Three Levels of the NLME Hierarchy

Why an NLME Model Has Three Levels

A pharmacokinetic model can describe a concentration profile for one person, but a population analysis must also represent how profiles differ between people and how measurements vary around each person’s profile. These are related but distinct sources of information. Confusing them can lead to interpreting a measured concentration as if it were a parameter, or treating a subject-specific prediction as if it were the population’s average profile.

The hierarchy separates population-level parameters, subject-specific parameters, and individual observations. The population level describes the population distribution of subject parameters. A subject-specific parameter set determines that subject’s model-predicted profile. The observation model then describes how measured data relate to that profile.

These levels are not three names for the same quantity. A population parameter such as a parameter of the distribution of clearance values is not an individual’s clearance. An individual clearance is not a concentration measurement. Instead, the individual clearance enters a structural model that predicts concentrations, and the observation model connects those predictions to the recorded concentrations.

From Subject Parameters to Observations

Write the structural prediction for subject \(i\) at observation \(j\) as \(f(x_{ij},\theta_i)\). For a continuous response, one possible observation model is additive normal error:

Here \(Y_{ij}\) is a random variable whose realized value is \(y_{ij}\), and \(\sigma^2\) is the conditional observation variance. This formula says that, given the subject’s parameters and design conditions, the expected observation is the structural prediction and the observation varies around it according to the stated error model. Other observation models may be appropriate for different response types or error patterns; normal additive error is an example, not a universal requirement.

The hierarchy also requires a population model for the \(\theta_i\). Its form and assumptions depend on the application. For example, it must assign a valid probability distribution to possible subject parameters. Subjects may have different parameter values, but those values are modeled as arising from a population distribution described by \(\psi\). The details of how that distribution is parameterized are a separate question; for now, the important point is that \(\psi\) describes the distribution, while \(\theta_i\) belongs to a particular subject.

Population-Level Parameters and Population Predictions

The population parameters govern the distribution of subject-specific parameters. To obtain a distribution for observations in a randomly selected subject, average the conditional observation distribution over the possible subject parameters. For one observation with design \(x\), the marginal density is

This expression follows from the law of total probability: the observation can have different conditional distributions for different values of \(\theta\), so its population distribution is a weighted average of those conditional distributions. The integral is over the subject-parameter space. For a discrete population distribution, the corresponding operation is a sum.

This distinction matters because averaging parameters and then predicting is generally not the same as averaging predictions. For a nonlinear structural model \(f\), in general \(E[f(x,\theta)]\ne f(x,E[\theta])\). The population-average prediction is obtained by averaging subject-level predictions over the parameter distribution, not automatically by plugging an average parameter into the structural model.

A Short Derivation: Why Observation Variability Has Multiple Sources

The hierarchy helps explain why observations across people can vary even when the same design is used. The following variance identity makes the distinction explicit. For a response \(Y\) and subject parameter \(\theta\), provided the variances exist,

To justify it, write \(Y-E[Y\mid x]\) as the sum of \(Y-E[Y\mid\theta,x]\) and \(E[Y\mid\theta,x]-E[Y\mid x]\). Squaring and taking expectations, the cross-product has expectation zero because \(E[Y-E[Y\mid\theta,x]\mid\theta,x]=0\). The first squared term averages the conditional observation variance; the second is the variance of conditional means across subject parameters. Thus total variability includes within-subject observation variation and variation in the subject-level expected response. Their magnitudes depend on the model and design; this identity does not say they are equal or that one can be ignored.

Assumptions, Diagnostics, and Alternatives

A three-level description is a framework, not a guarantee that a particular model is adequate. Before using it, specify what is assumed at each link. Common working assumptions include a population distribution that is appropriate for the subject parameters, an observation model with a suitable mean and variance structure, and conditional independence of observations when the likelihood is written as a product across observations. Independence is not automatic: repeated measurements can have additional serial structure, in which case the observation model must account for it. If subjects are modeled as independent draws from the target population, their parameter vectors are also treated as independent conditional on the population parameters.

Diagnostics should be interpreted at the level they address. Subject-level plots of observations against structural predictions can reveal systematic profile patterns; residuals plotted against time or prediction can reveal an observation model that misses trends or changing spread. Comparing profiles across subjects can expose a population model that fails to represent between-subject patterns. A single unusual measurement may concern observation error, while a persistent subject-specific pattern may indicate a mismatch in the structural model or in the population description. These clues are not definitive by themselves, and inspection should respect the design and the assumed error model.

The main alternatives answer different questions. A pooled model that uses one common parameter set for all subjects can describe a shared profile but does not represent a distribution of subject parameters. Fitting each subject separately can describe individual profiles, but it does not by itself provide a common population distribution and can be unstable when a subject has sparse data. The hierarchical model links population description, subject parameters, and measurements in one structure. That connection is useful when the target includes both population behavior and individual profiles, but it depends on adequate data and defensible assumptions at each level.

A limitation follows directly from the hierarchy: observed data do not reveal each layer without modeling assumptions. Sparse observations may provide little information about a subject’s parameters; variability attributed to the population layer can be difficult to distinguish from observation variation if the design or model is inadequate. A flexible population model cannot repair an incorrect structural prediction, and a good fit to one subject does not establish that the population distribution is appropriate.