Why Preserve Cohort and Batch Composition?
A multi-omics subtype analysis may combine samples from several cohorts or laboratory batches. Those groups can differ in sample size, recruitment, measurement platform, or technical processing. In a resampling analysis, an ordinary bootstrap draws samples from the pooled data. Its resamples therefore vary not only in which individuals are represented, but also in the proportions drawn from each cohort or batch. Stratified resampling fixes those proportions at chosen values, making it possible to ask a more focused question: how stable is the clustering when the cohort or batch composition is held constant?
That question is not automatically better than the unstratified one. The two designs represent different resampling targets. If the goal is stability for the observed mix of cohorts, fixing the observed mix can be useful. If the goal is to reflect uncertainty in the mix of cohorts in a wider target population, allowing the mix to vary may be appropriate. Neither design removes technical effects or establishes that a stable cluster is a biological subtype.
Specify the Resampling Target First
Let \(h=1,\ldots,H\) index mutually exclusive strata, and let \(N_h\) be the number of observed samples in stratum \(h\). The total sample size is \(N=\sum_h N_h\), and the observed stratum weight is \(W_h=N_h/N\). A clustering pipeline is rerun on each resample, including the preprocessing steps that should be estimated from that resample. The earlier material on refitting preprocessing within resamples and end-to-end stability is relevant here: a design that resamples samples but freezes data-adaptive preprocessing may not represent the intended analysis.
Write \(S_h\) for a stability summary associated with stratum \(h\), and \(S\) for an overall summary. A common overall target weights stratum-specific summaries by prespecified weights \(w_h\):
The weights need an interpretation. Setting \(w_h=W_h\) describes the observed cohort mixture. Equal weights give each stratum equal influence, which is a different target when stratum sizes differ. External target-population proportions may also be used when they are specified and defensible. The choice should be made before comparing estimates; it should not be selected because it produces the most favorable stability result.
In a stratified bootstrap, draw \(N_h\) samples with replacement from each stratum \(h\). This preserves the observed stratum counts in every resample. In a stratified subsampling design, draw a prespecified number \(m_h\) without replacement within each stratum; this also preserves the chosen composition, though the resample size and its interpretation differ from a bootstrap. The draw must occur at the correct observational unit. If several rows belong to one participant, for example, resampling rows independently would not preserve participant-level dependence.
Use the cohort or batch structure that matters to the intended stability question. If both matter and are crossed, consider joint cohort-by-batch cells, provided the cells are adequately populated.
State whether the target is the observed mixture, an equal-stratum summary, or another prespecified composition. For a bootstrap that fixes observed composition, draw \(N_h\) times within stratum \(h\).
Apply the specified preprocessing and clustering pipeline to each draw, align cluster labels where needed, and calculate the chosen stability summary. Use the same summary definition when comparing resampling designs.
Worked Calculations
Worked Example: Preserving a Small Cohort
Suppose a hypothetical integrated data set has 10 participants: 8 from cohort A and 2 from cohort B. An ordinary bootstrap draws 10 participants with replacement from the pooled data. What is the probability that a pooled bootstrap replicate contains no participant from cohort B?
Each draw has probability \(8/10=0.8\) of coming from A. The 10 draws are independent under the ordinary bootstrap, so:
Thus about 10.74% of these pooled bootstrap replicates contain no cohort B participant at all. Under a stratified bootstrap, 8 draws are made from A and 2 from B, so each replicate includes both cohorts by design. This is useful if the target is stability conditional on retaining the observed 8:2 composition. It would be inappropriate to claim that stratification has demonstrated B’s clustering is stable: only two distinct B participants were observed, and bootstrap duplicates do not provide new independent information.
Worked Example: Weighting Stratum-Specific Stability
In a hypothetical study, 80 participants come from cohort A and 20 from cohort B. A prespecified within-cohort agreement summary is 0.90 for A and 0.60 for B. Calculate the observed-mixture summary and the equal-cohort summary.
The observed weights are \(W_A=80/100=0.80\) and \(W_B=20/100=0.20\). Weighting by the observed mixture gives:
Equal weighting instead gives:
Both calculations are arithmetically valid, but they answer different questions. The first gives cohort A more influence because it is larger in the observed data. The second treats each cohort as equally important. Reporting 0.84 without stating the weights could make a reader mistake an observed-mixture summary for an equal-cohort result.
Worked Example: Composition Variation in an Unstratified Resample
Consider the same observed 80:20 cohort mixture, and suppose, for illustration, that cohort-specific stability summaries remain fixed at 0.90 and 0.60. An ordinary bootstrap of 100 participants may produce a replicate with 70 participants from A and 30 from B. What mixture-weighted summary does that replicate imply?
Its replicate weights are 0.70 and 0.30, so the calculation is:
A replicate with 90 from A and 10 from B would instead give \((0.90)(0.90)+(0.10)(0.60)=0.87\). These different values arise from composition alone under the stated simplifying assumption that the stratum-specific summaries do not change. In a real analysis, the clustering and its within-stratum stability may also change when the sampled individuals change. Consequently, a difference between stratified and unstratified estimates is not automatically evidence of a changed biological structure; it may partly reflect the resampling design.
Why Stratification Removes One Source of Variation
The composition effect can be justified directly. Suppose an unstratified resample has \(n\) draws and the chance of drawing from stratum \(h\) is \(W_h\). Let \(W_h^*\) be the proportion from stratum \(h\) in that resample. If the stratum-specific summaries \(S_h\) are held fixed for this calculation, the resample’s mixture summary is \(S^*=\sum_h W_h^*S_h\). The vector of resampled stratum proportions has a multinomial distribution, so its mean is \(E(W_h^*)=W_h\), and its covariance entries are \(\operatorname{Cov}(W_h^*,W_j^*)=(W_h\mathbf{1}\{h=j\}-W_hW_j)/n\). Substitution gives:
The variance term is the variation in the mixture summary caused by random stratum proportions under these fixed-\(S_h\) assumptions. When stratified resampling fixes each stratum’s contribution to \(W_h\), this particular composition-variation term is removed. That does not mean the full stability estimate has zero variability: resampled individuals and refitted analyses can still change the summaries. Nor does the calculation imply that stratification is always preferable. It establishes only what composition fixing does under the stated simplification.
Design Choices, Diagnostics, and Limits
When cohorts and batches are crossed, resampling within joint cohort-by-batch cells can preserve both structures more exactly than separately preserving their margins. But joint stratification can create sparse or empty cells. A cell with too few distinct participants cannot support a credible bootstrap assessment of its own reproducibility. Combining cells may be necessary, but it changes what is being held fixed and should be justified by the scientific design rather than chosen solely to avoid an inconvenient result.
When batch is nested within cohort, cohort and batch effects cannot be cleanly distinguished from the observed structure alone. Resampling within the nested groups preserves that structure; it does not identify whether a cluster is associated with biology, cohort, or batch. Examine cluster membership against cohort and batch, inspect whether putative subgroups occur across technical groups, and check that each relevant group has enough samples to contribute. A subgroup found almost entirely in one batch may be reproducible under resampling within that batch and still reflect a technical artifact.
A pooled stability statistic may obscure weak performance in a small stratum. Report stratum-specific summaries alongside an overall result, and state the weights used in that overall result. If a stability summary compares partitions, its label alignment and calculation must be consistent across resamples; the earlier material on aligning cluster labels and partition-comparison measures gives the relevant tools. A co-clustering summary can also be restricted to pairs within a stratum when the question concerns within-cohort reproducibility, but it then does not summarize cross-stratum pairs.
Useful warning signs include a stratum absent from many unstratified replicates, very small or empty joint cells, sharply different stability summaries across cohorts, and clusters that track batch labels closely. Also check the number of distinct individuals represented in each draw: repeated bootstrap copies can make the nominal resample size look adequate while the effective number of distinct samples is small. These diagnostics reveal design limitations; they do not by themselves prove that a subtype is technical or biological.
| Design | What is held fixed? | Main interpretation |
|---|---|---|
| Unstratified bootstrap | No stratum counts are fixed; counts vary according to the pooled sample. | Stability under resampling from the pooled observed mixture, including variation in its composition. |
| Stratified bootstrap | Chosen stratum draw counts, commonly the observed counts. | Stability conditional on the specified composition, with samples varying within strata. |
| Stratified subsampling | Prespecified within-stratum sample sizes, drawn without replacement. | Stability under smaller within-stratum subsets; not the same resampling distribution as a bootstrap. |
A common mistake is to argue, “Stratified resampling preserves the batches, so a stable cluster must be a true subtype.” The reasoning confuses reproducibility under a design with biological interpretation. Stratification controls how batch composition varies across resamples; it does not remove batch effects, balance biological variables within batches, or establish an independent biological explanation. The correction is to describe the target precisely, report the subgroup’s distribution across cohorts and batches, and treat technical association as a limitation requiring further investigation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A study has 60 samples from cohort A and 40 from cohort B. What counts are drawn in a stratified bootstrap that preserves the observed composition?
- Within-stratum stability is 0.80 for A and 0.50 for B. Calculate the observed-mixture summary and the equal-cohort summary.
- Why can a pooled bootstrap produce replicates with different stability summaries even if within-stratum summaries are unchanged?
- Batch is nested within cohort, and each discovered cluster occurs mainly in one batch. What can stratified resampling establish, and what can it not establish?
Solutions. (1) Draw 60 samples with replacement from A and 40 with replacement from B in each replicate. (2) The observed weights are 0.60 and 0.40, giving \((0.60)(0.80)+(0.40)(0.50)=0.48+0.20=0.68\). Equal weights give \((0.50)(0.80)+(0.50)(0.50)=0.40+0.25=0.65\). (3) The pooled bootstrap changes the proportions drawn from each stratum, so the weighted summary changes with those proportions; the clustering itself may change as well. (4) It can assess stability conditional on the preserved nested structure. It cannot separate cohort from batch effects or show that a batch-associated cluster is a biological subtype.