Evaluating Subgroup Prevalence Uncertainty

Why Subgroup Prevalence Matters

An adaptive design can perform differently depending on how common a subgroup is among people who may enroll. If the subgroup is less prevalent than assumed, fewer subgroup participants may be available for interim decisions and final estimates. The trial may need more total enrollment to reach a subgroup target, or it may reach its enrollment cap with too little subgroup information. These consequences are relevant even when the treatment effects assumed within the subgroup do not change.

The key is to evaluate the design over a set of plausible subgroup prevalence values, rather than treating one planning value as certain. For each value, calculate or simulate the consequences using the same design rules and the same other assumptions. This isolates the effect of prevalence and makes it possible to see which operating characteristics are sensitive to it.

Definition: Let \(p\) denote the prevalence of a specified subgroup among eligible people who enroll. A prevalence scenario is an assumed value of \(p\) used to evaluate design performance. Prevalence uncertainty means that the value relevant to the future trial is not known exactly; it is distinct from the random variation in subgroup counts that occurs even when \(p\) is fixed.

For example, \(p=0.15\) may represent a scenario in which 15% of enrolled participants belong to the subgroup. It does not mean that exactly 15% of every trial’s participants will belong to it. If subgroup status is independently observed for each enrollee with the same probability \(p\), then the count among \(N\) enrollees is random. If recruitment deliberately targets subgroup members, or subgroup prevalence changes over time or across sites, the simple model below may not apply without modification.

From Prevalence to Enrollment Requirements

Start with a simple recruitment model: each enrollee independently belongs to the subgroup with probability \(p\), and this probability stays constant during accrual. Let \(K\) be the subgroup count among \(N\) total enrollees. Under these assumptions, \(K\sim\operatorname{Binomial}(N,p)\), with mean \(Np\) and variance \(Np(1-p)\). Thus, the expected subgroup count is not a guaranteed count.

E(K)=Np,\qquad \operatorname{Var}(K)=Np(1-p)

If the design needs at least \(r\) subgroup participants and accrual continues until that target is reached, let \(T\) be the total number enrolled by the time the \(r\)th subgroup participant arrives. Under the same independent, constant-prevalence assumptions, \(T\) is the sum of \(r\) independent waiting times, each with a geometric distribution counting trials through the first subgroup enrollee. Each waiting time has mean \(1/p\) and variance \((1-p)/p^2\). Adding the means and variances gives:

E(T)=\frac{r}{p},\qquad \operatorname{Var}(T)=\frac{r(1-p)}{p^2},\qquad SD(T)=\sqrt{\frac{r(1-p)}{p^2}}

These are planning quantities, not guarantees. A recruitment cap changes the question: instead of asking how many participants will be needed to reach \(r\), ask for the probability of reaching \(r\) by the cap \(C\). If the binomial model holds, reaching the target by \(C\) is exactly the event that at least \(r\) of the first \(C\) enrollees are subgroup members.

P(T\le C)=P(K\ge r) =\sum_{k=r}^{C}\binom{C}{k}p^k(1-p)^{C-k}

Worked Example: Total Enrollment to Obtain a Subgroup Target

Suppose a hypothetical trial needs \(r=100\) subgroup participants. Compare scenarios in which subgroup prevalence is \(p=0.15\) and \(p=0.10\). Assume independent subgroup status with constant prevalence and no subgroup-targeted recruitment.

At \(p=0.15\), the expected total enrollment to obtain 100 subgroup participants is:

E(T)=\frac{100}{0.15}=666.7\text{ participants}

The variance and standard deviation of total enrollment are:

\operatorname{Var}(T)=\frac{100(1-0.15)}{(0.15)^2} =\frac{85}{0.0225}=3777.78\text{ participants}^2 $$ $$ SD(T)=\sqrt{3777.78}\approx61.46\text{ participants}

At \(p=0.10\), the corresponding calculations are \(E(T)=100/0.10=1000\) participants and \(\operatorname{Var}(T)=100(0.90)/(0.10)^2=9000\) participants\(^2\), giving \(SD(T)=\sqrt{9000}\approx94.87\) participants. Lower prevalence therefore increases both the expected total enrollment and its variability. These summaries do not tell us the probability of reaching the target under a particular cap; that requires the binomial tail probability.

Worked Example: Probability of Meeting a Subgroup Target by a Cap

In a hypothetical recruitment plan, the subgroup prevalence is assumed to be \(p=0.20\), the target is \(r=2\) subgroup participants, and the total enrollment cap is \(C=10\). Assume each enrollee’s subgroup status is independent and has the same probability.

The probability of meeting the target is one minus the probabilities of getting zero or one subgroup participant:

P(K\ge2)=1-P(K=0)-P(K=1) $$ $$ =1-(0.8)^{10}-\binom{10}{1}(0.2)(0.8)^9 $$ $$ =1-0.107374-0.268435=0.624191

The probability of reaching the target by the cap is about 0.6242, or 62.42%. The expected subgroup count is \(10(0.20)=2\), but that expectation does not imply the target will be met: in this scenario, the chance of meeting it is well below 100%.

Evaluate Selection Performance and Precision Separately

Prevalence can also affect adaptive selection. A selection rule may use interim subgroup results to choose a treatment, expand an arm, or allocate future participants. Fewer subgroup participants can make those decisions noisier or leave less information for an interim comparison. Evaluate the complete adaptive rule under each prevalence scenario, including its stopping rules, allocation changes, and selection criterion. Keep the treatment-outcome assumptions and other scenario inputs fixed when the goal is to isolate prevalence sensitivity.

A useful selection operating characteristic is the probability that the rule selects the best treatment for the subgroup under a specified set of true treatment effects. This probability is conditional on the scenario assumptions; it is not a probability that a treatment is best. Simulations estimate it by the proportion of simulated trials in which the rule selects the treatment that is best under those assumed effects.

Worked Example: Comparing Subgroup Selection Across Prevalence Scenarios

Consider a hypothetical simulation with 10,000 trials per scenario. The assumed treatment effects make Treatment A the true best treatment for the subgroup in both scenarios. The adaptive rule selects Treatment A in 7,100 trials when \(p=0.10\) and in 8,100 trials when \(p=0.20\).

The estimated selection probabilities are:

\hat{P}_{p=0.10}(\text{select best})=\frac{7100}{10000}=0.7100 $$ $$ \hat{P}_{p=0.20}(\text{select best})=\frac{8100}{10000}=0.8100

For a simulated proportion \(\hat{p}_{\text{sim}}\) based on \(M\) independent replicates, the Monte Carlo standard error is \(\sqrt{\hat{p}_{\text{sim}}(1-\hat{p}_{\text{sim}})/M}\). The two estimates have Monte Carlo standard errors \(\sqrt{0.71(0.29)/10000}\approx0.00454\) and \(\sqrt{0.81(0.19)/10000}\approx0.00392\). In this hypothetical output, selection performance is higher at the larger prevalence. The difference is a comparison of scenario-specific design performance under the stated assumptions, not evidence that changing prevalence alone would cause the same difference in a real trial.

Precision of subgroup estimates is another distinct outcome. For a continuous outcome with standard deviation \(\sigma\), the standard error of a subgroup sample mean based on \(n_g\) subgroup participants is \(\sigma/\sqrt{n_g}\), under independent sampling with common variance. For a difference in means between two randomized treatments, the standard error depends on the subgroup sample size in each treatment arm. For binary outcomes, the corresponding variance depends on the assumed response probabilities as well as those sample sizes.

Worked Example: Approximate Precision Under Two Prevalence Scenarios

Suppose a hypothetical trial enrolls \(N=400\) participants and randomizes equally to two treatments. The subgroup prevalence is either \(p=0.10\) or \(p=0.20\). Assume subgroup status is independent of treatment assignment, and consider a binary outcome with assumed response probabilities 0.30 and 0.50 in the two treatments. For a planning approximation, use the expected subgroup count per arm, \(Np/2\).

At \(p=0.10\), expected subgroup sample size is \(400(0.10)/2=20\) per treatment. The approximate standard error of the difference in response proportions is:

SE\approx\sqrt{\frac{0.30(0.70)}{20}+\frac{0.50(0.50)}{20}} =\sqrt{0.0105+0.0125} =\sqrt{0.023}\approx0.1517

At \(p=0.20\), expected subgroup sample size is \(400(0.20)/2=40\) per treatment, so:

SE\approx\sqrt{\frac{0.30(0.70)}{40}+\frac{0.50(0.50)}{40}} =\sqrt{0.00525+0.00625} =\sqrt{0.0115}\approx0.1072

The higher-prevalence scenario has better approximate precision because it supplies more subgroup participants. These calculations plug in expected sample sizes. In actual trials the subgroup counts are random, and the realized standard error may differ; the simulation should calculate precision from the realized data and report how often estimates are unavailable or too imprecise for the design’s intended use.

Assumptions, Diagnostics, and Alternatives

A prevalence assessment is only as useful as its assumptions. Plausible scenario values should reflect the population eligible for enrollment, not automatically the general population. If eligibility criteria, sites, referral pathways, or recruitment periods change the mix of participants, use scenarios that reflect those mechanisms. If subgroup membership is measured imperfectly, the prevalence of true membership and the prevalence of recorded membership may differ; the simulation must represent whichever one drives enrollment and decisions.

Useful diagnostics include plotting subgroup counts by prevalence scenario, checking the fraction of trials that meet subgroup targets, and examining the distribution—not only the mean—of subgroup sample sizes. For adaptive rules, inspect selection probability, allocation, stopping, and subgroup precision by scenario. Confirm that the simulation’s generated subgroup counts agree with the intended prevalence model, and that the adaptive decision rule uses the subgroup data specified in the protocol. Large Monte Carlo error can obscure small differences between scenarios, so report simulation uncertainty alongside estimated operating characteristics.

The independent, constant-prevalence binomial model is a transparent starting point, not a universal recruitment model. With changing site mix or time-varying prevalence, model the relevant strata or accrual periods explicitly and evaluate the resulting overall subgroup count. With quota-based recruitment, subgroup counts may be fixed or constrained rather than binomial, so the enrollment calculations above are not appropriate as stated. A single prevalence estimate with a confidence interval can summarize uncertainty from earlier data, but scenario-based design evaluation is often clearer when several plausible values are needed and no defensible probability distribution over them has been specified. Assigning scenario weights answers a different question and should not be implied by an unweighted scenario comparison.

Common pitfall: “The expected subgroup count at \(N=400\) and \(p=0.10\) is 40, so the trial will have 40 subgroup participants.” The correction is that 40 is the mean under the binomial model, not a guaranteed count. Recruitment risk and estimate precision depend on the distribution of subgroup counts and on the chance of falling below a relevant target.

Putting the Assessment Together

For each plausible prevalence scenario, preserve the same design rules and other assumptions unless the purpose is to examine their interaction. Then evaluate the outputs that matter to the design: enrollment required or the chance of meeting subgroup targets, adaptive selection performance, and the precision or availability of subgroup estimates. Report scenario-specific results rather than hiding material differences in one pooled summary. This makes clear whether the design remains usable across the prevalence values considered and where additional recruitment safeguards or design changes may be worth examining.

Specify plausible prevalence values.
Use values relevant to the eligible recruitment population and state what evidence or uncertainty motivates them.

Check the recruitment model.
Decide whether constant, independent subgroup membership is credible, or whether accrual patterns, sites, or quotas need explicit representation.

Evaluate each scenario.
Apply the same adaptive design and calculate enrollment, selection, and precision outcomes for each prevalence value.

Inspect uncertainty and failures.
Show subgroup-count distributions, target-attainment rates, estimate availability, and Monte Carlo error, not just averages.

Check Your Understanding

Answer each question using the stated model. Full solutions follow the questions.

Under independent recruitment with \(p=0.25\), what is the expected number of subgroup participants among \(N=200\) enrollees?
A design needs \(r=30\) subgroup participants. What is the expected total enrollment to reach the target if \(p=0.10\)?
Why does an expected subgroup count equal to a target not guarantee that the trial reaches the target?
In a simulation of 2,500 trials, the rule selects the best treatment 1,875 times. Calculate the estimated selection probability and its Monte Carlo standard error.

Solutions. (1) \(E(K)=Np=200(0.25)=50\) subgroup participants. (2) \(E(T)=r/p=30/0.10=300\) total participants, under the independent, constant-prevalence model. (3) The count is random: even if \(E(K)=r\), some trials will have fewer than \(r\) subgroup participants and others more; the mean does not specify the probability of meeting the target. (4) The estimated probability is \(1875/2500=0.7500\). Its Monte Carlo standard error is \(\sqrt{0.75(0.25)/2500}=\sqrt{0.000075}\approx0.00866\), or about 0.866 percentage points.