Choosing the Time Scale for Survival MSMs

Why the Clock Matters

A survival model needs a clock: the variable that indexes when risk is assessed. For a survival marginal structural model (MSM), the clock is not a cosmetic choice. It determines which people are compared in a risk set and what a hazard means. A hazard indexed by time since treatment assignment describes risk as follow-up proceeds; one indexed by attained age describes risk among people who have reached the same age; one indexed by calendar time describes risk during the same historical period.

These clocks can be translated for an individual, but they generally produce different risk sets across individuals. Consequently, simply changing the name of the model’s time variable does not preserve the model’s interpretation. The choice should follow the scientific question and the way follow-up begins.

A weighted survival MSM uses weights to address treatment assignment and, when appropriate, censoring under its stated identification assumptions. The time scale does not replace those assumptions. In particular, consistency, treatment exchangeability and positivity, and appropriate handling of censoring remain necessary for a causal regime interpretation. Weights also do not make different clock-specific risk sets identical.

How the Three Clocks Define Risk

For follow-up time \(t\), a participant enters at baseline, at \(t=0\), and contributes to the risk set at \(t\) if the participant is still event-free and uncensored just before \(t\). In continuous-time notation, this is represented by \(T_i\geq t\) and \(C_i\geq t\), where \(C_i\) is the censoring time. In discrete-time models, the corresponding risk set at the start of an interval contains participants who have not yet had the event or been censored by that interval’s start.

On the age clock, participant \(i\) enters at age \(A_{0i}\), not at a common age. At age \(a\), the person can contribute only if \(A_{0i}\leq a\), and must still be event-free and uncensored. On the calendar clock, entry occurs at calendar time \(C_{0i}\); a person cannot contribute to a calendar-time risk set before enrollment. Both clocks therefore involve delayed entry whenever baseline ages or enrollment dates differ.

Proof. Since \(Q^g=a_0+T^g\) when \(A_0=a_0\), the events \(Q^g>a\) and \(T^g>a-a_0\) are identical. Taking their conditional probabilities gives the first equality. Since \(R^g=c_0+T^g\) when \(C_0=c_0\), the events \(R^g>c\) and \(T^g>c-c_0\) are identical, giving the second equality. These are strict-survival identities, so they remain valid even when event times can have atoms. \(\square\)

The result is conditional on baseline age or enrollment time. It does not say that a marginal age-clock survival curve is the follow-up survival curve with its horizontal axis relabelled: people have different baseline ages, and the composition of the age risk set changes as people enter and experience events. The same qualification applies to calendar time.

Worked Risk-Set Comparisons

Interpretation, Assumptions, and Diagnostics

With time since baseline, the hazard at \(t\) is the instantaneous event rate among those who have remained event-free through \(t\) years since baseline, under the regime represented by the MSM. This is often natural when baseline is treatment assignment or a well-defined eligibility event and the scientific question concerns risk after that point. It requires a meaningful and consistently defined baseline.

With attained age, the hazard at age \(a\) concerns people who have entered the study by age \(a\) and remain at risk at that age. This clock is useful when risk is strongly related to age and cohorts have different baseline ages. But it is not automatically a population hazard for everyone of age \(a\): the observed risk set may reflect who enrolled, and delayed-entry assumptions matter. If entry depends on unobserved future risk, the risk set can be selected.

With calendar time, the hazard at date \(c\) concerns participants observed and at risk during that period. It can be important when background risk, diagnosis, or care changes over calendar time. It is less directly a measure of risk after treatment initiation because participants at the same calendar date may have very different follow-up durations. A calendar-time comparison also does not by itself remove confounding by historical changes in who receives which treatment.

A useful derivation clarifies why the three hazards need not agree. Conditional on a fixed baseline age \(a_0\), the age clock is a shift \(a=a_0+t\). A time shift has derivative one, so an infinitesimal interval of age \(da\) corresponds to the same duration \(dt=da\). Thus, conditional on that baseline age and regime, the local hazard at age \(a\) corresponds to the follow-up hazard at \(t=a-a_0\), when those hazards are defined on the same conditional population. But across a cohort with varying \(A_0\), an age risk set pools people at different \(t=a-A_0\), and the survivors’ distribution of baseline ages can change with \(a\). The pooled marginal hazard therefore need not be a simple relabelling of the follow-up marginal hazard.

The choice also interacts with model specification. A baseline hazard indexed by one clock encodes a different pattern of risk from a baseline hazard indexed by another. Do not interpret a treatment-regime coefficient or a fitted curve without stating its clock and its risk set. In weighted analyses, inspect the number of participants and total weight at risk over the clock, entry and exit times, and the distribution of follow-up time or baseline age among those contributing. Warning signs include impossible contributions before enrollment, abrupt risk-set changes unexplained by events or censoring, sparse weighted support in some periods, and treatment groups represented by very different calendar eras.

The principal alternatives are complementary rather than universally ranked. Follow-up time directly answers post-baseline questions; attained age aligns risk by age; calendar time aligns risk by historical period. A single analysis may need to account for more than one source of time variation, but choosing a primary clock does not guarantee that other time-related variation is adequately modeled. The selected clock should be justified by the target interpretation, entry mechanism, and observed support, not by whichever fit is easiest to compute.

Common Mistakes and Limits

• Using inclusive survival events by accident. If survival is defined as \(P(T>t)\), the event is strict. For translated clocks, use \(Q^g>a\) if and only if \(T^g>a-a_0\). Replacing both inequalities with inclusive ones can change probabilities when event times have atoms.
• Putting everyone in an age or calendar risk set from its beginning. A person who has not yet reached the age or calendar date of entry cannot contribute before then. Respect delayed entry.
• Assuming a translated individual clock implies equal marginal hazards. Individual times translate exactly, but pooled risk sets can contain different mixes of baseline ages, enrollment dates, and surviving participants.
• Treating clock choice as confounding control. Choosing attained age or calendar time does not by itself establish exchangeability or remove treatment confounding; those are separate identification and analysis questions.

No clock is universally best. Even with correct time alignment, inference is limited by the study’s entry process, positivity, measurement of event and censoring times, and the adequacy of the weighted model. A well-chosen time scale clarifies the target hazard; it does not make that hazard causal without the assumptions supporting the MSM.