Why Survival Data Need More Than One Row per Person
A one-row-per-person data set can describe a baseline covariate and the observed event or censoring time. It cannot, by itself, show that a covariate changed during follow-up and which value applied while a person was at risk. Start-stop data provide a row structure for that information: each row records a participant’s covariate values over a particular interval of observed follow-up.
The distinction between a person and a row is essential in multiple imputation. One participant may contribute several rows, but those rows are not several independent participants. They are pieces of that person’s survival record. The layout identifies where covariate values apply; it does not, on its own, determine how missing values should be imputed.
The notation \((a_{ij}, b_{ij}]\) means that the interval excludes its left endpoint and includes its right endpoint. A row with start 4 and stop 9 therefore represents time after day 4 through day 9. This convention permits adjacent rows, such as \((0,4]\) and \((4,9]\), to meet without counting a time point in both intervals. In ordinary single-event follow-up, a participant’s event or censoring time is the stop time on the final row.
What the Row Says About a Participant
In the usual start-stop representation for a single event, define \(\delta_{ij}=1\) if the event occurs at the end of row \(j\) for participant \(i\), and \(\delta_{ij}=0\) otherwise. An event indicator of zero does not mean that the participant was not at risk during the row. It means that the event did not occur at that row’s stop time. A participant censored at the end of follow-up has a final row with \(\delta_{ij}=0\).
For a participant who experiences the event, the event indicator is zero on earlier rows and one on the final row. The record ends at the event: there should be no later at-risk rows for that participant in this single-event setup. For a participant who is censored, all their rows have event indicator zero. The event indicator is attached to a row, but the event remains an outcome for the participant, not a new event for every row.
A covariate recorded on a row is the value assigned to that participant for the interval represented by that row. The representation treats that value as applying throughout the interval. It does not claim that the covariate was measured continuously, or that it could not have changed between measurement times. Those are substantive and measurement assumptions, not properties guaranteed by the data format.
Worked Example: Reading a Participant’s Changing Treatment Record
Suppose a hypothetical participant is followed from day 0 until an event on day 12. Treatment status changes at day 5. The table records the status applying over each interval.
| Participant | Start \(a_{ij}\) (days) | Stop \(b_{ij}\) (days) | Treatment | \(\delta_{ij}\) |
|---|---|---|---|---|
| A | 0 | 5 | No | 0 |
| A | 5 | 12 | Yes | 1 |
The first row covers \((0,5]\), and the second covers \((5,12]\). The intervals are adjacent: the first ends where the second begins, so there is neither an overlap nor an uncovered stretch of follow-up. Their lengths are \(5-0=5\) days and \(12-5=7\) days, for a total of 12 days.
The participant is untreated on the first row and treated on the second. The event occurs at the stop time of the second row, so that row has \(\delta_{ij}=1\); the earlier row has \(\delta_{ij}=0\). The table therefore describes one participant, one event, and two interval records—not two participants or two events.
Rows, Risk Sets, and Time Boundaries
Start-stop records make it possible to determine which participants are under observation and at risk just before a particular time. For an interval \((a_{ij},b_{ij}]\), its row contributes to the risk set at time \(t\) when \(a_{ij}<t\leq b_{ij}\). If participant \(i\) has consecutive, nonoverlapping rows covering their observed follow-up, at most one of their rows contributes at a given time.
Here \(I(\cdot)\) is 1 when the condition inside the parentheses is true and 0 otherwise. Under continuous, nonoverlapping follow-up represented by adjacent rows, \(Y_i(t)\) is 1 when participant \(i\) is at risk just before time \(t\), and 0 otherwise. The row contributing to \(Y_i(t)\) also supplies the covariate value assigned to that time. This is why the interval boundaries and covariate values must be aligned correctly.
The endpoint convention matters when an event occurs at a row boundary. Under \((a_{ij},b_{ij}]\), an event at \(b_{ij}\) belongs to that row. A subsequent row beginning at \(b_{ij}\) represents time after the boundary. Thus, if a covariate changes at a boundary, the data convention must specify which value applies to the event at that exact time. With this interval convention, an event at the boundary is assigned to the interval ending there; the new value applies to the following interval.
Worked Example: Identifying the Risk Set at a Time
Consider three hypothetical participants observed on the following intervals. Participant A is censored at day 8; participant B has an event at day 7; participant C has an event at day 10.
| Participant | Start (days) | Stop (days) | Event indicator |
|---|---|---|---|
| A | 0 | 8 | 0 |
| B | 0 | 7 | 1 |
| C | 3 | 10 | 1 |
At \(t=5\), the risk-set condition is \(a_i<t\leq b_i\). For A, \(0<5\leq8\), so A contributes. For B, \(0<5\leq7\), so B contributes. For C, \(3<5\leq10\), so C contributes. The risk set just before day 5 contains A, B, and C.
At \(t=7\), A satisfies \(0<7\leq8\), B satisfies \(0<7\leq7\), and C satisfies \(3<7\leq10\). All three are in the risk set just before day 7. B’s event indicator is 1 at the end of its row, so B experiences the event at day 7 and is no longer at risk after that time. At \(t=8\), the interval for A still includes the endpoint, but A is censored at that time; A is not at risk after day 8. The endpoint convention captures the time immediately before exit without extending follow-up beyond the observed stop.
Why This Layout Matters for Multiple Imputation
When a covariate varies over follow-up, the imputation data must preserve the information about which value belongs to which at-risk interval. A start-stop data set makes that alignment explicit. It also makes clear which fields are person-level and which can vary from row to row. A baseline covariate may be repeated on all rows for one participant, while a time-varying covariate may differ across that participant’s rows.
Repeated rows have consequences for interpreting the data supplied to an imputation procedure. The number of rows is not the number of independent participants. Subject identifiers link the records, and duplicated values for a baseline field are repeated representations of one participant-level value. Treating each row as an unrelated person would discard that structure and could give participants with more rows disproportionate influence. A start-stop layout alone does not resolve how an imputation model should account for repeated records; it only makes the record structure visible.
The survival outcome information also needs to remain coherent across rows. Within the ordinary single-event setup, event status is 1 on at most one terminal row per participant and 0 on earlier rows. The final stop time represents that participant’s event time or censoring time. If the same person’s final time or event status is mistakenly copied as though each row were a separate outcome, the data no longer describe the intended follow-up.
Worked Example: Distinguishing Participants From Rows in an Imputation File
Suppose a hypothetical file contains one baseline severity score and one time-varying laboratory value for two participants. Participant P contributes two intervals and participant Q contributes three. Their records are:
| Participant | Start (weeks) | Stop (weeks) | Baseline severity | Laboratory value | Event indicator |
|---|---|---|---|---|---|
| P | 0 | 2 | 4 | 10 | 0 |
| P | 2 | 6 | 4 | 13 | 1 |
| Q | 0 | 1 | 2 | 8 | 0 |
| Q | 1 | 3 | 2 | 9 | 0 |
| Q | 3 | 5 | 2 | 11 | 0 |
There are five rows but only two participants. The severity scores, 4 for P and 2 for Q, are participant-level values repeated across that person’s records. The laboratory values vary by interval: P’s recorded value changes from 10 to 13, and Q’s from 8 to 9 to 11. P experiences an event at week 6, so the last P row has event indicator 1. Q has no event during the recorded five weeks, so all Q rows have event indicator 0; the final stop at week 5 marks the end of observed follow-up.
For an imputation file, the participant identifier and row boundaries are needed to interpret these records. The repeated severity values do not constitute five independent baseline measurements, and the laboratory values cannot be assigned to a participant without their interval. The table specifies the structure that an imputation analysis must respect; it does not prescribe a particular imputation model or treatment of missing values.
Assumptions, Diagnostics, and Alternatives
A start-stop file is meaningful only when its intervals match the intended follow-up and its covariate assignments are defensible. Before using it, check the following conditions and data features:
- Valid intervals: every row has \(a_{ij}<b_{ij}\), and the start and stop times use the same time scale and units.
- Coherent sequencing: rows are ordered within participant. For uninterrupted observed follow-up, each next interval starts where the previous one stops; a genuine gap should be represented and explained rather than silently filled.
- No unintended overlap: a participant should not be represented as at risk in two rows at once. Overlapping intervals can count a person more than once at a time.
- Correct terminal row: for ordinary single-event follow-up, the event indicator is 1 on the event-ending row only. If no event is observed, all row indicators are 0 and the final stop is the censoring time.
- Covariate timing: the value on each row must be the value assigned to that interval under the study’s measurement and recording convention. A value observed after an interval cannot simply be treated as though it were known throughout that earlier interval.
- Consistent participant fields: values intended to be fixed at the participant level should not vary across that participant’s rows unless a documented data rule explains the difference.
These checks help distinguish a structural error from a legitimate feature of the data. A gap may be appropriate if a person is genuinely not under observation or not at risk during that period, but unexplained gaps in otherwise continuous follow-up may indicate omitted rows or incorrect boundary values. Similarly, repeated event indicators of 1 for one person may signal a coding error in a single-event analysis, while a different event structure would require a different representation.
The main alternative is a one-row-per-person survival file. It is suitable for a covariate measured only at baseline, but it cannot directly attach different covariate values to distinct at-risk periods. A conventional longitudinal file can store repeated measurements, but its rows do not necessarily define the intervals over which a person contributes to a survival risk set. Start-stop data combine interval boundaries with covariate values and event status, making their survival-time meaning explicit. These formats serve different purposes; the choice is determined by the question and observed data, not by a preference for more or fewer rows.
A further limitation is the piecewise-constant representation: a row assigns one covariate value to an interval even if the underlying quantity changes between recorded measurements. The validity of that assignment depends on the measurement schedule and scientific context. Start-stop formatting cannot recover unobserved changes or correct an unsuitable measurement assumption. Nor does it guarantee that the imputation model preserves the dependence among a participant’s records.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A row has start 2 weeks, stop 6 weeks, and event indicator 0. What interval does it represent, and what does the zero mean?
- One participant has rows \((0,3]\), \((3,7]\), and \((7,9]\), with event indicators 0, 0, and 1. When and how many times does the event occur?
- At time \(t=4\), does a row \((4,8]\) contribute to the risk set under the stated convention? Explain.
- A data set has 420 rows from 160 participants. Which count is the number of independent participant identifiers represented, and why is the row count different?
- For a single-event analysis, one participant has two rows with event indicator 1. Name a structural check that should be made.
Solutions. (1) It represents \((2,6]\), or time after week 2 through week 6. The zero means no event occurred at the row’s stop; it does not mean the participant was absent from the risk period. (2) The event occurs once, at week 9, the stop time of the final row. (3) No. The risk-set condition is \(a<t\leq b\); for this row \(4<4\) is false, so it does not contribute at week 4. (4) There are 160 participant identifiers, because multiple rows can belong to the same person; 420 counts interval records. (5) Check whether the event indicator has been duplicated or assigned incorrectly: in ordinary single-event follow-up it should be 1 on only the event-ending row.