What an Individual-Prediction Plot Shows
An observed-versus-individual-predicted concentration plot places each measured concentration against the model prediction for that same subject and sampling time. It is a useful check of how well the fitted model can describe the observed records after accounting for estimated differences among subjects. Its apparent fit, however, has to be interpreted differently from the fit in an observed-versus-population-predicted plot: an individual prediction uses information from the subject’s own observations.
For observation \(j\) from subject \(i\), let \(y_{ij}\) denote the observed concentration. A population prediction (PRED) is computed from the population model and the subject’s covariates, with the subject’s random effects set to their population-typical values. An individual prediction (IPRED) is computed with the subject-specific empirical Bayes estimates (EBEs) of those random effects. The plot pairs \(y_{ij}\) with the corresponding IPRED for each record.
A point close to the identity line means that the individual prediction is close to that observed concentration on the scale plotted. It does not establish that the model predicts new subjects or future observations accurately. IPREDs are usually conditional on estimated subject effects obtained using that subject’s observed concentration record, so the plot is an in-sample, subject-adapted diagnostic.
Constructing and Reading the Plot
Use the same observations and the same fitted model to create the population-prediction and individual-prediction plots. Match every prediction to its observed record; a prediction at one time after dose cannot be paired with an observation at another time. Label both axes with concentration units, and show the identity line over the range of the data. If subjects contribute repeated samples, distinguish subjects where useful: many records from one person do not represent many independent subjects.
With observed concentration on the vertical axis and IPRED on the horizontal axis, a point above the identity line has a positive raw difference \(y_{ij}-\mathrm{IPRED}_{ij}\). A point below the line has a negative difference. A systematic displacement or pattern may flag a problem, but the plot alone does not identify its cause. For example, a curve or changing spread can reflect structural-model mismatch, a misspecified residual error model, concentration-dependent error, or features of the sampling design. A roughly balanced cloud around the line is reassuring about conditional description, not proof that all model assumptions are correct.
Interpret the plot alongside the subject-level data, dosing and sampling records, and residual diagnostics. In particular, raw differences on a concentration scale can have different practical meaning at low and high concentrations. Do not infer that residuals have constant variance simply because the points appear close to the line. The choice of a linear or logarithmic diagnostic axis also affects visual appearance and is considered separately.
Worked Example: Pairing Observations With Both Predictions
Suppose a hypothetical subject has three concentration measurements, all in mg/L. The fitted model gives the following population and individual predictions. The values are illustrative, not from a real study.
| Sampling record | Observed \(y_{ij}\) (mg/L) | PRED (mg/L) | IPRED (mg/L) |
|---|---|---|---|
| 1 | 3.5 | 4.0 | 3.7 |
| 2 | 7.2 | 8.0 | 7.6 |
| 3 | 11.4 | 12.0 | 11.7 |
For record 1, the point in the individual-prediction plot is \((3.7,3.5)\) when the horizontal axis is IPRED and the vertical axis is observed concentration. Its difference from the identity line is \(3.5-3.7=-0.2\) mg/L. For records 2 and 3, the corresponding differences are \(7.2-7.6=-0.4\) mg/L and \(11.4-11.7=-0.3\) mg/L. All three points lie below the identity line, so all three observations are smaller than their IPREDs.
The sum of squared raw differences is \(0.2^2+0.4^2+0.3^2=0.29\;(\mathrm{mg/L})^2\). For PRED, the differences are \(3.5-4.0=-0.5\), \(7.2-8.0=-0.8\), and \(11.4-12.0=-0.6\) mg/L, giving \(0.5^2+0.8^2+0.6^2=1.25\;(\mathrm{mg/L})^2\). The individual predictions are closer by this unweighted summary for these records. That comparison describes the example; it is not a general test of model adequacy or a substitute for the model’s residual-error weighting.
Why IPRED Often Looks Better
In a population model, the subject-specific random effects describe deviations from typical population parameters. EBEs are estimated using both the population distribution of those effects and the subject’s observed data. Consequently, IPREDs adapt the model prediction toward the observations used to estimate the EBEs. This adaptation is expected to improve the apparent fit compared with PRED, which does not use the subject-specific estimated effects.
A simple normal model shows the source of this adaptation. Suppose a subject’s latent effect \(\eta\) has prior distribution \(\eta\sim N(0,\omega^2)\), and a data summary \(z\) satisfies \(z\mid\eta\sim N(\eta,\sigma^2)\). Under these assumptions, the conditional distribution of \(\eta\) given \(z\) is normal, with mean and variance
The mean is a weighted version of the observed summary \(z\), pulled toward the population mean zero. The weight on \(z\) is between zero and one when both variances are positive. More informative subject data (smaller \(\sigma^2\)) give a larger weight to \(z\); less informative data give a stronger pull toward zero. This derivation is an illustration under a scalar Gaussian model. In a nonlinear population PK model, the EBE calculation may use a more complex likelihood and the resulting IPRED need not follow this exact formula.
Worked Example: A Simple Empirical Bayes Update
Consider the illustrative normal model above with \(\omega^2=0.16\), \(\sigma^2=0.25\), and an observed subject summary \(z=0.60\). The conditional mean effect is
The data summary is \(0.60\), but the estimated effect is \(0.2341\), closer to the population mean of zero. To see how this can affect a prediction, suppose a hypothetical concentration relationship is \(C=10\exp(\eta)\) mg/L. The typical prediction at \(\eta=0\) is \(10\exp(0)=10.00\) mg/L. Using the conditional mean effect gives \(10\exp(0.2341)=12.64\) mg/L, while using \(z\) directly would give \(10\exp(0.60)=18.22\) mg/L. Thus the individual prediction adapts toward the subject’s data but retains a pull toward the population. This simple calculation illustrates the principle; actual IPREDs come from the fitted PK model and its EBE procedure.
Comparing the Same Subjects With PRED and IPRED
The two plots answer related but distinct diagnostic questions. PRED asks how observations compare with predictions based on typical population parameters and available covariates. IPRED asks how observations compare with predictions after subject-specific effects have been estimated from those subjects’ records. A better-looking IPRED plot is therefore expected and is not, by itself, evidence that the population model is well specified.
| Feature | Population-prediction plot | Individual-prediction plot |
|---|---|---|
| Subject-specific random effects | Set to typical values | Uses subject-specific EBEs |
| Information from the subject’s observations | Not used to estimate that subject’s effects | Used in estimating the subject’s effects |
| Main diagnostic emphasis | Population-level description with covariates | Conditional, subject-adapted description |
| Interpretation of closer points | Typical predictions are closer to observations | Adapted predictions are closer to observations |
Worked Example: Separating Typical and Adapted Fit
Two hypothetical subjects each contribute one concentration observation, measured in mg/L. Subject A has an observed concentration of 5, with PRED 8 and IPRED 5.8. Subject B has an observed concentration of 15, with PRED 12 and IPRED 14.2.
For subject A, the absolute difference from PRED is \(|5-8|=3\) mg/L, while the difference from IPRED is \(|5-5.8|=0.8\) mg/L. For subject B, the corresponding differences are \(|15-12|=3\) mg/L and \(|15-14.2|=0.8\) mg/L. Across these two records, the sum of absolute differences is \(3+3=6\) mg/L for PRED and \(0.8+0.8=1.6\) mg/L for IPRED.
The population predictions miss the two subjects in opposite directions, whereas the EBEs adapt each prediction toward its subject’s observation. The smaller IPRED differences show improved conditional description of these same records. They do not show that the model would predict new subjects’ concentrations with the same accuracy, because the new subjects’ EBEs would not yet have been estimated from their own observations.
Diagnostics, Limitations, and Common Mistakes
Before interpreting the cloud, check that the model fit and prediction calculations completed as intended, each observation is matched to its own prediction, and concentrations and units are consistent. Examine whether a few subjects contribute most of the points. If the plot looks excellent but the population-prediction plot shows substantial departures, ask whether the apparent improvement is mainly due to subject adaptation. Compare patterns within subjects as well as across the pooled records; pooling can hide subjects whose predictions are systematically high or low.
An individual-prediction plot has important limitations. It is conditional on EBEs, which are estimates rather than known true individual parameters. With sparse or uninformative sampling, an individual’s data may provide little information about their random effects, so EBEs are pulled toward typical values. Conversely, when estimating the EBEs, those same observations contribute to the fit being displayed. Neither situation makes the plot useless, but both limit what can be inferred from proximity to the identity line. A plot also cannot alone distinguish a structural-model problem from residual error, data problems, or influential observations.
A common mistake is to conclude, “The IPRED plot lies close to the identity line, so the model predicts concentrations accurately.” The reasoning overlooks how the predictions were obtained: each subject’s observed data helped determine the EBEs used for that subject’s IPREDs. The correction is to describe the plot as evidence about the model’s conditional in-sample fit, then assess population-level fit and residual behavior separately. For claims about performance on new data, use an evaluation that withholds or otherwise does not reuse the observations being predicted.
Another mistake is to treat every plotted record as an independent subject-level validation. Repeated concentrations within one subject share estimated effects and are not independent replicates of the population. Report which level is being assessed, and avoid treating a large number of within-subject points as equivalent to a large number of independent subjects. Individual-prediction plots complement population-prediction and residual diagnostics; none is a stand-alone pass/fail test.
Match each observed concentration with PRED and IPRED for the same subject and sampling record, using consistent concentration units.
Look for systematic offsets or patterns, and remember that proximity describes agreement on the plotted scale.
Use PRED to inspect typical population predictions and IPRED to inspect subject-adapted conditional predictions from the same observations.
Interpret the plot with residual diagnostics, subject-level patterns, and the information used to estimate EBEs; do not equate an attractive IPRED plot with independent predictive validation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- On a plot with IPRED on the horizontal axis and observed concentration on the vertical axis, what does a point above the identity line mean?
- Why can IPREDs look closer to observed concentrations than PREDs for the same subjects?
- In the simple normal update, let \(\omega^2=0.09\), \(\sigma^2=0.21\), and \(z=0.50\). Calculate \(E(\eta\mid z)\).
- Does a close observed-versus-IPRED plot establish accurate predictions for new subjects? Explain.
- Why should repeated observations from one subject not be treated as independent subject-level validations?
Solutions. (1) The observed concentration is greater than IPRED, so \(y_{ij}-\mathrm{IPRED}_{ij}>0\). (2) Each subject’s observations help estimate that subject’s EBEs, which are then used to calculate IPREDs; this subject-specific adaptation generally brings predictions toward the data. (3) \(E(\eta\mid z)=\frac{0.09}{0.09+0.21}(0.50)=\frac{0.09}{0.30}(0.50)=0.1500\). (4) No. The displayed IPREDs use subject-specific estimates derived from those subjects’ observed data, so the plot assesses conditional in-sample fit rather than independent prediction for new subjects. (5) Repeated observations share subject-specific effects and the same subject’s information contributes to their predictions; they are not independent new subjects.