Risk Sets For Visit-Specific Dropout

Why Visit-Specific Risk Sets Matter

A percentage of participants missing at a visit can describe how much follow-up has been lost, but it does not by itself answer a transition question: among participants still under follow-up immediately before this visit, what fraction drop out now? The denominator for that question is the risk set at the visit. Defining it correctly is essential because people who drop out at the current visit have no outcome recorded there, but they still belong in the denominator for the current-visit dropout probability.

This tutorial concerns scheduled visits and monotone dropout: once a participant drops out, their later outcomes are not observed. Temporary missed visits and other departures from monotone dropout require different accounting. The earlier material on monotone dropout versus intermittent missingness distinguishes these patterns.

Thus the risk set at visit \(t\) contains two groups: participants observed at \(t\), and participants whose first unobserved visit is \(t\). The latter are the visit-\(t\) dropout events. In particular, the risk set is generally larger than the set with an observed outcome at that visit. If \(r_t\) is the number in the risk set and \(d_t\) is the number who drop out at that visit, then, when status is known for everyone in the risk set, the number with an observed outcome at \(t\) is \(r_t-d_t\).

Visit-Specific Probabilities and Their Relationship

The visit-specific dropout probability is conditional on having entered that visit’s risk set. For a cohort with complete follow-up-status information, it is estimated by the number of dropout events at the visit divided by the number at risk immediately before it.

Here \(q_t\) is the population probability of dropping out at visit \(t\), conditional on still being under follow-up immediately before that visit. The estimate \(\hat q_t\) is a descriptive sample proportion; by itself, it does not explain why participants dropped out. Its interpretation depends on the visit definition and on classifying dropout events consistently.

The probability of remaining under follow-up through visit \(t\), in the same unconditional cohort, is built from the probabilities of surviving each visit’s dropout transition. This multiplication follows from the chain rule: to remain through \(t\), a participant must avoid dropout at visits 1 through \(t\), with each successive probability conditional on reaching that visit.

The corresponding estimated cumulative dropout proportion by visit \(t\), when everyone begins follow-up and there is no separate censoring, is \(1-\prod_{j=1}^{t}(1-\hat q_j)\). It is not generally equal to \(\hat q_t\): the latter is conditional on entering visit \(t\), whereas the cumulative proportion uses the original cohort as its reference.

Conditioning on Recorded Observation Histories

Participants who enter a visit can have different recorded histories. A visit-specific probability can therefore be calculated within a history group. Let \(H_{i,t-1}\) denote participant \(i\)’s recorded outcome history through visit \(t-1\), for a participant who remains under follow-up to visit \(t\). For a specified history \(h\), the conditional probability and its sample estimate are

The denominator \(r_t(h)\) counts participants with history \(h\) who enter the visit’s risk set. The numerator \(d_t(h)\) counts those among them who drop out at that visit. This is a way of describing dropout probabilities across recorded histories; it does not establish that the history caused dropout, nor does it show that the recorded history is sufficient to explain dropout.

Deriving the Risk-Set Calculation

The denominator in \(\hat q_t=d_t/r_t\) follows directly from the conditional probability being estimated. Within a risk set, every participant satisfies \(T_i\ge t\), so the event \(T_i=t\) is the visit-\(t\) dropout event among those who reached the visit. In a fully observed cohort, the number in that conditioning set is \(r_t\), and the number satisfying the event is \(d_t\). The sample conditional proportion is therefore \(d_t/r_t\).

The same counting gives the link to the number still under follow-up for the next visit. Participants in \(R_t\) either have \(T_i=t\), or have \(T_i>t\). These groups do not overlap and together make up \(R_t\). Consequently, \(r_t=d_t+r_{t+1}\) and \(\hat q_t=1-r_{t+1}/r_t\), provided follow-up status is complete and the next risk set contains precisely those who avoided dropout at visit \(t\).

This justification depends on the event definition. If an outcome is collected at visit \(t\) and the participant withdraws afterward, that participant has an observed outcome at \(t\); the withdrawal should not be counted as dropout before the visit-\(t\) outcome. The study’s visit and withdrawal records must support the chosen classification.

Worked Example: Correcting a Denominator Error

Suppose a hypothetical cohort begins with 80 participants. Eight drop out before their visit-1 outcome is observed, leaving 72 with observed visit-1 outcomes. At visit 2, 18 of those 72 drop out before the visit-2 outcome, while 54 have an observed visit-2 outcome. Assume there is no administrative censoring and all statuses are known.

The visit-2 risk set consists of all 72 participants who remained under follow-up after visit 1. Thus the correct estimate is

The 54 participants with observed visit-2 outcomes are not the visit-2 denominator: they exclude the 18 dropout events being counted. Dividing by 54 would give \(18/54=0.3333\), which is not the conditional probability of dropout among participants at risk before visit 2. The cumulative dropout proportion by visit 2 is instead \((8+18)/80=0.325\). It also agrees with the risk-set product: \(1-(72/80)(54/72)=1-54/80=0.325\).

Checks, Comparisons, and Limitations

A risk-set calculation is useful only if the visit status and denominator correspond to the stated question. Check that the following conditions are reasonable before interpreting these estimates:

• Monotone dropout is a suitable description. If a participant misses a visit but returns later, treating that absence as permanent dropout gives the wrong later risk sets.
• Dropout is classified relative to the outcome at the visit. An outcome recorded before withdrawal is not a missing outcome at that visit.
• Status is known for people in the risk set. If some are lost to administrative follow-up before the visit, they cannot simply be counted as known dropouts or known non-dropouts. The simple ratio needs adjustment or a more explicit account of censoring.
• Counts reconcile. Under the stated complete-status setup, \(r_{t+1}=r_t-d_t\), and \(r_t-d_t\) should equal the number observed at visit \(t\). A failure points to inconsistent classification, incomplete records, or a process that is not monotone.
• Group-specific denominators are adequate. Small risk sets can yield unstable proportions; a zero or near-zero denominator does not support a precise probability estimate.

The visit-specific probability answers a different question from the fraction missing at a visit using the original cohort as denominator. The original-cohort fraction is cumulative; the risk-set proportion is conditional on reaching that visit. Both can be informative, but they should not be interchanged. In particular, a declining risk set can make a later conditional proportion appear substantial even when the number of later events is modest.

The simple risk-set ratio also differs from an analysis that models dropout using several covariates, or an analysis that accounts explicitly for administrative censoring. Those approaches require additional assumptions and are not provided by the raw ratio. Risk-set probabilities describe observed transitions under the chosen classification; they do not establish why dropout occurred or recover outcomes that were not observed.