Why Prevalence Drift Matters
An enrichment strategy depends on how often the target marker occurs among people who can actually enter the trial. That prevalence need not remain constant during recruitment. Sites may serve different populations, sites may open or close, referral patterns may change, or the mix of screened patients at a site may shift over calendar time. A forecast based on one pooled prevalence can therefore miss both the number of screens needed to reach an enrollment target and the eventual balance of marker-positive and marker-negative participants.
Prevalence drift is a change in the distribution of the marker, not necessarily a change in the marker's biological meaning or in treatment effect. Its consequences are nevertheless important: it changes the rate of positive-subgroup accrual, the information available for subgroup comparisons, and—when subgroup effects differ—the treatment effect averaged over the all-comer trial population. The analysis should keep these consequences distinct.
The population in that definition matters. Prevalence among everyone screened, among eligible people, and among randomized participants can differ. A trial should state which denominator is being monitored. If screening records are incomplete or the assay is performed only after eligibility assessment, the observed denominator may not represent everyone screened.
Estimating and Diagnosing Drift
For a group of \(n\) screened participants with observed marker status and \(x\) positives, the estimated prevalence is \(\hat{p}=x/n\). If marker statuses are independent Bernoulli observations with a common probability \(\pi\) in that group, then \(\hat{p}\) is unbiased and its standard error is estimated by \(\sqrt{\hat{p}(1-\hat{p})/n}\). This standard error describes sampling variation; it does not account for site clustering, measurement changes, or a changing composition within the group.
A useful first display is a table or plot of prevalence by site and calendar period, alongside the corresponding denominators. A pooled estimate can hide important differences. For example, the overall prevalence may fall simply because a larger share of screens now comes from a low-prevalence site, even if each site's own prevalence is stable. Conversely, a within-site change can be concealed by a shift toward a high-prevalence site.
For a time trend, one possible model is logistic regression:
where \(t_i\) is calendar time for screened participant \(i\), and \(\gamma_{s(i)}\) represents the participant's site. A nonzero \(\beta_1\) describes a time trend conditional on the modeled site effects. The model assumes the log odds change linearly with time on the chosen scale and that the site structure and observations are adequately represented. If each site contributes data in only one time period, site and time effects cannot be separated from these data alone. A fitted trend is not a remedy for that lack of overlap.
No single test diagnoses every operational problem. Inspect site-period denominators, missing marker results, assay or laboratory changes, and the timing of site openings. Large fluctuations from small denominators can be ordinary sampling noise; persistent changes across adequately sized periods are more concerning. Repeatedly testing for change as data accumulate can inflate false alarms, so operational triggers should be specified in advance or interpreted as surveillance signals rather than as unadjusted confirmatory tests.
Worked Example: Comparing Two Recruitment Periods
Suppose a hypothetical trial screens 200 people in an early period and observes 80 marker-positive results. In a later period it screens 250 people and observes 75 positives. Assume the same assay and eligibility-independent screening process in both periods, and treat the two groups as independent for this comparison.
The estimated prevalences are \(80/200=0.400\) and \(75/250=0.300\). Their estimated standard errors are:
The estimated decline is \(0.400-0.300=0.100\), or 10.0 percentage points. Under independent binomial sampling, the standard error of this difference is:
A normal-approximation 95% confidence interval is \(0.100\pm1.96(0.04517)\), which is approximately \(0.0115\) to \(0.1885\). Under the stated assumptions, the data are consistent with a later prevalence between about 1.15 and 18.9 percentage points lower than the early prevalence. This comparison does not establish that calendar time caused the change: a change in site mix or screening practice could also explain it.
Accrual Forecasts Require the Joint Enrollment Probability
To forecast marker-positive enrollment, define the probability of interest per screened person. Let \(E\) indicate eligibility and \(R\) indicate enrollment. The probability that a screen yields an enrolled marker-positive participant is \(q=P(M=1,E=1,R=1\mid\text{screened})\). Expected positive enrollment from \(S\) similar screens is \(Sq\). If probabilities vary by site or period, the expected total is the sum of the corresponding probabilities over those screens.
A factorization is valid only when its conditional probabilities match the sequence of events. For example:
This expression does not assume that eligibility, marker status, and enrollment are independent; each factor is conditional on the preceding events. If enrollment decisions differ by marker status, the final factor must be specific to marker-positive eligible people. Using an overall enrollment probability in its place can misstate positive accrual.
Worked Example: Forecasting Positive Enrollment Across Two Periods
Consider hypothetical screening in two periods, with 1,000 people screened in each. In period 1, 60% are eligible; among eligible people, 40% are marker-positive; and 75% of eligible marker-positive people enroll. In period 2, the corresponding probabilities are 50%, 30%, and 80%.
For period 1, the probability of an enrolled positive per screen is \(0.60(0.40)(0.75)=0.180\), so expected positive enrollment is \(1{,}000(0.180)=180\). For period 2, it is \(0.50(0.30)(0.80)=0.120\), giving \(1{,}000(0.120)=120\). The expected total over both periods is \(180+120=300\) enrolled marker-positive participants.
If the trial needs 360 positives, 60 more are required after these screens. If period 2 conditions persist, the projected number of additional screens is \(60/0.120=500\). This is an expectation-based forecast, not a guarantee: actual counts vary, and the forecast should be updated if eligibility, prevalence, or enrollment behavior changes. Multiplying 0.40 by 0.75 would be incorrect here because 0.40 is prevalence among eligible people while the screen-to-eligibility probability is also needed.
When future site mix is changing, use site- and period-specific forecasts rather than extrapolating a single pooled prevalence. A practical forecast can sum expected positive enrollment across planned screens at each site using its estimated \(q\). Scenarios for plausible changes in site mix, eligibility, and enrollment are often more informative than a single point projection. Forecast uncertainty should be communicated, especially when decisions depend on reaching a threshold by a deadline.
Subgroup Information and All-Comer Effects
Prevalence drift changes the number of participants available for marker-specific treatment comparisons. With \(n\) randomized participants and prevalence \(\pi\), the expected marker-positive count is \(n\pi\), and the expected marker-negative count is \(n(1-\pi)\). If randomization is 1:1 within each marker stratum, each treatment arm is expected to receive about half of each subgroup count. Actual information also depends on outcome variability, missing outcomes, allocation, and the estimand; the count alone is not a complete precision calculation.
The overall treatment effect can also depend on the marker prevalence in the population being described. For a risk difference, suppose the marker-specific causal effects are \(\delta_+\) for marker-positive people and \(\delta_-\) for marker-negative people, and these effects are constant across the populations being compared. In a target population with prevalence \(\pi\), the all-comer risk difference is the weighted average:
This follows because the population's treatment-arm risk is a mixture of the marker-positive and marker-negative risks. Subtracting the two treatment-arm mixtures gives the same prevalence-weighted mixture of the two subgroup risk differences. Thus, if subgroup effects differ, changing prevalence changes the all-comer effect even when neither subgroup effect changes.
Worked Example: Prevalence Changes the All-Comer Risk Difference
Suppose, hypothetically, the treatment risk difference is 6 percentage points in the marker-positive subgroup and 1 percentage point in the marker-negative subgroup, with those subgroup effects stable over recruitment. In an early population with prevalence \(0.40\), the all-comer risk difference is:
percentage points. In a later population with prevalence \(0.30\), it is:
percentage points. The difference arises from population composition, not from a change in either assumed subgroup effect. If equal numbers were recruited from these two periods, the combined prevalence would be \((0.40+0.30)/2=0.35\), and the corresponding all-comer risk difference would be \(0.35(6)+0.65(1)=2.75\) percentage points. This weighting applies to the specified risk-difference estimand under the stated stable-effect assumptions; other effect measures, such as a hazard ratio, generally cannot be obtained by this simple weighted average.
Choosing a Monitoring Approach and Recognizing Failure
The method should match the question. A period-by-site table is transparent and useful for operational surveillance. A regression model can summarize a trend while adjusting for site, provided time and site are sufficiently represented in the data. A pooled prevalence is simple but can be misleading if recruitment composition changes. These approaches are complementary: a model should not replace inspection of the underlying counts and denominators.
| Approach | Useful for | Main limitation |
|---|---|---|
| Pooled prevalence | A concise summary for a stable, well-defined screened population | Can conceal time trends and site-mix changes |
| Site-by-period estimates | Finding where and when prevalence changes | Small cells can be noisy; many comparisons invite overinterpretation |
| Time-trend model with site terms | Estimating a conditional trend when sites contribute across time | Depends on model form and adequate overlap of site and time |
| Scenario-based accrual forecast | Planning screens under changing recruitment conditions | Forecasts depend on assumptions about future site mix and conditional probabilities |
Investigate a drift signal before treating it as a biological change. Check for altered assay platforms or thresholds, changes in who is screened, incomplete results, site-specific referral patterns, and changes in eligibility or enrollment rules. A sudden jump at the exact time an assay process changed is not equivalent to a gradual within-site trend under an unchanged process. If observations are clustered by site or recruitment team, simple binomial standard errors may be too small; account for clustering or present the estimates as descriptive with that limitation stated.
Prevalence monitoring should also be separated from treatment-outcome monitoring. A blinded prevalence summary can often inform recruitment forecasts without inspecting treatment effects. If prevalence information triggers a change in recruitment policy or treatment allocation, the operational rule and its inferential consequences need advance consideration. In particular, changing who enters the trial can change the population represented by an all-comer estimate. Generalization requires specifying the target population and its marker distribution, rather than assuming the enrolled mixture is universal.
Common Pitfalls
- Combining mismatched probabilities. Multiplying prevalence among eligible people by an enrollment rate that includes a different denominator does not give enrolled positives per screen. Define the joint event and factor it only with matching conditional probabilities.
- Reading a pooled change as a within-site trend. An increase in the share of screens from low-prevalence sites can lower the pooled estimate even when each site's prevalence is constant. Examine site-period counts.
- Calling random variation drift. Small groups can have unstable estimates. Consider denominators and uncertainty, not just the direction of successive percentages.
- Assuming the all-comer effect is fixed when prevalence changes. If subgroup effects differ, the weighted all-comer risk difference changes with the target population's prevalence, even when subgroup effects remain constant.
Check Your Understanding
Try each problem, then compare your reasoning with the solutions below.
- A hypothetical period has 500 screens, 70% eligibility, 25% marker prevalence among eligible people, and 80% enrollment among eligible marker-positive people. What is the expected number of enrolled positives?
- Why can pooled prevalence decline when no site's own prevalence changes?
- With subgroup risk differences of 4 percentage points for positives and 2 percentage points for negatives, calculate the all-comer risk difference at prevalence 0.25.
- Name one reason a time-trend model with site effects may fail to distinguish site differences from temporal drift.
Solutions. (1) The enrolled-positive probability per screen is \(0.70(0.25)(0.80)=0.140\), so the expected count is \(500(0.140)=70\). (2) The share of screens contributed by low-prevalence sites may increase, changing the weighted pooled prevalence despite stable site-specific prevalences. (3) \(0.25(4)+0.75(2)=1.0+1.5=2.5\) percentage points. (4) If each site contributes only in a distinct time period, site and time are confounded in the observed data, so their effects cannot be separated without additional assumptions or data.