Pairwise Covariate Plots for Dependence and Coverage

Why Examine Covariates in Pairs?

A study can have broad coverage of each covariate considered separately and still contain few participants with particular combinations of covariates. For example, age and body weight may each span a wide range, but older participants might mostly be heavier, leaving few older, lighter participants. That joint pattern matters when describing the study population and considering what comparisons the data can support.

Pairwise covariate displays are exploratory tools for describing dependence, clustering, overlap, and gaps in observed combinations. They do not establish that one covariate causes another, and they do not decide which covariates belong in a pharmacokinetic model. The aim here is to diagnose the structure and coverage of the observed covariate data before model development.

Choose the Display for the Covariate Types

For two continuous covariates, a scatterplot places one covariate on each axis, with one point per participant. Inspect the overall direction and shape, clusters, gaps, unusual points, and whether the spread changes across the range. Use meaningful units and clear axis labels. A transparent or lightly jittered display can help when many participants have the same recorded values.

For two categorical covariates, use a two-way count table, a heat map, or a mosaic-style display. Counts show how much data support each combination; row or column percentages show composition relative to a stated denominator. A zero count is a genuine gap in the observed combination space, while a small positive count indicates sparse coverage. Percentages without counts can hide how little information a cell contains.

For a continuous and a categorical covariate, show the continuous values separately by category, for example with side-by-side boxplots or jittered points overlaid with a summary. This can reveal overlap, shifts, skew, and within-category spread. Always show group sizes: an apparently stable distribution based on a few participants deserves more caution than one based on many.

When the full set of covariates is large, a scatterplot matrix can provide a compact overview of continuous pairs. It should complement, not replace, targeted plots with readable scales and sample counts. For each pair, state the number of participants with usable values for both covariates. If missingness differs across pairs, the plots may describe different subsets of the analysis population.

Correlation as a Summary, Not a Substitute for the Plot

For paired quantitative observations \((x_i,y_i)\), \(i=1,\ldots,n\), the sample Pearson correlation \(r\) summarizes the direction and strength of their linear association. Let \(\bar{x}\) and \(\bar{y}\) be the sample means. Provided neither variable is constant, define:

The value is unitless, between \(-1\) and \(1\). A positive value indicates that larger values of one variable tend to occur with larger values of the other in a linear pattern; a negative value indicates a tendency for larger values of one to occur with smaller values of the other. Values near zero indicate little linear association, not necessarily an absence of other dependence.

Pearson correlation is a descriptive calculation: it does not require normally distributed covariates. Its interpretation as a summary of linear association is most useful when the plot is reasonably linear and not dominated by influential outliers. If making population-level inferences about correlation, additional sampling and distributional assumptions are needed; an exploratory plot and coefficient alone do not provide such an inference.

The limitation follows directly from the formula: it summarizes centered cross-products and cannot encode the full shape of a relationship. A curved pattern can have correlation near zero, and a single extreme point can strongly alter \(r\). Rank correlation, commonly Spearman correlation, is Pearson correlation applied to the ranks of the observations. It can summarize a monotone association that is not linear, but it also compresses the data to one number and should be interpreted alongside the plot.

Read Sparse Combinations Directly

A count display makes joint coverage visible without assuming a particular association measure. For continuous covariates, a plot can show gaps, but a two-dimensional grid may help quantify them. Any such grid depends on chosen cutpoints: report those cutpoints, retain counts, and avoid implying that values within a bin are identical. Binning is a display choice, not a new measurement of the covariate.

Data Structure, Diagnostics, and Common Pitfalls

In population PK data, covariates such as baseline age, baseline weight, or genotype are often participant-level variables, while concentration records may be repeated within a participant. For a pairwise display of participant-level covariates, use one appropriately defined record per participant. Repeating the same baseline covariate on every concentration row would give participants with more records greater visual and numerical weight. For time-varying covariates, decide which time point or summary is scientifically relevant, or display within-participant variation separately rather than treating repeated rows as independent participants.

Before interpreting a pairwise display, check whether both variables are measured on the same analysis population and whether missing values have been handled transparently. Confirm the units, category labels, and any transformations. Look for data errors, but do not remove an unusual observation merely because it weakens a pattern; verify it against source information and retain it unless a documented data-quality rule justifies correction or exclusion.

A major pitfall is treating a small Pearson correlation as evidence that two covariates are unrelated. A U-shaped relationship can have little linear correlation despite clear dependence in the scatterplot. The correction is to inspect the shape and, where appropriate, consider rank correlation as a supplementary summary. Conversely, a large absolute correlation is not a causal result and does not by itself dictate model structure.

Another pitfall is assigning arbitrary numbers to categories and calculating Pearson correlation as though those codes were measurements. Coding categories as 1, 2, and 3 creates an artificial order and spacing unless those properties are genuinely meaningful. Use count displays for two categorical variables and grouped displays for mixed variable types.

Pairwise summaries also have limits. They can reveal pairwise structure but may miss relationships that occur only among three or more covariates. Sparse cells can arise from real study design, recruitment patterns, or the chosen categorization. Correlation can be affected by restricted ranges, outliers, and measurement error. These displays describe the observed data and their coverage; they do not establish causation, resolve confounding, or perform covariate selection.

A Practical Pairwise Review