Why the Record Structure Matters
In survival data, a row is not merely a collection of values: it represents a person’s observed follow-up under a defined time origin and a particular record structure. That structure must distinguish who is being followed, when follow-up begins and ends, whether an event was observed, and when each covariate value was measured. If these fields are confused, chained-equation imputation can use information from the wrong person or from a time that had not yet occurred.
A simple data set may have one row per participant, which can represent baseline covariates and a single follow-up endpoint. When covariates vary over follow-up, a person may instead contribute multiple rows, each describing an interval. Those rows are linked by the participant identifier; they are not separate participants. This distinction is especially important in MICE, where the analyst must decide which variables belong to the person and which are indexed by follow-up time.
Fields That Must Not Be Confused
Every analysis needs a clearly defined time origin, such as enrollment or randomization, and a time scale, such as weeks since enrollment. Follow-up times are measured relative to that origin. In a basic one-row record, let \(T_i\) denote the time to the event of interest for participant \(i\), and let \(C_i\) denote the time at which observation ends if the event has not been observed. The recorded follow-up endpoint is \(Y_i=\min(T_i,C_i)\). Define \(\delta_i=1\) when the event is observed by the end of follow-up and \(\delta_i=0\) when follow-up ends without observing that event. These definitions describe the record; they do not, by themselves, specify a model for the event process.
Thus, \(Y_i\) is an observed endpoint time, not necessarily the time at which the event occurred. When \(\delta_i=1\), the endpoint is the observed event time under this convention. When \(\delta_i=0\), it is the censoring time: the participant was observed up to that point without the event being recorded. In practical data, a tied event and censoring time needs a prespecified coding convention; the expression above assigns the event when \(T_i=C_i\).
A covariate measurement time is different from either endpoint. It records when a value was obtained, not when follow-up ended or when an event happened. A baseline value may be measured at the time origin; a later value becomes available only from its measurement time onward. For interval records, the analyst must specify how a measurement is represented within intervals. If each interval is to have one covariate value that is available at its start and treated as constant during that interval, an interval must not straddle a change in that value.
The basic record assumptions are therefore explicit: the participant identifier uniquely identifies a person; the time origin and units are common or otherwise documented; event status follows a defined coding rule; and each covariate value is attached to its actual measurement time. For an interval representation, intervals must also be ordered and nonoverlapping, and their boundaries must agree with the chosen endpoint convention. These are data-structure requirements, not assumptions that make censoring statistically independent of event time.
Worked Example: Reading a One-Row Survival Record
Suppose a hypothetical study follows participants from enrollment, with time measured in weeks. Participant P14 experiences the event at week 8. Participant P27 has no observed event and last has confirmed follow-up at week 10. For this illustration, event is coded 1 and no observed event is coded 0.
| Participant ID | Observed follow-up, \(Y_i\) (weeks) | Event indicator, \(\delta_i\) | Meaning of endpoint |
|---|---|---|---|
| P14 | 8 | 1 | Observed event at week 8 |
| P27 | 10 | 0 | Follow-up ended at week 10 without observed event |
For P14, \(T_{14}=8\) weeks and the record indicates \(\delta_{14}=1\). For P27, the available record gives \(C_{27}=10\) weeks and \(\delta_{27}=0\); it does not establish that the event occurred at week 10. In both rows the participant ID is a label for joining records, not a quantitative predictor. The table has one row per person because it records only a single endpoint and does not represent covariate changes during follow-up.
Interval Records and Measurement Timing
When a covariate changes during follow-up, one endpoint row cannot show which value applies at each time. An interval record makes the time span explicit. A common notation is \((a_{ik},b_{ik}]\), where \(a_{ik}\) is the interval start and \(b_{ik}\) is its end for person \(i\) on interval \(k\). The endpoint may be included in the final interval according to the data convention; the notation must be consistent with event coding. For a start-of-interval value, the covariate attached to an interval should have been measured no later than \(a_{ik}\), and should not be presented as applying before it was measured.
For example, suppose a hypothetical participant has a covariate measured at weeks 0, 4, and 6, with values 5.2, 5.8, and 6.9 units. If follow-up ends with an event at week 7 and the interval value is defined as available at the interval start and constant within the interval, the measurement times must be interval boundaries. The correct split is \((0,4]\), \((4,6]\), and \((6,7]\). The week-6 value applies to \((6,7]\), not to the interval beginning at week 4.
Worked Example: Aligning Measurements With Follow-Up Intervals
A hypothetical participant is enrolled at week 0, has a covariate measured as 5.2 units at week 0, 5.8 units at week 4, and 6.9 units at week 6, and experiences the event at week 7. Assume values are used from their measurement time onward and are constant until the next measurement. Use intervals that do not cross measurement times.
| Participant ID | Interval (weeks) | Covariate value available at interval start | Event at interval end? |
|---|---|---|---|
| P31 | (0, 4] | 5.2 units, measured at week 0 | No |
| P31 | (4, 6] | 5.8 units, measured at week 4 | No |
| P31 | (6, 7] | 6.9 units, measured at week 6 | Yes |
The interval lengths are \(4-0=4\), \(6-4=2\), and \(7-6=1\) week, so the intervals cover the observed follow-up from week 0 through week 7 without overlap or a gap. The week-6 value is used only after week 6. If the record instead had one interval \((4,7]\) with 6.9 units labelled as available at the interval start, it would be temporally wrong: that value was not measured until week 6. Splitting at week 6 resolves the mismatch and keeps the start-of-interval rule intact.
The event is recorded at the end of the final interval, while the covariate measurement times remain separate information. This record says nothing about the covariate after week 7, because follow-up ended then. It also does not imply that the values are known without measurement error; it only describes their recorded timing and interval assignment.
Building Records for MICE
Before fitting imputation models, define the unit of each variable. A participant-level characteristic, such as treatment assignment or a baseline covariate, has one value per person. A time-varying covariate has values indexed by person and measurement time, or by person and interval if represented in interval form. An event indicator and observed follow-up endpoint are person-level survival descriptors in a basic record. A participant ID links these pieces and is generally not an imputation target.
In MICE, the data layout affects which conditional models are being fitted. If interval rows are stacked into a table, the repeated rows for one person are dependent, and the person-level outcome descriptors repeat across those rows. Treating every interval row as a new independent participant can give people with more intervals disproportionate influence and can misrepresent uncertainty. Conversely, collapsing a genuinely changing covariate to one value per participant discards its timing. Neither layout is universally correct: it must match the scientific variables and the intended analysis.
A practical record-building sequence is:
Choose a stable participant identifier, a time origin, and units such as days or weeks since enrollment.
Store observed follow-up time and the event indicator, with the event/censoring convention documented.
Keep measurement time distinct from the endpoint, and do not use a value before it was measured.
Use a person-level row for a single fixed record; use multiple linked rows when the representation must preserve changes over follow-up.
Verify unique IDs at the person level, ordered interval boundaries, consistent status coding, and plausible measurement-time alignment.
Worked Example: Detecting a Record-Structure Error
Consider a hypothetical interval table with three rows for participant P42. Follow-up starts at week 0 and ends at week 9 with no observed event. A baseline covariate is measured at week 0; another value is measured at week 5. The intended intervals are \((0,5]\) and \((5,9]\). Suppose an analyst instead creates \((0,6]\) and \((5,9]\), attaching the week-5 value to the second interval.
The interval lengths are \(6-0=6\) weeks and \(9-5=4\) weeks, but their overlap is \((5,6]\), one week. The total of the two lengths is \(6+4=10\) weeks, while the observed follow-up spans only \(9-0=9\) weeks. That excess of \(10-9=1\) week identifies the overlap. The intervals should be \((0,5]\) and \((5,9]\), whose lengths are \(5\) and \(4\) weeks, summing to \(9\) weeks.
The participant still contributes one person, not two independent participants. The no-event status belongs to the person’s endpoint at week 9; it does not indicate that the participant was event-free at every instant beyond week 9. The time-5 measurement belongs to the later interval under the stated convention. This example checks record integrity, not the statistical model used to impute a missing covariate.
Diagnostics, Alternatives, and Limitations
Several checks can reveal that a survival record is not ready for imputation:
- Identifier checks: each person-level table should have one record per participant; interval tables may repeat IDs, but each repeated ID must link to the same person.
- Time checks: follow-up times should be nonnegative and use the documented origin and units. Interval starts and ends should be ordered and should not overlap.
- Status checks: the event indicator should use the documented coding, and event records should have an endpoint interpreted as an event time under that convention.
- Measurement-time checks: a covariate value must not be assigned to a period before its measurement. Check specifically for intervals that cross measurement times when one start-of-interval value is being used.
- Missingness checks: distinguish a missing recorded covariate from a value that is not defined because the participant had not yet entered follow-up or had already left it.
The main representation alternatives are a single row per person and multiple rows per person. The single-row form is compact and makes person-level identity clear, but it cannot faithfully encode a sequence of covariate values without additional columns or a summary that loses timing. The interval form retains timing and allows values to be associated with follow-up spans, but it introduces repeated dependent rows and demands careful boundary rules. A separate measurement table can preserve irregular measurement times directly, but it requires explicit linkage to person and time when preparing variables for imputation.
A record layout alone does not solve every survival-data problem. It cannot make censoring independent of event time, recover unmeasured covariates, or determine whether a particular imputation model is compatible with the substantive analysis. Those questions require additional modeling choices. The earlier material on longitudinal imputation is useful for recognizing repeated measurements, but survival records add an endpoint and event-status structure that must remain distinct from the covariate measurement schedule.
Common Mistake
A common but incorrect argument is: “The value measured at week 6 is the most recent value in the interval \((4,7]\), so it can label the whole interval.” That reasoning uses information measured at week 6 to represent weeks 4 through 6, when the value was not yet available. If the representation requires a value available at the interval start and constant within each interval, split \((4,7]\) at week 6: use the earlier value for \((4,6]\), then the week-6 value for \((6,7]\). If the data format instead permits a value to change within a row, it must explicitly record that change time; it should not label the later value as available at the earlier interval start.
Check Your Understanding
Answer each question using the record definitions above. Solutions are included after the questions.
- A participant’s event indicator is 0 and recorded follow-up is 12 weeks. What does the endpoint represent, and what does it not establish?
- A covariate is measured at weeks 0 and 3, and follow-up ends at week 8. Under the start-of-interval, constant-value convention, what intervals and values should be used?
- Why should a participant ID not be treated as a quantitative imputation predictor simply because it is stored as a number?
- An interval table has rows \((0,4]\) and \((3,7]\) for the same person. Identify the structural problem and calculate the overlap.
- When is one row per person insufficient for representing a covariate in a survival record?
Solutions. (1) The endpoint is the time observation ended without an observed event; it does not establish that the event occurred at week 12 or what happened after follow-up. (2) Use \((0,3]\) with the week-0 value, then \((3,8]\) with the value measured at week 3. The intervals split at the second measurement so that no value is assigned before it is available. (3) The ID identifies a person; numerical differences between ID codes have no measurement meaning, so modeling them numerically can create arbitrary associations. (4) The intervals overlap on \((3,4]\), of length \(4-3=1\) week. A nonoverlapping partition would need a boundary rule and corrected intervals. (5) One row per person is insufficient when the analysis needs to retain distinct covariate values or measurement times during follow-up; a time-indexed or interval representation is then needed.