Continuous-Time Correlation for Unequally Spaced Samples

Why elapsed time matters

Repeated pharmacokinetic samples are not always collected at equal intervals. A residual correlation that depends only on how many observations apart two measurements occur can therefore misrepresent their dependence: two adjacent measurements might be separated by minutes in one part of a schedule and hours in another. A continuous-time model instead makes correlation a function of the actual elapsed time.

The focus here is the exponential correlation function. It gives a simple, interpretable way to describe residual dependence that decays as observations become farther apart. The model concerns within-subject residuals after accounting for the structural prediction and, when appropriate, the subject-specific random effects. It is not itself a model for between-subject or between-occasion variation.

The correlation is 1 at zero time separation and decreases toward 0 as the separation increases. It remains positive at every finite time gap under this model. The parameter \(\tau\) is the time gap at which the correlation has fallen to \(e^{-1}\), approximately 0.3679. A larger \(\tau\) means slower decay and more persistent residual correlation; a larger \(\lambda\) means faster decay.

Deriving the continuous-time form

One standard justification is a stationary Ornstein–Uhlenbeck residual process. Suppose \(U_i(t)\) has mean zero and follows the stochastic differential equation $$ dU_i(t)=-\lambda U_i(t)\,dt+\sqrt{2\lambda}\,\sigma_e\,dW_i(t), $$ where \(W_i(t)\) is Brownian motion, \(\lambda>0\), and \(\sigma_e^2\) is the stationary residual variance. Assume separate subjects have independent processes, and that the process is in its stationary distribution. These assumptions imply a constant variance over time and an exponential correlation depending only on the time gap.

For \(h\geq 0\), the solution over an interval of length \(h\) is

The integral is independent of \(U_i(t)\), because it uses Brownian increments after time \(t\). Its mean is zero. Thus the covariance between \(U_i(t+h)\) and \(U_i(t)\) is determined by the first term:

Dividing by the common standard deviation \(\sigma_e\) at each time gives \(\operatorname{Corr}\{U_i(t+h),U_i(t)\}=e^{-\lambda h}\). Replacing \(h\) by \(|t-s|\) yields the stated function for any pair of times. This derivation also shows why the process is a continuous-time analogue of a decaying, first-order dependence model: over a gap \(h\), the earlier residual contributes to the later residual with multiplier \(e^{-\lambda h}\).

For \(n_i\) observation times \(t_{i1},\ldots,t_{in_i}\), define the residual correlation matrix \(R_i\) by

This construction gives a valid positive-semidefinite correlation matrix for any set of observation times, including irregularly spaced times. If residual variances differ by observation, let \(s_{ij}\) be the residual standard deviation specified by the observation model. Then the residual covariance is

In matrix form, this is \(D_iR_iD_i\), where \(D_i\) is diagonal with entries \(s_{ij}\). The correlation function determines the standardized dependence; the residual variance model determines the scale. If all residual standard deviations are the same, \(s_{ij}=\sigma_e\), the covariance becomes \(\sigma_e^2(R_i)_{jk}\).

Using the correlation with residual variances

The exponential function specifies correlation, not covariance. To obtain a covariance, multiply the correlation by the standard deviations of the two residuals. This distinction matters when the observation model has concentration-dependent residual variability: two residuals can have the same correlation with other pairs but different covariances because their scales differ.

Assumptions, checks, and alternatives

The exponential structure is useful only when its assumptions are reasonable. In the basic formulation, residuals have mean zero after conditioning on the model predictions and random effects; the standardized process is stationary over the time range being modeled; dependence is determined by elapsed time through one positive decay rate; and residual processes are independent across subjects. If a single process is meant to describe one dosing occasion, observations from distinct occasions should not automatically be treated as continuous parts of the same process. The cross-occasion dependence must match the design and model assumptions.

Practical checks include plotting appropriately standardized residuals against time and examining residual pairs grouped by their time separation. Under the model, pairs with similar time gaps should have broadly similar correlation patterns, regardless of where in the schedule they occur. Estimated or empirical correlations should generally decline with increasing gap. These checks are informative rather than definitive: noisy residual estimates, few subjects, and unevenly represented time gaps can obscure the pattern.

Warning signs include a clear change in residual variance over time after standardization, correlations that depend on the location in the schedule as well as the gap, or substantial negative correlations that the positive exponential function cannot represent. A fitted decay rate near zero implies correlations close to 1 over the observed time range and can make the covariance structure difficult to distinguish from a shared offset. A very large rate implies near-independence except for extremely close samples. In either case, the observation schedule may contain too little information to estimate the decay scale reliably.

The main simple alternative is independent residual error: its correlation is 1 for a measurement with itself and 0 for distinct measurements. It is appropriate when remaining residual dependence is negligible, but can understate dependence when nearby measurements share short-term fluctuations. A discrete-time autoregressive structure instead makes dependence a function of integer observation lag, often using a common lag parameter. That can be suitable for regularly spaced samples, but treats equal index gaps alike even when the actual time gaps differ. The continuous-time exponential model naturally handles irregular schedules, at the cost of assuming a particular stationary decay shape.

The model is parsimonious, but that simplicity is also a limitation. It cannot represent oscillating or negative correlation, multiple decay time scales, or nonstationarity without extensions. Nor does a fitted residual correlation automatically identify its scientific source: it summarizes remaining dependence under the specified mean, random-effect, and variance models. It should not be used as a substitute for checking whether those other parts of the model are adequate.