Defining Dropout Time In A Visit-Based Study

Why Define Dropout Time?

In a longitudinal study, outcomes are scheduled repeatedly, but not every participant contributes an outcome at every visit. A visit-specific indicator records whether each outcome was observed. A dropout-time variable compresses a participant’s sequence of observation indicators into one boundary: the first scheduled visit at which outcomes are no longer observed. This makes the timing of persistent loss of observation precise, provided the observation pattern really has that form.

The boundary convention matters. Some descriptions call the last observed visit the dropout time; others use the first visit with no observed outcome. Both can describe the same boundary, but their visit numbers differ by one. Here, dropout time means the first scheduled visit at which the outcome is not observed because observation has ended. The last observed visit is therefore the visit immediately before it. Stating this convention avoids off-by-one errors in summaries and calculations.

Definition: Suppose outcomes are scheduled at visits $j=1,\ldots,J$. For participant $i$, let $R_{ij}=1$ if the outcome at visit $j$ is observed and $R_{ij}=0$ otherwise. Under a pattern in which observation, once ended, does not resume, define the dropout time $D_i$ as the first visit index with no observed outcome: $$ D_i=\min\{j\in\{1,\ldots,J\}:R_{ij}=0\}. $$ If outcomes are observed at every scheduled visit, set $D_i=J+1$ as a sentinel value meaning “no dropout during the scheduled visits.” If $D_i\leq J$, the last observed visit is $D_i-1$.

This definition concerns the observation process, not the outcome values. For example, if $D_i=4$, it means the outcome is observed through visit 3 and not observed from visit 4 onward; it does not mean that the outcome at visit 4 is zero or that the participant’s underlying outcome stops existing. The definition also does not, by itself, explain why observation ended.

The $J+1$ convention is useful because every participant receives a value, including someone with complete scheduled follow-up. It must be interpreted as a coding device rather than as a real visit. If a study has five visits, $D_i=6$ means outcomes were observed at all five, not that a sixth outcome was collected.

From Indicators to a Boundary

A single dropout time summarizes a sequence of indicators only when the sequence consists of observed outcomes up to a boundary, followed by unobserved outcomes. The indicator sequence then has the form $1,\ldots,1,0,\ldots,0$. This condition is sometimes called a monotone observation pattern. It is an assumption about the pattern being represented; it should not be silently imposed on a sequence that contains later observed outcomes after a missed visit.

Result: If participant $i$'s outcomes are observed at every visit before $D_i$ and unobserved at every scheduled visit from $D_i$ onward, then $$ R_{ij}=\begin{cases} 1,&j<D_i,\\ 0,&j\geq D_i, \end{cases} \qquad j=1,\ldots,J. $$ Equivalently, $R_{ij}=I(j<D_i)$, where $I(\cdot)$ equals 1 when its condition is true and 0 otherwise.

The representation follows directly from the definition. If $j<D_i$, visit $j$ occurs before the first visit with no observed outcome, so its outcome is observed and $R_{ij}=1$. If $j\geq D_i$, visit $j$ is at or after the boundary; under the stated no-resumption condition, its outcome is unobserved and $R_{ij}=0$. When $D_i=J+1$, every scheduled index satisfies $j<D_i$, so all $J$ indicators equal 1. This also shows why the sentinel value represents complete scheduled follow-up consistently.

Worked Example: Finding the Boundary From Visit Indicators

Suppose five visits are scheduled, and a hypothetical participant has observation indicators $R_{i1},\ldots,R_{i5}=(1,1,1,0,0)$. Find the dropout time and the last observed visit.

The first zero occurs at visit 4, so the definition gives $D_i=4$. The last observed visit is one visit earlier: $$ D_i-1=4-1=3. $$ Thus the outcomes at visits 1, 2, and 3 are observed, while outcomes at visits 4 and 5 are not. The pattern satisfies the no-resumption condition needed for one dropout time to summarize it.

Worked Visit-Based Examples

A visit number has meaning only relative to the schedule. It is helpful to state both the visit index and the scheduled time, especially when visits are not equally spaced or when participants begin follow-up at different calendar dates.

Worked Example: Scheduled Visits at Fixed Weeks

In a hypothetical study, visits are scheduled at weeks 0, 4, 8, 12, and 16, indexed as visits 1 through 5. Participant A has outcomes recorded at weeks 0, 4, and 8, and no outcomes at weeks 12 or 16 because participation ended after week 8. Determine $D_A$ and the last observed visit.

The first scheduled outcome that is not observed is at week 12, which is visit 4. Therefore, $$ D_A=4,\qquad \text{last observed visit}=D_A-1=3. $$ In calendar-time language, the last observed outcome is at week 8, and the boundary begins at the scheduled week-12 visit. The arithmetic is consistent with the visit sequence: visits 1–3 have $R_{Aj}=1$, and visits 4–5 have $R_{Aj}=0$. This conclusion assumes the lack of later outcomes reflects the end of observation rather than a temporary missed visit followed by a return.

Worked Example: Complete Scheduled Follow-Up

A second hypothetical participant has an observed outcome at each of the same five scheduled visits. Find the dropout-time value under the stated convention.

There is no $j$ in $\{1,2,3,4,5\}$ with $R_{ij}=0$, so the set in the minimum definition is empty. The complete-follow-up rule assigns $$ D_i=J+1=5+1=6. $$ The participant’s last observed scheduled visit is visit 5, not visit 6. The value 6 is a sentinel indicating that no dropout boundary occurred within the five-visit schedule.

The definition also makes it possible to describe boundaries at different points in follow-up without changing the indexing rule. A participant whose outcomes stop after an earlier visit has a smaller first-unobserved visit index than a participant whose outcomes stop later. The coding records when observation is no longer available, not how many calendar days each participant was followed unless the visit schedule maps those indices to calendar time.

Worked Example: Follow-Up Ends at Different Scheduled Visits

A hypothetical study has six scheduled visits, indexed 1 through 6. Participant B has outcomes observed at visits 1–2 and none at visits 3–6. Participant C has outcomes observed at visits 1–5 and none at visit 6. Assume in each case that observation ended and did not resume. Compare their dropout times and last observed visits.

For participant B, the first unobserved visit is 3: $$ D_B=3,\qquad D_B-1=3-1=2. $$ B’s last observed outcome is at visit 2.

For participant C, the first unobserved visit is 6: $$ D_C=6,\qquad D_C-1=6-1=5. $$ C’s last observed outcome is at visit 5. Thus $D_C>D_B$, reflecting a later boundary on the common six-visit schedule. The numbers do not establish that the participants had the same reason for ending observation; they summarize the location of the boundary only.

What the Dropout-Time Value Does—and Does Not—Say

Dropout time is a convenient summary, but it is not a complete description of every observation history. If a participant misses visit 3 but has an outcome recorded at visit 4, the indicator sequence contains a zero followed by a one. Taking the first zero would give a boundary at visit 3, yet that boundary would incorrectly imply that visit 4 and all later outcomes were unobserved. Such a pattern cannot be faithfully summarized as a permanent dropout time under the definition above. The distinction between persistent dropout and intermittent missingness is treated separately; for now, the key check is whether observed outcomes resume after a missing visit.

A boundary can also arise because the study’s observation window ends, rather than because an individual has stopped participating. If data collection ends for a participant after visit 4, the first unavailable scheduled outcome may be visit 5. That index alone does not identify the reason. A dropout-time label should be used for a genuine end to participant-level observation, not assigned automatically whenever the record ends. Reasons and study-end timing require information beyond the sequence of indicators.

There are several common ways to encode the same boundary. The first-unobserved-visit convention used here gives $D_i$. A last-observed-visit convention gives $D_i-1$ when dropout occurs within the schedule. The full vector $(R_{i1},\ldots,R_{iJ})$ retains the visit-by-visit record rather than compressing it. Use the full vector when the pattern itself matters, and state the convention whenever reporting a single dropout-time value. These representations should not be mixed without an explicit conversion.

Write down the schedule.
Index the scheduled visits in order and, where useful, note their calendar times.

Inspect the observation indicators.
Locate the first visit with $R_{ij}=0$, and check whether later outcomes remain unobserved.

Assign and interpret the boundary.
If the pattern is observed outcomes followed by no further observed outcomes, set $D_i$ to the first unobserved visit. If all scheduled outcomes are observed, set $D_i=J+1$.

Check the meaning of the endpoint.
Do not infer why observation ended from $D_i$ alone, and do not treat a temporary missing visit as a permanent boundary.

Common Mistakes and Practical Checks

Using the last observed visit as $D_i$. Under this tutorial’s convention, $D_i$ is the first unobserved visit. If the last observed visit is 3, the dropout time is 4, not 3.
Forgetting the complete-follow-up code. When all $J$ visits are observed, use $D_i=J+1$; do not call the final scheduled visit the first unobserved visit.
Reading a boundary as a cause. $D_i$ says where the observation sequence ends. It does not say whether the reason was participant withdrawal, a study endpoint, or another event.
Applying a single boundary to a pattern with a return. If an outcome is observed after a missed visit, the sequence is not represented by one permanent dropout time. Check the indicators rather than forcing them into the formula.
Ignoring the visit schedule. A visit index is not automatically a duration. State the scheduled times when comparing boundaries in calendar time.

These checks are useful diagnostics: inconsistent indicators, a later observed outcome after the proposed boundary, or confusion between a visit number and elapsed time signals that the dropout-time representation needs review. The main alternative is to retain the complete indicator vector, which avoids compressing the pattern but is less compact. Neither representation by itself explains the missingness mechanism or supplies unobserved outcome values.

Key takeaway: With visits indexed $1,\ldots,J$, define dropout time as the first visit with no observed outcome, and use $J+1$ when every scheduled outcome is observed. The last observed visit is one less than the dropout time when dropout occurs within the schedule. Confirm that observation does not resume and do not infer the reason for an endpoint from its index alone.

Check Your Understanding

Answer each question using the first-unobserved-visit convention. Full solutions follow the questions.

With four scheduled visits, a participant has indicators $(1,1,0,0)$. What are $D_i$ and the last observed visit?
With $J=3$, a participant has outcomes observed at all three visits. What is the assigned dropout-time value, and what does it mean?
A participant has indicators $(1,0,1,0)$. Why is a single permanent dropout time not an adequate representation of this sequence?
With six scheduled visits, one participant’s first unobserved outcome is at visit 2 and another’s is at visit 5. Which participant has the later observation boundary?

Solutions. (1) The first zero is at visit 3, so $D_i=3$; the last observed visit is $3-1=2$. (2) Assign $D_i=J+1=3+1=4$. It is a sentinel for complete observation at all three scheduled visits, not an observed fourth visit. (3) The outcome at visit 3 is observed after the visit-2 outcome is missing. A first-zero rule would set $D_i=2$, incorrectly implying no later outcomes are observed, so the sequence is not a permanent dropout pattern. (4) The participant with the first unobserved outcome at visit 5 has the later boundary, since $5>2$, assuming both are on the same visit schedule.