How Sampling Times Change Parameter Correlations

Why Sampling Times Affect Parameter Correlations

A sampling schedule determines which parts of a concentration-time profile are observed. If the observed concentrations respond in similar ways to changes in two parameters, the data may not clearly distinguish those parameters. Their estimates can then be correlated: across repeated data sets, a change in one estimate tends to accompany a change in the other. A schedule that captures different features of the profile may change the strength, or even the direction, of that relationship.

Here, parameter correlation means correlation between parameter estimates from the same fitted model, conditional on the model and its assumptions. It is not automatically the correlation between individual patients’ parameter values, such as a random-effects correlation. Those are different quantities and answer different questions. Always identify which correlation matrix is being interpreted.

Definition: For estimated parameters $\hat{\theta}_1$ and $\hat{\theta}_2$, their estimated correlation is $$ \hat{\rho}_{12} = \frac{\widehat{\operatorname{Cov}}(\hat{\theta}_1,\hat{\theta}_2)} {\sqrt{\widehat{\operatorname{Var}}(\hat{\theta}_1)\widehat{\operatorname{Var}}(\hat{\theta}_2)}}. $$ It is dimensionless and lies between $-1$ and $1$. A positive value indicates that the estimates tend to vary in the same direction; a negative value indicates that they tend to vary in opposite directions. The value describes joint uncertainty under the fitted model, not a causal relationship.

A useful local explanation comes from linearizing the model near the fitted parameter values. Let $J$ be the matrix of derivatives of predicted concentrations with respect to the parameters, and let $W$ represent the observation weights. Under a locally linear model with independent, correctly specified residual errors and a well-behaved likelihood, the approximate information matrix is $J^{T}WJ$; the parameter covariance is approximately its inverse, with any residual-scale factor included as appropriate. Changing sampling times changes the rows of $J$, and therefore can change both the variances and covariances of the estimates.

\widehat{\operatorname{Cov}}(\hat{\theta}) \approx (J^{T}WJ)^{-1}

This approximation is local: it works best when the model is adequately specified, the fitted solution is not on a problematic boundary, and the likelihood is reasonably close to quadratic around its optimum. Population models use more elaborate likelihood calculations, but the same intuition applies: observations at different times contribute different information about model parameters. The earlier material on sampling windows and parameter precision motivates which profile features a schedule can capture; correlation adds the question of whether parameters are being estimated separately or in combination.

Reading Correlations Through the Concentration-Time Model

Consider a hypothetical one-compartment model after an intravenous bolus dose:

C(t)=\frac{\text{Dose}}{V}\exp\left(-\frac{CL}{V}t\right)

Here $V$ is volume of distribution and $CL$ is clearance. Early concentrations are strongly influenced by the concentration scale, which includes $V$. The terminal decline reflects the elimination rate $CL/V$. If observations mainly characterize that rate, several combinations of $CL$ and $V$ can produce similar slopes: increasing both together can leave their ratio nearly unchanged. Their estimates may therefore be positively correlated. This is a model-based explanation for a possible pattern, not a rule that every schedule must produce the same sign or magnitude.

For oral dosing, early samples can help describe absorption while later samples contribute to the elimination phase. If a schedule misses the transition between those features, changes in absorption and elimination may produce similar predictions at the observed times. Correlation can then increase. The pattern depends on the model, dose regimen, residual-error assumptions, and which parameters are estimated together.

Worked Example: Clearance and Volume Under a Terminal-Phase Schedule

Suppose a hypothetical fit reports estimates on the log scale for clearance and volume. The estimated covariance matrix is

\widehat{\operatorname{Cov}} \begin{pmatrix} \widehat{\log CL}\\ \widehat{\log V} \end{pmatrix} = \begin{pmatrix} 0.040 & 0.051\\ 0.051 & 0.090 \end{pmatrix}.

The diagonal entries are the estimated variances; the off-diagonal entry is the covariance. The correlation is

\hat{\rho}_{\log CL,\log V} = \frac{0.051}{\sqrt{0.040 \times 0.090}} = \frac{0.051}{\sqrt{0.0036}} = \frac{0.051}{0.060} =0.850.

This hypothetical fit has a strong positive correlation between the log-parameter estimates. If its sampling schedule is concentrated in the terminal phase, the pattern is consistent with clearance and volume partly compensating through their ratio $CL/V$. The correlation does not say that the parameters are biologically correlated across patients, nor does it establish that either estimate is imprecise by itself. It describes their estimated joint uncertainty.

Worked Example: Adding Samples That Characterize Absorption

Consider a hypothetical oral-dose analysis estimating absorption rate constant $k_a$ and clearance $CL$. Both are represented on the log scale. A sparse schedule with observations mostly after the early absorption period gives the following estimated covariance matrix:

\begin{pmatrix} 0.160 & 0.060\\ 0.060 & 0.040 \end{pmatrix}.

The estimated correlation is

\hat{\rho}_{\log k_a,\log CL} = \frac{0.060}{\sqrt{0.160 \times 0.040}} = \frac{0.060}{\sqrt{0.0064}} = \frac{0.060}{0.080} =0.750.

Now suppose an alternative hypothetical schedule, with the same number of samples, adds observations during the early rise and around the peak. Its fitted covariance matrix is

\begin{pmatrix} 0.100 & 0.012\\ 0.012 & 0.040 \end{pmatrix}.

For that schedule,

\hat{\rho}_{\log k_a,\log CL} = \frac{0.012}{\sqrt{0.100 \times 0.040}} = \frac{0.012}{\sqrt{0.004}} \approx \frac{0.012}{0.06325} =0.190.

The smaller positive correlation is consistent with early observations adding a feature of the profile that responds differently to absorption than later observations do. These matrices are hypothetical fit outputs, not values calculated from raw concentrations. The comparison illustrates how to interpret such outputs; it does not guarantee that adding early samples will always reduce correlation or improve every parameter’s precision.

When Correlation Changes Sign—or Barely Changes

The sign of a correlation is specific to the parameterization and the data pattern. A negative correlation can arise when increasing one parameter can be offset by decreasing another to preserve predictions at the observed times. For example, the concentration scale and rate of decline in a one-compartment profile may trade off differently depending on whether samples cover the early distribution of concentrations, the terminal slope, or only a narrow middle region. A positive correlation can arise when both parameters move together to preserve a feature such as a ratio. Neither sign has a universal biological interpretation.

Correlation also depends on scale. A correlation between log-transformed parameters describes the association of their log-scale estimates. A positive rescaling of a parameter, such as expressing clearance in different volume-per-time units, does not change the correlation. A nonlinear transformation can change it, however, so reports should state the parameterization when it matters.

Some parameter pairs remain strongly correlated despite changing sampling times. For example, with oral concentration data alone, bioavailability $F$ and volume $V$ may be difficult to separate when the observed concentration scale mainly reflects $F/V$. Moving samples around can describe the time course more fully without necessarily resolving that scale confounding. A different data source or design feature may be needed; sampling time alone cannot guarantee separate estimation.

Worked Example: Distinguishing a Weaker Correlation From Better Precision

Two hypothetical schedules estimate log-clearance and log-volume. Schedule A gives variances $0.010$ and $0.040$, with covariance $0.018$. Schedule B gives variances $0.090$ and $0.160$, with covariance $0.012$. Calculate each correlation and compare the uncertainty.

For Schedule A,

\hat{\rho}_A = \frac{0.018}{\sqrt{0.010 \times 0.040}} = \frac{0.018}{0.020} =0.900.

For Schedule B,

\hat{\rho}_B = \frac{0.012}{\sqrt{0.090 \times 0.160}} = \frac{0.012}{\sqrt{0.0144}} = \frac{0.012}{0.120} =0.100.

Schedule B has the smaller absolute correlation, but its estimated variances are much larger: $0.090$ versus $0.010$ for log-clearance, and $0.160$ versus $0.040$ for log-volume. Thus, the smaller correlation alone does not show that Schedule B estimates either parameter more precisely. Schedule A has more strongly linked estimates but smaller marginal variances in this hypothetical comparison. Correlation and precision must be read together.

Diagnostics, Comparisons, and Limitations

Before interpreting a reported correlation, check what matrix it came from and whether the estimates support the approximation. A correlation based on the covariance matrix of fixed-effect estimates is not interchangeable with a random-effects correlation matrix. Also check that the covariance matrix is symmetric and has nonnegative variances; a computed correlation outside $[-1,1]$ indicates inconsistent inputs or a numerical problem.

Very large absolute correlations, especially values near $1$ or $-1$, can flag that two estimates are strongly linked under the fitted model. They do not by themselves prove that the model is unidentifiable. Conversely, a modest pairwise correlation does not establish that the entire multivariate fit is reliable: several parameters can have complex joint relationships that are not captured by one pair. Check convergence, parameter boundaries, sensitivity to starting values, and whether the estimated covariance is stable. If the likelihood is notably asymmetric or the estimates are near boundaries, a local covariance approximation may be a poor summary.

For schedule comparisons, keep the model, dose information, assumed error model, participant structure, and total sampling effort fixed where possible. Then compare the same parameter pairs across schedules and inspect marginal uncertainty as well as correlations. Rich schedules may separate profile features within participants; balanced sparse schedules can distribute those features across participants. Either approach can change correlations, but the result depends on which times are observed and how the data are modeled.

Approach	What it contributes	Important limitation
Parameter-correlation matrix	Quick summary of pairwise joint uncertainty near the fitted solution	Does not show all multivariate structure or prove biological association
Comparing candidate schedules	Shows how alternative observation times change the estimated parameter relationships	Comparisons depend on shared modeling assumptions and the scenario used
Likelihood profiles or simulation-based assessment	Can reveal asymmetry or behavior not well summarized by local correlations	Requires additional computation and careful choice of scenarios

The central limitation is that a correlation is conditional on the model and data. If the structural model or residual-error assumptions are wrong, the fitted covariance matrix may give a misleading picture. Correlation is a diagnostic and interpretive aid, not a substitute for checking predictions, model assumptions, and parameter uncertainty.

Common Pitfall

Wrong reasoning: “The correlation between clearance and volume is $0.90$, so clearance and volume are strongly positively correlated in the population, and the schedule estimates them poorly.” This confuses parameter-estimate correlation with between-subject association, and it treats correlation as a direct measure of precision.

Correction: State which estimates or random effects are being correlated. For a parameter-estimate correlation, say that the two estimates have strongly linked uncertainty under this fit. Then inspect their variances or standard errors separately, and consider whether the schedule observes features that can distinguish the parameters. A high correlation can coexist with relatively small marginal uncertainty, and a low correlation can coexist with large uncertainty.

Key takeaway: Sampling times change which concentration-time features inform the fit, so they can change the direction and strength of parameter-estimate correlations. Interpret each correlation in the context of the parameterization, model, and schedule, and never use it alone as a measure of precision or as evidence of a biological relationship.

Check Your Understanding

Answer each question, then compare with the solutions below.

A fitted covariance matrix has variances $0.025$ and $0.100$, and covariance $-0.025$. What is the estimated correlation, and how should its sign be interpreted?
Why might observations concentrated in the terminal phase produce positively correlated estimates of $CL$ and $V$ in a one-compartment model?
An alternative schedule has a lower absolute parameter correlation but larger variances for both estimates. Does it necessarily give more precise estimates?
Does a correlation of $0.80$ between estimated fixed effects show that the corresponding individual random effects are positively correlated?
Name two checks to make before interpreting a near-$1$ parameter correlation.

Solutions. (1) $\hat{\rho}=-0.025/\sqrt{0.025 \times 0.100}=-0.025/0.050=-0.500$. The estimates tend to move in opposite directions under the fitted model. (2) Terminal slopes inform the ratio $CL/V$; increasing both can preserve that ratio, creating a positive relationship in their estimates. (3) No. Correlation is not precision; compare the marginal variances or standard errors as well. (4) No. Fixed-effect estimate correlation and between-subject random-effects correlation are distinct quantities. (5) Check the covariance matrix and parameterization, convergence and boundary behavior, and stability to starting values or alternative assessment such as a likelihood profile.