Stratified Virtual Populations by Genotype

Why Stratify a Virtual Population?

A PBPK virtual population may need enough subjects in every genotype group to characterize subgroup exposures, even when some genotypes are uncommon. Sampling each virtual subject’s genotype according to its population frequency can leave a rare group with very few simulated subjects. A stratified design controls how many subjects are simulated in each group. The key is to keep two quantities distinct: the simulation allocation and the genotype frequency in the target population.

The allocation determines how much computational effort is spent on each genotype. It does not, by itself, determine that genotype’s contribution to population exposure. To recover population summaries after using deliberately chosen subgroup sizes, combine the subgroup results using target-population genotype frequencies. These weights should correspond to the population the analysis is intended to represent.

The exposure \(Y\) must be defined consistently across groups—for example, AUC over the same dosing interval and expressed in the same units. A subgroup simulation should represent the intended distribution of virtual subjects conditional on belonging to that genotype group, including the specified PBPK assumptions and relevant covariates. The earlier material on preserving dependence among virtual-subject covariates is relevant when constructing those within-group subjects. This tutorial focuses on how to allocate the groups and combine their simulated exposure results.

Choosing Controlled Group Sizes

A proportional allocation chooses \(n_g\) approximately in proportion to \(q_g\). It is efficient when the main aim is to represent the target population directly and every group will still have enough subjects. A controlled or disproportionate allocation instead sets group sizes to meet analysis needs, such as a minimum number in each genotype group. This can improve the precision of subgroup-specific simulation summaries, but population-level results must then be reweighted.

For integer counts, a practical allocation begins with a total simulation budget \(N\) and planned counts \(n_g\) satisfying \(\sum_g n_g=N\). If counts are chosen by rounding \(Nq_g\), the rounded counts may not sum exactly to \(N\); assign the remaining subjects in a documented way. Alternatively, specify minimum group sizes and distribute the remaining budget according to a chosen rule. In either case, record \(q_g\) separately from \(n_g\). A group assigned zero subjects cannot contribute a simulated subgroup distribution, regardless of its target frequency.

Frequency-Weighted Means and Variance

For each group, the simulated mean exposure is \(\bar{Y}_g\), the arithmetic mean of its \(n_g\) simulated exposure values. If the subgroup simulation adequately represents the conditional exposure distribution for genotype \(g\), the target-population mean is estimated by weighting these subgroup means by their target frequencies:

This is not generally the same as the unweighted mean across all simulated subjects. That unweighted mean gives each simulated subject equal weight, and therefore weights group \(g\) in proportion to \(n_g\). It is appropriate as a target-population estimate only when the allocation proportions match the target frequencies (or by coincidence when the subgroup means make the results equal).

Population variability reflects both exposure variation within genotype groups and differences between their mean exposures. If \(\sigma_g^2\) is the conditional exposure variance in group \(g\), then the population variance has the decomposition below. In a simulation, replace each \(\sigma_g^2\) by the corresponding subgroup sample variance \(s_g^2\) to estimate the within-group component.

Why the Variance Decomposition Holds

Let \(Y\) be an exposure drawn from the target population and let \(G\) denote its genotype group. Within group \(g\), the conditional mean is \(\mu_g\) and the conditional variance is \(\sigma_g^2\). Write the deviation from the population mean as \(Y-\mu_{\mathrm{pop}}=(Y-\mu_g)+(\mu_g-\mu_{\mathrm{pop}})\). Squaring and taking the conditional expectation within group \(g\), the cross-product has expectation zero because \(E(Y-\mu_g\mid G=g)=0\). Thus:

Averaging this conditional quantity across groups with probabilities \(q_g\) gives the stated result. The decomposition depends on using the same target frequencies in both terms, and on the genotype groups forming a complete partition of the target population. It shows why averaging only the subgroup variances understates total population variability whenever subgroup means differ.

Combining Exposure Distributions

A population exposure distribution is a mixture of the conditional distributions in the genotype groups. For any threshold \(y\), its cumulative distribution function is the weighted sum of the subgroup cumulative distribution functions:

where \(F_g(y)=P(Y\leq y\mid G=g)\). The same principle gives a population probability of exceeding a threshold by weighting the corresponding subgroup probabilities. To estimate a population quantile, first form the weighted mixture distribution and then find the value at which its cumulative probability reaches the desired level. In general, a population quantile is not the frequency-weighted average of subgroup quantiles.

Diagnostics, Alternatives, and Limitations

Several checks help identify a stratified analysis that is not answering its intended question. Confirm that the genotype groups are exhaustive and non-overlapping, that the target frequencies sum to one, and that each group’s simulated exposures use the same outcome definition and units. Check that every group with positive target frequency has simulated subjects. Compare allocation proportions \(n_g/N\) with target frequencies \(q_g\); a difference is not automatically an error, but it signals that unweighted pooled summaries will generally target the wrong mixture. Also inspect subgroup exposure summaries and empirical distributions for implausible values, unexpectedly sparse tails, or computational failures.

Proportional sampling is a direct alternative when a cohort is intended to mirror target frequencies. It simplifies unweighted summaries but may provide little information about rare groups. Controlled stratification improves subgroup coverage and allows deliberate group comparisons, while requiring explicit weighting for population summaries. A third approach is to simulate within groups and then generate a frequency-proportional resample for a particular analysis. That can make the target mixture visually or computationally convenient, but it does not add information to the original subgroup simulations.

Frequency weighting cannot correct a poorly represented conditional distribution: a subgroup with too few or unrepresentative virtual subjects can still yield unstable estimates, especially for tail probabilities and quantiles. Nor can weighting repair incorrect target frequencies, omitted genotype groups, or a PBPK model that does not adequately represent the intended population. Results describe the specified target mixture under the model and assumptions; they are not, by themselves, evidence that genotype causes an exposure difference. Further evaluation of genotype–physiology dependence is a separate question.