Population and Individual Predictions Across Tools

Why Prediction Labels Need Interpretation

A column labelled “prediction” does not, by itself, say which parameter values produced it. A prediction may use typical population parameters, or it may use a subject-specific estimate of the random effects. Those answers can differ substantially, even when they come from the same fitted model and the same time point. Correctly identifying the prediction target matters when checking model fit, comparing software output, or preparing a simulation.

The central distinction is between the model’s prediction for a typical subject and its prediction conditional on a particular subject’s data. An empirical Bayes estimate is one way to obtain subject-specific random-effect values for the latter. It is not itself a concentration prediction. In mrgsolve, predictions are generated by running a model with numerical inputs; the software does not automatically turn an estimation tool’s prediction column or empirical Bayes estimates into a simulation.

Model Notation and the Three Quantities

Write the model prediction for subject \(i\) at time \(t_{ij}\) as \(f(t_{ij};\theta,\eta_i,x_i)\), where \(\theta\) denotes population parameters, \(\eta_i\) is the subject’s vector of random effects, and \(x_i\) denotes relevant covariates or other subject-specific inputs. The observation model also includes residual error; the prediction of the model’s expected response is distinct from an observed response that includes residual error.

The typical-subject population prediction is conventionally calculated with \(\eta_i=0\). If \(\hat{\theta}\) is the fitted population parameter estimate, this is \(f(t_{ij};\hat{\theta},0,x_i)\). The individual prediction based on an empirical Bayes estimate \(\hat{\eta}_i\) is \(f(t_{ij};\hat{\theta},\hat{\eta}_i,x_i)\). Software may use labels such as PRED and IPRED for these concepts, but the precise output definition should be checked in the model and output documentation.

The empirical Bayes estimate is informed by both the population distribution of random effects and the subject’s observed data. Depending on the method and reporting convention, it may be a conditional mode or a conditional mean. It is a shrunken estimate: when the subject’s data provide limited information, the estimate is generally pulled toward the population value of zero. It is not a directly observed subject characteristic, nor is it necessarily the subject’s true random effect.

A further distinction is important: setting \(\eta_i=0\) gives the model prediction for a typical subject, not generally the average prediction across the population. The latter averages \(f(t;\theta,\eta,x)\) over the random-effects distribution. For a nonlinear function \(f\), that average usually does not equal \(f(t;\theta,0,x)\).

Worked Example: Typical and Individual Concentration Predictions

What the Labels Mean Across Tools

The conceptual distinction is common across NONMEM, Monolix, and nlmixr2, but exact output names, table contents, and calculation options can depend on the model code, estimation method, and reporting settings. Interpret the formula that generated a column rather than relying on its name alone.

In NONMEM, PRED and IPRED are commonly used for population and individual predictions, respectively, while ETA estimates are reported separately. Confirm how the model code defines each item: derived parameters, covariate relationships, and residual-error conventions affect what a reported prediction represents. In Monolix, distinguish its population prediction output from its individual prediction output and from the estimated individual parameters or random effects. In nlmixr2, inspect the result object and the prediction-related columns generated for the fitted model; do not infer from a column label alone that it represents an EB estimate or a prediction of a particular type.

For all three tools, the key questions are the same: Were random effects set to zero or estimated for each subject? Were the population parameters fixed at their fitted estimates? Are covariates and dose histories subject-specific? Is the value a model prediction, an observed value, or a random-effect estimate? These checks make comparisons meaningful even when output naming differs.

From Estimation Output to mrgsolve Simulation

mrgsolve calculates model outputs by simulating a coded model with supplied numerical inputs. It does not require an estimation tool’s PRED or IPRED column as a simulation input. Instead, provide the model and the values that define the desired simulation: structural parameters, covariates, dose and event records, and any subject-specific effects that the model uses. For a typical-subject simulation, use the population parameter values and set the random effects to zero, if the model is parameterized that way. For an EB-based subject-specific simulation, supply that subject’s EB estimates or the corresponding transformed individual parameters, consistently with the model code.

A model may also generate random effects for a simulated population, but that is a different target from using fitted EB estimates. Random effects drawn from a population distribution represent simulated subjects; EB estimates represent estimates conditional on particular observed subjects. In either case, ensure that the simulated model implements the same parameter relationships and transformations as the estimation model. Transferring a number without preserving its scale or meaning can change the prediction.

Justification: Why Typical and Average Predictions Differ

The distinction between a typical prediction and a population-average prediction follows directly from averaging the nonlinear model. In the example, let \(k=CL/V\) and let \(CL=6\exp(\eta)\), where \(\eta\) varies across subjects. At a fixed positive time \(t\), the concentration prediction is proportional to \(\exp[-(6/V)\exp(\eta)t]\). The typical-subject prediction evaluates this expression at \(\eta=0\); the population average instead averages the expression over the distribution of \(\eta\).

The expectation in the second expression is an average of a nonlinear function, so it cannot generally be replaced by evaluating that function at the average random effect. In symbols, \(E_\eta[f(t;6,\eta)]\ne f(t;6,E[\eta])\) in general. Even when \(E[\eta]=0\), the right-hand side is the typical prediction, not necessarily the population average. The distinction disappears in special cases, such as a prediction function that is linear in the random effect under the relevant averaging scheme, but should not be assumed.

This matters when a question asks for a typical curve, a curve conditional on observed subjects, or an average response across a population. Those are different targets. A population-average prediction may be obtained by integrating over the random-effects distribution or by averaging predictions from representative random-effect draws; it is not obtained simply by labelling the zero-random-effect curve “the population average.”

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to compare prediction columns with a small hand calculation for one subject and one time point. Verify the population parameter values, covariates, dose history, random-effect setting, and transformations. If two tools disagree, first check whether one prediction uses zero random effects and the other uses individual estimates. Then check whether the reported individual quantity is an \(\eta\), a transformed parameter, or a model response.

Individual predictions are useful for examining fit conditional on subjects’ data, but they are not independent validation predictions: the same observations used to estimate the individual effects may also be used to assess their fitted predictions. Population predictions avoid conditioning on each subject’s estimated effects but may not closely track every subject’s observed trajectory. A population-average prediction answers a different question again. The appropriate alternative depends on the target, not on which output column looks most familiar.

• Failure mode: treating an EB estimate as a response. An \(\eta\) is a random effect, often dimensionless; transform it through the model to obtain an individual parameter, then evaluate the structural model to obtain a response prediction.
• Failure mode: double-transforming an effect. Check whether the input is on the \(\eta\) scale or already a transformed parameter scale.
• Failure mode: calling the zero-effect curve an average curve. It is conventionally the typical-subject prediction; averaging over random effects is a separate calculation.
• Failure mode: comparing unlike predictions across tools. Confirm the random-effect setting, covariates, dose events, parameter estimates, and prediction definitions before interpreting differences.

The terminology also has limits. “Individual prediction” is used with some variation across software and workflows, and empirical Bayes estimation procedures need not produce identical subject-effect estimates across tools. Different estimates can arise from estimation and reporting choices even when the structural model is intended to match. For cross-tool comparison, define the mathematical target explicitly and verify it with the model equations and inputs.