Landmark Cox Models for Residual Survival

Why Model Residual Survival?

At a clinical landmark, the question may concern not only whether an event occurs by one chosen future horizon, but also how the rate of subsequent events varies over time. A Cox model can describe that association without requiring a parametric form for the baseline hazard. The time scale begins at the landmark, so the modeled duration is residual event time: time remaining after the patient has reached the landmark without the event.

The analysis applies to the patients who are eligible at that landmark, not to the original cohort as if everyone had been followed from the same starting point. The earlier material on landmark times and eligible risk sets, conditional risk estimands, and building a landmark dataset establishes how to identify this analysis population and its available records.

Definition: Let \(s\) be a fixed landmark time, \(T_i\) the event time for patient \(i\), and \(C_i\) the censoring time. A patient is eligible if they have not had the event by \(s\) and meet the study's other prespecified eligibility requirements. For each eligible patient, define the observed residual follow-up time \(Y_i=\min(T_i,C_i)-s\) and event indicator \(\Delta_i=I(T_i\leq C_i)\). Thus \(Y_i\) is elapsed time after the landmark, and \(\Delta_i=1\) denotes an observed event during that follow-up.

Only eligible patients enter the landmark dataset. Patients whose event occurred before or at the landmark are not at risk for a subsequent first event in this analysis. A patient censored after the landmark contributes follow-up up to that censoring time, with \(\Delta_i=0\); censoring is not counted as an event. Covariates \(Z_i\) are the values selected for use at the landmark, such as a clinical measurement recorded there. The representation of those covariates is a separate modeling choice.

Model and Interpretation

For eligible patient \(i\), let \(\lambda_i(u\mid Z_i)\) denote the instantaneous event hazard at elapsed time \(u\) after the landmark, conditional on the patient's landmark covariates and continued event-free follow-up to \(u\). A landmark Cox model specifies

\lambda_i(u\mid Z_i)=\lambda_{0,s}(u)\exp(\beta^\top Z_i), \qquad u>0.

Here \(\lambda_{0,s}(u)\) is an unspecified baseline hazard for the population eligible at landmark \(s\), and \(\beta\) is a vector of regression coefficients. The subscript \(s\) emphasizes that the baseline hazard and analysis population pertain to this landmark. The model does not require a particular distribution for residual survival time, but it does assume that the covariate effects multiply the hazard by a constant factor over elapsed time.

Interpretation: Holding the other included covariates fixed, a one-unit increase in covariate \(Z_j\) is associated with a residual event hazard multiplied by \(\exp(\beta_j)\) at any given elapsed time \(u\), among patients eligible at the landmark who remain event-free to \(u\). For a binary covariate, \(\exp(\beta_j)\) compares the hazard in the group coded 1 with that in the group coded 0. This is a conditional association, not by itself a causal effect or an absolute risk difference.

The phrase “at any given elapsed time” is central: the proportional-hazards assumption says that the hazard ratio does not vary with \(u\). The model permits the baseline hazard to change over time in an arbitrary way. Resetting time to zero changes the time origin used for the analysis; it does not assert that patients have identical histories before the landmark or erase the fact that eligibility depends on reaching it.

Assumptions and the Partial Likelihood

The Cox model is appropriate only when its analysis structure and assumptions are defensible. The event definition and landmark must be specified consistently, the covariates must be available as intended at the landmark, and each patient's residual time and event indicator must correctly reflect follow-up after that landmark. Patients are treated as independent observations unless dependence is explicitly handled by the analysis.

Proportional hazards: covariate hazard ratios are constant over elapsed time, conditional on the modeled covariates.
Noninformative censoring, conditional on modeled information: among patients with the same modeled information, censoring should not selectively remove those with systematically different unobserved future event times. Independent administrative censoring is a common supporting scenario.
Suitable covariate and functional-form specification: the included covariates and their modeled effects must adequately represent the intended associations.
Adequate event information: the event count and risk-set composition must support estimation of the coefficients. Sparse events or covariates that nearly separate event groups can produce unstable estimates.

For distinct event times, the Cox partial likelihood follows by considering, at each event time, the relative hazards of the patients then at risk. Let \(R(u)\) be the eligible patients with \(Y_i\geq u\). If patient \(i\) has an event at \(u\), then, conditional on one event occurring in that risk set at that instant, the probability that \(i\) is the one who experiences it is proportional to their hazard:

\frac{\lambda_i(u\mid Z_i)}{\sum_{k\in R(u)}\lambda_k(u\mid Z_k)} = \frac{\lambda_{0,s}(u)\exp(\beta^\top Z_i)} {\sum_{k\in R(u)}\lambda_{0,s}(u)\exp(\beta^\top Z_k)} = \frac{\exp(\beta^\top Z_i)} {\sum_{k\in R(u)}\exp(\beta^\top Z_k)}.

The baseline hazard cancels from this ratio. Multiplying the event contributions gives the partial likelihood, which can be maximized to estimate \(\beta\):

L(\beta)=\prod_{i:\Delta_i=1} \frac{\exp(\beta^\top Z_i)} {\sum_{k\in R(Y_i)}\exp(\beta^\top Z_k)}.

This derivation assumes distinct event times for the displayed expression. When recorded event times are tied, standard tie-handling methods such as the Efron or Breslow method are used; the chosen method should be reported. The partial likelihood estimates covariate associations, but does not directly estimate the baseline hazard through this expression.

Worked Example: Constructing Residual Follow-Up

Suppose the landmark is month 6. The following hypothetical records describe four patients who were event-free and eligible at that time. Times in the original records are measured from study entry.

Patient	Event or censoring time	Status	Residual time \(Y_i\)	\(\Delta_i\)
A	Month 9, event	Event	9 − 6 = 3 months	1
B	Month 11, censoring	Censored	11 − 6 = 5 months	0
C	Month 8, event	Event	8 − 6 = 2 months	1
D	Month 14, censoring	Censored	14 − 6 = 8 months	0

For patient C, the observed event occurs two months after the landmark, so the Cox event time is \(Y_C=2\), not month 8. Patient B contributes five months of event-free follow-up after the landmark and is censored at that point; their record is not an event at month 5. These calculations assume that all four were event-free and eligible at month 6 and that their recorded times and status are correct. A patient who had the event before month 6 would not be added to this residual risk set.

Worked Example: Interpreting a Fitted Hazard Ratio

In a hypothetical landmark analysis, a biomarker is standardized so that one unit represents one standard deviation. Suppose its fitted coefficient is \(\hat\beta=0.550\), with standard error \(SE=0.200\). Assume the proportional-hazards model and its inference are suitable for this example.

The estimated hazard ratio for a one-standard-deviation higher biomarker value, holding the other modeled covariates fixed, is

\widehat{HR}=\exp(\hat\beta)=\exp(0.550)=1.733.

The estimated residual event hazard is therefore 1.73 times as high for a one-standard-deviation increase, at a given elapsed time after the landmark and among eligible patients still event-free then. This is a 73.3% higher estimated hazard, not a claim that the probability of an event is 73.3% higher.

For a Wald 95% confidence interval, first calculate the coefficient-scale interval:

0.550\pm 1.96(0.200)=0.550\pm0.392=(0.158,0.942).

Exponentiating both endpoints gives the hazard-ratio interval:

\big(\exp(0.158),\exp(0.942)\big)=(1.171,2.565).

Under the fitted model and the Wald approximation, this procedure produces intervals that cover the underlying hazard ratio in 95% of repeated samples. The interval is compatible with hazard ratios from 1.17 to 2.57 for each one-standard-deviation increase. It does not give a 95% probability that this particular interval contains the parameter. The Wald statistic is \(z=0.550/0.200=2.75\), with a two-sided \(p\)-value of approximately 0.006. This is evidence against a zero coefficient under the model, not evidence that the association is causal.

Worked Example: A Small Partial-Likelihood Calculation

Consider a hypothetical set of four eligible patients with a binary landmark covariate \(Z\), where 1 means high and 0 means low. Patient A has \(Z=1\) and an event at residual time 1; B has \(Z=0\) and an event at time 4; C has \(Z=1\) and an event at time 2; D has \(Z=0\) and is censored at time 5. Assume independent patients, accurate follow-up, no tied events, and the stated Cox model.

At time 1, all four patients are at risk. At time 2, A has had an event, so B, C, and D remain at risk. At time 4, only B and D remain at risk; D is censored at time 5. Write \(b\) for the binary covariate coefficient. The partial likelihood contributions are

L(b)= \underbrace{\frac{\exp(b)}{2\exp(b)+2}}_{\text{A's event at time 1}} \underbrace{\frac{\exp(b)}{\exp(b)+2}}_{\text{C's event at time 2}} \underbrace{\frac{1}{2}}_{\text{B's event at time 4}}.

For illustration, evaluate the likelihood at \(b=\log(2)\), so that \(\exp(b)=2\):

L(\log 2)=\frac{2}{2(2)+2}\times\frac{2}{2+2}\times\frac{1}{2} =\frac{2}{6}\times\frac{2}{4}\times\frac{1}{2} =\frac{1}{12}\approx0.08333.

This calculation shows how event contributions depend on the composition of the risk set at each event time; censored patients contribute to denominators while they remain at risk, but do not supply event contributions. The value \(1/12\) is the partial likelihood at the specified coefficient, not a \(p\)-value or a probability that the model is true. In an actual analysis, the coefficient is obtained by maximizing the partial likelihood rather than selecting a value from this illustrative calculation.

Diagnostics, Alternatives, and Limits

Diagnostics should focus on the assumptions that give the coefficient its interpretation. Examine proportional-hazards diagnostics, such as scaled Schoenfeld residuals over elapsed time, for evidence that a covariate effect changes with time. Compare observed and fitted patterns and inspect influential event contributions, especially when few events occur. Also examine risk-set sizes and covariate distributions: a coefficient based on very few events in one group can be unstable even when the software returns an estimate.

Check that the time origin, eligibility, event coding, and censoring dates were constructed consistently. A large share of censoring does not automatically invalidate the model, but informative censoring can bias the association. Diagnostics cannot prove that censoring is noninformative, since the relevant future event times after censoring are unobserved. The model's results are also sensitive to the covariate representation and omitted predictors; an adequate proportional-hazards diagnostic does not establish that the model is otherwise correctly specified.

Common pitfall: Resetting time to zero at month 6 does not mean that the analysis includes every original patient at time zero. Only patients who meet the landmark eligibility criteria enter the risk set. Including patients who had the event before month 6, or subtracting month 6 from their event time and treating that as residual follow-up, answers a different and generally invalid question.

A fixed-horizon landmark logistic model, covered in the earlier material on fixed-horizon landmark logistic models, targets event status by a specified future time and does not model when events occur within that period. The landmark Cox model uses event and censoring times after the landmark and imposes proportional hazards, while leaving the baseline hazard unspecified. A Cox analysis with time measured from study entry uses a different time origin and risk-set construction; its coefficients need not represent the same association among landmark-eligible patients. The choice should follow the question and the eligible population, not just which model is familiar.

A landmark Cox model describes residual hazard associations, not by itself a patient's absolute event probability. Its interpretation is conditional on reaching the landmark and remaining event-free until each elapsed time under consideration. Therefore it does not automatically generalize to people who were ineligible at that landmark, and survival selection may make eligible patients differ from the original cohort. It also relies on a proportional effect over residual follow-up; if that assumption fails, a single hazard ratio may conceal meaningful time variation. These limitations matter when reporting estimates or comparing them with results from models built around different populations or time origins.

Check Your Understanding

Answer each question using the landmark population and residual-time scale.

A patient is eligible at month 8 and has an event at month 13. What is the residual event time?
In a fitted model, a binary covariate has coefficient \(\hat\beta=-0.40\). Calculate and interpret its hazard ratio.
At residual time \(u\), which patients belong in the Cox risk set \(R(u)\)?
Does a hazard ratio of 1.5 mean the probability of an event is 50% higher? Explain.

Solutions. (1) \(13-8=5\) months after the landmark. (2) \(\exp(-0.40)=0.6703\); holding other modeled covariates fixed, the group coded 1 has an estimated residual event hazard 0.670 times that of the group coded 0 at a given elapsed time, among eligible patients still event-free then. (3) Eligible patients whose observed residual follow-up time satisfies \(Y_i\geq u\); they have not yet had an observed event or been censored before \(u\). (4) No. It describes a 50% higher hazard at a given time under the proportional-hazards model, not a 50% higher event probability. Absolute probabilities also depend on the baseline hazard and elapsed time.

Landmark Cox Models for Residual Survival

What You'll Learn