Preserving Joint Structure During Subsampling

Why Subsampling Can Change a Virtual Population

An existing virtual-patient ensemble may contain more individuals than are needed for a simulation or analysis. Subsampling reduces computational cost, but a small subset can differ from its source ensemble in important ways. Even when every patient has the same chance of selection, the subset may have different subgroup proportions, correlations, or joint phenotype frequencies simply because it contains fewer individuals.

The goal is not necessarily to make every sample statistic identical to the source ensemble. It is to select a subset that retains the features important for the intended use, while avoiding avoidable selection bias. The earlier material on why matching marginals can distort joint structure is relevant here: matching separate variable distributions does not automatically preserve how those variables occur together. Subsampling requires the same attention to joint structure.

A selection design should be specified before interpreting the resulting subset. Simple random sampling gives each subset of size \(n\) the same probability of selection. Stratified sampling first divides the ensemble into defined subgroups and selects a specified number from each. Constrained selection chooses a subset subject to requirements such as fixed subgroup counts or joint-cell counts. These designs answer different needs; none guarantees that every feature of the source ensemble will be retained.

What Random Subsampling Preserves in Expectation

Under simple random sampling without replacement, each individual has inclusion probability \(n/N\). Thus, for a category represented by \(K\) of the \(N\) individuals, the expected number selected is \(nK/N\). This is an expectation over repeated subsets, not a promise that one particular subset will contain that number.

The variability of a category count can be substantial in a small subset. If the category proportion in the finite ensemble is \(p=K/N\), then its count \(X\) in a simple random sample without replacement has a hypergeometric distribution, with variance and standard deviation:

The factor \((N-n)/(N-1)\) is the finite-population correction: sampling without replacement produces less variability than independent sampling with replacement. The formula describes a single category count. Correlations and other joint summaries are calculated from several counts together, so they can fluctuate even when each individual has equal selection probability.

Stratified Selection for Subgroup Proportions

When subgroup composition matters, stratification makes the subgroup counts part of the design rather than leaving them to chance. Define mutually exclusive strata, decide the desired count from each, and then select individuals within each stratum. Proportional allocation uses the source proportions: if stratum \(h\) contains \(N_h\) individuals, its proportional quota is \(n_h=nN_h/N\), subject to integer rounding.

Stratification fixes the selected counts in the specified strata, but does not fix associations within a stratum or between variables that were not used to define the strata. If an important joint pattern is not represented by the stratification variables, its frequency can still drift. If quotas are not proportional to the source composition, the subset may be deliberately disproportionate; estimates intended to represent the source then need appropriate stratum weights.

Constrained Selection for Joint Structure

When particular combinations of characteristics must be retained, selection constraints can target joint cells directly. For a cross-classification, let \(x_{ab}\) be the selected count in cell \((A=a,B=b)\). Counts must be nonnegative integers, sum to \(n\), and satisfy any specified margins or cell quotas. Feasibility matters: the required count in a cell cannot exceed the number of available individuals in that cell.

A constrained design can be implemented by defining a feasible set of subsets, then selecting within that set. For example, an optimization objective may penalize deviations from target subgroup or joint-cell counts while enforcing availability limits. The earlier material on optimization-based virtual population selection provides context for such objectives. Regardless of the algorithm, inspect whether the constraints express the intended scientific priorities, and whether the selected individuals remain diverse within the allowed cells.

Choosing a Design and Checking Its Limits

Before using a subset, verify the selected count, subgroup proportions, and the joint features that motivated the design. Check that requested counts were feasible and that the implementation actually met the constraints. For random or within-stratum random selection, compare observed counts with their expected sampling variation rather than treating every difference as a design failure. Also check for duplicated patients, accidental exclusions, or a small number of individuals dominating a weighted or optimization-based selection.

A frequent mistake is to conclude that a random subset preserves correlations because each patient had an equal chance of selection. Equal inclusion probabilities support unbiasedness over repeated samples; they do not make the realized subset’s correlation equal to the source correlation. The correction is to distinguish long-run expectation from one subset’s finite-sample result, quantify plausible count variation where useful, and constrain or stratify only the features that genuinely need tighter control.

A second pitfall is to equate exact subgroup proportions with preserved joint structure. Fixing the number of patients in each age group, for instance, does not fix how disease status and treatment response co-occur within those age groups. Conversely, exact joint-cell quotas preserve those particular cell frequencies but do not guarantee the distribution of continuous variables within cells. Every selection design has a scope: state what it controls, and do not infer preservation of features it did not constrain.