Why Check NPDEs Against Time?
A model can produce an NPDE distribution that looks acceptable overall while missing an important part of the concentration-time profile. For example, positive NPDEs early after a dose and negative NPDEs later can cancel in an overall mean, even though the model's predictions are systematically misaligned with time. A time-pattern check asks whether the errors show structure across the dosing interval, not just whether their overall distribution appears reasonable.
For this purpose, time should have a clear clinical and model-related meaning. In a single-dose setting, it is commonly elapsed time since dose. With repeated dosing, it may be time since the most recent dose or time within a specified dosing interval; the choice should match the question and account for dose history. Calendar time, study day, and time since dose are not interchangeable. A pattern against study day, for example, may reflect changing treatment or study procedures rather than absorption or elimination.
NPDEs are transformed prediction errors intended to be approximately standard normal under the predictive model. Their calculation includes decorrelation to account for modeled dependence among repeated observations. That background does not make each plotted value independent in every practical sense: the data still come from a finite set of subjects, and multiple observations may be contributed by the same subject. Temporal plots are therefore diagnostic summaries, not automatic hypothesis tests.
Plotting and Summarizing Temporal Structure
Begin with a scatterplot of NPDE against the chosen time variable. Show the observations rather than only a summary, and include a horizontal reference at zero. A reference band at \(-2\) and \(2\) can help orient the reader to the scale, but it is not a formal simultaneous confidence band, nor does an individual point outside it by itself establish model failure. If many observations are plotted, use transparency or small plotting symbols to reveal overlap.
A useful display can add a smooth curve to reveal gradual structure. The curve is descriptive: it highlights local changes without requiring that the true pattern be linear. Binned means or medians can provide a more stable visual summary when observations are numerous or times are irregular. Bins should be chosen before interpreting the pattern when possible, should reflect meaningful phases or adequate data support, and should not conceal sparse regions. Report the number of observations and, where relevant, the number of subjects contributing to each bin.
For a binned mean, the basic summary is the arithmetic average of the NPDEs in that bin. A value above zero means the errors in that bin are positive on average; a value below zero means they are negative on average. This sign interpretation should be tied to the precise convention for the prediction error used in the NPDE calculation. In the usual observed-minus-predicted orientation, positive errors indicate observations tending to be higher than predicted, while negative errors indicate observations tending to be lower.
Worked Example: A Phase-Specific Shift in NPDEs
Suppose six hypothetical NPDEs from observations at different times after a dose are as follows. The first three are in the early absorption phase; the last three represent later profile observations.
| Time after dose (h) | NPDE | Phase group |
|---|---|---|
| 0.25 | 0.2 | Early |
| 0.5 | 0.6 | Early |
| 1 | 1.1 | Early |
| 2 | 0.5 | Later |
| 4 | -0.4 | Later |
| 8 | -0.9 | Later |
The early-phase mean is
The later-phase mean is
The early values are positive and increase to 1.1. The later values are mostly negative, but the value at 2 h is positive (0.5); they are not all negative. This small example suggests an early-versus-later pattern that could be consistent with a phase-related mismatch. It is far too small to establish a reliable trend or its cause, and the group labels do not replace inspection of the actual times.
Why a Trend Is Not an Overall Distribution Test
The overall NPDE mean can be close to zero even when errors vary systematically with time. If early positive errors are balanced by later negative errors, the pooled mean may hide the shape of the mismatch. Similarly, an overall check of distributional shape does not establish that the NPDE distribution is stable across time. The temporal check adds a conditional question: does the location or spread of the errors change as time changes?
One way to express the mean-location target is \(E(Z\mid t)=0\), where \(Z\) denotes an NPDE and \(t\) is the relevant time variable. This condition is a useful diagnostic target, not a complete definition of model adequacy. A time-dependent spread can occur even when the conditional mean remains near zero. Conversely, a smooth mean near zero does not rule out tail, skewness, or other distributional changes across time.
A simple linear slope can summarize a roughly monotone trend, but a zero slope does not rule out a curved pattern: positive NPDEs at both ends and negative NPDEs in the middle can have little net linear slope. A flexible smooth curve and carefully chosen phase summaries can reveal such structure. The summaries are complementary, not competing tests of a single scalar quantity.
Worked Example: Comparing NPDE Means Across Time Bins
Consider four hypothetical subjects, each contributing one NPDE in each of three prespecified time bins. The bins represent early, middle, and late times after dose.
| Subject | Early | Middle | Late |
|---|---|---|---|
| A | -0.2 | 1.0 | -0.8 |
| B | 0.1 | 0.8 | -0.5 |
| C | 0.3 | 0.4 | -0.1 |
| D | 0.6 | 0.2 | 0.2 |
The bin means are calculated by adding the four values in each column and dividing by four:
The pattern rises from 0.20 to 0.60 and then falls to -0.30. This points to a possible middle-time positive shift followed by a late-time negative shift, rather than a simple monotone trend. With only four subjects, these means are descriptive and unstable. They should be displayed with the data and subject counts, not presented as strong evidence. Because each subject contributes three observations, treating all 12 values as independent when calculating uncertainty would ignore the repeated-measures structure.
A Short Derivation: What a Linear Trend Measures
When a linear summary is useful, let \(z_i\) be an NPDE observed at time \(t_i\), for \(i=1,\ldots,n\). The least-squares slope \(\hat{\beta}_1\) is the value that minimizes the sum of squared residuals from a line with an intercept. Centering time and NPDE gives the slope formula:
The denominator is the sum of squared deviations in time. The numerator is the sum of paired deviations: it is positive when observations above their mean NPDE tend to occur at times above the mean time, and negative when higher NPDEs tend to occur earlier. Thus the slope summarizes average linear change in NPDE per unit of time. It does not establish causation, demonstrate that the relation is truly linear, or account automatically for subjects contributing repeated observations.
Worked Example: A Descriptive Linear Slope Across Time
Suppose a hypothetical smooth summary is represented by mean NPDEs of 0.7, 0.9, 0.3, -0.4, and -0.6 at times 0.5, 1, 2, 4, and 8 h, respectively. To quantify the overall linear direction, use the slope formula above. The mean time is \(\bar{t}=3.1\) h and the mean NPDE is \(\bar{z}=0.18\).
The cross-product sum can be evaluated as \(\sum t_i z_i-n\bar{t}\bar{z}\). Here, \(\sum t_i z_i=0.35+0.90+0.60-1.60-4.80=-4.55\), while \(n\bar{t}\bar{z}=5(3.1)(0.18)=2.79\). The numerator is therefore \(-4.55-2.79=-7.34\). The denominator is
Thus the fitted slope is
This negative slope summarizes a decline in the mean NPDE over the time range. It does not mean that the expected value falls by exactly 0.1973 at every hour: the values are not evenly spaced in time, and the pattern may be curved. These five points are summaries, not independent subject-level observations, so this calculation is descriptive rather than an inferential test.
Interpreting Phase Patterns and Diagnosing Problems
A positive early-time pattern followed by negative later-time NPDEs may indicate that observed concentrations tend to exceed predictions early and fall below predictions later, under the usual observed-minus-predicted orientation. Possible explanations include a mismatch in absorption rate, lag time, or the predicted profile shape. A trend concentrated in the elimination phase may instead prompt review of the disposition profile or time-dependent residual variability. These are hypotheses for model investigation, not conclusions that can be identified from the NPDE plot alone.
Before attributing a pattern to model structure, check the time variable, data processing, and prediction setup. Confirm that time is measured relative to the intended dose; verify dose times, nominal versus actual sample times, and treatment of repeated dosing; and ensure that the predictive simulations used the same relevant design features as the observations. Examine whether one subject, one study site, or a small number of densely sampled profiles dominates the apparent trend. Overlaying subject identifiers or using subject-level summaries can reveal such concentration.
For uncertainty around a temporal summary, resampling subjects rather than individual observations is often a more defensible approach when subjects contribute repeated data. A subject-level bootstrap resamples whole subjects, preserving each selected subject's observation pattern. Any resulting interval still depends on the sampling design, the number of subjects, and the bootstrap procedure; it is not a substitute for checking the plot and assumptions.
State whether time means time since dose, time within the dosing interval, or another defined clock, and explain why that scale addresses the diagnostic question.
Plot NPDEs against time with a zero reference, checking coverage of the time range, overlap, and contributions from individual subjects.
Use bins or a smooth to make broad structure visible; report support in each region and avoid interpreting sparse-bin summaries as precise estimates.
Check dose and sampling records, simulation design, and phase-specific model predictions before treating a pattern as evidence for a particular mechanism.
Alternatives, Limitations, and Common Mistakes
A time plot of NPDEs is useful because it directly displays standardized predictive errors against time. A visual predictive check (VPC) instead compares observed concentration quantiles with predictive quantiles across time bins, retaining the concentration scale and often making distributional discrepancies easier to relate to the data. These approaches answer related but different questions; neither replaces the other in every setting. Formal distribution tests for NPDEs address overall distributional properties, while a temporal display asks whether errors change with time. A good overall distribution result cannot guarantee the absence of a temporal pattern.
Temporal checks have limits. Uneven sampling can leave some phases poorly represented; a smooth near the edge of the observed time range may be especially unstable. Bin boundaries can change the apparent summary, and averaging can conceal opposing subject-specific patterns. NPDEs also depend on the predictive model and the simulation design used to obtain them. Therefore, a temporal trend is a diagnostic signal that guides follow-up, not a stand-alone model-selection rule or proof of a specific pharmacokinetic mechanism.
- Mistake: looking only at the pooled mean. Positive and negative phase-specific errors can cancel. Correction: inspect NPDE against time and summarize relevant phases separately.
- Mistake: treating every observation as independent. Repeated observations from one subject do not become independent simply because NPDEs were decorrelated for their calculation. Correction: display subject contributions and use subject-level methods when uncertainty estimates are needed.
- Mistake: calling every local fluctuation a trend. A few scattered values can produce a wavy smooth. Correction: examine data density, subject contributions, and stability under reasonable display choices.
- Mistake: assigning a unique mechanism from the plot. A phase pattern may have several explanations. Correction: describe the observed direction first, then check data and model assumptions before proposing a cause.
Check Your Understanding
Answer the questions, then compare your reasoning with the short solutions.
- What temporal pattern would be expected in NPDEs under a well-calibrated predictive model, and why is an overall mean near zero not sufficient to establish it?
- In a plot against time since dose, early NPDEs are mostly positive and late NPDEs mostly negative. State a careful interpretation under the usual observed-minus-predicted convention.
- Why can treating all repeated observations as independent give misleading uncertainty for a time-bin mean?
- Mean NPDEs in three bins are 0.1, 0.7, and -0.5. What pattern do these summaries suggest, and what information is needed before drawing a firm conclusion?
- Why might a smooth trend and a binned summary lead to different impressions?
Solutions. (1) The errors should show no systematic change in location or spread with the relevant time variable. A pooled mean can be near zero because positive errors in one phase cancel negative errors in another. (2) Observations tend to be higher than predictions early and lower later, suggesting a time- or phase-related predictive discrepancy; the plot alone does not identify its cause. (3) Several values from one subject share that subject's profile and are not independent sampling units; uncertainty should account for the subject-level clustering. (4) The summaries suggest a rise followed by a fall, not a simple monotone pattern. Check the number of subjects and observations per bin, the raw points, sampling times, and whether a few subjects dominate. (5) Bins average observations within chosen boundaries, whereas a smooth weights nearby points according to a smoothing rule. Different boundaries, sparse regions, or local curvature can therefore change how the pattern appears.