Power and Sample Size for Weighted Logrank Designs

Why Power Depends on the Effect Over Time

A weighted logrank design is intended to detect a treatment effect that may change over follow-up. Its power therefore depends not just on an overall hazard ratio, but on when events occur, how the treatment effect changes over time, and which event times receive the greatest weight. A weight that emphasizes the period of benefit can improve sensitivity to that particular alternative; it can also reduce sensitivity if benefit occurs elsewhere.

For a fixed-look calculation, weighted drift and variance give a useful analytic approximation to power and the required number of events. For a group sequential design, the statistics at the interim looks are correlated, and power is the probability of crossing at least one efficacy boundary. That probability must account for the full sequence of looks, using a multivariate normal approximation or simulation.

Assumptions and the Planning Approximation

The planning approximation below assumes randomized treatment allocation with treatment fraction \(q\), a prespecified weight function, independent censoring conditional on treatment and modeled covariates, and a specified alternative hazard ratio \(HR(t)=\lambda_T(t)/\lambda_C(t)\). It also assumes enough events for the weighted score to be approximately normal. The event-time distribution used in the integrals must describe the design’s anticipated event pattern; under larger effects, that distribution can differ between arms. The approximation is most reliable when the alternative is not so large that this difference materially changes the risk sets and event mix.

Let \(f_D(t)\) be a normalized event-time density over the planned follow-up under a chosen reference model, so that \(\int f_D(t)\,dt=1\). Write \(\beta(t)=\log HR(t)\), and define

Here \(J\) is the benefit-oriented weighted effect averaged over the event-time distribution, and \(C\) is the corresponding average squared weight. Under the planning approximation, with \(D\) total events and a stable allocation fraction \(q\),

Thus the approximate one-sided power at a single final analysis with critical value \(z_{1-\alpha}\) is \(1-\Phi(z_{1-\alpha}-\mu_D)\), where \(\Phi\) is the standard normal cumulative distribution function. Solving for the event target for power \(1-\gamma\) gives

This formula shows why a favorable average effect alone is not enough: weights that increase the drift can also increase the variance through \(C\). It is the ratio \(J/\sqrt C\), rather than \(J\) by itself, that determines the approximate standardized signal per square root of an event.

From a Fixed Look to Sequential Power

At interim look \(k\), let \(U_k\) and \(V_k\) denote the cumulative weighted score and its variance estimate. Under the null, independent increments of the score imply that standardized statistics at ordered looks have approximate correlations \(\operatorname{Corr}(Z_i,Z_j)\approx\sqrt{V_i/V_j}\) for \(i<j\). Under a specified alternative, calculate the mean at each look from the cumulative weighted drift through that look, divided by \(\sqrt{V_k}\). The sequence is then approximated by a multivariate normal vector with those means and the corresponding covariance matrix.

If \(b_k\) is the efficacy boundary at look \(k\), one-sided sequential power is the probability that at least one statistic crosses its boundary:

The boundaries must be those calibrated for the design’s null type I error and the prespecified analysis plan. A fixed-look formula using \(z_{1-\alpha}\) does not account for repeated opportunities to stop early. For sample-size planning, evaluate sequential power over candidate event targets and increase or decrease \(D\) until the desired power is reached. When the alternative changes substantially over time, each look’s mean should be computed from its own cumulative drift rather than assuming it is proportional to the square root of information.

Checking the Effect Shape and Weight

The weighted summaries can be inspected directly when the alternative is piecewise constant. This is useful both for checking arithmetic and for seeing which periods drive the planned sensitivity. The following calculation treats the event-time fractions as specified planning inputs; in an actual design they should be justified by the assumed hazards, follow-up, and censoring.

Estimating Sequential Power by Simulation

Simulation is often preferable when the hazard changes sharply, censoring is substantial, allocation among those still at risk shifts over time, or the analytic normal approximation is questionable. A simulation should reproduce the planned randomization, event and censoring distributions, prespecified weights, information-based analysis times, and boundaries. Each simulated trial is analyzed using the same weighted score and stopping rule intended for the real design. Power is estimated by the fraction of simulated trials that cross an efficacy boundary.

Diagnostics, Alternatives, and Limitations

Before relying on a power estimate, check that the design inputs describe the intended analysis. In particular, verify that the weight function is prespecified and implemented consistently at every look, the assumed event-time fractions are compatible with the hazard and follow-up assumptions, and the sequential boundaries match the specified type I error. In simulation, check that event counts, allocation, censoring, and look timing behave as intended. Under the null, a separate simulation can assess whether the full stopping rule has the intended type I error; the alternative simulation estimates power, not type I error.

A common mistake is to calculate a weighted logrank event target from a fixed-look formula and call it the required group sequential sample size. That calculation omits the probability of crossing at earlier looks and the correlation among looks. Use it as an initial estimate, then evaluate the actual sequential crossing probability. Another mistake is to choose weights after inspecting the observed treatment differences: data-adaptive weights can invalidate the intended type I error unless the adaptation is handled by a valid prespecified procedure.

The main alternatives answer different questions. A logrank design with constant weights is a natural comparator when proportional hazards are plausible or when a broad average effect is the target. A weighted logrank design can focus sensitivity on a prespecified effect pattern, but its power may be poor for a different pattern. Simulation can accommodate complex designs more directly than the analytic approximation, but it depends on the correctness of the simulated model and can have material Monte Carlo error if too few trials are run. No method removes the need to examine how conclusions change under plausible alternatives; the calculated power is conditional on the modeled hazards and design assumptions.