Population Predictions Versus Individual Predictions

Why the Prediction Target Matters

A pharmacokinetic model can produce several different predictions for the same formulation and sampling time. One may describe the model’s typical subject, another may use one participant’s observed concentrations to estimate that participant’s random effects, and a third may average predictions over the population distribution of random effects. These are different targets, not interchangeable computational versions of the same quantity.

The distinction matters when model predictions are used to describe exposure or compare formulations. The model may predict concentrations from which AUC or Cmax can be derived, but the interpretation of those derived quantities still depends on whose predictions are being summarized. A typical-subject prediction is not automatically a population average, and a subject-specific prediction is not automatically an estimate of the population’s average response.

A Model That Makes the Distinctions Explicit

Let \(C_{if}(t)\) denote concentration for individual \(i\), formulation \(f\), at time \(t\). A general nonlinear mixed-effects model can be written as

Here \(h\) is the structural concentration model, \(x_i\) represents measured individual characteristics, \(\beta\) contains fixed effects, \(b_i\) contains individual random effects, and \(\varepsilon_{if}(t)\) is residual error. The random effects have population distribution \(G\), with mean zero and covariance matrix \(\Omega\). This formulation assumes that the structural model is appropriate, the random-effects distribution is specified adequately, and the residual model describes remaining observation error. The way predictions are formed does not make a misspecified model valid.

For clarity, the definitions below refer to the model’s structural prediction \(h\), before adding a new residual error. A prediction of a future measured concentration can also account for residual error, but that adds observation-level variability and is a different target.

The empirical Bayes estimate \(\widehat b_i\) is typically obtained from the conditional distribution of the random effects given the individual’s data and the fitted model. It is shrunk toward the population mean when that individual’s data are limited or noisy. The marginal prediction instead averages across the distribution of individuals, including their random-effect variation.

The zero-random-effect prediction is often called a typical-subject prediction. It is a useful reference, but the individual with \(b_i=0\) need not exist in the sample, and this prediction is generally not the population mean on the concentration scale.

Why Marginal and Fixed-Effects Predictions Can Differ

If the model is linear in a random effect and the response is also on that linear scale, averaging over a zero-mean random effect can leave the mean unchanged. Nonlinear transformations change that result. In pharmacokinetic models, concentrations are often positive and a model may be specified for log concentration. Even if the random effect has mean zero on the log scale, exponentiating a typical log prediction does not generally give the arithmetic mean concentration.

For example, suppose the log concentration at a specified formulation and time is

The fixed-effects prediction on the concentration scale is \(\exp(\eta)\), corresponding to \(b=0\). To obtain the marginal arithmetic mean, average \(\exp(\eta+b)\) over the distribution of \(b\). Since a normal random variable \(b\) with mean zero and variance \(\omega^2\) satisfies \(E[\exp(b)]=\exp(\omega^2/2)\), it follows that

Thus the marginal arithmetic mean exceeds the fixed-effects prediction whenever \(\omega^2>0\) in this model. The fixed-effects prediction is the median of this lognormal concentration distribution, while the marginal prediction is its arithmetic mean. The result depends on the model and scale: it should not be generalized to every possible structural model without checking how the random effects enter.

Empirical Bayes Predictions Use Individual Data

An empirical Bayes prediction answers a conditional question: given the fitted population model and the available observations for a particular individual, what random-effect value is estimated for that person, and what model prediction follows from it? It is not simply the observed concentration copied back into the model. The estimate balances the individual’s data against the population distribution, with the balance determined by the relative sizes of between-individual and residual variability.

A simple normal model makes this shrinkage visible. Suppose \(d_{ij}=y_{ij}-\mu\) are residuals from the fixed prediction on a log scale, and

with independent residuals conditional on \(b_i\). For known variance components and \(n_i\) observations, the conditional mean of \(b_i\) is

The multiplier is between zero and one. When residual variance is large relative to random-effect variance, the individual data provide weak evidence for a departure from zero, so the estimate is strongly shrunk. With more observations, \(\sigma^2/n_i\) decreases and the data can support a less-shrunk estimate. In a real analysis the variance components are estimated, so the resulting empirical Bayes prediction also inherits uncertainty from fitting the population model.

Marginal Predictions Average Across Individuals

A marginal prediction integrates the structural prediction over the population distribution of random effects. It answers what the model predicts after averaging across individuals, for a specified formulation, time, and covariate profile. When the model is nonlinear in the random effects, the integral may not have a closed form and may need numerical quadrature or another integration method. The distribution being averaged over must be the fitted population distribution, not the set of empirical Bayes estimates treated as if they were known population values.

Choosing a Prediction for a Bioequivalence Question

The appropriate prediction is determined by the estimand: the quantity the analysis is intended to describe. The earlier material on model-based bioequivalence estimands establishes that the target must be specified before interpreting a model contrast. Prediction type is one part of that specification.

A fixed-effects contrast is often useful for describing a model’s typical-subject formulation effect. An empirical Bayes contrast can describe how the model predicts a particular participant under different formulations, if the design and data support that individual-level target. A marginal contrast is relevant when the question concerns the modeled population average. These choices do not by themselves establish bioequivalence; the intended target and decision rule must both be appropriate to the scientific question.

For a paired crossover comparison, care is needed in constructing individual predictions. If the aim is to compare formulations for the same individual, that individual’s shared random effects should be held consistently across the two formulation predictions. If a formulation-by-subject random effect is part of the model, its role also has to match the target. Otherwise, a calculated difference may mix formulation change with an inconsistent change in the assumed individual. Prediction calculations should follow the model structure rather than silently changing random effects between conditions.

Diagnostics, Failure Modes, and Limitations

Predictions are only as credible as the model and the data supporting them. Useful checks depend on the target:

• Fixed-effects prediction: check whether the chosen covariate values and times are within the range adequately represented by the data. A neat prediction at an unsupported time or covariate profile may be extrapolation, not reliable evidence.
• Empirical Bayes prediction: inspect how much information each individual contributes and whether estimates are excessively shrunk. Very sparse data can make individual predictions resemble the population typical prediction, even when the true individual response differs.
• Marginal prediction: verify the random-effects distribution, covariance structure, and integration calculation. In a nonlinear model, replacing the integral with the prediction at mean random effects can produce a biased target.
• All predictions: assess structural fit and residual patterns. A wrong absorption or disposition structure can yield misleading concentrations and derived exposure quantities even if numerical calculations are exact.

A frequent mistake is to call \(\exp(\eta)\) the population mean concentration whenever the model is written for log concentration. This is the typical-subject prediction and, under the normal random-intercept model, the median; the population arithmetic mean is \(\exp(\eta+\omega^2/2)\). A second mistake is to interpret empirical Bayes estimates as direct observations of each person’s true parameters. They are conditional estimates, typically shrunk and uncertain. A third is to assume that the marginal mean is always the right target: it is right only when the scientific question calls for that population average.

These prediction types are not substitutes for uncertainty intervals. A plug-in predicted curve does not show uncertainty in fixed effects, random effects, residual error, or model structure. Likewise, prediction differences do not automatically give a formal equivalence conclusion. They are summaries generated under a model, and their scope should be stated precisely.