Allocating Sample Size Across Assays

What Is Being Allocated?

A sample-size plan for an integration study must specify more than a total number of measurements. It must say how many subjects are assigned to each assay, whether those assignments refer to the same people, and how many measurements are expected to be usable. These distinctions matter because 300 subject-assay records could be 100 subjects measured on three assays, or 300 different subjects measured on one assay each. The record totals match; the opportunities to analyze measurements jointly do not.

The earlier material on matched and unmatched omics sampling distinguishes shared subjects from separate assay samples. Here the emphasis is on allocating limited recruitment and assay resources, then translating that allocation into counts of records and linked subjects. Counts describe design capacity; they are not, by themselves, guarantees of statistical precision or evidence that one design is preferable for every research question.

For a fully matched plan with \(m\) subjects on all \(K\) assays, the planned number of subject-assay records is \(Km\), and the planned number of complete subject rows is \(m\). If assay sample sets are disjoint, the planned number of records is \(\sum_{j=1}^{K} n_j\), but the number of complete multi-assay subjects is zero. With partial overlap, neither count can be inferred from the \(n_j\) values alone: subject identities or overlap counts are needed.

Budget and Capacity Constraints

Allocation begins with the resources that actually limit the study. These may include a total budget, separate assay-specific budgets, recruitment capacity, or a maximum number of specimens a laboratory can process. If assay \(j\) has budget \(B_j\), its maximum affordable sample is \(\lfloor B_j/c_j\rfloor\), where \(\lfloor x\rfloor\) is the greatest integer no larger than \(x\). For a fully matched plan, every subject consumes resources on every assay. Therefore the matched sample cannot exceed the tightest assay-specific capacity:

If instead there is one total budget \(B\), a matched subject costs \(\sum_{j=1}^{K}c_j\). Thus the maximum matched sample, before accounting for recruitment or other constraints, is:

These expressions follow directly from the resource constraint. A matched plan with \(m\) subjects requires \(m c_j\leq B_j\) for every assay, so \(m\leq B_j/c_j\) for every \(j\), and the integer \(m\) must be no larger than the smallest of those limits. With a single budget, the required total is \(m\sum_j c_j\leq B\), which gives the second expression. An unmatched plan can allocate resources differently across assays; its counts must satisfy the corresponding assay-specific constraints \(n_jc_j\leq B_j\), or the single total constraint \(\sum_j n_jc_j\leq B\).

Cost-Aware Comparisons

Equal numbers of records need not cost the same amount. Nor does the cheapest plan necessarily meet the scientific objective. A matched plan spends resources on each assay for every shared subject; an unmatched plan can direct more resources toward selected assays. The allocation should therefore be compared using both its counts and the constraints that produced them. If the objective requires joint measurements, maximizing the total record count alone is not an appropriate substitute for maximizing complete subjects.

Planned Counts Versus Usable Counts

A planned assay measurement may not yield a usable record. For a simple expectation calculation, let \(q_j\) be the probability that an attempted measurement on assay \(j\) is usable. If each of \(n_j\) measurements has this same probability, the expected number of usable records on assay \(j\) is \(n_jq_j\). The expected total across assays is:

For a matched allocation with \(m\) subjects, define \(q_{\text{all}}\) as the probability that a subject has usable measurements on every assay. The expected number of complete subjects is \(m q_{\text{all}}\); the expected number of records belonging to these complete subjects is \(Km q_{\text{all}}\). These are not the same as the expected total number of usable records, \(m\sum_j q_j\), because subjects with only some usable assays contribute to the latter but not to a complete-case analysis.

If the usability events for different assays are independent within a subject, then \(q_{\text{all}}=\prod_{j=1}^{K}q_j\). Independence is a substantive assumption, not a default fact. Shared specimen quality, processing batches, or subject characteristics may make assay failures dependent. When independence is not justified, estimate or specify the joint probability \(q_{\text{all}}\) directly rather than multiplying the marginal probabilities.

Checking Overlap and Allocation Performance

For partial overlap, let \(S_j\) denote the set of subjects assigned to assay \(j\). The number of complete subjects is the size of the intersection \(\left|\bigcap_{j=1}^{K}S_j\right|\), not the smallest \(n_j\) unless the smallest assay group is entirely contained in every other group. For two assays, the overlap is \(|S_1\cap S_2|\); for three or more, pairwise overlaps alone generally do not determine the all-assay intersection. In addition, planned overlap and usable overlap are different: a subject assigned to all assays is not a complete usable subject if one or more records fail.

Practical checks should be done before interpreting any allocation table:

• Confirm that each subject is counted once per assay, using stable subject identifiers rather than row positions.
• Tabulate the number assigned to each assay, the unique-subject count, and the complete intersection; for partial designs, report pairwise overlaps as well when relevant.
• Separate planned counts from observed usable counts, and state whether a complete row requires usability on every assay.
• Compare the observed assay failure pattern with the assumptions used to estimate usable counts. If failures cluster by subject, batch, or specimen, a product of marginal success rates may misstate complete-case yield.
• Recalculate counts after exclusions or withdrawals. Removing subjects can reduce assay-specific samples and cross-assay overlap by different amounts.

A useful alternative to fully matched or fully disjoint sampling is a partially matched plan: some subjects receive several assays, while additional subjects receive only selected assays. It can preserve a core of linked data while allowing larger assay-specific samples. Its merit cannot be assessed from total records alone; the allocation must report the size of the shared core and the samples outside it. These counts describe what is available for different analyses, without determining which analysis is suitable.

Trade-Offs and Common Pitfalls

Matched allocation directly supports analyses that require measurements from the same subjects across assays, but each additional assay increases the cost per matched subject. If usable measurements are required on every assay, failures in any one assay reduce the complete-case count. Unmatched allocation may support larger or more balanced assay-specific samples under the same resources, but separate samples do not provide subject-level linkage. A partial-overlap design falls between these extremes and requires explicit overlap accounting.

These are design trade-offs, not a universal ranking. The number of records is not the same as the number of independent subjects, and neither count alone determines precision, bias, or statistical power. Those properties depend on the research question, measurement variability, dependence structure, and analysis plan. In particular, a larger number of records from repeated assays on fewer matched subjects should not be casually treated as the same number of independent subjects.

A further limitation is that expected usable counts are planning summaries, not guarantees. They rely on assumed or estimated usability probabilities and may be inaccurate if assay performance changes with specimen quality, recruitment site, or processing conditions. Sensitivity calculations using plausible lower and higher success rates can show how fragile a plan is, but they do not remove uncertainty. If the complete-case target is essential, the plan should make its assumptions and contingency for shortfall explicit.