Predictive Checks for Correlated Concentration Profiles

Why Concentration Profiles Need Their Own Checks

A predictive check at each sampling time can show whether a model reproduces the distribution of concentrations at that time. But a concentration profile is more than a collection of separate values: the values from one individual occur in temporal order and are usually dependent. A model can match the concentration distribution at every scheduled time yet generate implausible rises, falls, or peak timing within individuals.

The target here is the within-individual trajectory: the ordered concentrations measured for one individual at their sampling times. A profile-level predictive check reduces each observed or simulated trajectory to one or more summaries and compares the observed summaries with those from simulated profiles. The summaries should be chosen to represent the temporal features relevant to the model assessment.

For a useful comparison, simulations should reproduce the observation design: the same sampling times, number of observations per individual, and relevant observation process. The model and simulation assumptions must also be stated. For example, if simulations use fixed fitted parameters, the check assesses predictions conditional on those parameter values; it does not represent parameter uncertainty. The simulation should include the model’s intended sources of variability, including between-individual variability and residual variability when these are part of the predictive target.

A profile summary is not automatically a good diagnostic. It must be defined for the available sampling design, have an interpretable direction and units, and be relevant to a plausible model failure. For example, adjacent changes are meaningful when the sampling times are comparable; with unequal intervals, raw changes can confuse elapsed time with trajectory behavior. In that setting, changes per unit time or summaries based on a common time grid may be more appropriate, provided interpolation does not create misleading precision.

Why Matching Marginals Does Not Match Dependence

The distinction can be stated mathematically. Suppose concentrations at two times have variances \(\sigma_1^2\) and \(\sigma_2^2\), with covariance \(\operatorname{Cov}(Y_1,Y_2)\). The variance of their change depends on that covariance:

Thus, two models can give the same marginal distributions at both times—and therefore the same \(\sigma_1^2\) and \(\sigma_2^2\)—but different distributions of changes because their within-individual covariance differs. Marginal checks do not determine the joint distribution across times.

A practical check proceeds by calculating the chosen summary for every observed profile and for every simulated profile. The simulated summaries form a predictive reference distribution. Compare the observed summaries with that distribution, looking both at their overall location and spread and at unusual individual profiles. For cohort-level displays, one can compare the observed distribution of profile summaries with the corresponding distribution across simulated data sets. These comparisons are diagnostic: a simulated envelope is not automatically a confidence interval for a population parameter, and an observed value inside it does not prove the model is correct.

Choosing and Calculating Profile Summaries

Different summaries target different aspects of a profile. The mean adjacent change describes average direction over the observed intervals; the variance of adjacent changes describes how uneven those changes are. Lag-one correlation summarizes whether successive changes tend to move together, although it may be unstable for short profiles. The observed peak time describes when the maximum recorded concentration occurs, not necessarily the true continuous-time peak. A trapezoidal area under the curve summarizes overall exposure over the observed interval.

A summary should be calculated in the same way for observed and simulated profiles. Where individuals have different numbers or times of observations, comparisons may require stratification by design, summaries defined on each individual's actual schedule, or a carefully justified common time grid. Do not treat observations at widely different intervals as equivalent adjacent changes without considering the elapsed time.

Diagnostics, Alternatives, and Limitations

A mismatch in a profile summary can help localize a predictive failure, but its interpretation depends on sampling and the summary’s sensitivity. If observed adjacent-change variability repeatedly lies beyond the simulated range, the model may under-represent short-term variation or temporal dependence. If peak times are systematically earlier or later than simulated peak times, the predicted timing may be inaccurate. A mismatch in AUC with reasonable peak timing instead points toward a different aspect of overall exposure. These are prompts for further model assessment, not unique diagnoses of a particular mechanism.

Useful diagnostics include plotting observed and simulated profile summaries on the same scale, displaying individual trajectories in temporal order, and checking whether a conclusion is driven by a few extreme profiles. Repeat summaries within relevant design groups when schedules differ. Very sparse profiles may not contain enough observations to estimate a within-profile correlation or locate a peak reliably; correlation estimates from only two or three time points can be especially unstable. For profiles with censoring or below-quantification observations, the summary must respect how those observations are represented in the predictive simulation rather than silently treating them as exact concentrations.

Profile-level checks complement rather than replace other predictive checks. A visual predictive check (VPC) is useful for time-specific distributions, but marginal agreement does not establish that within-individual sequences are realistic. Normalized prediction distribution errors (NPDEs), including checks for time patterns, provide a different residual-based view; a profile summary can make a selected temporal feature more directly interpretable. Joint predictive checks across PK endpoints examine dependence among endpoints, whereas this tutorial’s profile summaries examine dependence across times within a concentration trajectory. Each approach answers a different question.

The main limitations are that results depend on summary choice, summaries can discard information, and cohort-level comparisons may have limited resolution with few individuals. Correlation summaries need enough repeated measurements and may obscure nonlinear or changing dependence. A peak-time summary based on sparse sampling is only the time of the largest recorded value. AUC over a limited window says nothing directly about exposure outside that window. Therefore, use a small set of complementary summaries tied to a stated predictive concern, and inspect representative profiles rather than relying on a single scalar.

Common Pitfall

A second practical pitfall is to compare summaries calculated under different observation schedules. A raw concentration change over 12 hours is not directly comparable to one over 1 hour. Use a rate of change or a design-aware approach, and apply exactly the same definition to observed and simulated profiles.