Cox Models for Intermediate-to-Final Prediction

Why Use a Landmark Cox Model?

At an interim selection time, a trial may have intermediate outcomes available for some participants while the final time-to-event endpoint remains immature. The intermediate outcome can help predict which participants are likely to experience the final event later. A landmark Cox model formalizes this prediction: it relates information available at a chosen time to the hazard of an event occurring after that time.

The landmark matters. The analysis concerns participants who have reached the landmark without the event and whose relevant intermediate information is available there. The model predicts subsequent risk for this defined group; it does not, by itself, establish that changing the intermediate outcome would change final-event risk, or that an arm with better predictions has a causal treatment benefit.

Model, Interpretation, and Assumptions

A model with a common baseline hazard and a treatment-by-intermediate-outcome interaction is

where \(z\) is a specified coding of \(Z\), often centered or scaled, and \(\lambda_0(u)\) is the baseline hazard. For example, if \(Z\) is a score, one may define \(z=(Z-50)/10\), so that a one-unit increase represents a 10-point increase from 50. In the two-arm case, \(A=0\) is the reference arm and \(A=1\) the other arm. Additional treatment indicators are needed for more than two arms.

The hazard ratio for a one-unit increase in \(z\), comparing otherwise identical participants in arm \(A\), is \(\exp(\beta_Z+\beta_{AZ}A)\). The treatment-arm hazard ratio at a given \(z\) is \(\exp(\beta_A+\beta_{AZ}z)\). These are conditional hazard comparisons, not ratios of event probabilities. A positive coefficient indicates a higher hazard, conditional on the model covariates; it is not automatically evidence of a causal mechanism.

The model is useful only if its assumptions are credible for the prediction task. The proportional-hazards assumption says that each included covariate has a constant multiplicative effect on the hazard over time since the landmark. The functional form for continuous predictors must also be adequate; a linear term in \(z\) assumes linearity on the log-hazard scale. The censoring mechanism must be independent of future event time conditional on the modeled predictors, or otherwise handled appropriately. Participants must enter the analysis under a consistent landmark eligibility rule, and predictors must be measured by \(L\), not after it.

The common-baseline model assumes a shared baseline hazard across treatment arms after adjustment for the specified treatment terms. If that assumption is unsuitable, a stratified Cox model can allow a separate baseline hazard by arm. In that case, predictions use the estimated baseline for the participant's arm; treatment contrasts are not represented by a single common-baseline treatment coefficient. The model choice should reflect the intended prediction and the available event data.

From Cox Fit to Absolute Risk

Cox regression estimates regression coefficients using comparisons within risk sets at observed event times. For an absolute-risk prediction, the fitted linear predictor is not enough: an estimate of the baseline cumulative hazard is also needed. With the Breslow estimator, the baseline cumulative hazard at time \(u\) is the sum, over event times \(u_j\leq u\), of the number of events \(d_j\) divided by the risk-set sum of fitted relative hazards.

Here \(R(u_j)\) is the set still at risk immediately before event time \(u_j\), and \(\widehat\eta\) is the fitted linear predictor for the target profile. The predicted probability of an event by \(u\), conditional on being event-free at the landmark, is \(1-\widehat S(u\mid A,Z,W)\). It is a prediction under the fitted model for that profile, not a guarantee for an individual participant.

Treatment-Specific Prediction for Selection

For selection, predictions should be made at clearly defined predictor profiles or for a clearly defined target population. A useful comparison holds the intermediate outcome and other predictors fixed, then calculates each arm's predicted risk using its treatment-specific linear predictor. If the goal is to compare arms for a population rather than a single profile, predictions can be averaged over a specified distribution of covariates. The target population and averaging method should be fixed before interpreting the ranking.

The following example uses a common baseline hazard and a treatment term. It illustrates a descriptive model-based comparison, not a rule for retaining or dropping an arm.

Checks, Failure Modes, and Alternatives

A landmark analysis avoids using future information to predict from the landmark, but it does not automatically resolve every design issue. Check that eligibility is defined consistently, intermediate values are available at the landmark, and event and censoring times are measured from that point. Excluding participants who experienced the event before the landmark changes the target population: predictions apply to those event-free and eligible at \(L\), not to all randomized participants.

Useful diagnostics include examining scaled Schoenfeld residuals or other time-varying-effect checks for proportional hazards, checking nonlinear functional forms for continuous intermediate outcomes, and assessing calibration at clinically relevant prediction times. Compare predicted and observed event frequencies in suitable groups or use an appropriate validation procedure. Inspect whether the target profiles have covariate support in each arm; extreme extrapolation can produce precise-looking but unreliable predictions. The estimates can also be unstable when event counts are small relative to model complexity, particularly with interactions and many candidate predictors.

Censoring that depends on unobserved future risk can bias both fitted associations and absolute-risk predictions. A model that omits an important predictor may be poorly calibrated even if its treatment and intermediate-outcome coefficients look plausible. Missing intermediate outcomes can also alter the analyzed group; the analysis should state how missingness is handled rather than silently treating complete cases as representative.

A Cox model is not the only possible prediction approach. A parametric survival model specifies a distribution for event times and can yield smooth baseline-risk estimates, but depends on that distributional assumption. A flexible survival model may relax some functional-form restrictions but still requires validation and adequate data. A landmark Kaplan–Meier estimate can describe risk within groups, but does not adjust continuously for multiple predictors in the same way. The Cox model's attraction is its flexible baseline hazard and interpretable relative-hazard structure; its limitations include proportional-hazards assumptions and the need to estimate and validate baseline risk for absolute predictions.

Common Mistake

A common error is to read a lower predicted risk in one arm as proof that treatment causes the final endpoint to improve, or that the intermediate outcome is a valid surrogate. The correction is to distinguish three claims: the model describes a conditional association, it predicts risk under its fitted assumptions, and a causal treatment or surrogacy claim requires additional justification. Likewise, a hazard ratio is not an absolute risk difference; compute each arm's predicted survival or event risk at a stated time before comparing absolute risks.