From Longitudinal Records to Survival Person-Periods

Why Reshape Longitudinal Records?

A survival record often begins with one row per person containing an event or censoring time, event status, and a sequence of treatment and covariate measurements. To represent changing histories over follow-up, the record can be expanded into intervals: each row describes one person over a span of time, with the treatment and other information applicable to that span. The challenge is to expand the record without moving an event, duplicating time at a boundary, or assigning a treatment measured after an event to the event interval.

The resulting data are often called start-stop or person-period data. Here, “period” need not mean a fixed-width calendar period: intervals can have different lengths, with boundaries chosen to reflect treatment updates, measurements, or the end of follow-up. The construction is a data representation, not a model-fitting step.

The event-at-censoring-time convention is appropriate only when the recorded data establish that the event was observed at that time. If a data system cannot distinguish an event from censoring at a tied timestamp, the tie must be resolved using the study’s ascertainment rules rather than silently assigned. Likewise, treatment and covariate values need a timing convention: a value measured at a boundary should not be treated as available before it was recorded.

Constructing the Intervals

For a person with \(U_i>0\), make a strictly increasing sequence of cut points \(0=b_{i0}<b_{i1}<\cdots<b_{im_i}=U_i\). Include every time at which the history relevant to the rows changes during follow-up, and include the endpoint \(U_i\). The person then contributes \(m_i\) rows, with \(s_{ij}=b_{i,j-1}\) and \(e_{ij}=b_{ij}\). Do not include cut points after \(U_i\): follow-up ends at the event or censoring time.

A practical timing convention is that a treatment change recorded at \(b_{ij}\) becomes effective immediately after \(b_{ij}\). The row ending at \(b_{ij}\) therefore reflects the treatment in force before the change; the next row, if there is one, reflects the updated treatment. An event at \(b_{ij}\) is assigned to the row ending there, not to a row beginning there. This convention matches the \((s,e]\) interval definition. If a study instead defines a treatment change as effective before events at the same timestamp, the data need a finer ordering or a different explicit convention.

What the Representation Preserves

The key properties can be checked directly from the construction. They depend on positive-length follow-up: if \(U_i>0\), the strict sequence of cut points makes every row have positive length. When \(\Delta_i=1\), the observed event must also occur at a positive time under this construction.

Proof. The cut points are strictly increasing, so \(b_{ij}-b_{i,j-1}>0\) for every row. The adjacent intervals meet at their common boundary, with that boundary included in the interval ending there and excluded from the one starting there. Thus no boundary is counted twice, and their union is \((0,U_i]\). Their lengths telescope:

If \(\Delta_i=1\), the construction assigns \(Y_{ij}=1\) only to the final row, and \(b_{im_i}=U_i=T_i\); hence there is exactly one event row and its stop is the event time. If \(\Delta_i=0\), the rule assigns zero to every row. \(\square\)

This result also gives a simple reconstruction check. For an event record, the stop of the unique row with \(Y_{ij}=1\) must equal the original event time. For a censored record, there must be no event row, and the final stop must equal the original censoring time.

Boundary Cases and Data Checks

An event at a treatment-update time requires careful use of the endpoint rule. Suppose an update is recorded at time 4 and the event is also recorded at time 4. The event belongs to the interval ending at 4. Under the convention that the update takes effect just after time 4, the row ending at 4 contains the pre-update treatment. There is no post-update interval for this person because follow-up ends at the event. If the data indicate that treatment changed before the event at that timestamp, the timestamps alone do not encode that ordering; use recorded sequence information or state a scientifically justified tie rule.

An event at time zero is a genuine boundary exception, not an ordinary positive-length interval. If \(T_i=0<C_i\), then \(U_i=0\) and \(\Delta_i=1\). No interval can satisfy both \(s_{ij}<e_{ij}\) and \(e_{ij}\leq U_i=0\). Therefore the positive-length construction and the result above require \(U_i>0\). Keep a time-zero event as a separate zero-time record or represent it with a study-specific baseline-event convention; do not invent a positive-length interval or claim that it has a usual event-indicator row. A censored record with \(U_i=0\) likewise contributes no positive-duration interval and should be retained separately if it is meaningful for the analysis.

After reshaping, verify these properties person by person:

• Every interval has \(s_{ij}<e_{ij}\), and successive rows meet at their shared boundary.
• The first start is 0, the final stop is \(U_i\), and no row extends beyond \(U_i\).
• The row lengths sum to \(U_i\).
• An observed event with \(U_i>0\) has exactly one event row, at the original event time; a censored record has none.
• Treatment and covariate values agree with the measurement-time and boundary conventions, including at tied timestamps.

Gaps indicate omitted follow-up; overlaps can double-count time. Multiple event rows usually indicate that the event indicator was copied to every row rather than attached only to the terminal event interval. A row after an event or censoring time changes the observed follow-up record and must be corrected.

Choosing a Data Representation

One row per person is compact and can preserve the original event time and status, but it cannot directly display a changing treatment history as interval-specific values. Start-stop records make those changes explicit and preserve the exact endpoint when cut points are correctly chosen. Fixed-width bins are another possibility, but can shift event times to bin boundaries or obscure when treatment changed unless the binning rule is carefully specified. Exact start-stop intervals are preferable when the analysis needs the observed ordering and timing of history changes. They do not, by themselves, solve causal identification, determine appropriate weights, or choose a survival model.

The format is only as reliable as the source data and conventions. Coarsely recorded timestamps may not establish whether a treatment update preceded an event. Measurement error, missing history, and ambiguity about treatment effectiveness cannot be repaired by reshaping. Moreover, an interval row is not a new independent person: several rows from one person remain linked observations. The purpose here is faithful representation, not to justify treating rows as independent.