Conditional and Individual Prediction Plots

Why Compare Observations With Conditional Predictions?

A population prediction describes the model's expected response for a subject with typical random effects. It is useful for asking whether the population model captures the broad pattern in the data. But a population prediction does not use a subject's estimated random effects to represent that subject's individual trajectory. An observed-versus-conditional-prediction plot makes that subject-specific comparison: each observed response is paired with the prediction conditional on the fitted model and the subject's information.

This distinction matters when a model allows subjects to differ in parameters such as clearance or volume. A population prediction can miss a subject's observations in a consistent direction even when the model's subject-specific predictions track that subject more closely. Conversely, an apparently close conditional fit does not by itself show that the population model is adequate: the subject-specific estimates were informed by the same observations being plotted.

Definitions and Assumptions

For subject \(i\) at observation time \(t_{ij}\), consider a nonlinear mixed-effects model with a structural prediction \(f(t_{ij}, x_{ij}, \theta, \eta_i)\). Here \(x_{ij}\) denotes known design or covariate information, \(\theta\) denotes population parameters, and \(\eta_i\) denotes subject \(i\)'s random effects. An observation model relates the measured response \(Y_{ij}\) to this structural prediction and to residual variability. For example, under an additive observation model, \(Y_{ij}=f(t_{ij},x_{ij},\theta,\eta_i)+\epsilon_{ij}\), where \(\epsilon_{ij}\) has mean zero under the model.

The word “conditional” signals that the subject-specific prediction depends on information used to estimate that subject's random effects. In the conventional plug-in definition above, \(\hat{\eta}_i\) is estimated from the fitted model and that subject's observations. Software may offer related quantities, such as a prediction averaged over a conditional distribution of random effects. A report should state which quantity it uses; values from different definitions need not agree.

The plot assumes that observed responses and predictions refer to the same response variable, are on the same scale, and are matched to the same subject, occasion, time, and record type. Model parameters and random-effect estimates must come from a defined fit, and the plotted observations should be the records represented by those predictions. If observations are censored, below quantification limits, or otherwise transformed, they cannot be treated as ordinary measured points without accounting for that status. These requirements concern a valid comparison; they do not imply that the model itself is correct.

Constructing and Reading the Plot

Place IPRED on the horizontal axis and the corresponding observed response on the vertical axis. Draw the equality line, \(y=x\), using the same numerical scale and units on both axes. A point above the line has an observed value greater than its conditional prediction; a point below has an observed value smaller than its conditional prediction. The signed vertical difference is observed minus IPRED. This description identifies the point's position; interpretation of residual patterns belongs to the relevant residual diagnostics.

Points close to the equality line indicate close agreement for those plotted records on the chosen scale. A wide spread indicates that conditional predictions and observations differ substantially for some records. A concentrated cloud around the line is not, by itself, proof of adequate population behavior, correct uncertainty, or predictive performance in new subjects. The plot is a model-fit diagnostic conditional on the analysis data, not an independent validation exercise.

Why the Two Predictions Differ

PRED and IPRED differ because PRED evaluates the structural model at the typical random-effect value, whereas IPRED uses a subject-specific estimate. A local expansion of \(f\) around \(\eta_i=0\) makes the relationship explicit. For a scalar random effect and a small \(\hat{\eta}_i\), a first-order Taylor approximation gives:

The first term is PRED; the second term is the approximate subject-specific adjustment. With multiple random effects, the adjustment is the gradient with respect to the random-effect vector multiplied by that vector of estimates. This is a local approximation, not an exact equality in a nonlinear model. It explains why predictions can separate when the estimated random effects are nonzero and the response is sensitive to them.

Conditional estimates are generally informed by the subject's data as well as the population distribution assumed by the model. Consequently, IPRED can move toward the observations used to estimate \(\hat{\eta}_i\). The amount of movement depends on the data, the model, and the estimation method; it is not guaranteed that every IPRED will be closer to its observation than PRED. Nor should the plot be described as an independent test of the model.

Interpreting the Plot in Context

Interpretation begins with the plot's purpose and construction, not with a visual verdict. Confirm the response, units, axis scales, record selection, and prediction definition. Then consider whether points tend to lie near the equality line and whether their positions differ among subjects, times, occasions, or ranges of the response. These are clues for follow-up, not a substitute for examining the model and data. A few distant points may reflect measurement error, a data-processing issue, a subject-specific mismatch, or model limitations; the plot alone cannot identify the cause.

Comparison with the observed-versus-population-prediction plot clarifies what changes. The population plot evaluates typical-subject predictions against observations. The conditional plot instead uses subject-specific estimates, so it asks whether the fitted model can represent records more closely after conditioning on each subject's data. Use the two plots as complementary views. If the conditional plot is much closer to the equality line, that may reflect meaningful subject-level information, but it may also reflect the reuse of observations during estimation. The population plot remains important for evaluating whether typical predictions align with the observed population data.

Diagnostics, Failure Modes, and Limitations

A useful diagnostic check is to verify the data-to-prediction join before interpreting a figure. Each plotted record must pair an observation with the correct subject's estimate and the matching time, occasion, and response. Duplicate joins can create repeated points; mismatched identifiers can pair an observation with another subject's prediction. A plot can look plausible despite either error, so automated checks of record counts, missing predictions, identifiers, and units are valuable.

Sparse subject data require particular caution. When a subject has few informative observations, the estimate \(\hat{\eta}_i\) is typically pulled toward the population distribution; conditional predictions may then be close to PRED. This is often described as shrinkage of individual estimates. A close or narrow-looking cloud can therefore coexist with limited information about individual differences. Conversely, many observations per subject do not guarantee a good model if the structural or observation model is misspecified.

Use a common scale on both axes when drawing the equality line. If either axis is logged, make clear whether the plotted values are transformed and ensure that the equality line represents equality on that transformed scale. Log scales are not appropriate for zero or negative values without a justified transformation. Restricting axes can hide discrepancies, and pooling many records can obscure subjects with few observations or distinct trajectories; inspect the point distribution and retain enough subject and record context to understand it.

A frequent mistake is to say, “IPRED is closer to the observations, so the model has been independently validated.” The reasoning fails because the conditional estimates use information from the same subjects, often including the observations on the plot. The correction is to say that the conditional plot describes in-sample, subject-specific agreement under the stated estimation procedure. Assessing prediction for new subjects requires an appropriate validation design, not merely replacing PRED with IPRED.

Another pitfall is treating PRED and IPRED as labels that mean exactly the same thing in every software package. The conventional definitions in this tutorial are explicit, but implementations and report pipelines may calculate or name related predictions differently. Record the formula, estimation method, data records, and scale used. This reproducibility information is necessary for comparisons across reports and model runs.

Practical Workflow

This workflow differs from simply choosing whichever plot looks best. The population-prediction plot emphasizes typical predictions across the population; the conditional plot emphasizes the fitted representation of individual subjects. Neither alone shows that the model will predict future observations accurately, and neither explains a discrepancy without further investigation.