Nonproportional Hazards in Seamless Survival Trials

Why Nonproportional Hazards Change the Confirmatory Question

A survival test is not merely a way to obtain a p-value. Its construction determines which differences between treatment and control contribute most to the evidence. This matters when hazards are nonproportional: the treatment effect may change over time, and a single hazard ratio may not summarize that pattern. In a seamless trial, the stagewise tests must also represent a coherent confirmatory question, even when early-stage evidence informed the transition to the confirmatory stage.

For example, a treatment might initially delay benefit because of a biological lag, then reduce the event hazard later. A conventional log-rank test gives events equal weight over time, so early and late differences can offset one another. A weighted log-rank test changes that balance. The weight function must therefore be chosen for a clinical reason and specified before examining the confirmatory-stage outcomes.

Constructing a Stagewise Weighted Survival Test

Let \(T\) denote the treatment group and \(C\) the control group. At event time \(t\), let \(Y_T(t)\) and \(Y_C(t)\) be the numbers at risk immediately before \(t\), and let \(Y(t)=Y_T(t)+Y_C(t)\). Let \(dN_T(t)\) and \(dN_C(t)\) be the observed event counts at \(t\), and \(dN(t)=dN_T(t)+dN_C(t)\). Under equal survival distributions, the expected treatment-group event count at that time is \(Y_T(t)dN(t)/Y(t)\).

A weighted log-rank score subtracts that expected count from the observed count and weights the difference. We use the sign convention that a positive \(Z_w\) favors treatment: the score itself is negated because fewer treatment-group events than expected is favorable.

For untied event times, the usual null variance estimate is

For tied events, the variance calculation should use the appropriate finite-population correction for the tied risk set; the formula above is the familiar large-sample form. Under the null of equal survival distributions, randomization, independent censoring within groups, and adequate event information, \(Z_w\) is approximately standard normal. The weight function should be fixed in advance or be a prespecified predictable function of the pooled history. A common choice is \(w(t)=1\), the ordinary log-rank test. Other weights can emphasize earlier or later events.

Each stage uses its own participants and its own stagewise score and p-value. The weight rule, event-time horizon, censoring conventions, direction of benefit, and hypothesis must be aligned across stages. If an early-stage treatment is selected from several candidates, its p-value must also be valid for that selection procedure; an unadjusted p-value for the winner is not automatically confirmatory.

What the Weighting Means for the Treatment Effect

With \(w(t)=1\), each event contributes according to the risk-set comparison at its time. A weight that is larger later makes late event differences more influential; a weight that is larger earlier does the reverse. This can improve sensitivity to a prespecified pattern, but it also changes the scientific question. If treatment helps late and harms early, a late-emphasis test can favor treatment even when an unweighted test does not.

That is why a test statistic and an effect estimate must not be treated as interchangeable. A weighted log-rank test provides a test of equality of survival distributions against a chosen weighted direction. It does not, by itself, produce a readily interpretable treatment-effect quantity such as “months gained.” The protocol should name the confirmatory test and its weighting rationale, and should report an effect estimate that directly communicates the intended clinical contrast.

One useful effect measure is the difference in restricted mean survival time (RMST) through a prespecified time \(\tau\). For a survival function \(S(t)\), RMST through \(\tau\) is the area under the survival curve from time 0 to \(\tau\). It answers a different, directly time-based question: how much survival time, on average, is accumulated within that horizon? If the confirmatory test is weighted log-rank, an RMST estimate can aid interpretation, but it does not turn the weighted test into an RMST test. The test and reported estimate should be clearly distinguished.

Combining Stagewise Evidence

A simple way to combine stagewise evidence is the inverse-normal method. For each stage \(k\), let \(p_k\) be a valid one-sided p-value in the direction favoring treatment, and define \(Z_k=\Phi^{-1}(1-p_k)\). If the stagewise tests are independent under the null and the weights \(a_k\) are fixed in advance, the combined statistic is

Its null calibration follows directly. Under the null, each \(Z_k\) is approximately standard normal; independent stage data make the \(Z_k\) independent. Therefore the weighted sum has mean zero and variance \(\sum_k a_k^2\). Dividing by its standard deviation gives an approximately standard normal statistic. For a one-sided test at level \(\alpha\), reject \(H_0\) when \(Z_{\mathrm{comb}}>z_{1-\alpha}\), where \(z_{1-\alpha}\) is the \(1-\alpha\) standard-normal quantile.

This calculation assumes the stagewise p-values test the same directional confirmatory claim and are valid after any selection or adaptation. The simple formula also assumes independent stagewise evidence and fixed combination weights. Reusing participants across stages, choosing weights based on interim results, or selecting among several treatments can change the null distribution. Such designs require a procedure calibrated for that dependence or adaptation; the fixed-weight formula should not simply be applied as if nothing changed.

Design Checks, Alternatives, and Common Mistakes

Before the trial begins, the analysis plan should make the test reproducible: state the weight function, the time origin, the event definition, the analysis horizon, censoring assumptions, the direction of benefit, the stagewise p-value method, and the rule for combining evidence. Check that event accrual and follow-up will provide information at the times the weights emphasize. A late-emphasis test is uninformative if follow-up ends before enough late events occur.

Diagnostics should examine whether the observed survival curves and estimated hazards suggest substantial time variation, whether censoring differs in ways that undermine independent censoring, and whether the prespecified weights align with the intended clinical pattern. These checks help explain results; they do not license choosing a new weight after seeing confirmatory outcomes and then reporting its ordinary p-value. If a prespecified weighted test and the observed pattern disagree, report that discrepancy rather than silently changing the confirmatory test.

A common mistake is to inspect the survival curves, select the weight that gives the smallest p-value, and call that result confirmatory. The reasoning is that the selected test “best matches” the data. The correction is that selection among tests uses the data twice and generally inflates the type I error unless the selection is included in the calibration. Prespecify the test and its clinical rationale, or use an analysis that accounts for the data-dependent choice.

Other limitations remain. Weighted log-rank tests can have low power if their weights emphasize the wrong period or if treatment effects change direction in a way that cancels even after weighting. Large-sample normal approximations may be unreliable with few events. In addition, neither a stagewise combination rule nor a nonproportional-hazards test repairs informative censoring, an invalid selection adjustment, or an ill-defined endpoint. The confirmatory conclusion is only as credible as these design and analysis assumptions.