Subject Identity Creates Dependence Across Concentrations

Why Subject Identity Matters

Repeated pharmacokinetic profiles are clustered by subject: measurements from one person share more than a sampling schedule. A subject may have persistently higher or lower concentrations than another because of stable differences in factors such as body size, physiology, or drug handling. Consequently, two concentrations measured from the same subject can move together, even when they are taken at different times. This is a statement about dependence, not a claim that one concentration causes the other.

A simple way to study this idea is to consider two sampling occasions and retain the identity of the subject for each pair. If subjects with relatively high concentrations at the first occasion also tend to have relatively high concentrations at the second, the paired observations contain a within-subject association. Looking only at the two overall averages cannot show that association.

A Simple Model for Paired Concentrations

Let \(C_{ij}\) denote the concentration for subject \(i\) on occasion \(j\), where \(i=1,\ldots,n\) indexes subjects and \(j=1,2\) indexes the two occasions. Concentration is measured in a common unit, such as mg/L. A useful introductory model separates an occasion average, a stable subject-specific deviation, and a measurement-level residual:

Here, \(\mu_j\) is the average concentration at occasion \(j\), \(b_i\) is subject \(i\)'s stable deviation from the population average, and \(\epsilon_{ij}\) is the remaining occasion-specific deviation. The \(b_i\) term is shared by both measurements from the same subject. That shared term is the basic mechanism by which subject identity can create dependence.

Under these assumptions, concentrations from the same subject have a covariance contribution from \(b_i\). The occasion-specific means do not contribute to covariance because they are constants, and the residuals contribute none across occasions because they are independent. Thus:

The variance of either measurement is \(\tau^2+\sigma^2\). If the variances are the same at both occasions, the correlation is therefore:

This correlation is between 0 and 1 under the stated model. When stable differences between subjects are large relative to occasion-specific variation, the paired measurements tend to be strongly associated. When occasion-specific variation is large relative to stable differences, the association is weaker. A positive correlation does not mean the concentrations are equal; the means \(\mu_1\) and \(\mu_2\) may differ substantially.

Worked Examples

How to See and Describe Dependence

For two occasions, a scatterplot with one point per subject is a direct first check: plot occasion 1 against occasion 2, retaining each pair. A positive trend indicates that high values on one occasion tend to accompany high values on the other. A display of paired points connected by subject can also make it easier to see whether subjects maintain their relative ordering, while showing the direction and size of occasion changes.

A descriptive correlation is useful because it summarizes linear association on a unitless scale. It does not capture every possible dependence pattern. For instance, a curved relationship can be strong even when the linear correlation is near zero. Also, two occasion-specific means can differ while the paired correlation remains high: the means describe average levels, whereas correlation describes how paired deviations vary together.

A short derivation clarifies why the shared subject term is sufficient to create covariance in the simple model. Expand the covariance after subtracting the occasion means:

The three zero terms follow from the stated independence assumptions. If residuals at the two occasions are themselves correlated, the final term need not be zero. The covariance would then include that residual covariance in addition to the shared subject variance. This is one reason the simple formula is an explanatory starting point rather than a complete description of every profile.

Diagnostics, Alternatives, and Limits

Signs that the simple account may be inadequate include a paired scatterplot with curvature, markedly different spreads at the two occasions, or subject trajectories whose changes vary systematically rather than appearing as a shared level shift plus independent residual variation. A concentration assay may also have time-dependent measurement error, and subject differences may evolve across the profile. With more than two occasions, inspect the pattern at each occasion and how association changes with the time separation; one constant correlation need not fit every pair.

The main descriptive alternative is to analyze each occasion separately. That can summarize the average concentration at each occasion, but it discards which two values belong to the same subject and therefore cannot describe within-subject association. A paired summary instead keeps subject identity and can describe both the distribution of within-subject changes and the relationship between paired levels. These summaries answer different questions. Neither a scatterplot nor a correlation by itself explains the pharmacokinetic mechanism that produced the pattern.

The model also has limitations. It assumes one stable subject deviation shared equally across the two occasions, equal residual variances, and independent residuals. Actual concentrations may have subject-specific time trends, unequal variability, nonlinear time courses, or dependence that changes with the interval between samples. Correlation can also be affected by the range of subjects and occasions included; it is not an intrinsic constant of a drug independent of the data and design. The introductory material on repeated pharmacokinetic profiles as clustered data establishes why observations should be kept linked to subjects; the paired formulation here makes one specific consequence explicit.