[C\mid T]E[Z\mid T]]\). Subtract \(E[C]E[Z]\) from both sides. By iterated expectation, \(E[E[C\mid T]]=E[C]\) and \(E[E[Z\mid T]]=E[Z]\), so the remaining mean-product term is exactly \(\operatorname{Cov}(E[C\mid T],E[Z\mid T])\). This explains why the pooled plot and the time-stratified plots may tell different descriptive stories.
Worked Example: A Pooled Association Created by Sampling Time
Consider eight hypothetical observations. The covariate is body weight in kilograms and concentration is in milligrams per liter. Four observations were collected in an early window and four in a late window.
| Window | Body weight (kg) | Concentration (mg/L) |
|---|---|---|
| Early | 40 | 9 |
| Early | 50 | 11 |
| Early | 60 | 11 |
| Early | 70 | 9 |
| Late | 80 | 1 |
| Late | 90 | 3 |
| Late | 100 | 3 |
| Late | 110 | 1 |
Within the early window, the means are \(55\) kg and \(10\) mg/L. The sum of products of deviations is \((-15)(-1)+(-5)(1)+(5)(1)+(15)(-1)=0\), so the within-window covariance, using the divisor \(4\), is \(0\). Within the late window, the means are \(95\) kg and \(2\) mg/L; the same calculation gives covariance \(0\).
Across all eight observations, the means are \(75\) kg and \(6\) mg/L. The time-specific means differ: the early window has lower mean weight but higher mean concentration. The covariance of the two time-specific means, weighted equally because each window has four of the eight observations, is
Because both within-window covariances are zero, the covariance decomposition gives an overall covariance of \(-80\ \text{kg}\cdot\text{mg/L}\). The pooled plot would show a negative association, although neither time window has a linear within-window association in these data. The display should therefore show time, and the pooled trend should not be described as evidence that greater body weight is associated with lower concentration at comparable sampling times.
Assessing Whether a Pattern Persists Within Time
Time-stratified plots help separate two questions: do concentration and covariate vary together within comparable sampling times, and do the time windows themselves differ in concentration? A trend that appears in several well-populated time windows is more consistent with a within-time association than a trend that is visible only in the pooled cloud. Even then, the plot remains descriptive. Other differences between subjects or occasions may account for the pattern, and the plotted observations may be repeated within subjects.
Worked Example: A Similar Trend in Two Time Windows
In a second hypothetical dataset, three observations in each of two sampling-time windows have body weights of 40, 60, and 80 kg. Concentrations in the early window are 8, 7, and 6 mg/L; in the late window they are 4, 3, and 2 mg/L, respectively. Consider the descriptive least-squares slope within each window.
For the early window, the means are \(60\) kg and \(7\) mg/L. The slope is the sum of cross-products divided by the sum of squared covariate deviations:
For the late window, the means are \(60\) kg and \(3\) mg/L. The concentration deviations are again \(1,0,-1\), so the slope calculation is also \(-40/800=-0.0500\) mg/L per kg. The time-stratified display would show a downward pattern within each window, alongside lower concentrations in the late window. That pattern is compatible with a within-time association in these observations, but it does not establish that body weight causes concentration to change. It also does not account for repeated observations or other covariates.
Coverage, Diagnostics, and Failure Modes
A time-stratified plot is most informative when the covariate ranges overlap across sampling times. If low covariate values occur only early and high values only late, the plot cannot show what concentrations would look like at the same time across that covariate range. This is a limitation of the observed design, not something a smoother or a different axis scale can fix. Check the covariate coverage by time window, using the coverage displays and pairwise covariate plots as useful context.
Look for an apparent trend that changes direction across time windows, a trend restricted to one sparse panel, isolated points that dominate the range, or a curved pattern hidden by a straight reference line. Also check whether one subject contributes many more records than others. If conclusions change substantially when a heavily sampled subject is visually distinguished, the pooled point cloud may be emphasizing that subject rather than a broad between-subject pattern. Document units, time-window definitions, the number of records, and whether the covariate is baseline or occasion-specific.
Worked Example: When Time and Covariate Coverage Do Not Overlap
Suppose a hypothetical dataset contains two early observations at 40 and 50 kg with concentrations of 10 and 8 mg/L, and two late observations at 90 and 100 kg with concentrations of 2 and 1 mg/L. The early mean covariate and concentration are \(45\) kg and \(9\) mg/L; the late means are \(95\) kg and \(1.5\) mg/L. The difference in mean concentration is \(1.5-9=-7.5\) mg/L, while the time windows also differ by \(50\) kg in mean body weight.
There are no observations at similar weights in both windows. A pooled plot would show lower concentrations at higher weights, but time and weight are completely separated in these data. There is no within-window range of weights from which to assess a weight trend, and the plot cannot distinguish a time-related difference from a weight-related one. A correct interpretation is that this dataset does not provide overlap to assess their separate descriptive patterns; it is not that the plot proves either explanation.
Choosing Among Plotting Approaches
A pooled scatterplot is compact and useful for spotting the overall observed pattern, but it can conceal time structure. Coloring points by sampling-time window keeps all records in one frame and makes separation apparent, although colors may become hard to interpret when there are many windows or substantial overplotting. Faceting by time makes each window easier to inspect, but unequal axis limits can make visual comparisons deceptive; use common scales when comparing slopes or ranges across panels. Summaries within time bins can reduce clutter, but hide individual observations and may obscure sparse coverage or outliers. For exploratory work, it is often useful to inspect both the individual points and a carefully labeled summary rather than relying on a summary alone.
These displays answer different questions rather than competing to identify one definitive relationship. The pooled plot describes the observed joint distribution. The time-colored or faceted display asks whether the pattern is also visible within sampling-time strata. Neither replaces a model that accounts for the study design, repeated observations, and relevant covariates. Those analyses require assumptions and diagnostics beyond what a scatterplot can establish.
Common Mistakes
- Reading the pooled trend as a covariate effect. The wrong reasoning is that a downward pooled cloud proves concentration decreases as the covariate increases. The correction is to examine sampling-time patterns and coverage; the pooled association may arise from differences between time windows.
- Treating repeated records as independent subjects. Several points may belong to one subject. Make the repeated structure visible where possible and describe the display as observation-level unless it uses subject-level summaries.
- Using time windows with no meaningful overlap. A time-stratified plot cannot separate time and covariate patterns where the data provide no comparable observations. Show the lack of coverage and state the limitation.
- Overinterpreting a smoother. A smooth line is a visual aid, not proof of a causal or population-level relationship. Check the points, time strata, and sample sizes behind it.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A pooled plot shows lower concentration at higher body weight, but early samples have lower weights and higher concentrations than late samples. What should you examine before interpreting the pooled trend?
- Why can several concentration records from one subject not automatically be treated as several independent subjects?
- In a time-stratified plot, the same downward concentration-versus-covariate pattern appears in three adequately populated windows. What does this support, and what does it not establish?
- Low covariate values occur only at early times and high values only at late times. What conclusion about separate time and covariate patterns is justified?
Solutions. 1. Compare the pattern within sampling-time windows and check covariate overlap; the pooled association may reflect differences between windows. 2. Records from one subject share subject-level characteristics and are correlated, so the record count is not the independent-subject count. 3. It supports a consistent descriptive within-window association in the observed data, but does not establish causality or rule out other explanations. 4. The observed design lacks overlap, so the plot cannot distinguish separate time and covariate patterns over those ranges.