Enrichment With Overlapping Biomarkers

Why Overlapping Biomarkers Need Explicit Population Definitions

A trial may measure several baseline biomarkers, any of which could help identify patients likely to benefit from treatment. When the biomarkers overlap, a participant can be positive for more than one marker. As a result, “the A-positive group” and “the B-positive group” are not necessarily separate sets of people. Adding their sizes as if they were disjoint overstates the number eligible or enrolled, and comparing their treatment effects as if they were independent ignores the participants shared by both groups.

The practical starting point is to define eligibility and analysis populations as sets of participants. Then represent overlapping groups using mutually exclusive combinations of biomarker status. This preserves clear enrollment totals and makes it possible to describe how each subgroup effect is assembled from its component cells.

The labels “or” and “and” have different consequences for eligibility. Eligibility under \(A\) or \(B\) means membership in the union \(A\cup B\): a participant qualifies if positive for at least one marker. Eligibility under \(A\) and \(B\) means membership in the intersection \(A\cap B\): a participant must be positive for both. A protocol should state which rule applies, how biomarker status is determined, and how missing or indeterminate results are handled.

The usual counting identity follows by separating the two positive groups into disjoint parts. The intersection is counted once in \(|A|\) and once in \(|B|\), so it must be subtracted once to obtain the number of unique people in their union:

Here \(|G|\) is the number of distinct participants in group \(G\). For more than two biomarkers, the mutually exclusive status combinations still provide a reliable counting framework, though there are more cells. The union count then follows inclusion–exclusion: add the single-marker counts, subtract pairwise intersections, add triple intersections, and continue with alternating signs. In practice, a table of disjoint combinations is often easier to audit.

Worked Example: Counting Unique Eligible Participants

Define the Trial Population and the Analysis Groups

An eligibility population describes who can enter the trial; an analysis group describes whose outcomes are summarized for a particular estimand. They may coincide, but they need not. For example, a trial could enroll participants who are positive for A or B, then report results separately for A-positive, B-positive, and all enrolled participants. Every double-positive participant belongs to both marker-specific groups and to the overall eligible group.

Let \(S\) denote a participant’s baseline biomarker cell, and let \(G\) be a prespecified analysis group formed from one or more cells. Let \(Y(1)\) and \(Y(0)\) denote the outcome that would be observed for that participant under treatment and control, respectively. For a specified outcome and time point, the treatment effect in group \(G\) is the mean contrast:

This definition makes the target population explicit. The effect in A-positive participants is not automatically the effect in B-positive participants, nor is either one automatically the effect in everyone eligible. A subgroup’s treatment effect is an average over the biomarker cells making up that group. The weights reflect the distribution of those cells in the target group.

For example, with two markers, the A-positive group consists of A-only and both-positive participants. If the proportions in the target A-positive population are \(w_{A\text{-only}}\) and \(w_{\text{both}}\), where they sum to one, then:

This is a weighted average of cell-specific mean treatment effects. It is not, in general, the unweighted average of the cell effects. The weights depend on which population is being described and can differ between A-positive and B-positive groups.

Worked Example: Combining Cell-Specific Treatment Effects

Why Overlapping Subgroup Estimates Are Related

The A-positive and B-positive groups in the example each include the both-positive participants. Therefore, data from those participants contribute to both subgroup summaries. The estimates are not based on disjoint samples and should not be treated as statistically independent. Their dependence matters for uncertainty calculations and formal comparisons, but the immediate design lesson is simpler: identify shared membership before interpreting separate subgroup results.

Randomization within biomarker strata, as in the earlier material on randomization within biomarker strata, helps ensure treatment and control comparisons are made within the intended baseline groups. It does not make overlapping analysis groups disjoint. Nor does stratified randomization change the definition of the target population. For an overall or subgroup comparison, the analysis must still use a clearly stated population and a suitable method for combining the cell-specific information.

A raw pooled treatment-control difference can also be affected by different biomarker-cell proportions in the two arms. If cell-specific outcomes differ at baseline and the arms contain different proportions of the cells, a pooled difference can reflect both treatment effects and composition differences. A transparent approach is to report cell-specific treatment-control contrasts and state the weights used to summarize them. When the target population is fixed, use the same target-population weights for both treatment conditions.

Worked Example: Enrollment Accounting and Allocation

Assumptions, Diagnostics, and Limitations

Interpreting randomized treatment contrasts within overlapping biomarker groups requires more than correct arithmetic. Biomarker status should be defined using measurements made before randomization, and the assay rules should assign each participant consistently to the intended cell. The eligibility rule and analysis groups should be specified before examining outcomes. Treatment should be randomized as planned, outcome definitions and time points should be consistent, and the treatment comparison should respect the randomization scheme. A causal interpretation additionally relies on standard trial conditions, including consistency of treatment assignment, adequate outcome ascertainment, and no interference between participants.

Useful checks include the following:

• Verify that disjoint biomarker-cell counts sum to the number of unique screened or enrolled participants in the relevant population.
• Check that each participant has one—and only one—baseline status combination, with a stated rule for missing or indeterminate results.
• Confirm that the eligibility rule matches the intended union, intersection, or other prespecified combination.
• Report both unique enrollment and subgroup membership counts when groups overlap; label denominators clearly.
• Compare treatment and control composition across cells, and verify that summaries use the intended target-population weights.
• Check whether the both-positive cell is small. Sparse cells can make cell-specific summaries imprecise and can make overall subgroup results sensitive to a few observations.

A common failure is to add the A-positive and B-positive counts and call the sum the number enrolled. The correction is to subtract the intersection, or preferably to build the total from disjoint cells. Another mistake is to report an A-positive and a B-positive effect as two independent findings because they have different labels. Since the groups share participants, the estimates are dependent; their uncertainty and any formal comparison must account for that overlap. Finally, a difference between two subgroup point estimates is not, by itself, evidence of a true treatment-effect difference.

The main alternative is to report results in the mutually exclusive biomarker cells. Those summaries avoid duplicated participants and show the component effects directly, but may be unstable when cells are small. Overlapping groups, such as A-positive and B-positive, can answer clinically meaningful population questions and may have larger sample sizes, but share data and have different compositions. A prespecified overall eligible population is useful for a trial-level summary, yet it can conceal variation across its component cells. These are complementary descriptions, not interchangeable populations.

Overlap also limits what can be inferred from a simple table. Cell-weighted subgroup summaries describe treatment effects in the populations defined by the observed biomarker composition. They do not establish that a biomarker is predictive, prove a biological mechanism, or guarantee that results transport to a population with different marker prevalence. Those questions require additional modeling and inferential work.