Why estimate attrition while keeping treatment assignments masked?
A trial’s statistical design typically specifies how many participants need an observed primary outcome. The number who must be enrolled is larger if some participants are expected not to provide that outcome. If the loss rate is higher than planned, the trial may end with fewer usable outcomes than required, reducing precision or power. A blinded interim estimate can help update the enrollment target without revealing which treatment each participant received.
The key quantity is the proportion of participants who will lack the primary outcome at the analysis time. It is not necessarily the proportion who have already withdrawn, missed a visit, or have no recorded value at an interim review. A participant whose outcome has not yet had time to be collected is still being followed, not yet known to be missing.
“Blinded” here means that the attrition calculation uses participants pooled across randomized groups, without treatment labels. It does not mean that no interim information is examined: the number with and without an ascertainable primary outcome is used. The calculation estimates overall attrition in the pooled trial population, not attrition separately in each treatment group.
Who belongs in the denominator?
The denominator \(m\) should include participants whose primary outcome was due to be collected by the interim review, according to the prespecified outcome window. Participants still within their follow-up period should not be counted as outcome-missing merely because their value is not yet available. Including them would mix genuine non-ascertainment with ordinary delay and could overstate attrition.
The definition must also be consistent with the planned analysis. For example, if the primary outcome is measured at 12 weeks, the interim attrition review should identify participants who have reached the 12-week assessment window, and then determine which of those participants lack an outcome under the trial’s data rules. A participant who discontinued treatment but completed the 12-week assessment has an observed primary outcome and is not attrition for this calculation. Conversely, the reason for a missing value does not make it observed.
The simple estimate \(\hat r=x/m\) is most directly justified when eligible participants can be treated as independent observations with a common probability of lacking the outcome, and when interim eligibility and missingness are classified consistently. It is an estimate of a pooled rate; it need not describe every site or randomized group equally well. The interim data must also be sufficiently mature to distinguish missing outcomes from outcomes that are merely pending.
Worked Example: Counting Mature Follow-Up Correctly
At a masked interim review, 150 participants have been enrolled. By the review date, 120 have passed the prespecified primary-outcome assessment window. Of those 120, 18 lack the primary outcome. The other 30 have not yet reached that window.
Only the 120 participants whose outcome window has passed belong in the attrition denominator. Thus,
The estimated pooled attrition rate among participants with mature follow-up is 15%, and the estimated completion proportion is 85%. The 30 participants still awaiting their assessment are not classified as missing on these facts. Counting all 150 as eligible would give \(18/150=0.12\), but that would answer a different question and understate the observed rate among participants whose outcomes were due.
From a completion rate to an enrollment target
Let \(n_{\mathrm{obs}}\) denote the planned number of participants who need an observed primary outcome, and let \(N\) denote the enrollment target. If the completion probability is \(q=1-r\), then, under a common completion probability, the expected number with observed outcomes among \(N\) enrolled participants is \(Nq\). Setting this expectation equal to the required number gives the usual plug-in inflation rule.
Because enrollment must be an integer, the total is rounded up, not to the nearest integer. This rule targets the required number in expectation under the estimated completion rate; it does not guarantee that at least \(n_{\mathrm{obs}}\) outcomes will be observed. If the allocation ratio requires integer counts in each group, round the group targets in a way that preserves that ratio and meets the required group-specific numbers. The final total can therefore exceed the rounded pooled calculation.
Worked Example: Inflating a Planned Total
Suppose a trial needs 400 participants with an observed primary outcome, and the blinded interim estimate is \(\hat r=0.15\). Assume the follow-up population and outcome-ascertainment rules at the interim review are relevant to the participants still to be enrolled.
The estimated completion proportion is \(1-0.15=0.85\). The enrollment calculation is
Rounding up gives an enrollment target of 471. Under a 15% attrition rate, the expected number with observed outcomes is \(471(0.85)=400.35\), which is at least 400 in expectation. This calculation is an enrollment adjustment, not a change to the number of observed outcomes the analysis requires.
Why the inflation formula works—and what it assumes
The justification is an expectation calculation. If each enrolled participant has probability \(q\) of providing the primary outcome, let \(I_i=1\) when participant \(i\) provides it and \(I_i=0\) otherwise. The total number with observed outcomes is \(C=\sum_{i=1}^{N} I_i\). If each participant has the same completion probability \(q\), linearity of expectation gives the following result; independence is not needed for this expectation, although dependence affects variability.
Substituting the interim estimate \(\hat q=1-\hat r\) for the unknown \(q\), then solving \(N\hat q=n_{\mathrm{obs}}\), yields the plug-in enrollment rule. Its interpretation depends on the estimated completion proportion being reasonably applicable to future enrollees. If attrition changes over calendar time, differs by site, or depends on factors that change in the remaining enrollment, the pooled estimate may not transport well.
The simple derivation also highlights a limitation: \(E(C)=n_{\mathrm{obs}}\) does not mean \(P(C\ge n_{\mathrm{obs}})=1\). There is sampling variation in how many participants complete follow-up. If the design requires a high probability of reaching a target number of outcomes, an expectation-based inflation alone may be insufficient; a more conservative enrollment calculation may be needed. The choice should be prespecified rather than improvised after seeing interim data.
Worked Example: Preserving a Fixed Allocation Ratio
A trial requires 300 observed primary outcomes, with equal numbers from its two randomized groups. The masked interim estimate is \(\hat r=0.15\). The pooled calculation is \(300/0.85=352.94\), so at least 353 participants would be needed by the simple total-count rule. But an odd total cannot be divided equally between groups.
The target is 150 observed outcomes per group. Applying the same pooled rate to each group gives
Rounding up separately gives an enrollment target of 177 per group, or 354 total. The expected number with outcomes is \(177(0.85)=150.45\) per group. Thus, the allocation constraint increases the total above the pooled calculation of 353. This calculation assumes that the pooled rate is a reasonable planning rate for both groups; masking does not establish that assumption.
Diagnostics, alternatives, and limits
Before using a pooled estimate, check that the data and the rule answer the intended question. A review should verify the outcome window, the eligible denominator, the count of truly missing outcomes, and whether the interim participants have follow-up comparable to that expected for future participants. A change in assessment procedures or follow-up duration can make early and later attrition rates incomparable. A sharp difference in missingness across sites or calendar periods is also a warning that one pooled rate may conceal meaningful variation.
A pooled rate has an important limitation: because treatment labels are hidden, it cannot reveal whether attrition differs between randomized groups. If group-specific completion probabilities are \(q_1\) and \(q_2\), and the planned allocation proportions are \(\pi_1\) and \(\pi_2\), the expected overall completion proportion under that allocation is \(\pi_1q_1+\pi_2q_2\). A pooled interim rate can estimate the overall rate represented by the interim allocation, but it cannot identify \(q_1\) and \(q_2\) separately. If differential attrition is substantial or changes over time, a pooled update may not provide adequate protection against shortfalls in one group.
The main alternatives answer different questions. A rate from prior studies or the original protocol avoids relying on a noisy interim estimate, but may not reflect the current trial’s retention. A blinded pooled interim estimate uses current trial experience while withholding treatment labels, but cannot diagnose treatment-specific attrition. An arm-specific estimate can reveal group differences, but requires unmasking and is not the procedure described here. These approaches are not interchangeable: the appropriate choice depends on what information is available and what the re-estimation rule permits.
A plug-in estimate can also be unstable when few participants have reached the outcome window. For example, one additional missing outcome changes the estimate by \(1/m\), so small \(m\) can produce large changes in the enrollment target. The rate itself has sampling uncertainty; treating \(\hat r\) as if it were known exactly can give a false sense of precision. A prespecified minimum amount of mature follow-up, a conservative planning rate, or a prespecified constraint on the update can reduce sensitivity to an unstable estimate. Such safeguards do not remove uncertainty and should be justified before the interim review.
Finally, increasing enrollment addresses the number of outcomes observed; it does not by itself correct bias caused by missing outcomes. If outcome availability is related to participants’ unobserved outcomes, the final analysis may still need assumptions or methods for missing data. Nor does masking prove that attrition is unrelated to treatment or outcome. The estimate is a planning input, not an analysis of the consequences of missingness.
Common mistake
Wrong reasoning: “There are 18 missing outcomes among the 150 enrolled participants, so the attrition rate is 12%; participants still being followed count as missing until their outcome appears.” This treats delayed ascertainment as permanent non-ascertainment. Correction: define the assessment window first, include only participants whose outcome was due, and calculate missing outcomes divided by that eligible denominator. In the example, the rate is \(18/120=15\%\), not \(18/150=12\%\).
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At an interim review, 200 participants have reached the primary-outcome window; 26 lack the outcome. Another 40 participants are still within their follow-up window. What are \(\hat r\) and \(\hat q\), and which participants form the denominator?
- A study needs 510 observed primary outcomes. If the estimated attrition rate is 15%, what enrollment target follows from the pooled plug-in rule before any allocation-ratio adjustment?
- Why can’t a blinded pooled attrition estimate establish that the two treatment groups have equal attrition?
- State one reason why the number of enrolled participants with observed outcomes might be smaller than the required number even when enrollment is set to \(n_{\mathrm{obs}}/(1-\hat r)\).
- Name one diagnostic that could indicate that a single pooled attrition rate is not a good planning summary.
Solutions. (1) The denominator is the 200 participants whose outcome window has passed; the 40 still in follow-up are not yet classified as missing. Thus, \(\hat r=26/200=0.13\) and \(\hat q=1-0.13=0.87\). (2) \(N_{\mathrm{target}}=510/0.85=600\), so the pooled rule gives 600 participants before any allocation-ratio adjustment. (3) Pooling removes the treatment labels, so it gives an overall rate but not separate estimates of each group’s rate. (4) The actual number completing follow-up varies around its expectation, and the estimated rate is uncertain; the plug-in formula does not guarantee a minimum realized count. (5) Examples include sharply different missing-outcome rates across sites or calendar periods, or evidence that follow-up procedures changed.