Prediction Without Re-Estimation

What It Means to Predict Without Re-Estimation

External validation asks what a development model predicts for subjects who were not used to estimate it. In this tutorial, the model parameters remain fixed at their development estimates: external concentrations do not alter the population parameter estimates, the random-effects covariance matrix, or the residual-error model. The external subjects’ known covariates, doses, and sampling times can still be supplied to the model. For a subject-conditional prediction, that subject’s observed concentrations can also inform the subject-specific random effects, while the population model itself remains locked.

This distinction is important because two different predictions can be made for the same external subject. A population prediction averages over the model’s distribution of subject-to-subject random effects. A conditional prediction uses the subject’s observed concentrations to update what is predicted for that subject. Neither prediction re-estimates the fixed development-model parameters. The predictions answer different questions, so they should not be treated as interchangeable.

Model and Prediction Targets

Write a general population PK model for subject \(i\) at observation \(j\) as

Here \(Y_{ij}\) is the observed concentration, \(t_{ij}\) is the sampling time, \(D_i\) denotes the dose history, \(X_i\) denotes covariates, \(\theta\) contains the fixed structural and covariate parameters, \(\eta_i\) is the subject-specific random effect, and \(\epsilon_{ij}\) is residual error. The development model also specifies distributions for random effects and residual error, such as \(\eta_i\sim N(0,\Omega)\), with \(\Omega\) the random-effects covariance matrix. A locked prediction uses the development estimates of \(\theta\), \(\Omega\), and the residual-error parameters.

A prediction must state its target. It may concern the latent concentration described by the structural model, or a future measured concentration that includes residual error. For a future observation at time \(t^\ast\), with the external subject’s covariates and dosing history denoted by \(X_i,D_i\), a population prediction is the model-based expectation over random effects (and, for a measured concentration, residual error):

A subject-conditional prediction additionally conditions on the external subject’s observed concentration data \(Y_{i,\mathrm{obs}}\). It averages over the distribution of that subject’s random effects after seeing those observations, rather than simply replacing the random effect with a single estimate:

The development parameters remain fixed in both expressions. The conditional prediction changes because information about \(\eta_i\) has changed, not because the population model has been refitted. In practice, a software procedure may report a plug-in prediction using an empirical Bayes estimate of \(\eta_i\). That is a point approximation to the conditional calculation, and it is not generally identical to integrating over the full conditional distribution of \(\eta_i\).

A prediction at \(\eta_i=0\), often called a typical-subject prediction, is another distinct quantity. It evaluates the structural model at the mean random effect. In a nonlinear model, that value need not equal the population-average prediction. Nor does a typical-subject prediction automatically include residual error.

Conditioning on External Concentrations

Observed concentrations can provide information about a subject’s random effect. They do not, by themselves, require refitting the population model. The distinction is easiest to see in a simple log-concentration model. Suppose

with independent \(\eta\) and \(\epsilon\). The development values of 6, \(\Omega\), and \(\sigma^2\) are fixed. After observing \(Z_{\mathrm{obs}}\), the distribution of \(\eta\) changes. In this conjugate normal example, completing the square in the prior density times the likelihood gives

The conditional mean is a weighted version of the observed log-concentration deviation from \(\log 6\). The weight is between zero and one: noisy observations contribute less information relative to the prior random-effects distribution. This derivation relies on a normal prior, normal log-scale error, independence, and known fixed variances. More complex PK models generally require numerical integration or approximation to obtain the corresponding conditional distribution.

Applying the Locked Model to an External Design

Prediction requires using the external subject’s actual known inputs, not silently replacing them with development-sample averages. These inputs include dose amounts and times, sampling times, covariate values, and any protocol features represented in the model. The fixed parameters remain the same for every external subject; predicted concentrations can differ because their inputs differ.

Assumptions, Checks, and Alternatives

A locked prediction is interpretable only if the model and external inputs are sufficiently compatible. Before calculating predictions, verify that the external dataset uses the same units and variable definitions, and that dose histories and sampling times are represented correctly. Check that the model’s structural assumptions and covariate relationships are intended to apply to the external setting. These checks do not establish that the model is adequate; they identify avoidable discrepancies that would make its predictions incoherent.

For conditional predictions, confirm that each concentration used for conditioning belongs to the subject and dosing history being predicted. A concentration measured after a target time cannot be used for a genuinely prospective prediction at that earlier time without changing the prediction question. Likewise, using an observation both to condition a prediction and to describe that same observation as an independent future prediction is circular. Conditioning must be documented, including which observations were available at the prediction time.

Useful implementation checks include verifying units, reproducing a few predictions by hand, confirming the parameter file is the locked development version, and checking that any numerical integration or optimizer reports convergence. For conditional calculations, inspect whether the subject-specific estimate is being driven by data or is effectively shrunk toward the population distribution. Extreme estimates, unstable results from small changes in starting values, or predictions outside plausible ranges can signal sparse information, numerical problems, or a mismatch between data and model assumptions. These checks diagnose computation and applicability; they do not justify changing parameters while still calling the result prediction without re-estimation.

The main alternatives answer different questions. A typical-subject prediction evaluates the model at \(\eta_i=0\); it is simple but is not generally the population mean. A population prediction integrates over random effects and answers what the model predicts on average for subjects with the stated inputs. A conditional prediction uses external observations to focus on one subject, while retaining the development population model. Re-estimating parameters with external data is model updating, not prediction with a locked model. It may be useful for a different objective, but it must be distinguished from an external prediction made with unchanged parameters.

A common mistake is to call a prediction “population-based” after setting every random effect to zero and to interpret it as the average concentration in the population. In a nonlinear model, generally \(E\{f(\eta)\}\ne f\{E(\eta)\}\). The correction is to state whether the calculation uses \(\eta=0\) or integrates over the random-effects distribution, and to include residual error only when the target is a measured concentration. Another mistake is to treat conditional use of external concentrations as re-estimation. Conditioning updates information about an individual random effect under fixed population parameters; re-estimation changes one or more population-model parameters using external data.