Why Prediction-Correct Across Dose Levels?
A visual predictive check (VPC) compares observed outcome quantiles with quantiles from datasets simulated under a fitted model. When concentrations from several dose levels are placed in the same time bin, their scales may differ because the model predicts different concentrations at different doses. A raw VPC can therefore show wide quantiles that reflect the dose mix, making it harder to see whether the model describes the data relative to its own dose-specific predictions.
A prediction-corrected VPC (pcVPC) rescales observations and simulations within each time bin to a common prediction reference. The correction is model-based: records expected to have higher concentrations receive a smaller multiplier, while records expected to have lower concentrations receive a larger one. It is not simply a division by dose. The model’s predicted concentration can depend on dose, time, covariates, and the model’s nonlinear structure.
The median is a common reference choice because it is resistant to unusually high or low predictions. Other reference choices may be used, but they should be stated and applied consistently. A bin is usually defined over time after dose; it may also be stratified by dose or another design feature. If records are pooled across dose levels, the reference prediction is calculated across the records in that pooled bin.
Assumptions and the Logic of the Correction
Prediction correction is most interpretable when concentrations are positive and the main differences across records in a bin act approximately as multiplicative changes in expected concentration. This is plausible in many settings where dose changes concentration scale while the time-course shape is broadly comparable. The correction then removes the predicted scale difference, so that the corrected values can be compared on a common reference scale.
The pcVPC does not require every subject to have the same predicted concentration, and it does not assume that concentration is proportional to dose. It does require credible model predictions for the records being corrected. In particular, \(P_{ij}\) must be positive and sufficiently far from zero for the ratio \(P_b^{\mathrm{ref}}/P_{ij}\) to be stable. A correction factor based on a very small prediction can greatly magnify both the observed value and its residual variation.
The correction also changes the residual scale. Under a multiplicative error model, rescaling both the expected concentration and its variability can preserve a useful comparison. Under predominantly additive error, multiplying by \(C_{ij}\) also multiplies the absolute residual, potentially distorting the error pattern. The pcVPC is therefore a diagnostic display, not a transformation that makes every error model or dose-response structure comparable automatically.
Worked Calculation Across Two Dose Levels
The calculation proceeds record by record. First obtain typical predictions using the fitted model and each record’s design information. Then calculate the bin reference and correction factors. Finally, multiply observed and simulated concentrations by those factors before calculating quantiles.
Worked Example: Correcting a Bin With Two Dose Levels
Hypothetical data. At one time after dose, four records come from subjects given 10 mg or 100 mg. The fitted model’s typical predictions and observed concentrations are shown below. Concentrations and predictions are in mg/L.
| Record | Dose (mg) | Typical prediction \(P_{ij}\) (mg/L) | Observed \(Y_{ij}\) (mg/L) |
|---|---|---|---|
| 1 | 10 | 2.0 | 2.4 |
| 2 | 100 | 20.0 | 24.0 |
| 3 | 10 | 2.0 | 1.8 |
| 4 | 100 | 20.0 | 22.0 |
The prediction values are positive, and the subjects share the same time bin. The bin reference is the median of \(2.0,20.0,2.0,20.0\) mg/L. In sorted order these are \(2.0,2.0,20.0,20.0\), so \(P_b^{\mathrm{ref}}=(2.0+20.0)/2=11.0\) mg/L.
For a 10 mg record, \(C_{ij}=11.0/2.0=5.5\); for a 100 mg record, \(C_{ij}=11.0/20.0=0.55\). The corrected observations are therefore:
The two dose levels now share a reference prediction of 11.0 mg/L. Their corrected observed values are not forced to equal that reference: the values still show observed deviations from the model’s typical predictions. For example, the 2.4 mg/L observation is 20% above its 2.0 mg/L prediction, and its corrected value, 13.2 mg/L, is 20% above the reference.
Calculating Simulated Quantiles
For each simulated replicate, retain the simulated concentration at each corresponding record and multiply it by that record’s correction factor. In this example, the factors are fixed by the fitted model’s typical predictions and the observed design. Using the same factors for observed and simulated records ensures they are compared on the same scale.
In a full pcVPC, for each time bin and each simulation replicate, calculate the desired quantiles of the corrected simulated concentrations. Repeating this across many simulated datasets gives a distribution of each simulated quantile. A central simulation quantile and an interval summarizing its replicate-to-replicate variation can then be plotted against the corresponding corrected observed quantile. The interval reflects simulation variability under the fitted model and design; it is not a confidence interval for a model parameter.
Worked Example: Comparing an Observed Median With Simulated Medians
Continue with the four records and factors \(5.5,0.55,5.5,0.55\) from the previous example. The sorted corrected observations are \(9.9,12.1,13.2,13.2\) mg/L. With four values, define the sample median as the mean of the two middle values:
Consider three hypothetical simulated replicates. The simulated raw values are listed in record order. The corrected values use the same record-specific factors.
| Replicate | Simulated raw values (mg/L) | Corrected values (mg/L) | Corrected median (mg/L) |
|---|---|---|---|
| 1 | 2.0, 20.0, 2.4, 24.0 | 11.0, 11.0, 13.2, 13.2 | \((11.0+13.2)/2=12.1\) |
| 2 | 1.6, 18.0, 2.6, 22.0 | 8.8, 9.9, 14.3, 12.1 | \((9.9+12.1)/2=11.0\) |
| 3 | 2.2, 22.0, 1.8, 18.0 | 12.1, 12.1, 9.9, 9.9 | \((9.9+12.1)/2=11.0\) |
For replicate 2, for example, the first corrected value is \((1.6)(5.5)=8.8\) mg/L and the second is \((18.0)(0.55)=9.9\) mg/L. The three simulated medians are 12.1, 11.0, and 11.0 mg/L. Their median is 11.0 mg/L, and their range is 11.0 to 12.1 mg/L. The observed median, 12.65 mg/L, is above all three simulated medians in this tiny illustration. Three replicates and four records are far too few for a stable predictive check; the calculation demonstrates the method, not a reliable decision rule. In practice, many simulated replicates are used, and the observed quantile is assessed against the simulated quantile distribution.
Reading the pcVPC and Checking Its Behavior
A pcVPC commonly displays corrected observed quantiles and corresponding simulated quantiles over time, with a band representing variation across simulation replicates. If observed quantiles repeatedly fall outside the simulated bands or show a systematic time pattern relative to them, the model may not reproduce the observed concentration distribution. A single crossing or isolated departure is not, by itself, proof of model failure: binning, finite sample size, and simulation variability all affect the display.
Before interpreting the plot, check the construction:
- Prediction calculation: Confirm that typical predictions use the correct dose, time, covariates, and prediction scale. An incorrect prediction at a record produces an incorrect correction factor.
- Bin contents: Inspect how many observations and dose levels enter each bin. Very sparse bins yield unstable observed quantiles; uneven dose composition can also make adjacent bins difficult to compare.
- Correction factors: Look for very small predictions or extreme factors. Large factors magnify measurement noise and can make a few records dominate a bin.
- Simulation alignment: Apply factors to the corresponding simulated records and preserve the study’s dose and sampling design. Do not calculate one factor for observations and a different rule for simulations.
- Error structure and scale: Check whether a multiplicative correction is sensible for the outcome and residual model. For concentrations with an important additive error component, the corrected spread may be hard to interpret.
A common mistake is to say, “The 100 mg observations are about ten times higher, so divide every concentration by dose.” That is only justified under a suitably proportional dose-response relationship. For nonlinear pharmacokinetics, concentration need not increase in proportion to dose. The correction instead uses the fitted typical prediction for each record, which can represent nonlinear dose effects—but it is only as credible as that fitted model.
When to Use Another View
A raw VPC keeps concentrations in their original units and shows the model’s ability to reproduce the distribution actually observed across the dose mix. It is often useful alongside a pcVPC, especially when absolute concentration calibration matters. A dose-stratified VPC compares records within dose groups, avoiding the need to pool different doses into one corrected distribution, but it can become imprecise when each group has few observations. A pcVPC is useful when a pooled view is desired and model-based scaling is defensible; it should not replace views that answer different questions.
The correction can conceal a dose-related discrepancy if the same model predictions used to construct the factors are wrong. It can also obscure differences in time-course shape if dose changes absorption or elimination behavior rather than only concentration scale. Therefore, inspect the predictions and design, use adequate bins, and interpret the corrected plot together with uncorrected or stratified displays where relevant. The pcVPC assesses predictive adequacy on a model-defined reference scale; it does not establish that the model is biologically correct.
Worked Example: Why Dose Division Can Mislead
Suppose a hypothetical nonlinear model predicts 2.0 mg/L after a 10 mg dose and 12.0 mg/L after a 100 mg dose at the same time. The expected concentration increases sixfold, not tenfold. Assume the bin reference is the median of these two predictions, \(P_b^{\mathrm{ref}}=(2.0+12.0)/2=7.0\) mg/L, and the corresponding observations are 2.4 and 13.2 mg/L.
The pcVPC factors are \(7.0/2.0=3.5\) and \(7.0/12.0=0.5833\). Corrected observations are \((2.4)(3.5)=8.4\) mg/L and \((13.2)(0.5833)\approx7.70\) mg/L. Relative to their predictions, the observations are respectively 20% above and 10% above; the corrected values preserve that difference around the shared 7.0 mg/L reference.
By contrast, dividing by dose gives \(2.4/10=0.24\) and \(13.2/100=0.132\) mg/L per mg. That comparison assumes dose-proportional concentration, contrary to the stated predictions, and changes the scale in a way that does not preserve the model-relative deviations. Prediction correction accounts for the fitted nonlinear prediction rather than imposing proportionality to dose.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one bin, the typical predictions are 4 and 16 mg/L. What is the median reference prediction, and what are the correction factors?
- An observation is 3.0 mg/L and its typical prediction is 2.5 mg/L. If the bin reference is 10 mg/L, calculate its corrected value and explain its relation to the reference.
- Why should the same record-specific correction factor be applied to the corresponding observed and simulated concentrations?
- Give one reason why dividing every concentration by dose may be inappropriate in a pcVPC across dose levels.
- What should be investigated if corrected observed quantiles repeatedly lie above the simulated quantile bands?
Solutions. (1) The median is \((4+16)/2=10\) mg/L; the factors are \(10/4=2.5\) and \(10/16=0.625\). (2) The factor is \(10/2.5=4\), so the corrected value is \((3.0)(4)=12.0\) mg/L. It is 20% above the 10 mg/L reference, matching the observation’s 20% excess over its typical prediction. (3) A common factor puts observations and simulations on the same model-defined scale; different correction rules would confound the comparison. (4) Concentration may not be proportional to dose, particularly with nonlinear pharmacokinetics, so dose division can impose an incorrect scale. (5) Check model predictions, dose and time patterns, bin sizes, correction factors, simulation alignment, and whether the residual structure makes multiplicative correction appropriate. A persistent pattern can indicate model misfit, but should be judged against sampling and simulation variability.