Time-to-Event Estimates After Adaptive Stopping

Why stopping changes survival inference

In a time-to-event trial, interim monitoring is often based on the accumulating number and timing of events. The event history determines both the evidence about treatment and the information available at each analysis. If the trial stops when interim evidence crosses a boundary, the final data are not simply a fixed-size sample: the decision to stop is linked to the observed evidence. A conventional estimate can still summarize the observed treatment contrast, but its ordinary confidence interval need not have its advertised coverage under that stopping rule.

This tutorial uses a two-arm randomized trial and a Cox proportional-hazards model. The treatment indicator is \(x_i=1\) for a participant assigned to treatment and \(x_i=0\) for control. The parameter \(\theta\) is the log hazard ratio, so the hazard ratio is \(\exp(\theta)\); negative values favor treatment when the event is harmful. The focus is how to use the observed event and censoring structure, then how to account for the rule that ended observation.

Score and information from event risk sets

Assume a proportional-hazards model in which participant \(i\)'s hazard at time \(t\), conditional on treatment, is \(h_0(t)\exp(\theta x_i)\). Here \(h_0(t)\) is an unspecified baseline hazard. The standard partial-likelihood analysis assumes independent participants, correct treatment and covariate recording, proportional hazards over the analyzed period, and non-informative censoring conditional on the modeled covariates and treatment. In a randomized trial, treatment assignment is independent by design, but that alone does not guarantee proportional hazards or non-informative censoring.

For a single event at \(t_j\), the partial-likelihood contribution for the event participant \(i_j\) is their relative hazard among the people at risk. The score contribution and observed information contribution are:

where \(\bar{x}_j(\theta)\) and \(\overline{x^2}_j(\theta)\) are the risk-set averages of \(x\) and \(x^2\), weighted proportionally to \(\exp(\theta x)\). For a binary treatment indicator, \(x^2=x\), so if \(p_j(\theta)\) is the weighted treatment fraction in the risk set, then \(U_j=x_{i_j}-p_j(\theta)\) and \(I_j=p_j(\theta)\{1-p_j(\theta)\}\). Contributions are summed over event times. Tied events require a specified standard method, such as the Breslow or Efron approximation; the simple single-event expressions below do not resolve ties.

From the observed score to an estimate

The Cox partial-likelihood estimate \(\hat{\theta}\) is ordinarily obtained by solving \(U(\theta)=0\). A useful approximation, especially for explaining an interim estimate, is one Newton step from a specified starting value \(\theta_0\):

The second expression is the usual large-sample model-based standard error, using observed information evaluated at the fitted estimate. A one-step value is not automatically the exact partial-likelihood maximum. If the score changes substantially between \(\theta_0\) and its root, additional Newton iterations or direct maximization are needed. The estimate is reported on the log-hazard-ratio scale; exponentiating it gives the estimated hazard ratio.

Conventional intervals and the stopping rule

A conventional large-sample Wald interval uses the approximate normal distribution of the estimator under a fixed analysis plan. On the log scale it is \(\hat{\theta}\pm z_{1-\alpha/2}SE(\hat{\theta})\); exponentiating both endpoints gives an interval for the hazard ratio. This calculation does not automatically account for the fact that the analysis was performed because an interim statistic triggered stopping.

For the hypothetical estimate above, a nominal 95% Wald interval is:

Under the usual fixed-sample approximation, the interval is for the log hazard ratio, and its exponentiated form is for the hazard ratio. It should not be interpreted as a 95% probability that the fixed parameter lies in this realized interval. After adaptive stopping, its nominal coverage is not guaranteed merely because the usual standard error was computed correctly.

The reason is selection on interim evidence. Under a fixed analysis time, a score is assessed without conditioning on having crossed a stopping boundary. Under an efficacy stopping rule, the observed score is known to have passed that rule. The distribution relevant to inference can therefore differ from the fixed-look reference distribution. Risk-set construction still uses the observed event and censoring history, but the uncertainty calculation must also reflect the rule governing when that history was examined.

Constructing an adaptation-aware interval

A general approach is to invert tests whose reference distributions include the stopping mechanism. For each candidate log hazard ratio \(\theta_0\), specify the null model, compute the score process from the event risk sets at each planned look, and evaluate the probability of the observed result under the actual stopping rule. Depending on the inferential target, this can be conditional on the stopping event or unconditional over all possible stopping paths. The accepted candidate values form a confidence set; exponentiating its endpoints gives the corresponding hazard-ratio set.

The event and censoring records matter at every step. Each event changes the score and information according to its risk set; censoring alters later risk sets. The number of events alone is generally insufficient to reconstruct the Cox score, because the treatment composition of each risk set and the assignment of events to treatment groups also matter. A survival-adaptive analysis must therefore retain the analysis-time data needed to reproduce event ordering, risk-set membership, tie handling, and the interim decision.

In practice, the design's boundary and its information schedule should be used as specified. A conditional interval and an unconditional interval answer different coverage questions, as emphasized in earlier material on confidence-interval coverage and conditioning. Neither is universally preferable: the target depends on whether coverage is required given the observed stopping path or averaged over all paths generated by the design. Test inversion gives a principled construction, but the exact calculation can require numerical evaluation of the joint distribution of sequential statistics.

Diagnostics, alternatives, and limitations

Before interpreting an adaptive survival estimate, check whether the model and data support the calculation. Important diagnostics include:

• Risk-set reconstruction: verify entry, event, and censoring times, treatment assignments, and the exact risk-set convention at each event. Check that censoring before an event removes a participant from subsequent risk sets.
• Proportional hazards: assess whether a constant log hazard ratio is plausible over time. Strong time variation can make a single \(\theta\) misleading even if the numerical fit converges.
• Censoring: examine whether loss to follow-up could depend on prognosis after accounting for modeled variables. Informative censoring can bias the score-based estimate and invalidate its usual interpretation.
• Information and events: sparse events, near-separation, or risk sets containing only one treatment group can produce unstable estimates or little information. A large enrollment count does not compensate automatically for few informative events.
• Stopping documentation: confirm that the recorded rule, boundary, analysis times, and any changes to the plan match the rule used for the adaptation-aware calculation.

The main alternatives are not interchangeable. The ordinary Cox Wald interval is simple and uses observed information, but ignores the stopping rule unless its reference distribution has been adjusted. A design-based, stagewise, or conditional test-inversion interval incorporates the stopping mechanism, but relies on a correctly specified sequential model and can be computationally demanding. Reporting both an unadjusted descriptive estimate and an adaptation-aware interval can be useful, provided their distinct inferential roles are explicit. These procedures do not repair a misspecified proportional-hazards model, informative censoring, or an inaccurately recorded adaptation.