Calculating Weighted Logrank Power Under a Lag

Why calculate power under a lag?

When a treatment effect begins only after a delay, a single hazard ratio can conceal the pattern that matters for a weighted logrank test. Early events may contribute little or nothing to the expected score for benefit, while later events contribute a systematic signal. Power therefore depends not only on the size of the later effect, but also on how many participants remain at risk then and on the weight assigned to those events.

The calculation below uses the piecewise change-point hazard model from the earlier material on modeling delayed effects. The onset time and the weight function are treated as prespecified. The aim is to calculate the approximate mean and variance of the weighted logrank score under that alternative, then use them to approximate power.

Model, risk sets, and score approximation

Write the intervals as \(j=1,\ldots,J\), with interval \(j\) of length \(\Delta_j\). Within that interval, let \(\lambda_{0j}\) and \(\lambda_{1j}\) be the control and treatment hazard rates, and let \(Y_{0j}\) and \(Y_{1j}\) be representative expected numbers at risk. The treatment effect is delayed until a specified change point: before it, \(\lambda_{1j}=\lambda_{0j}\); after it, the hazards may differ. For a beneficial treatment, a common post-lag specification is \(\lambda_{1j}=\theta\lambda_{0j}\), where \(0<\theta<1\).

The expected risk sets must be calculated separately by arm. Under independent, non-informative censoring within each arm, the expected number at risk at time \(t\) depends on the number enrolled in that arm, its survival function, and its censoring pattern. With a common entry time, for example, it is \(n_g S_g(t)G_g(t)\) for arm \(g\), where \(S_g\) is survival and \(G_g\) is censoring survival. With staggered entry, expected risk sets also account for the entry-time distribution. Different hazards or censoring patterns can make the risk-set allocation ratio differ from the enrollment allocation ratio; it must not simply be assumed to stay fixed.

For the interval approximation, define the harmonic risk-set quantity \(A_j\) and the pooled event rate \(q_j\) as follows:

If \(w_j\) is the weight in interval \(j\), the first-order expected score contribution is \(w_j A_j(\lambda_{1j}-\lambda_{0j})\Delta_j\). The corresponding null-variance contribution is approximately \(w_j^2 A_j q_j\Delta_j\). Thus the total mean \(\mu_w\), variance \(V_w\), and signed noncentrality parameter \(\delta_w\) are approximated by:

These expressions use a large-sample normal approximation and representative risk sets and hazards within each interval. With time-varying risk sets, the corresponding continuous-time expressions integrate the risk-set and hazard terms over time. In either form, the variance used for the usual logrank power approximation is the null variance. A standard two-sided level-\(\alpha\) test has approximate power \(P(Z<-z_{1-\alpha/2})+P(Z>z_{1-\alpha/2})\), where \(Z\) is approximately \(N(\delta_w,1)\). For a prespecified one-sided test for treatment benefit, power is approximately \(\Phi(-z_{1-\alpha}-\delta_w)\).

Why the score mean has this form

A short derivation clarifies both the sign and the risk-set factor. In a small time increment \(dt\), the expected treatment events are \(Y_1\lambda_1dt\), while total expected events are \((Y_0\lambda_0+Y_1\lambda_1)dt\). Under the null, the expected treatment share of those events is \(Y_1/(Y_0+Y_1)\). The alternative mean of the unweighted score increment is therefore:

Multiplying by \(w(t)\) gives the weighted mean increment. Under the null, the treatment event count has the event-allocation variance associated with the two risk sets; to first order this is \(\{Y_0Y_1/(Y_0+Y_1)\}\lambda\,dt\). Replacing the common null event rate by the pooled rate \(q(t)\) gives the usual planning approximation to the variance under the alternative. The weight enters squared because variance scales with the square of a score contribution.

Unequal risk sets and weight scaling

Equal randomization at enrollment does not guarantee equal numbers at risk at later times. The following calculation uses unequal expected risk sets to show how they enter the score and variance directly. A weight that is multiplied by the same positive constant in every interval changes the score and its standard deviation by that constant, so it does not change \(\delta_w\). Changing the relative weights across intervals can change power.

How a late-emphasizing weight changes power

The practical comparison is usually between different relative time weights, not two constant multiples of the same weights. A late-emphasizing weight can improve power when the effect is truly delayed: it downweights early events that add variance without a treatment-related mean score, and upweights later events that contribute in the direction of benefit. But later risk sets may be small, and late weighting also squares the weights in the variance. The net effect must be calculated rather than assumed.

Assumptions, diagnostics, and limitations

A planned calculation is credible only if its inputs represent the intended design and alternative. Check that the delay, post-lag hazard ratio, baseline hazard pattern, accrual, and censoring assumptions are stated in time units and match the intended follow-up. Calculate expected risk sets from arm-specific survival and censoring patterns. In particular, do not multiply a single pooled risk-set curve by the enrollment allocation fractions when treatment and control survival or censoring differ.

Useful diagnostic checks include confirming that hazards and weights are defined on the same intervals, that \(A_j\) is positive whenever both arms have participants at risk, and that the sign of every mean contribution agrees with the chosen score convention. Under treatment benefit and positive weights, a post-lag interval should contribute a nonpositive mean to the treatment-minus-control score. Recalculate the result using finer time intervals if hazards, risk sets, or weights change substantially within an interval. Large differences between coarse and fine partitions warn that the representative-value approximation is inadequate.

The calculation is an approximation, not a guarantee of realized power. It relies on a large-sample normal approximation, a specified hazard and censoring model, and a variance approximation based on the null logrank variance. Sparse events, heavy censoring, very small risk sets, delayed entry, or strong departures from the assumed event pattern can impair accuracy. Simulations using the planned accrual, censoring, event process, and exact test implementation can be useful when the normal approximation is doubtful.

The main alternative within this calculation is to use ordinary logrank weights rather than a weight tailored to the prespecified delay. Ordinary weights are less dependent on a particular timing pattern, but can spend variance on early events that carry little mean signal under a lag. Late-emphasizing weights can target the delayed pattern, but may lose power if the effect begins earlier than expected or if few participants remain at risk late. A weighted logrank power calculation quantifies the consequences under its assumed model; it does not make the weight robust to an incorrectly specified effect pattern.

Common mistake

A frequent error is to assume that equal randomization means equal expected risk sets at every follow-up time, then use that assumed ratio in the score calculation. The correction is to calculate each arm’s expected risk set from its own survival, censoring, and entry patterns. Another error is to multiply the variance by \(w_j\), rather than \(w_j^2\); the score is multiplied by \(w_j\), so its variance is multiplied by \(w_j^2\). Finally, a larger absolute score mean alone does not establish higher power: compare \(|\mu_w|/\sqrt{V_w}\), because weighting changes both mean and variance.