Hierarchical Priors for Between-Subject Variation

Why Model Between-Subject Variation?

A QSP model may describe the same pathway in every subject while allowing parameters such as receptor abundance, clearance, or pathway potency to differ. A calibration that assigns a separate unrestricted value to every subject can be unstable when each subject has limited data. Conversely, forcing all subjects to share one value can hide real biological heterogeneity. A hierarchical prior represents both levels: variation among subject-specific parameters and uncertainty about the population distribution that describes that variation.

The central distinction is between individual uncertainty about a particular subject’s parameter and population uncertainty about quantities such as the population mean and between-subject standard deviation. Data from one subject directly inform that subject’s parameter; under a shared population model, they can also inform the population distribution, which in turn informs other subjects.

A Hierarchical Model for a QSP Parameter

For a positive subject-specific parameter, such as a clearance rate, define a dimensionless log parameter \(\eta_i\) for subject \(i\) by \(\theta_i=\theta_{\mathrm{ref}}\exp(\eta_i)\), where \(\theta_{\mathrm{ref}}\) is a fixed reference value with the same units as \(\theta_i\). A simple population model is \(\eta_i\mid\mu,\omega\sim N(\mu,\omega^2)\). The parameters \(\mu\) and \(\omega\) describe the population distribution on the log scale.

Here \(N\) is the number of subjects. The half-normal hyperprior places probability only on nonnegative values of \(\omega\); its scale \(s_\omega\) must be chosen to represent plausible population spread. The prior on \(\mu\) and the hyperprior on \(\omega\) should be stated on the scale used in the model and checked for biological plausibility. Other hyperprior families can be appropriate, but they must also respect the parameter’s constraints and be justified.

The population mean \(\mu\) is the mean of the log-relative parameter, not the arithmetic mean of \(\theta_i\). Conditional on \(\mu\) and \(\omega\), the median of \(\theta_i\) is \(\theta_{\mathrm{ref}}\exp(\mu)\), while its arithmetic mean is \(\theta_{\mathrm{ref}}\exp(\mu+\omega^2/2)\). Thus, reporting \(\exp(\mu)\) as the arithmetic population mean would generally be incorrect when \(\omega>0\).

In a mechanistic calibration, each subject’s data \(D_i\) have a likelihood determined by the QSP model output using that subject’s parameter vector. To keep the hierarchy clear, write the likelihood for subject \(i\) as \(p(D_i\mid\eta_i,\psi)\), where \(\psi\) denotes other model quantities. A basic assumption is conditional independence of subjects’ data given their respective parameters and shared quantities. In this tutorial’s numerical illustrations, simple normal likelihoods are used to make the updating transparent; real biomarker observations require an observation model suited to their measurement process.

This expression shows the two-way connection in the model: each subject’s data inform \(\eta_i\), and the collection of subject parameters informs \(\mu\) and \(\omega\). In a full analysis, the subject parameters are not known values plugged into a population calculation. Their uncertainty is retained when estimating the population quantities.

How an Individual’s Data Update Their Parameter

The hierarchy provides a population-informed prior for each subject. For a transparent derivation, suppose the subject-level quantity \(x_i\) is observed with a normal likelihood \(x_i\mid\eta_i\sim N(\eta_i,\tau^2)\), where \(\tau\) is known. Conditional on fixed \(\mu\) and \(\omega\), the prior is \(N(\mu,\omega^2)\). Multiplying the two normal densities and collecting the terms involving \(\eta_i\) gives a normal posterior. Its precision is the sum of the prior and likelihood precisions.

Equivalently, the posterior mean is a precision-weighted average of the population location and the subject’s data value. If the subject data are precise, \(\tau^2\) is small and \(x_i\) receives more weight. If the population distribution is narrow, \(\omega^2\) is small and the subject estimate is pulled more strongly toward \(\mu\). This is called partial pooling: an individual estimate is informed by both that subject’s data and the shared population model.

How Subjects Update the Population

Population quantities learn from the full set of subjects, but the strength of that learning depends on the amount of information about each subject. A simple calculation illustrates this distinction. Suppose each subject has a noisy log-scale summary \(x_i\), with \(x_i\mid\eta_i\sim N(\eta_i,\tau^2)\), and \(\eta_i\mid\mu,\omega\sim N(\mu,\omega^2)\). If \(\omega\) and \(\tau\) are fixed, integrating over the unobserved \(\eta_i\) gives \(x_i\mid\mu\sim N(\mu,\omega^2+\tau^2)\). The marginal variance includes both between-subject variation and uncertainty in the subject summary.

With a normal prior \(\mu\sim N(m_0,s_0^2)\), these marginal normal observations yield a normal posterior for \(\mu\). This conjugate calculation is useful for seeing the population update, although in a full QSP analysis subject likelihoods need not reduce to normal summaries and \(\omega\) is usually uncertain too.

Assumptions, Diagnostics, and Alternatives

The normal hierarchy above assumes that subjects are exchangeable with respect to the modeled population: before observing their data, subject labels do not determine different distributions for \(\eta_i\). It also assumes the chosen log-scale distribution is a reasonable representation of heterogeneity and that the likelihood correctly links each subject’s parameter to that subject’s observations. Exchangeability does not mean subjects are biologically identical. It means systematic differences are not represented unless the model includes them.

Check whether the model’s assumptions are plausible rather than interpreting a fitted hierarchy mechanically. Useful checks include whether the prior permits biologically plausible subject values, whether subjects’ data show patterns inconsistent with one common population distribution, and whether the fitted spread is informed by the data or mainly determined by the hyperprior. The introductory material on prior predictive checks gives a way to assess implications of the prior before conditioning on data. With few subjects, the data may provide little information about \(\omega\), so a seemingly precise description of heterogeneity can be misleading.

• Unmodeled structure: If treatment arm, disease severity, or another covariate systematically shifts the parameter, a single exchangeable distribution may misrepresent that structure. A population model with relevant covariates may be more appropriate.
• Confounding of variation sources: Sparse measurements can make it difficult to distinguish true between-subject differences from uncertainty in each subject’s parameter. Do not interpret all apparent spread as biological variability.
• Distribution mismatch: A normal distribution on the log scale implies a lognormal distribution on the original scale. Strongly skewed or multimodal biological populations may not be represented adequately by this assumption.
• Hyperprior influence: When the number of subjects is small or individual data are weak, posterior conclusions about \(\omega\) can depend substantially on its hyperprior. State that limitation and examine whether the prior allows plausible variation.

A hierarchical model is not the only option. Complete pooling imposes one shared parameter across subjects and cannot represent between-subject variation. No pooling estimates every subject separately and does not let subjects inform one another; with sparse data, individual estimates may be unstable. Hierarchical partial pooling is a middle ground, but it is appropriate only when subjects can reasonably be treated as draws from a shared population after accounting for modeled differences.

A common mistake is to treat the estimated population location as every subject’s true parameter, or to call \(\omega\) a measurement-error standard deviation. The correction is to keep the levels distinct: \(\mu\) and \(\omega\) describe a distribution across subjects, while \(\eta_i\) describes subject \(i\). Measurement uncertainty belongs in the likelihood; it can affect how well \(\eta_i\) is known, but it is not interchangeable with population spread.