Correlation Between Interindividual And Interoccasion Effects

Why Correlation Between Random-Effect Levels Matters

A subject-level random effect represents a persistent difference between a person and the population. An occasion-level random effect represents a difference associated with one treatment occasion for that person. These effects need not be unrelated. For example, subjects with relatively high typical clearance might also show systematically higher or lower clearance on a particular occasion, relative to their own typical value. A model that assumes zero correlation rules out such a pattern; a model that estimates correlation allows it, but asks more of the data.

The key issue is not merely whether each variance can be estimated. A covariance model must distinguish the persistent component, the occasion-specific component, and their association. Repeated occasions are important, but repetition alone does not guarantee that all three quantities are separately identifiable.

A Joint Model for Subject and Occasion Effects

Consider a positive PK parameter \(\theta_{ij}\), such as clearance, on the log scale. A simple repeated-occasion model is

Here \(\theta_{\mathrm{pop}}\) is the typical value on the original scale, while \(\eta_i\) and \(\kappa_{ij}\) are mean-zero effects on the log scale. A joint normal model for one subject–occasion pair can be written

The subscripts \(I\) and \(O\) denote interindividual and interoccasion variability. Thus \(\omega_I^2=\operatorname{Var}(\eta_i)\), \(\omega_O^2=\operatorname{Var}(\kappa_{ij})\), and the covariance is \(\rho\omega_I\omega_O\). The matrix must be positive semidefinite, which requires \(-1\leq\rho\leq1\); a nondegenerate bivariate normal model uses \(-1<\rho<1\).

A complete repeated-occasion specification must also state how occasion effects relate across occasions. One coherent construction is

where \(\omega_I>0\), and the \(\varepsilon_{ij}\) are mutually independent and independent of \(\eta_i\). This construction gives the stated marginal variance and covariance for each pair \((\eta_i,\kappa_{ij})\). It also makes clear that, when \(\rho\neq0\), occasion effects for the same subject are not marginally independent: they share a component associated with \(\eta_i\). The model must account for that dependence rather than silently treating all occasion effects as mutually independent.

What the Correlation Does to Repeated Occasions

The random deviation in log clearance on occasion \(j\) is \(d_{ij}=\eta_i+\kappa_{ij}\). Its variance within one occasion follows by expanding the variance of a sum:

The covariance term appears twice because \(\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)\). Positive correlation increases the variability of the sum; negative correlation can reduce it. Consequently, a fitted total variability cannot be interpreted as the simple sum of the two component variances unless the covariance is zero.

Under the conditional construction above, the cross-occasion covariance of the total deviations is also informative. Substituting the construction into \(d_{ij}\) gives

For two distinct occasions \(j\neq k\), the independent \(\varepsilon\) terms have zero covariance with each other and with \(\eta_i\). Therefore,

The within-occasion variance exceeds this cross-occasion covariance by \((1-\rho^2)\omega_O^2\), the variance of the occasion-specific innovation in this construction. Correlation therefore changes both the total variability on an occasion and the dependence between a subject’s occasions.

When Can the Correlation Be Distinguished?

The assumptions needed for interpreting \(\rho\) include a suitable random-effect distribution, a correct mapping from effects to the PK observations, and a specified dependence structure across occasions. The design must also provide information that separates the three covariance parameters. At minimum, subjects generally need repeated occasions to reveal persistent versus occasion-varying patterns, and the occasions must contain informative concentration data for the parameter being modeled. If occasion effects are almost entirely hidden by sparse or uninformative sampling, their variance and covariance will be weakly informed.

There is a more fundamental limitation in a simple scalar model. If the observed data inform only the distribution of \(d_{ij}=\eta_i+\kappa_{ij}\), then within-occasion variance and cross-occasion covariance provide only two distinct covariance quantities. The model has three unknowns: \(\omega_I^2\), \(\omega_O^2\), and \(\rho\). Repeated measurements can estimate those two quantities precisely without identifying all three components. Additional information is needed—for example, other well-informed PK parameters whose random effects distinguish subject and occasion patterns, a design that separately informs the persistent and occasion components, or a justified constraint on part of the covariance model.

This is structural identifiability, not simply sample size. More subjects or more precise concentrations can reduce uncertainty when the model is identifiable, but cannot create a third independent covariance feature when the likelihood contains only two. Conversely, a rich design does not guarantee a useful estimate if the model is misspecified or the correlation is weakly informed.

Model Choice, Diagnostics, and Common Pitfalls

A diagonal structure sets \(\rho=0\), reducing the number of covariance parameters and often improving stability. It is appropriate when zero association is a defensible assumption or when the data cannot support estimation of the covariance. A correlated structure is more flexible, but should be used only when the data and design provide information to estimate the additional parameter. Fixing \(\rho\) to a scientifically justified value is another possibility, but it is an assumption and should be reported as such. The choice is not resolved by selecting whichever model returns a converged fit.

Useful checks include whether the estimated covariance matrix is positive definite, whether the correlation estimate is close to \(1\) or \(-1\), and whether its uncertainty is large or strongly dependent on other covariance estimates. Inspect likelihood profiles or other uncertainty summaries, and assess whether materially different covariance values produce nearly indistinguishable fits. Simulation-based checks using the actual design can reveal whether the proposed parameters can be recovered: simulate data at plausible values, refit the model, and examine bias, variability, and convergence.

Empirical Bayes estimates can help visualize subject and occasion patterns, but their shrinkage and estimation error mean their sample correlation is not automatically an estimate of the population correlation. The likelihood-based random-effects model, with appropriate uncertainty assessment, is the basis for inference. A numerical correlation reported by software is not evidence that the data identified it.

• Pitfall: adding variances and ignoring covariance. The incorrect calculation is \(\operatorname{Var}(\eta_i+\kappa_{ij})=\omega_I^2+\omega_O^2\) for all models. The correction is to include \(2\rho\omega_I\omega_O\), unless the model explicitly sets \(\rho=0\).
• Pitfall: treating repeated occasions as proof of identifiability. Repetition supplies cross-occasion information, but a scalar total may still provide too few distinct covariance features to estimate three parameters. Check identifiability under the full observation model and design.
• Pitfall: interpreting correlation as causation. A nonzero \(\rho\) describes an association between random effects in the specified model. It does not establish a mechanism causing one effect to change the other.