Hierarchical Structure of Population PD Models

Why Population PD Models Are Hierarchical

A concentration-effect relationship describes how a person's response changes as drug concentration changes. In a population, however, the same concentration need not produce the same response for every person. A population PD model must therefore represent both the systematic concentration-effect relationship and the differences between individuals. It must also distinguish those persistent differences from variation in the measurements made on a particular individual.

The hierarchy makes those roles explicit. Population parameters describe typical values or shared relationships. Covariates explain modeled differences associated with measured subject characteristics. Individual random effects represent remaining between-subject differences. The structural PD model maps an individual's parameters and concentration to a response, and the observation model connects that response to the recorded data.

A Nonlinear Concentration-Effect Model

Consider a hypothetical study with subject \(i\) measured at occasion \(j\). Let \(C_{ij}\) be the drug concentration used as input to the PD model, and let \(Y_{ij}\) be the observed response. One possible structural model is an Emax relationship:

Here, \(E_{ij}\) is the underlying response predicted for subject \(i\) at occasion \(j\). The parameter \(E_{0,i}\) is that subject's baseline response, \(E_{\max,i}\) is the maximum drug-related change from baseline, and \(EC_{50,i}\) is the concentration producing half of that maximum change. The model assumes \(EC_{50,i}>0\) and uses nonnegative concentrations. If the observed response decreases with concentration, the direction of change must be represented consistently, for example by a negative \(E_{\max,i}\).

The individual parameters can, in turn, be connected to the population level. For illustration, suppose the covariate-adjusted typical value of \(E_{\max}\) for subject \(i\) is \(\theta_{\max}m(z_i;\beta)\), where \(\theta_{\max}\) is a population parameter, \(z_i\) is a measured covariate, and \(m(z_i;\beta)\) is a specified covariate function. Represent the subject's deviation from that value by \(\eta_{\max,i}\):

This equation shows the hierarchy without prescribing a particular covariate function or a general choice of random-effects distribution. The covariate function specifies a systematic relationship shared across subjects; \(\eta_{\max,i}\) allows subject-specific deviation around that relationship. The exponential ensures a positive \(E_{\max,i}\) when \(\theta_{\max}\) and \(m\) are positive. Other PD parameters can be represented with their own population and subject-level components when supported by the model and data.

For an illustrative model, one might assume \(\eta_{\max,i}\sim N(0,\omega_{\max}^2)\) and \(\epsilon_{ij}\sim N(0,\sigma^2)\), with subject effects independent between subjects and observation errors independent across observations and independent of the subject effects. These are modeling assumptions, not automatic properties of PD data. In this example, \(E_{\max,i}\) is the individual parameter; \(\epsilon_{ij}\) is observation-level error. The same \(\eta_{\max,i}\) applies to all observations on subject \(i\), while a new \(\epsilon_{ij}\) enters at each observation.

From Individual Responses to the Likelihood

The model can be understood by conditioning first on the individual effect. Given the population parameters, covariates, random effect, and concentrations, the model calculates each \(E_{ij}\). Under the illustrative normal observation model, the conditional density for a single observation is

If the observation errors are conditionally independent within a subject, the subject's conditional likelihood is the product of these densities over that subject's observations. The marginal likelihood for the subject averages that conditional likelihood over the distribution of the subject effect:

The integral matters: the subject effect is not separately observed. Estimation must account for uncertainty about it rather than treating a fitted individual deviation as known data. This conditional-to-marginal distinction is a core feature of nonlinear mixed-effects modeling. It also clarifies why a collection of observations from one subject is not equivalent to the same number of independent subjects: those observations share the same \(\eta_{\max,i}\).

This derivation relies on the stated conditional observation model and random-effects distribution. If the observation errors are dependent, the product factorization is not valid as written. If the concentration values are uncertain, treating them as fixed inputs does not account for that uncertainty; a joint PK/PD formulation may instead model concentrations and responses together. The earlier material on choosing an estimation strategy addresses the implications of staged and joint estimation.

How the Hierarchy Changes Predictions

A typical prediction sets the subject effect to its reference value and uses the population parameter adjusted for the specified covariates. A subject-specific prediction uses that subject's effect estimate as well. A population-level prediction averages over the distribution of subject effects. Because the concentration-effect function is nonlinear, these predictions need not agree.

Covariates add another layer. Their effect enters the subject's typical parameter before the individual deviation is applied. A covariate relationship and a random effect therefore answer different questions: the former describes a systematic modeled difference associated with measured characteristics; the latter represents unexplained subject-to-subject variation conditional on the covariate model.

Assumptions, Diagnostics, and Alternatives

The hierarchy is useful only when its assumptions are plausible and its components can be informed by the data. In the illustrative specification, key assumptions include a correctly specified concentration-effect relationship, a suitable covariate function, a distribution for subject effects, and a conditional observation model. Repeated measurements must be linked to the correct subject, and the assumed independence structure must match the design. If concentrations are treated as known inputs, that conditional analysis also assumes their uncertainty is negligible for the PD inference or has been handled elsewhere.

Useful checks follow the hierarchy rather than treating all discrepancies as interchangeable. Plot observed responses against concentration and compare them with model predictions to look for systematic misspecification, such as curvature or an inadequately represented baseline. Examine residuals against time, concentration, and predicted response for trends or changing spread. Compare individual predictions with observations by subject to detect persistent subject-level patterns that the random-effects structure may not capture. Also check whether parameter estimates are plausible, whether uncertainty is large, and whether the model has difficulty separating parameters that produce similar concentration-effect curves.

These checks can suggest where a model is failing but cannot prove that a particular hierarchy is correct. For instance, residual trends may arise from a misspecified structural relationship, an unmodeled time pattern, or an inappropriate observation model. A persistent subject-level discrepancy may point to an omitted covariate or an inadequate random-effects structure. A problem should not be “fixed” by assigning every mismatch to a different layer without scientific and data-based justification.

A conditional PD analysis treats concentrations supplied to the PD model as fixed inputs; a joint PK/PD analysis models the PK and PD components together. The conditional approach can be appropriate when the concentration information is sufficiently precise for the question at hand. A joint approach can propagate PK uncertainty into PD estimation, but it requires a defensible PK model and additional information. Neither approach removes the need to distinguish population parameters, subject effects, structural response, and observation error.