Age-Stratified Diagnostics for Pediatric PK Models

Why assess fit by age?

A pediatric PK model can describe the overall concentration data reasonably well while systematically overpredicting in one age range and underpredicting in another. Such patterns matter: developmental changes in clearance, body size, dosing, and sampling can leave age-specific discrepancies that disappear when all observations are pooled. A pooled diagnostic is therefore not sufficient evidence that predictions are adequate across pediatric ages.

This tutorial focuses on three complementary views of fit: residual summaries by age, prediction-corrected visual predictive checks (pcVPCs) by age and time, and age-specific exposure summaries. Each answers a different question. Residuals help locate directional errors in the observed data; a pcVPC compares observed distributions with distributions expected under the model; and exposure summaries translate concentration-level discrepancies into a clinically interpretable PK quantity. None alone establishes that a model is adequate.

The earlier material on body-size scaling, maturation, neonatal PK, and nonlinear PK provides the developmental and structural context for these checks. Here the emphasis is on evaluating whether a fitted model captures that context in the data.

Residuals within age bands

Definition: For observation \(j\) from individual \(i\), let \(DV_{ij}\) be the observed concentration and \(PRED_{ij}\) the model-predicted concentration for the observation. The signed raw residual is \(DV_{ij}-PRED_{ij}\): a positive value means the observation exceeds the prediction. A standardized residual expresses the discrepancy relative to its model-based variability. For example, a weighted residual is scaled using an estimated residual covariance; a conditional weighted residual (CWRES) is a model-based standardized residual that accounts for the fitted model structure. Under an adequate model and appropriate approximation, standardized residuals should generally be centered near zero and have a scale near one. They are diagnostics, not independent observations guaranteed to follow a standard normal distribution exactly.

Assign each record to a prespecified age band using age at the time of sampling, or another explicitly justified age definition. Bands should reflect developmental questions and contain enough individuals and observations to make summaries interpretable. A useful first display is standardized residual against age, with age-band-specific means or medians and a zero reference line. Also inspect residuals against time after dose, predicted concentration, dose, and relevant covariates. Age-specific patterns may be localized to a particular sampling period or concentration range rather than appearing as a simple offset across the whole band.

The sign convention matters. A negative mean residual \(DV-PRED\) indicates that observations tend to be below predictions, consistent with overprediction; a positive mean indicates underprediction. Standardization permits a rough comparison across observations with different predicted variability, but should not be mistaken for a correction of model misspecification. A useful interpretation combines the direction, magnitude, distribution, and context of the residuals.

Worked Example: Opposing Residual Patterns Across Ages

Suppose a hypothetical analysis reports CWRES for three age bands. The values are: neonates, \(1.2,\ 0.8,\ 1.0\); infants, \(-0.6,\ -0.9,\ -1.5\); and older children, \(0.2,\ -0.1,\ -0.1\). Calculate each band mean and the pooled mean.

For neonates, the mean CWRES is \((1.2+0.8+1.0)/3=3.0/3=1.00\). For infants, it is \((-0.6-0.9-1.5)/3=-3.0/3=-1.00\). For older children, it is \((0.2-0.1-0.1)/3=0/3=0.00\). The pooled mean is \((3.0-3.0+0)/9=0.00\).

The pooled mean is exactly zero, but it hides a directional pattern: neonatal observations tend to exceed predictions, while infant observations tend to fall below them. The magnitude of the band means warrants investigation, though this tiny hypothetical sample cannot establish a stable age effect. A plot of individual residuals and the distribution within each band would help reveal whether the pattern is consistent or driven by a few records.

Why pooled residual summaries can mislead

The cancellation in the example follows directly from the pooled mean. If age band \(g\) contains \(n_g\) observations and has mean residual \(\bar{r}_g\), then the pooled mean is a sample-size-weighted average:

\bar{r}_{\mathrm{pooled}}= \frac{\sum_g n_g\bar{r}_g}{\sum_g n_g}.

Thus, positive and negative age-specific means can offset one another. Even when their magnitudes do not match exactly, a large band can dominate the pooled result and obscure a smaller but important band. This is why age-stratified displays should accompany, rather than replace, an overall goodness-of-fit review.

Repeated concentration measurements from the same child are also correlated. A band with many samples from a few participants should not be interpreted as if it provided the same independent evidence as samples from many participants. Consider the number of individuals contributing, their sampling schedules, and whether patterns recur across individuals. When uncertainty bands or formal comparisons are used, the method should respect the subject-level structure—for example, by resampling or simulating individuals rather than treating every concentration as independent.

CWRES and related residuals have limitations. Their distribution can be affected by parameter estimation, model complexity, and shrinkage of individual random effects. A residual plot can be uninformative if the fitted model absorbs individual deviations or if a band has little data. Examine the type of residual and how it was computed; avoid applying a normality interpretation mechanically. A residual trend is a warning to investigate, not a diagnosis of its cause.

Prediction-corrected visual predictive checks

Definition: A visual predictive check (VPC) compares observed concentration summaries with distributions of corresponding summaries from data simulated under the fitted model. In a prediction-corrected VPC, each concentration is adjusted for the individual prediction level within its time bin. If \(PRED_{ij}\) is the population prediction for record \(ij\), and \(M_b\) is the median of those predictions in bin \(b\), a common correction is \(pcDV_{ij}=DV_{ij}(M_b/PRED_{ij})\). Apply the same record-specific correction to simulated concentrations when constructing their VPC. The correction places observations on a more comparable prediction scale; it does not repair an incorrect model.

For pediatric assessment, define age bands and time-after-dose bins before interpreting the plot. Within each age band and time bin, compare observed pcDV quantiles—often the median and lower and upper quantiles—with the corresponding simulated quantiles and their simulation-based uncertainty intervals. The simulations should reproduce relevant design features such as dosing, covariates, sampling times, and the number of individuals. The VPC then asks whether the observed distribution is plausible under the model for that part of the design.

Prediction correction is useful when predictions vary substantially across individuals in a bin, for example because body size or dose differs. Without correction, a wide range of predictions can make a pooled concentration distribution hard to interpret. However, correction is based on the model’s predictions. If the model gets the time-course shape wrong, or if a bin combines distinct phases of absorption and elimination, a correction factor cannot make the comparison valid. Use sufficiently informative bins, while avoiding so many bins that each contains too little information.

Worked Example: Calculating a Prediction-Corrected VPC Value

In a hypothetical 3-to-under-6-year age band and one time bin, five population predictions are \(2,\ 3,\ 4,\ 5,\ 6\) mg/L. The corresponding observations are \(2.4,\ 3.3,\ 4.4,\ 6.0,\ 7.2\) mg/L. Calculate the prediction-corrected observations using the median prediction as the reference, then compare their median with a simulated interval.

The median prediction is \(M_b=4\) mg/L. The correction factors \(M_b/PRED\) are \(4/2=2\), \(4/3=1.333\), \(4/4=1\), \(4/5=0.8\), and \(4/6=0.6667\). Therefore the corrected observations are \(2.4(2)=4.8\), \(3.3(1.333)=4.40\), \(4.4(1)=4.40\), \(6.0(0.8)=4.80\), and \(7.2(0.6667)=4.80\) mg/L. Their sorted values are \(4.40,\ 4.40,\ 4.80,\ 4.80,\ 4.80\), so their median is \(4.80\) mg/L.

Suppose the simulated 90% interval for the VPC median in this bin is \(3.2\) to \(4.7\) mg/L. The observed median, \(4.80\) mg/L, is just above that interval. This is a localized indication that concentrations may be higher than the model predicts in this age-time bin. It is not, by itself, proof of systematic bias: the analyst should inspect the observed lower and upper quantiles, other bins, the number of contributing children, and whether the simulated uncertainty is calibrated to the study design.

Age-specific exposure summaries

Concentration-level displays can be complemented by exposure summaries, such as AUC or \(C_{\max}\), when those quantities are estimable for the data and relevant to the diagnostic question. One simple comparison is the ratio of an observed-data-derived exposure estimate to its model-predicted counterpart for each individual, summarized within age bands. A ratio above one means the observed-data estimate is higher than the prediction; a ratio below one means it is lower. State the exposure definition, units, dose interval, and estimation method.

The data-derived and model-predicted quantities must be comparable. For example, comparing AUC over a dosing interval requires the same interval and compatible dose conditions on both sides. With rich sampling, a prespecified noncompartmental estimate may be informative. With sparse sampling, individual exposure estimates can be highly uncertain and may depend strongly on the PK model being evaluated. In that setting, report the estimation approach and uncertainty, and do not present a model-derived individual exposure as independent confirmation of that same model.

Worked Example: Summarizing AUC Ratios by Age

In a hypothetical evaluation, four children in one age band have comparable dose-interval AUC estimates (in mg·h/L) from a prespecified data-based method and corresponding model predictions. The observed-data estimates are \(24,\ 22,\ 25,\ 21\), and the predictions are \(20,\ 20,\ 20,\ 20\). Calculate the individual observed-to-predicted ratios and their arithmetic mean.

The ratios are \(24/20=1.20\), \(22/20=1.10\), \(25/20=1.25\), and \(21/20=1.05\). Their arithmetic mean is \((1.20+1.10+1.25+1.05)/4=4.60/4=1.15\). In this small sample, the observed-data AUC estimates average 15% higher than the model predictions. This is a descriptive age-band discrepancy, not evidence that every child’s true exposure is 15% higher: individual estimation uncertainty, representativeness, and sample size remain important.

If the same calculation in another band gave a mean ratio near 1.00, that contrast would motivate checking age-specific concentration patterns and model assumptions. It would not by itself identify whether the cause is clearance, absorption, dose history, or another feature of the analysis.

Putting the diagnostics together

Use the diagnostics as complementary checks, not as three votes on a single pass-or-fail decision. Residuals are convenient for locating signed discrepancies at individual records. A stratified pcVPC evaluates whether the observed distributions are compatible with simulated distributions across age and time. Exposure summaries show whether discrepancies persist in a quantity integrated over time or expressed as a peak. Agreement across these views strengthens the case that a pattern deserves attention; disagreement can reveal sensitivity to the diagnostic scale or data limitations.

Specify the diagnostic strata.
Choose age definitions and time bins in advance where possible. Record the number of individuals and observations in each cell, and check whether dose, body size, or sampling schedules differ substantially across bands.

Inspect signed residual patterns.
Plot residuals against age, time, and predicted concentration. Summarize their direction and spread by band, while recognizing within-child dependence and the limitations of standardized residuals.

Compare observations with simulations.
Construct an age- and time-stratified pcVPC using simulations that reflect the study design. Check medians and tails, not only whether a single observed curve falls inside a band.

Check exposure consequences and investigate.
Compare appropriately defined exposure summaries by age, with uncertainty and estimation method stated. If diagnostics disagree or show a pattern, examine data quality, model structure, covariates, and sampling before drawing a conclusion.

An uncorrected VPC is a useful alternative when prediction levels are sufficiently similar within bins or when preserving the original concentration scale is important. It is simpler to interpret in that setting, but can be dominated by between-individual prediction differences when those differences are large. Residual plots provide record-level detail that a VPC quantile plot may hide, but depend more directly on residual definitions and approximations. Exposure summaries are clinically interpretable but compress time-course information and can be unreliable when sampling is sparse. The appropriate choice depends on the question and data; these methods are generally most informative when used together.

Common pitfalls and limitations

Relying on a pooled mean. A mean near zero can result from cancellation across age bands. Inspect the band-specific distributions and their contributors.
Reading a residual sign backwards. With \(DV-PRED\), negative values indicate observations below predictions, not above them. Confirm the convention used by the software.
Overinterpreting one VPC bin. Sparse bins, multiple comparisons, and simulation variability can produce isolated deviations. Look for coherent patterns across time and age, and check the amount of information behind each bin.
Assuming prediction correction fixes model form. The scaling adjusts prediction levels; it cannot correct a wrong concentration-time shape or unsuitable binning.
Treating samples as independent children. Repeated observations are clustered within individuals. Show subject counts and use subject-aware uncertainty procedures where appropriate.
Calling an exposure ratio a causal effect. An observed-to-predicted ratio is a diagnostic comparison under specified data and model assumptions; it does not establish why the discrepancy occurred.

Age-stratified diagnostics are limited by the study’s age coverage, sample size, sampling design, and quality of the exposure estimates. A lack of visible bias in a sparsely sampled age band is not strong evidence of adequate fit there. Conversely, a discrepancy does not automatically imply that a maturation function or other specific mechanism is wrong. Diagnostics identify where predictions may be inadequate; resolving the cause requires scientific review of the data and model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A band has residuals \(DV-PRED=-0.4\) and \(-0.8\) mg/L. What is their mean, and which direction of discrepancy does it indicate?
Two age bands have mean standardized residuals \(0.9\) and \(-0.9\), with equal sample sizes. What is the pooled mean, and why is it insufficient as a fit summary?
For one VPC record, \(PRED=5\) mg/L, the bin median prediction is \(4\) mg/L, and \(DV=6\) mg/L. What is the prediction-corrected observation?
A band’s mean observed-to-predicted AUC ratio is \(0.85\). State its descriptive interpretation and one important limitation.

Solutions. (1) The mean is \((-0.4-0.8)/2=-0.60\) mg/L. Both observations are below their predictions, so the model overpredicts these concentrations in this small sample. (2) The pooled mean is \((0.9+(-0.9))/2=0\). It hides opposing age-specific patterns, so inspect each band and the individual observations. (3) \(pcDV=6(4/5)=4.80\) mg/L. (4) The data-based AUC estimates average 15% below predictions in that band. This is descriptive; its reliability depends on the exposure estimation method, uncertainty, and number of independent children.