Why Covariate Values Need a Rule
A subject may contribute risk time to more than one transition, and the information available about that subject can change between transitions. A baseline measurement is available throughout follow-up, while a new measurement may become available during follow-up or when the subject enters a new state. The analyst must specify which value represents the covariate for each transition risk interval. That choice determines the regression records and can change the meaning of the fitted coefficient.
Consider a simple illness-death model: subjects begin in state 0 (well), may move to state 1 (ill), and may then move to state 2 (dead). Death from state 0 is also possible, and state 2 is absorbing. For a transition from state \(r\) to state \(s\), a transition-specific proportional-hazards model can be written as follows.
Here \(i\) indexes subjects, \(Y_{ir}(t)\) indicates whether subject \(i\) is in state \(r\) and at risk at time \(t\), and \(\lambda_{0,rs}(t)\) is the baseline hazard for transition \(r\to s\). The history \(\mathcal{H}_i(t-)\) contains information available immediately before \(t\); \(Z_{i,rs}(t)\) is the covariate vector used for that transition at that time. The transition-specific coefficient vector is \(\beta_{rs}\). A subject contributes records for a transition only while at risk for it.
A rule must be fixed in substantive terms before records are constructed. Values must be available by the time they are used: a value measured after a transition cannot be assigned to the risk time before that transition. If measurement and transition occur at the same recorded time, the data protocol must specify whether the measurement was available before the transition; absent such evidence, do not treat it as pre-transition information.
Three Ways to Assign Covariate Values
Carry forward a baseline value. The value measured at baseline is used throughout follow-up, including after a state change. This is appropriate when the intended predictor is baseline status, or when later measurements are unavailable or deliberately outside the scientific question. It does not assert that the underlying characteristic stayed constant; it defines the model in terms of baseline status.
Update over time. Use the most recent measurement that was available before each time. A change in the assigned value creates a new start-stop interval, even if the subject remains in the same state. This rule can represent changing exposure or disease severity, but it requires measurement times and a clear convention for which measurement applies between visits.
Reset at state entry. Use the value measured at entry to the current state for risk time in that state. “Reset” means replace the previously assigned value with the state-entry value; it does not mean automatically set the covariate to zero. The entry value can then be held fixed until another prespecified update or the subject leaves that state. This rule is useful when the relevant predictor is status at the start of a state-specific risk period.
These are choices about the predictor, not merely alternative data layouts. The same subject may have a different value in a transition record under each rule. Coefficients therefore answer different conditional association questions, even if the transition outcome and sample are unchanged.
Worked Example: Carrying a Baseline Value Across Transitions
Suppose a hypothetical subject has a baseline biomarker value of 6.2 units. The value is centered at 5 units, so the baseline covariate is \(Z=6.2-5=1.2\). The subject is in state 0 from time 0 to time 4, enters state 1 at time 4, and experiences the transition \(1\to2\) at time 9. For illustration, suppose the fitted transition-specific coefficients are \(\beta_{01}=0.30\) and \(\beta_{12}=0.50\) per one-unit increase in this centered biomarker.
Under a baseline carry-forward rule, the subject’s value is 1.2 on both risk intervals. The transition records are: state 0, time 0–4, at risk for transitions out of state 0, \(Z=1.2\); and state 1, time 4–9, at risk for \(1\to2\), \(Z=1.2\). The change of state does not change the assigned baseline value.
For transition \(0\to1\), the covariate contribution to the linear predictor is \(0.30(1.2)=0.36\), giving a relative hazard of \(\exp(0.36)=1.433\) compared with a subject at the centered reference value \(Z=0\), conditional on the model. For transition \(1\to2\), it is \(0.50(1.2)=0.60\), giving \(\exp(0.60)=1.822\) relative to that transition’s reference hazard. These are transition-specific comparisons, not overall risks of illness or death.
The record construction is consistent if the baseline value was measured before follow-up and the analysis intentionally targets baseline biomarker status. It would not be justified to describe these records as using the subject’s current biomarker at times 4 or 9 unless that status was actually measured and the rule used those updates.
Updated Measurements and State-Entry Values
For an updated covariate, records are split at each time its assigned value changes. Each interval uses the value known at its start and retains that value until the next update, transition, censoring, or other end of risk time. In the counting-process formulation, the covariate is predictable: its value at time \(t\) is determined from information available just before \(t\). This prevents the event being modeled from supplying its own predictor.
Worked Example: Updating a Measurement During Follow-Up
A hypothetical subject has a biomarker measured as 2 units at baseline, 4 units at time 3, and 3 units at time 6. The subject moves from state 0 to state 1 at time 4 and experiences \(1\to2\) at time 7. Assume each measurement applies from its measurement time until the next measurement. Suppose \(\beta_{01}=0.20\) and \(\beta_{12}=0.40\) per unit.
In state 0, the update at time 3 splits the record into intervals 0–3 with value 2 and 3–4 with value 4. The subject is at risk for outgoing transitions from state 0 on both intervals. In state 1, the interval 4–6 has value 4 and the interval 6–7 has value 3; the subject is at risk for \(1\to2\) on both. The value measured at time 6 does not apply retrospectively to times 4–6.
At the illness transition, the current covariate value is 4, so the covariate contribution for \(0\to1\) is \(0.20(4)=0.80\), with relative hazard \(\exp(0.80)=2.226\) compared with value 0. A one-unit increase corresponds to a transition hazard ratio of \(\exp(0.20)=1.221\). At the death transition, the assigned value is 3, giving contribution \(0.40(3)=1.20\) and relative hazard \(\exp(1.20)=3.320\) compared with value 0; a one-unit increase corresponds to \(\exp(0.40)=1.492\).
These calculations use the stated interval convention and hypothetical coefficients. They are coherent only if the measurements are available at the listed times, the value remains applicable between measurements as assumed, and the model uses the same time scale and transition risk sets as the records. If a biomarker is measured only when a subject becomes ill, treating it as continuously observed before illness would require additional justification.
A state-entry rule is different from unrestricted updating: it uses a particular measurement tied to entry and holds it fixed according to the prespecified rule. The following example compares that entry value with the baseline value for the same post-entry interval.
Worked Example: Replacing the Value at State Entry
Suppose a hypothetical subject has a score of 10 at baseline, enters illness state 1 at time 4 with a newly recorded score of 7, and remains at risk for \(1\to2\) until time 8. Let the score be centered at 5, and suppose the fitted \(1\to2\) coefficient is \(\beta_{12}=0.25\) per centered score unit. Assume the measurement at time 4 was available at state entry and the state-entry rule holds it fixed during this interval.
The state-entry value is \(7-5=2\). Its linear-predictor contribution is \(0.25(2)=0.50\), and its relative hazard compared with a score of 5 at entry is \(\exp(0.50)=1.649\). Under a baseline carry-forward rule instead, the assigned centered value would be \(10-5=5\); the contribution would be \(0.25(5)=1.25\), with relative hazard \(\exp(1.25)=3.490\) against the same reference.
The two calculations differ because they encode different predictors: baseline status versus status at illness entry. The result does not show that the subject’s hazard changed by a causal factor of \(3.490/1.649\). Nor should the score be set to zero at entry merely because the rule is called a reset. Zero is appropriate only if zero is the actual value under the chosen coding, such as a defined reference after centering.
Why the Records Change the Model
For a transition with a piecewise-constant covariate, the covariate enters the hazard throughout each interval. For example, over an interval \((a,b]\) with covariate value \(z\), the event-free contribution to the likelihood contains the integrated hazard \(\int_a^b \lambda_0(u)\exp(\beta z)\,du\). If \(z\) changes within the interval, using a single value for all of \((a,b]\) generally changes this integral and the event contribution. Splitting the interval at an update time allows each part to use its assigned value.
This equality holds when the covariate is \(z_1\) on \((a,c]\) and \(z_2\) on \((c,b]\). If the record instead assigns \(z_1\) across the full interval, it substitutes a different integrated hazard whenever \(z_1\ne z_2\) and the later interval contributes positive baseline hazard. Thus splitting is not just bookkeeping: it preserves the intended covariate history in the likelihood.
For each transition, state whether the predictor is baseline status, the latest available value, or a state-entry value.
List state entry and exit, measurement updates, censoring, and any other changes that end a risk interval.
Split records wherever the state, transition risk, or assigned covariate value changes; attach only values available before the interval’s event time.
Check that each row is in the correct transition risk set, uses the declared coding and time scale, and does not use information from the future.
Diagnostics, Alternatives, and Limitations
Before fitting, inspect a small selection of subjects by hand, including subjects with multiple measurements and state changes. Compare their dated measurement history with the generated start-stop records. After fitting, check that values are plausible in each transition risk set, that no interval extends beyond a transition or censoring time, and that a measurement update creates a split where the chosen rule requires one. Also check for an unexplained concentration of missing values or unusual values at state entry, which can reveal a data-collection or merge error.
The main alternatives differ in the question they encode. A baseline rule is simpler and avoids needing later measurements, but it does not represent changing status. An updated rule represents changes using observed measurements, but results depend on measurement timing and on the assumption that a value applies between visits. A state-entry rule focuses on status at entry to a state, but discards subsequent changes if none are incorporated. None is automatically best; the rule should reflect the scientific target and available data.
| Rule | Value used during risk time | Important check |
|---|---|---|
| Baseline carry-forward | Baseline value, including after a state change | Interpret as baseline status, not current status |
| Time-updated | Most recent value available before the time | Split at updates; do not use future measurements |
| State-entry | Value recorded on entry to the current state | Confirm entry timing and define how long it remains in use |
A common mistake is to call a covariate “time-varying” and then assign the subject’s final observed value to all earlier intervals. That uses future information and can make the modeled transition hazard depend on what happened later. The correction is to use only values available just before each risk time and split records at updates. Another mistake is to reset every covariate to zero on state entry. A state-entry rule replaces the prior value with the measured entry value; it does not impose zero unless zero is the substantive and statistical coding intended.
These rules alone do not resolve informative measurement, confounding, measurement error, or causal interpretation. For example, if clinicians measure a biomarker more often when a patient worsens, the observed update process may be related to prognosis. A transition-regression coefficient remains a conditional association under its model; selecting a covariate rule does not establish a causal effect. The rule also cannot recover values that were never measured without further modeling assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A subject’s baseline covariate is 3, and the subject later enters another state. Under a baseline carry-forward rule, what value is used after entry?
- A measurement changes from 5 to 8 at time 6. The subject remains at risk from time 2 to time 9. What intervals and values should be recorded under an update rule?
- A subject enters a state at time 4 with a newly measured score of 9. What does a state-entry rule mean, and does it require coding the score as zero?
- Why is it invalid to use a value measured at time 7 for an event at time 5 when constructing a time-updated predictor?
Solutions. (1) Use 3, because the rule defines the predictor as baseline status even after state entry. (2) Record time 2–6 with value 5 and time 6–9 with value 8, assuming the measurement at time 6 is available from then onward. (3) Use the measured entry value 9 for the specified post-entry risk time; zeroing is not required. (4) The time-7 measurement was not available before time 5, so its use introduces future information into the predictor and violates the required history convention.