Hierarchical Models for Population Design

Why Population Design Needs a Hierarchical Model

An individual sampling design asks when to measure one subject to learn about that subject’s pharmacokinetic parameters. A population design asks a different question: how should observations be allocated across subjects to learn about population-level quantities, such as typical clearance or the variability of clearance between subjects? Those questions are linked, but their data structures are not interchangeable. Several observations from one subject share that subject’s characteristics; they are not equivalent to the same number of observations from different subjects.

The hierarchical model makes the relationship explicit. It distinguishes the population parameters governing the distribution of individual characteristics from the individual deviations drawn from that distribution, and from residual measurement or model error at each observation. The distinction is essential when organizing a population information matrix: the population parameter vector has multiple kinds of components, while the repeated records within each subject form a single subject-level contribution.

Population Parameters and Individual Random Effects

Let subject \(i\) have a design \(\xi_i\), which specifies the relevant dosing conditions, covariates, and observation times. Let \(Y_{ij}\) denote the response for subject \(i\) at that subject’s \(j\)th observation, taken at time \(t_{ij}\). A general nonlinear mixed-effects model can be written as

Here \(f\) is the structural model, \(\boldsymbol{\theta}\) is a vector of population fixed effects, \(\boldsymbol{\eta}_i\) is the random-effect vector for subject \(i\), and \(\epsilon_{ij}\) is the residual error for that observation. For example, \(\boldsymbol{\theta}\) might contain typical values of clearance and volume, while components of \(\boldsymbol{\eta}_i\) describe how a particular subject’s clearance or volume differs from those typical values.

A common specification sets \(\boldsymbol{\eta}_i\sim N(\mathbf{0},\boldsymbol{\Omega})\), where \(\boldsymbol{\Omega}\) is the random-effects covariance matrix. Its diagonal entries are variances, and its off-diagonal entries describe covariance between random effects. The zero mean defines the fixed effects as the population reference values on the chosen model scale. The normal distribution is common, but it is a modeling assumption, not part of the definition of a random effect.

The population parameter vector may be written as \(\boldsymbol{\phi}=(\boldsymbol{\theta},\boldsymbol{\Omega},\boldsymbol{\psi})\), where \(\boldsymbol{\psi}\) contains residual-error parameters. A parameterization may instead use standard deviations and correlations to represent \(\boldsymbol{\Omega}\). Whichever representation is used, it must describe a valid covariance matrix, and the chosen components and parameterization must be stated when constructing an information matrix.

Residual Errors and the Nested Observation Structure

Conditional on a subject’s random effects, the structural model predicts that subject’s response at each design point. Residual errors describe the remaining observation-level deviations. A frequently used assumption is

with random effects and residual errors independent, and with different subjects’ random effects and residual-error processes independent. Here \(\boldsymbol{\epsilon}_i\) collects the residuals for all of subject \(i\)’s observations, and \(\mathbf{R}_i\) is their residual covariance matrix. If residual errors are assumed independent with common variance, \(\mathbf{R}_i=\sigma^2\mathbf{I}\). Other residual models can allow the variance to depend on the predicted concentration or can represent correlation across observations. The model must state which assumption is used; repeated records do not by themselves guarantee independent residual errors.

The nesting is visible in the subject-level response vector \(\mathbf{Y}_i=(Y_{i1},\ldots,Y_{in_i})^\mathsf{T}\). Every entry depends on the same \(\boldsymbol{\eta}_i\), while each entry has its own residual error. Consequently, two observations from the same subject can be statistically related even when their residual errors are conditionally independent: both depend on the same latent subject effect. Observations from different subjects instead have different random effects.

From Subject Records to Population Information

A population information matrix is organized around a parameter vector such as \(\boldsymbol{\phi}\), not just around the structural parameters for one subject. Thus, its rows and columns can correspond to typical fixed effects, random-effects variances or covariances, and residual-error parameters. A proposed design determines which subject records are available and which parts of the model those records can inform.

At the subject level, the required probability model describes the full vector \(\mathbf{Y}_i\) under \(\xi_i\), accounting for the random effects and residual errors. Since the random effects are unobserved, the relevant subject-level distribution for population inference must account for their distribution, rather than treating each \(\boldsymbol{\eta}_i\) as a known parameter. This distinction is crucial: information calculated conditional on known random effects is not automatically the information available for estimating population parameters from observed data.

When subjects are independent under their specified designs, their observed-data likelihood contributions multiply. The log likelihood therefore adds, and under the usual differentiability and expectation conditions, the expected information contributions add as well. If there are \(N\) subjects with the same design \(\xi\), then the total population information is \(N\) times the expected information contribution for one subject under that design. For differing designs, the contributions are summed using each subject’s own design. This aggregation is the basic connection between the nested data structure and a population information matrix.

Why the Additivity Result Holds

The information addition rule follows directly from independence. Let \(p_i(\mathbf{y}_i\mid\boldsymbol{\phi},\xi_i)\) denote the observed-data density for all records from subject \(i\), and let \(\ell_i(\boldsymbol{\phi})=\log p_i(\mathbf{y}_i\mid\boldsymbol{\phi},\xi_i)\). Independence across subjects gives the joint density as a product and hence the total log likelihood as a sum. Under regularity conditions that permit differentiation and expectation to be interchanged, the expected negative Hessian is the Fisher information.

The last equality uses linearity of expectation, as well as the fact that the Hessian of a sum is the sum of the Hessians. This result does not say that observations within a subject contribute independently. They are part of the same subject-level density; the additive units are independent subjects.

Assumptions, Diagnostics, and Design Consequences

The hierarchy is useful only when its assumptions are credible for the intended population. Before relying on a population information calculation, check that the model specifies the random-effect distribution and scale, the residual-error distribution and variance structure, and any within-subject residual correlation. Confirm that the subject-level designs and observation counts match the proposed sampling plan, and that the subjects are reasonably represented as independent sampling units. A shared clinic, family, or site effect may require an additional level rather than an assumption of independent subjects.

Several warning signs indicate that the structure or the information calculation may be failing:

• Residual patterns remain after accounting for the model. Trends over time, concentration-dependent spread, or serial correlation suggest that the residual model may be inadequate.
• Estimated random-effect variances approach zero or correlations approach their bounds. These may indicate weak information, an over-parameterized covariance model, or a genuine boundary case where regular information approximations need care.
• Information changes sharply under modest changes to nominal parameter values. The design may depend strongly on those values, so a single local calculation can be misleading.
• Too few independent subjects are available. Many repeated records on a small number of individuals do not necessarily compensate for limited information about between-subject variability.
• The matrix is singular or nearly singular. Some parameters may be confounded under the selected schedules, or the observations may not distinguish their effects.

Hierarchical modeling also differs from two tempting alternatives. A pooled model that treats every record as independent ignores shared subject effects and usually understates the dependence created by repeated measurements. A separate-parameter model that assigns each subject unrelated fixed parameters can describe individual records, but it does not by itself specify a population distribution from which between-subject variability can be estimated. The hierarchical model represents both levels, although its conclusions still depend on the chosen distributions and covariance structure.

A population information matrix is a local, model-based summary, not a guarantee that a design will perform well under every plausible model or parameter value. It does not correct misspecification, resolve sparse-data identifiability problems automatically, or replace checking the resulting estimation procedure. Its proper role is to compare designs within a stated model and parameterization, with sensitivity checks when assumptions or nominal values are uncertain.