Distinguishing Residual Correlation From Occasion Variability

Why covariance patterns matter

Two concentration measurements taken during the same dosing occasion can move together for different reasons. A shared occasion-level change can shift many measurements in that occasion, while serial residual correlation can make measurements close in time more alike than measurements farther apart. Both mechanisms may produce positive covariance for a pair of nearby samples. Treating one as the other can therefore misrepresent how variability is distributed across occasions and sampling times.

The distinction is about the pattern of covariance, not merely whether covariance is present. A shared occasion effect tends to contribute covariance across all samples influenced by that effect. Serial correlation tends to depend on the time gap between samples and is usually confined to a sampling sequence. Earlier material on serially correlated residuals and occasion variability introduces these components; here the focus is how their covariance signatures compare.

A covariance model for repeated samples

Let \(Y_{ikj}\) be the observation for subject \(i\), occasion \(k\), and sampling time \(t_{ikj}\). Let \(m_{ikj}\) denote its model-predicted mean after conditioning on the subject-level random effects. Define the conditional residual as \(r_{ikj}=Y_{ikj}-m_{ikj}\). For a vector of residuals from one occasion, a useful decomposition is

Here \(\mathbf b_{ik}\) is an occasion-level random effect with mean zero and covariance matrix \(\boldsymbol\Omega_{\mathrm{occ}}\); \(\mathbf Z_{ik}\) describes how that effect changes each observation. The vector \(\boldsymbol\epsilon_{ik}\) contains residual errors, with covariance matrix \(\mathbf R_{ik}\). This sum follows when the occasion effect and residual process are independent, conditional on subject effects. The covariance decomposition itself does not require normality, although a Gaussian covariance model is often used for likelihood-based fitting.

For a scalar occasion effect that adds the same amount to each observation, the corresponding row of \(\mathbf Z_{ik}\) is 1. If its variance is \(\tau^2\), it contributes \(\tau^2\) to every pairwise covariance within that occasion, regardless of the time gap. Other occasion effects need not act as a constant shift. For example, an effect on a structural parameter may change concentrations by different amounts at different times. In a local linear approximation, its covariance contribution has entries proportional to the product of the two time-specific sensitivities to that parameter.

A common serial residual model instead specifies, for two distinct samples in one occasion,

The correlation function \(\rho(\cdot)\) describes how residual correlation changes with separation in time. For example, an exponential correlation function is \(\rho(d)=\exp(-\phi d)\), where \(d\) is the time gap and \(\phi>0\) controls how quickly correlation declines. The residual variances can differ by sampling time; the formula accounts for that through their standard deviations. In a simple constant-variance case, \(\sigma_{\epsilon,j}=\sigma_\epsilon\) for all \(j\).

Worked examples with paired sampling times

Justifying the decomposition and using design information

The covariance sum follows directly from the model. Write one occasion’s residual vector as \(\mathbf r=\mathbf Z\mathbf b+\boldsymbol\epsilon\), with both component vectors having conditional mean zero. Expanding the covariance gives

Conditional independence of \(\mathbf b\) and \(\boldsymbol\epsilon\) makes the two cross-covariance terms zero. Substituting \(\operatorname{Cov}(\mathbf b)=\boldsymbol\Omega_{\mathrm{occ}}\) and \(\operatorname{Cov}(\boldsymbol\epsilon)=\mathbf R\) yields \(\mathbf Z\boldsymbol\Omega_{\mathrm{occ}}\mathbf Z^{\mathsf T}+\mathbf R\). If the components are dependent, those cross terms need not vanish, and the simple additive decomposition is not justified.

The design determines whether the component patterns can be distinguished. A single pair at one time gap provides only one off-diagonal covariance, which may be explained by many combinations of occasion and residual covariance. Samples at several time gaps offer more information about a declining residual correlation function. Replicated occasions provide a further contrast: a residual process restricted to each occasion does not, under the assumed independence, create cross-occasion covariance, whereas a shared subject-level effect could do so. The interpretation depends on conditioning appropriately on subject effects and on the assumption that occasion effects are independent across occasions.

Assumptions, diagnostics, and alternatives

Before interpreting covariance patterns, check the assumptions that give them meaning. The conditional mean model should adequately describe the systematic concentration-time trajectory; otherwise, residuals can share patterns because predictions are wrong. Subject-level effects must be accounted for when assessing occasion-level covariance. The form of the occasion effect must be plausible for the scientific mechanism, and the residual correlation function should reflect the sampling process. Independence between occasion effects and residual errors, and any assumed independence across occasions, should be stated rather than silently presumed.

Useful diagnostics compare residual covariance or correlation by time lag and by whether observations belong to the same occasion. Patterns that remain similar across a range of within-occasion time gaps suggest a shared contribution, especially if within-occasion residuals remain associated even at long gaps. A steady decline with time gap supports a serial component. Cross-occasion association after conditioning on subject effects challenges a model that assumes independent occasions and residual processes. These are clues, not definitive tests: finite data, unequal sampling schedules, changing residual variances, and uncertain mean predictions can obscure the ideal patterns.

The main alternatives are to fit an occasion effect alone, a serial residual structure alone, or both together. A model with only an occasion effect may miss a clear decline in correlation with time separation. A serial-only model may fail to account for a common within-occasion contribution that persists at long gaps. A combined model is more flexible, but can be difficult to estimate when there are few observations per occasion or too few replicated occasions. Compare whether the fitted covariance reproduces the observed within- and between-occasion patterns, and whether parameter estimates are stable and scientifically interpretable; a better numerical fit alone does not establish that a component has the correct scientific interpretation.

A scalar random intercept is only one possible occasion effect. If occasion variability acts through a parameter that changes the shape of the concentration-time curve, its covariance can vary with sampling time according to the model’s sensitivities. In that setting, a constant covariance contribution is not the right benchmark. Also, an occasion effect and serial correlation can be difficult to distinguish when the observed time gaps cover only a narrow range. No covariance pattern can compensate for a design that does not contain enough repeated information to identify the proposed components.

Common mistake

A common mistake is to see positive covariance between two samples from the same occasion and conclude that an occasion random effect must be present. That reasoning ignores serial residual correlation, which can also produce positive within-occasion covariance. The correction is to examine covariance across multiple time gaps and replicated occasions, and to compare the patterns implied by the candidate mechanisms. When the design supplies only one pair type, report that the sources are not separately identified by that covariance alone rather than assigning the entire pattern to one component.