Deriving Conditional Survival from a Landmark Cox Model

From a Landmark Cox Fit to Conditional Survival

A landmark prediction asks about future event-free survival among people who have reached a specified time without the event. A Cox model supplies a relative hazard through its covariates, but a survival probability also requires the baseline hazard over the prediction interval. The key calculation combines the covariate risk multiplier with the baseline survival accumulated after the landmark.

Let \(s\) denote the landmark time, \(u>0\) the prediction interval after that landmark, \(T\) the event time, and \(Z(s)\) the covariate vector available at the landmark. The event-free cohort at \(s\) consists of people with \(T>s\). We seek the probability of remaining event-free through \(s+u\), conditional on being event-free at \(s\) and on the landmark covariates.

The conditioning matters: this is not the probability of surviving from the study origin to \(s+u\). It starts with the selected group still event-free at \(s\). Earlier material on landmark Cox models establishes the residual-time setup and the eligible risk set; here the focus is how to obtain a survival probability from the fitted Cox model.

The Cox Model and the Survival Formula

Suppose a Cox proportional hazards model is fitted using absolute event time \(t\), with a fixed covariate vector \(Z(s)\) for prediction after the landmark. Its hazard is

where \(\lambda_0(t)\) is the baseline hazard, \(\beta\) is the coefficient vector, and \(\exp\{\beta^\top Z(s)\}\) is the covariate risk multiplier. The proportional hazards assumption means this multiplier does not vary with \(t\) over the modeled interval. The covariates used in this prediction are held at their landmark values.

Write \(\Lambda_0(t)=\int_0^t\lambda_0(v)\,dv\) for the cumulative baseline hazard and \(S_0(t)=\exp\{-\Lambda_0(t)\}\) for the baseline survival. Conditional on a fixed covariate vector, the model implies \(S(t\mid Z(s))=\exp\{-\Lambda_0(t)\exp(\beta^\top Z(s))\}\). To condition on being event-free at \(s\), take the survival probability through \(s+u\) divided by the survival probability through \(s\). This yields the landmark formula.

This is a ratio of baseline survival values over the post-landmark interval, raised to the covariate risk multiplier. Equivalently, the baseline cumulative hazard increment \(\Lambda_0(s+u)-\Lambda_0(s)\) is multiplied by \(\exp\{\beta^\top Z(s)\}\) before converting it to survival. The multiplier does not multiply the baseline survival probability itself.

Some landmark Cox models are fit directly on residual time \(v=T-s\), with baseline hazard \(\lambda_{0,s}(v)\). For that representation, define \(S_{0,s}(u)=\exp\{-\int_0^u\lambda_{0,s}(v)\,dv\}\). The corresponding prediction is

This is the same structure: \(S_{0,s}(u)\) is the baseline survival over the post-landmark interval, and can also be written as \(S_{0,s}(u)/S_{0,s}(0)\), since \(S_{0,s}(0)=1\). Do not substitute an absolute-time baseline survival into a residual-time formula, or vice versa, without matching the time scale and risk set used to fit the model.

Worked Predictions

Why the Formula Holds—and What Must Hold

The derivation relies on the fitted Cox model describing hazards appropriately for the people in the landmark risk set and across the prediction interval. Under a Cox model, conditional survival at a time is the exponential of negative cumulative hazard. Dividing survival at \(s+u\) by survival at \(s\) removes the hazard accumulated before the landmark:

Since \(S_0(t)=\exp\{-\Lambda_0(t)\}\), the ratio of baseline survivals is \(\exp\{-[\Lambda_0(s+u)-\Lambda_0(s)]\}\). Raising that ratio to \(\exp(\eta)\) gives the conditional survival expression. This derivation also shows why conditioning cancels pre-landmark cumulative hazard, while retaining the post-landmark increment.

The formula is meaningful when the following modeling conditions are reasonably supported:

• Proportional hazards: the covariate hazard ratio is constant over the prediction interval under the fitted model.
• Appropriate landmark cohort and covariates: the model is applied to people eligible at \(s\), and the prediction uses covariate information defined at \(s\).
• Non-informative censoring conditional on modeled information: censoring does not convey additional information about future event time once the modeled covariates and risk-set structure are accounted for.
• Estimable baseline survival: there is enough follow-up and risk-set support to estimate the baseline cumulative hazard through \(s+u\).

Censoring before the horizon is not itself an event. Cox estimation accommodates right-censored observations under its censoring assumptions; it does not turn a censored follow-up into evidence that the person survived event-free through the full horizon. Likewise, a fitted model cannot reliably provide a horizon-specific prediction beyond the time range with adequate follow-up support.

Diagnostics, Alternatives, and Limitations

Check that the time origin used in the baseline estimate matches the model fit. For an absolute-time fit, use \(S_0(s+u)/S_0(s)\); for a residual-time fit, use \(S_{0,s}(u)\). Confirm that \(S_0(s)>0\), that the ratio lies between zero and one, and that the resulting conditional survival is also between zero and one. A ratio greater than one often signals a reversed time order, inconsistent baseline estimates, or a mismatch in time scale.

Assess proportional hazards with suitable residual-based or time-interaction diagnostics, and examine whether covariate profiles at prediction time are represented in the landmark data. Also inspect the number at risk and censoring near the horizon. Sparse risk sets can make estimated baseline survival unstable even when the formula is algebraically correct. If the covariate effect changes over time, a single constant risk multiplier may produce misleading horizon predictions.

A fixed-horizon landmark logistic model directly models event status by a chosen horizon and can be convenient when that horizon is the only target, but it does not provide the same full residual survival curve. A landmark Cox model estimates survival across horizons subject to its hazard assumptions. Parametric survival models are another option when a defensible distributional form is available; they can extrapolate more readily, but their predictions depend on that additional distributional assumption. These approaches answer related but not identical modeling questions.

The displayed formula treats the fitted coefficients and baseline survival estimates as given. In practice, they are estimated from data, so prediction uncertainty includes uncertainty in both. The calculation does not by itself establish that predictions are calibrated or transportable to another population. Nor does it remove the selection inherent in conditioning on having survived event-free to the landmark. These are important qualifications when interpreting predictions, not reasons to alter the algebra.

Common Pitfall