Observed Versus Population-Predicted Concentration Plots

What the Plot Compares

An observed-versus-population-predicted concentration plot compares each measured concentration with the concentration predicted for that observation by the population model. It is a compact way to ask whether the model’s typical predictions are calibrated to the observations across the concentration range and across relevant subject groups.

Let \(C_{ij}\) be the observed concentration for subject \(i\) at observation \(j\), and let \(\widehat C^{\mathrm{pop}}_{ij}\) be the corresponding population prediction. The latter uses the fitted population parameters, the subject’s recorded covariates and dosing history, and the observation time, but sets the subject-specific random effects to their population-typical value of zero. Thus, it is not the same as an overall mean concentration: two observations can have different predictions because their times, regimens, or covariates differ.

The usual target is that observations be distributed around the identity line without a systematic trend or important subgroup displacement. This is a calibration target, not a claim that each observed concentration should equal its prediction. Measurement error and between-subject variability produce scatter even when a model is adequate.

Calibration, Scatter, and the Identity Line

Calibration means that predictions agree, on average, with the outcomes they are intended to predict. For a predictor representing the conditional mean concentration on the plotted scale, a simple calibration target is \(E(C_{ij}\mid \widehat C^{\mathrm{pop}}_{ij}=c)\approx c\). The identity line visualizes this target. In a real plot, the conditional average is not known; the point cloud provides only a noisy diagnostic of it.

A vertical departure from the identity line can be expressed as a signed difference:

Positive differences indicate underprediction and negative differences indicate overprediction. Their size must be interpreted in concentration units. A difference of \(0.5\ \mathrm{mg/L}\) may be substantial near a predicted concentration of \(1\ \mathrm{mg/L}\), but relatively small near \(50\ \mathrm{mg/L}\). Ratios can complement differences when proportional error is relevant, but they do not replace choosing a scale suited to the model and clinical question.

The cloud’s width also matters. Points widely dispersed around the identity line indicate that population predictions alone do not closely describe individual observations. That can reflect genuine between-subject variability, residual error, model misspecification, or sparse information about the subjects. A narrow cloud is not automatically evidence of a good model: the observations may cover only a limited concentration range, or a systematic offset may be present.

The identity line is a reference, not a fitted regression line. Fitting a line to the same points can summarize an apparent intercept and slope, but does not turn the plot into a validation study. Repeated observations from one subject are also not independent replicates. A dense cluster from a few subjects can dominate the visual impression, so inspect subject and group representation rather than treating every point as independent evidence.

Why Groupwise Bias Can Be Hidden

A satisfactory-looking overall cloud can conceal systematic errors in a clinically relevant subgroup. Groups may be defined by categorical covariates, such as formulation or renal-function category, or by other scientifically justified characteristics. The plot can be inspected with group symbols or colors, but the group definitions should be chosen for a reason; arbitrary subdivision can make chance fluctuations look meaningful.

A useful basic check is to compute signed differences within each group. A group mean near zero is not proof of adequate calibration: positive and negative differences can cancel, and a small group can give an unstable estimate. Examine the individual points, the range of predictions, and the number of subjects represented alongside any summary. When group sizes or prediction ranges differ substantially, an overall mean difference can be particularly misleading.

A Justification for the Identity-Line Target

The identity line follows directly from the definition of calibration. Suppose the population prediction \(P\) represents the conditional mean of the observed concentration \(C\) on the chosen scale. Then, under calibration, \(E(C\mid P)=P\). Subtracting \(P\) from both sides gives the expected signed difference at that prediction:

Thus, under this condition, the average signed difference should be zero at each prediction value, corresponding to observations centered on \(y=x\). The argument is conditional: it relies on the prediction representing the conditional mean on the plotted scale. For example, a model for log concentration does not automatically imply that an exponentiated prediction is the mean concentration; transformation and the error distribution matter. The plotted scale and the meaning of the model prediction must therefore be consistent.

This justification concerns average calibration, not individual agreement. Even if \(E(C\mid P)=P\), the conditional variance \(Var(C\mid P)\) can be large. The identity-line target alone says nothing about whether the observed scatter is clinically acceptable.

Assumptions, Diagnostics, and Limits

Before interpreting the plot, verify that each plotted pair refers to the same subject, observation time, dose history, and concentration units. Predictions must be generated from the fitted population model using the intended population-prediction convention. Data errors, mismatched records, or predictions calculated for the wrong time after dose can create apparent model failure that is not a model property.

The display is most informative when the dataset covers a meaningful range of predicted concentrations and includes adequate representation of subjects and relevant groups. Sparse sampling, a narrow concentration range, and few observations in a subgroup limit what can be seen. Repeated measurements from a subject share information and often have correlated errors; consequently, a visually dense plot should not be read as if it contained the same amount of independent evidence as an equally sized sample of unrelated subjects.

Warning signs include a consistent displacement above or below the identity line, curvature, changing scatter with concentration, or distinct group clusters. These patterns are prompts for investigation, not automatic diagnoses of a particular model defect. A trend may reflect a structural model issue, an observation-model issue, data handling, or a population not represented adequately by the fitted model. Use the plot alongside residual diagnostics, including the weighted-residual material, and check whether the pattern is plausible in the study context.

The main alternative displays answer different questions. Residual-versus-prediction plots show signed or standardized departures more directly and can make trends in residual spread easier to see. Group-stratified displays emphasize differences between selected groups. Observed-versus-individual-predicted plots use a different prediction target, so they should not be treated as interchangeable with this population-prediction plot. No one display summarizes all aspects of fit.

Most importantly, a plot using observations that contributed to model fitting is an in-sample diagnostic. The model may appear to agree because its parameters were estimated using those data, and sparse or noisy observations may conceal important shortcomings. Even an apparently good in-sample plot does not demonstrate accurate predictions for new subjects, new regimens, or settings outside the observed covariate and concentration ranges. Predictive performance requires evidence appropriate to that question, not visual agreement alone.

Common Pitfall

• Wrong reasoning: “The points have a strong upward trend and a high correlation, so the population predictions are accurate.”
• Correction: A high correlation describes how strongly the two variables move together, not whether their values agree. Check displacement from the identity line, the signed differences, concentration-dependent patterns, and groupwise behavior. Then remember that an in-sample diagnostic is not external predictive validation.
• Wrong reasoning: “The overall mean difference is close to zero, so no group is biased.”
• Correction: Opposing group differences can cancel. Inspect relevant groups and their representation before drawing an overall conclusion.