Comparing Cohort Designs for Population Exposure Prediction

Why Cohort Design Depends on the Target

A pharmacogenetic PBPK cohort is a computational representation of a target population. Its design affects which people are simulated, how their results are weighted, and how accurately a chosen population exposure summary is estimated. A cohort that estimates the overall mean efficiently may still represent an exposure tail or a rare genotype poorly. The choice should therefore start with the estimand—the population quantity of interest—and then account for genotype frequencies, within-genotype variability, and the simulation budget.

Let \(G\) denote genotype category, with target-population probability \(p_g=P(G=g)\), and let \(X\) denote the remaining subject characteristics, such as physiology or covariates. For a specified regimen and PBPK model, let \(Y=f(G,X)\) be an exposure metric, measured in stated units. The target mean exposure is an average over both genotype and the distribution of characteristics within each genotype.

These expressions assume that the genotype categories cover the target population, their probabilities are appropriate for that population, and each conditional distribution represents the target characteristics within its genotype. They also assume a specified PBPK model, regimen, and exposure definition. A simulated estimate describes the modelled target population under those inputs; it is not automatically an estimate of a clinical outcome beyond the model.

Three Ways to Construct a Cohort

In a random cohort, each virtual subject is sampled from the joint target population: genotype is sampled according to \(p_g\), then other characteristics are sampled from their distribution conditional on genotype. The ordinary sample mean gives each subject equal weight. This design is simple and naturally reflects population frequencies in expectation, but a finite cohort may contain few subjects from a rare genotype.

In a stratified cohort, the analyst fixes a sample size \(n_g\) for each genotype and samples subjects within each stratum. If genotype probabilities are known, the overall mean must use population weights, not the simulated proportions:

With independent, representative draws within strata, this estimator is unbiased for \(\theta\). Its variance is \(\sum_g p_g^2\sigma_g^2/n_g\). Stratification gives control over how many simulations represent each genotype, but an unweighted mean of all stratified subjects generally targets the wrong genotype mixture.

In a deterministic cohort, the analyst chooses a fixed set of representative inputs and weights rather than drawing subjects randomly. For example, points \(x_{gj}\) with weights \(w_{gj}\) can approximate the conditional distribution within genotype \(g\), where \(w_{gj}\geq0\) and \(\sum_jw_{gj}=1\). The estimate is

Such a design can be reproducible and avoid Monte Carlo sampling variability. It is nevertheless an approximation to the required population average unless the weighted points integrate the PBPK output exactly. Its approximation error is not an ordinary sampling standard error and cannot be removed just by applying a sample-mean standard error formula.

Why Stratification Can Improve a Mean Estimate

For an independent random cohort of size \(N\), each subject is drawn from the full population. The sample mean is unbiased and has variance \(\operatorname{Var}(Y)/N\). The law of total variance separates this into variation between genotype means and variation within genotypes:

In a stratified cohort, the genotype counts are fixed rather than fluctuating. Only within-genotype sampling contributes to the variance of the weighted estimator. This gives \(\operatorname{Var}(\hat{\theta}_{\mathrm{str}})=\sum_g p_g^2\sigma_g^2/n_g\). The comparison explains why stratification can be more precise for a mean, especially when genotype means differ or a random cohort would represent rare genotypes inconsistently. It does not guarantee that every particular stratified cohort is more accurate than every random cohort.

For a fixed total number of simulations \(N=\sum_g n_g\), the continuous allocation minimizing the stratified mean variance is \(n_g\propto p_g\sigma_g\), provided the \(\sigma_g\) values are known and simulations have equal cost. In practice, allocation must use whole numbers and may require minimum stratum sizes. If costs differ by genotype, the best allocation changes; a rare but expensive stratum may not receive the same number of simulations as under equal costs.

Allocating Simulations Under a Budget

Proportional allocation, \(n_g\approx Np_g\), makes the simulated genotype proportions resemble the target population and is a useful default when within-genotype variances are similar. For estimating a mean, Neyman allocation instead assigns simulations in proportion to \(p_g\sigma_g\). It places more simulations in strata that are both prevalent and variable. The allocation depends on the estimand: the variance-minimizing allocation for a mean need not be best for a tail probability or a rare-genotype estimate.

Choosing Designs for Distributions and Deterministic Integration

A population quantile or exceedance probability depends on the whole exposure distribution, not just its mean. For a threshold \(c\), the target exceedance probability is \(P(Y>c)=\sum_gp_gP(Y>c\mid G=g)\). A stratified estimate can estimate each conditional exceedance proportion and combine them using \(p_g\). To estimate a population quantile, first form the weighted population distribution and then find its quantile; averaging genotype-specific quantiles generally does not produce the population quantile. Random cohorts can estimate these summaries too, but rare strata and extreme tails may be represented by very few subjects. Deterministic weighted points can approximate a distribution, but a sparse set of points may poorly resolve a tail or make the estimated quantile jump abruptly.

A deterministic design is most defensible when points and weights are chosen to approximate the relevant input distribution and the resulting PBPK outputs are checked for sensitivity to that approximation. Evaluating the model only at average inputs is not generally equivalent to averaging the model output: PBPK relationships are often nonlinear.

Diagnostics, Trade-Offs, and a Common Pitfall

Before running a cohort, specify the estimand, target genotype probabilities, within-genotype sampling distributions, and simulation budget. Afterward, check the realized genotype counts and weights, the number of simulations in rare strata, and whether within-stratum covariates cover their intended ranges. For random cohorts, repeat simulations or increase cohort size to assess Monte Carlo stability. For stratified cohorts, inspect stratum-specific variability and confirm that the weighted estimate and its standard error use the target \(p_g\). For deterministic cohorts, compare estimates under denser or alternative weighted points; stability across such checks is evidence of numerical robustness, not proof that the approximation is exact.

Random sampling is simple and flexible, but may waste a fixed budget on overrepresented common strata when a rare stratum matters. Stratification controls representation and can improve precision for weighted summaries, but requires correct weights and adequate within-stratum sampling. Deterministic designs are reproducible and can be efficient for smooth averages, but require careful point selection and do not provide a conventional sampling-based uncertainty estimate. In all three designs, uncertainty in the PBPK model, genotype frequencies, or target covariate distributions is distinct from Monte Carlo variability and is not resolved merely by increasing the virtual-cohort size.

• Common mistake: Treating every subject in an intentionally oversampled stratified cohort as equally representative of the target population. This weights genotypes according to their cohort counts, not according to their population probabilities.
• Correction: For a population mean with known genotype probabilities, calculate \(\sum_gp_g\bar{Y}_g\). For distributional summaries, combine conditional distributions or proportions with those same target weights before calculating the summary.
• Another warning: A deterministic cohort has no random sampling error conditional on its chosen points, but it can still have substantial integration error. Repeating the same deterministic calculation does not diagnose that error.