Choosing Weights for Prespecified Delay Patterns

Choosing a Weight for an Anticipated Delay

The ordinary logrank test gives events similar influence throughout follow-up. When the treatment effect is expected to emerge only after a delay, events before onset may add variability without contributing much treatment signal. A weighted logrank test changes the influence of events over time. The design question is not simply whether to weight, but which prespecified time pattern best matches the scientific expectation about when the effect begins.

The introductory material on Fleming–Harrington weights explains how their parameters alter time emphasis. Here the emphasis is on choosing among those weights and piecewise alternatives for a stated delayed-effect scenario. The choice must be made using external knowledge or a prespecified design assumption, not by trying several weights on the observed treatment outcomes and reporting the most favorable result.

For a beneficial treatment that reduces the event hazard, it is convenient to define the score \(B_w\) so that positive values favor treatment: \(B_w\) is the negative of the usual treatment-minus-expected logrank score, with each event contribution multiplied by \(w(t)\). Its standardized statistic is \(T_w=B_w/\sqrt{V_w}\), where \(V_w\) is the variance under the null hypothesis. A larger positive \(T_w\) is evidence in the prespecified beneficial direction.

Assumptions and the Signal–Variance Trade-off

The comparison below concerns randomized time-to-event trials in which treatment assignment is independent of potential outcomes, censoring is noninformative for the event-time comparison (or handled appropriately by the design and analysis), and the chosen weight is fixed by the protocol. Event times are assumed to be observed with a valid risk-set construction; ties require a consistent standard logrank handling rule. Under the null, the weighted score is calibrated using its corresponding weighted variance. In a group sequential design, the same weight rule must be used at each look, and the stopping boundaries must be calibrated for that rule and its repeated looks.

A useful planning approximation divides follow-up into intervals. Let \(a_k\) be the anticipated mean beneficial score contribution in interval \(k\) before applying a weight, and let \(v_k\) be that interval’s null variance contribution. If the weighted score increments across intervals are approximately independent, then for interval weights \(w_k\),

The last expression is an approximate standardized signal, not an exact power calculation. It makes the design trade-off visible: increasing a weight where anticipated signal is strong can help, while increasing a weight where signal is absent can add variance without a matching increase in the mean.

For a small, time-varying hazard reduction, the direction and rough size of the anticipated interval signal can be related to the hazard ratio. If \(\theta(t)\) is the treatment-to-control hazard ratio and \(\pi\) is the treatment allocation fraction, then the expected beneficial score contribution per event under the alternative is approximately

When \(\theta(t)\) is near 1, this is approximately \(\pi(1-\pi)[1-\theta(t)]\). Thus, an interval with no anticipated treatment effect has little expected signal even though it can still contribute null variance. This is the basic reason a delayed-effect design may consider reducing the influence of early events.

A Short Justification for Matching Weight to Signal

The interval approximation also gives a simple benchmark for comparing candidate weights. Write \(x_k=w_k\sqrt{v_k}\) and \(y_k=a_k/\sqrt{v_k}\). Then the approximate standardized signal is the dot product of these vectors divided by the length of \(x\):

The inequality is Cauchy–Schwarz. Equality holds when \(x_k\) is proportional to \(y_k\), or equivalently when \(w_k\) is proportional to \(a_k/v_k\). This is an oracle benchmark: it assumes the signal pattern is known. In practice it is not a license to estimate an unrestricted weight from the trial’s observed outcomes. A real design chooses a simple, interpretable weight from a plausible prespecified scenario, recognizing that the true onset and effect size may differ.

Matching Fleming–Harrington and Piecewise Weights to Onset

Fleming–Harrington weights provide a smooth time emphasis based on pooled survival. A positive \(\rho\) emphasizes earlier event times relative to later ones, while a positive \(\gamma\) emphasizes later event times. For an anticipated delayed benefit, a late-emphasizing choice such as \((\rho,\gamma)=(0,1)\) may be more aligned with the scenario than an early-emphasizing choice such as \((1,0)\). These are relative emphases, not guarantees that early contributions are removed: a Fleming–Harrington weight generally remains nonzero before the anticipated onset.

A piecewise rule can express onset more directly. For example, \(w(t)=0\) before a prespecified delay time \(\tau\) and \(w(t)=1\) from \(\tau\) onward assigns no score contribution to earlier events. Other piecewise choices can use a gradual increase or a nonzero early weight. The cutpoint and values should have a clinical or design rationale. Time should be defined unambiguously, usually as time since randomization, and the same rule must be applied to all participants.

Design Checks, Comparisons, and Limitations

Before selecting a rule, state the anticipated onset pattern in words and translate it into interval-specific expected effects. Then compare a small number of plausible candidates using the same assumptions about allocation, event accrual, censoring, and follow-up. A useful planning table records each candidate’s interval weights, approximate mean score, variance, and standardized signal. If a claimed improvement comes from the mean alone, without accounting for the variance change, the comparison is incomplete.

The ordinary logrank test is a natural reference: it is simple and can be efficient when a common hazard ratio is plausible throughout follow-up. Fleming–Harrington weights offer a smooth, survival-based emphasis and avoid specifying a single sharp time cutpoint, but may not closely match an abrupt onset. Piecewise weights can represent a specified delay directly, but their performance can be sensitive to the cutpoint. Neither family is uniformly best. If timing is genuinely uncertain, a single specialized weight can be fragile; any approach that considers multiple candidate tests needs an appropriate prespecified method for controlling the overall type I error.

In a sequential trial, a weight choice is also part of the monitoring design. Weighted score trajectories and their covariance across looks need to match the statistic being monitored. Boundaries calibrated for an ordinary logrank statistic cannot simply be assumed valid for a different weighted statistic. The effect of unequal allocation, changing event information, and the planned analysis must also be reflected in design calculations.

Several diagnostics can reveal that the plan or analysis is going wrong. Check that the weight is evaluated using event time immediately before the event, that the time origin and cutpoints are applied consistently, and that the score and variance use the same weights. Compare observed survival patterns with the scientific onset assumption, but do not use unblinded outcome patterns to switch to a more favorable weight without a valid adaptation and error-control plan. A weighted test may have low power if the true effect starts earlier or later than anticipated, if the effect is weaker than planned, or if hazards cross rather than show the assumed delayed benefit. It also does not by itself provide a complete description of the treatment effect over time.

• Mistake: choosing the weight that gives the smallest p-value among several candidates after seeing treatment outcomes. Correction: prespecify the weight, or use a prespecified multiplicity-controlled strategy.
• Mistake: assuming a late-emphasizing Fleming–Harrington weight gives zero weight to all early events. Correction: inspect the actual function; if early events must be excluded, consider a justified piecewise rule.
• Mistake: selecting a very late cutpoint because the effect is described as delayed, without checking the assumed onset time. Correction: quantify anticipated signal and variance on both sides of the cutpoint.
• Mistake: comparing weighted tests using only their expected scores. Correction: compare standardized signal, which accounts for the weight-squared variance.