Worked NPDE for Repeated Concentrations

Why repeated concentrations need a joint check

A concentration profile contains several observations from the same subject. Those observations are generally related: a subject with concentrations above the population prediction at one time may also tend to be above it at another. A check that ranks each time point separately can show whether the model predicts each marginal distribution, but it does not by itself account for the joint predictive covariance of the profile. NPDE addresses this by first decorrelating the observed concentration vector using the covariance predicted by simulations, then converting each decorrelated component to a normal score.

The computation is subject-specific: the simulated vectors must follow the same dosing and observation schedule as the subject’s data. The estimated covariance describes variation across simulated profiles under that design. Whitening uses this covariance to remove linear covariance on the transformed scale; it is not a guarantee that all forms of dependence have been removed.

Whitening and the normal-score transformation

The centering and whitening steps have a precise covariance interpretation. If the predictive concentration vector has mean \(\mu_i\) and covariance \(\Sigma_i=L_iL_i^T\), then \(w_i=L_i^{-1}(y_i-\mu_i)\) has mean zero and covariance \(I\), provided those mean and covariance values are the true ones. This follows directly from the covariance rule for a linear transformation:

This result says that the transformed components have unit variances and zero pairwise covariances. It does not alone say that the components are normally distributed or independent. If the centered vector is jointly multivariate normal, the whitened vector is jointly \(N(0,I)\), so its components are independent standard normal variables. Without joint normality, zero covariance does not establish independence, and whitening can leave non-normal marginal distributions. The empirical rank and normal-score steps provide marginal checks, but their standard normal reference remains an approximation whose adequacy depends on the predictive distribution and on simulation and covariance-estimation accuracy.

In practice, \(\bar y_i\) and \(S_i\) are estimated from the simulated vectors, not known population quantities. For each component, the observed whitened value is compared with the corresponding transformed simulated values. The rank maps its relative position to a probability between zero and one; the inverse normal CDF maps that probability to a normal-score scale. Under a calibrated predictive distribution and sufficiently informative simulations, the scores should be compatible with a standard normal marginal reference. The plotting position above avoids infinite scores at the smallest and largest possible ranks. It does not eliminate the coarseness of a small number of simulations.

Worked repeated-profile calculation

The following hypothetical example uses two concentration times for one subject, with concentrations measured in mg/L. Four simulated profiles are deliberately used to make the arithmetic transparent; four is far too few for a stable practical NPDE analysis. The simulated mean is \(\bar y=(10,5)^T\) mg/L, and the sample covariance is

Assess calibration and within-subject dependence

NPDE diagnostics are usually examined across subjects and observation positions, with attention to the sampling design. For marginal calibration, inspect a histogram or normal quantile plot and summarize the NPDE mean and spread. Under a well-calibrated predictive model and adequate simulation, the marginal reference is approximately standard normal: mean near zero, variance near one, and no systematic departure in the distribution’s shape. These are diagnostic expectations, not requirements that finite-sample summaries equal zero and one exactly. Patterns by time, dose, or other relevant design features can reveal misfit hidden by an overall summary.

For repeated observations, also examine dependence after whitening and rank transformation. A trend or serial pattern in NPDE against time, or residual correlation between observation positions within subjects, can indicate remaining structure. The covariance transformation targets the simulated linear covariance, but it cannot guarantee independence in non-Gaussian data; the subsequent componentwise rank transformation is nonlinear and does not necessarily remove residual dependence. These diagnostics therefore assess whether the intended decorrelation is adequate rather than proving independence.

Assumptions, alternatives, and failure modes

A meaningful repeated-profile NPDE calculation requires simulations from the fitted predictive model at the subjects’ actual observation times and dosing histories, with the same outcome scale and observation process as the data. The simulated vectors must represent the intended predictive target; for example, a check using population-level simulations and one using subject-conditional predictions answer different questions. Earlier material on unconditional and conditional predictive checks distinguishes these targets. Each subject’s simulated covariance must be estimable and nonsingular. If the number of simulations is small relative to the number of repeated measurements, or the components are nearly redundant, the covariance estimate can be unstable or singular and whitening unreliable.

Common warning signs include extreme NPDEs concentrated at particular times, a mean or spread inconsistent with the reference across many subjects, visible skewness or heavy tails, dependence remaining across the profile, and results that change markedly with simulation count or covariance treatment. Inspect the simulated means, variances, and correlations as well as the final scores: an implausible covariance estimate can make the whitening step misleading. Ties, censoring, missing records, and very sparse schedules require attention to how the predictive distribution and rank calculation are defined; the simple continuous-outcome calculation shown here is not automatically appropriate for every observation process.

NPDE complements rather than replaces other predictive checks. A VPC compares observed and simulated summaries over time or strata and preserves an interpretable concentration scale. A prediction-corrected VPC can help compare profiles across design differences such as dose. NPDE supplies a subject-level numerical residual scale and explicitly uses the simulated within-profile covariance. Conversely, NPDE compresses a profile into transformed components and depends on estimated covariance and ranks; it may be less visually informative about concentration-scale discrepancies. Agreement among methods is reassuring, while disagreement calls for checking the prediction target, simulation design, data handling, and covariance estimate rather than declaring one summary definitive.