An End-To-End Event-Driven Logrank Design

From a Treatment Effect to a Sequential Design

An event-driven trial schedules analyses by the accumulation of outcome events rather than by calendar dates alone. This is natural for a time-to-event endpoint: the logrank statistic gains information as participants experience events, while enrollment and follow-up determine when those events occur. An end-to-end design therefore links three quantities: the treatment effect the study aims to detect, the total event information needed, and the rules for looking at the accumulating data.

The calculations below use a two-arm randomized trial. They are planning approximations, not substitutes for a protocol-specific design calculation. In particular, the event target depends on the assumed hazard ratio and allocation, whereas the eventual calendar time and number of randomized participants also depend on accrual, censoring, and follow-up.

Definition: Let the treatment-to-control hazard ratio be \(HR\), with \(HR<1\) representing a lower event hazard in the treatment arm. Let \(p\) be the proportion assigned to treatment, so \(1-p\) is assigned to control. The total number of observed events is \(D\). At information look \(j\), the cumulative logrank score is \(S_j=E_{T,j}-O_{T,j}\), where \(E_{T,j}\) and \(O_{T,j}\) are the treatment-arm expected and observed event counts. Positive scores favor treatment. With estimated variance \(V_j\), the standardized logrank statistic is \(Z_j=S_j/\sqrt{V_j}\).

The basic event-count approximation assumes proportional hazards: the treatment-to-control hazard ratio is constant over follow-up. It also assumes independent participants, correct randomization and event ascertainment, and censoring that is noninformative for the event process under the analysis. The event-count formula uses a large-sample approximation and the planned treatment fraction in the risk sets. If allocation or risk-set composition changes substantially, the approximation should be checked or replaced by a design-specific calculation.

Deriving the Target Event Information

For a standard logrank design, the approximate variance of the score under the null is \(V\approx p(1-p)D\). Under proportional hazards, the standardized statistic has approximate mean magnitude \(\sqrt{V}\,|\log(HR)|\) under the planned alternative. To attain two-sided type I error \(\alpha\) and power \(1-\beta\), set that mean approximately equal to the sum of the null critical value and the normal quantile for the target power. Solving for \(D\) gives the familiar event-count formula.

\sqrt{p(1-p)D}\,|\log(HR)| \approx z_{1-\alpha/2}+z_{1-\beta}

D \approx \frac{(z_{1-\alpha/2}+z_{1-\beta})^2} {p(1-p)\{\log(HR)\}^2}

This is a fixed-sample planning approximation. When interim monitoring is added, the rejection boundaries must also be specified. A sequential design may require more events than the fixed-sample calculation to achieve the same power, depending on the boundaries and analysis plan. The calculation below accounts for that distinction by choosing a conservative boundary and targeting the desired power at the final look alone.

Worked Example: Setting an Event Target for a Sequential Trial

Suppose a hypothetical two-arm trial uses 1:1 randomization and aims to detect \(HR=0.70\), with two-sided \(\alpha=0.05\) and 90% power. Plan three looks at approximately 50%, 75%, and 100% of the target information. For a transparent, conservative boundary rule, use the same two-sided threshold at each look and allocate \(\alpha/3\) to each look.

At each look, the per-tail error allowance is \(0.05/(2\times3)=0.008333\). The corresponding standard-normal critical value is \(c=z_{0.991667}\approx2.394\). Choose the event target so that the final test alone has about 90% power at this threshold. The required standardized mean is approximately \(c+z_{0.90}=2.394+1.2816=3.6756\). The opposite tail under the beneficial alternative is negligible at this mean.

Here \(p=0.50\), so \(p(1-p)=0.25\), and \(\log(0.70)=-0.3567\). Substitution gives:

D \approx \frac{(2.394+1.2816)^2} {0.25\{\log(0.70)\}^2} = \frac{13.510}{0.03180} \approx 424.8

Round up to a target of 425 events. Under the stated approximation, the final-only power is about 90%; allowing rejection at earlier looks can increase the overall probability of rejecting \(H_0\). This is not necessarily the smallest event target for 90% sequential power. A less conservative design would calculate power using the joint distribution of the sequential statistics and its chosen boundaries.

Turning Events into Looks and Participants

The planned event information at each look is set as a fraction of the final target. Event fractions are convenient planning targets, but they are not guaranteed to equal the realized information fractions. The logrank variance \(V_j\) is the direct measure of information for the statistic; it depends on the risk sets and events, not only on the cumulative event count.

Worked Example: Planning Interim Events and Approximate Enrollment

Continue the hypothetical design with a final target of 425 events and looks at 50%, 75%, and 100%. The cumulative event targets are \(0.50(425)=212.5\), \(0.75(425)=318.75\), and \(1.00(425)=425\). Rounding to whole events gives targets of 213, 319, and 425 events. Their event fractions are approximately 0.501, 0.751, and 1.000.

Suppose planning assumptions suggest that 65% of randomized participants will have an observed event by the final analysis. This event fraction must reflect the planned accrual and follow-up as well as censoring. The approximate total enrollment is:

N \approx \frac{D}{q} = \frac{425}{0.65} = 653.8

Thus, approximately 654 participants are needed, or 327 per arm with equal allocation. This is a planning approximation: the realized event fraction may differ, so the calendar duration or enrollment may need adjustment. If the 65% assumption does not already account for loss to follow-up and censoring, it is not appropriate to treat it as the expected event fraction.

At each analysis, use the accumulated data to calculate the cumulative score and its variance. For distinct event times without tied events, the usual logrank contributions can be written using \(Y_T\) and \(Y_C\), the numbers at risk in the two arms just before the event time, \(d\), the total events at that time, and \(d_T\), the treatment-arm events:

E_T = \sum_i d_i\frac{Y_{Ti}}{Y_{Ti}+Y_{Ci}}, \qquad O_T = \sum_i d_{Ti}, \qquad S=E_T-O_T

V = \sum_i d_i \frac{Y_{Ti}Y_{Ci}}{(Y_{Ti}+Y_{Ci})^2}, \qquad Z=\frac{S}{\sqrt{V}}

These variance contributions use the simple no-ties form. With tied events, an analysis should use the prespecified standard tie handling, including the corresponding variance calculation. The sign convention here makes positive \(Z\) favor treatment; software that reports the opposite score convention will reverse the sign, so confirm the orientation before applying boundaries.

Applying Sequential Boundaries

A boundary is a prespecified threshold that determines whether the trial stops or continues at a look. For the illustrative three-look rule above, compare the absolute value of each cumulative statistic with \(c=2.394\). Crossing the upper boundary indicates evidence favoring treatment; crossing the lower boundary indicates evidence favoring control. If neither boundary is crossed, continue to the next planned look, provided the trial has not stopped for another prespecified reason.

The threshold is conservative because it allocates the total error allowance equally across looks rather than using the dependence between cumulative statistics to spend alpha more efficiently. Under the null, if each look’s two-sided statistic is approximately standard normal, the probability of crossing any boundary is at most the sum of the three look-specific error probabilities, approximately \(3(0.05/3)=0.05\). This union-bound argument does not require the looks to be independent. More efficient alpha-spending or group-sequential boundaries use the joint behavior of the statistics and must be calibrated to the planned information times.

Worked Example: Making Decisions at Three Logrank Looks

Consider hypothetical cumulative logrank results at the planned event targets. The variance values are plausible realized values near the balanced-allocation approximation \(V\approx D/4\). The same prespecified two-sided boundary, \(c=2.394\), is used at each look.

At 213 events, suppose \(S_1=12.0\) and \(V_1=52.8\). Then \(Z_1=12.0/\sqrt{52.8}=12.0/7.266=1.651\). Since \(|1.651|<2.394\), neither boundary is crossed; continue.

At 319 events, suppose the cumulative values are \(S_2=19.0\) and \(V_2=79.4\). Then \(Z_2=19.0/\sqrt{79.4}=19.0/8.911=2.132\). Again, \(|2.132|<2.394\), so the trial continues.

At 425 events, suppose \(S_3=25.0\) and \(V_3=105.7\). Then \(Z_3=25.0/\sqrt{105.7}=25.0/10.281=2.432\). This exceeds the upper boundary, \(2.432>2.394\), so the sequential rule rejects \(H_0\) in favor of a treatment effect in the beneficial direction. The conclusion follows from the prespecified sequential rule; a nominal single-look p-value should not be presented as though the interim monitoring had not occurred.

Checks, Alternatives, and Limitations

An event-driven design is useful when events, rather than elapsed time, primarily determine the precision of the treatment comparison. It does not guarantee that event information will accumulate on schedule. At each look, check the observed event count, the actual logrank variance, treatment allocation in the risk sets, and the accuracy of event and censoring records. A planned event fraction is a practical trigger, but the realized information fraction \(V_j/V_{\mathrm{final}}\) is the more direct summary of statistical information.

The main alternatives differ in what they optimize. A fixed-sample logrank design avoids repeated efficacy looks but does not permit the same prespecified early stopping rule. Conventional group-sequential boundaries or alpha-spending boundaries can use the correlation across looks and are often less conservative than the Bonferroni rule used here; they require boundary calibration for the planned information times. A design based only on the fixed-sample event formula is simpler, but its stated power does not automatically carry over after interim monitoring is introduced.

The proportional-hazards assumption is central to interpreting the target \(HR\) and the event-count power calculation. If the treatment effect changes over time, the logrank comparison may not summarize the clinical effect as intended, and the planned power can be inaccurate. Other limitations include inaccurate event-fraction forecasts, delayed or incomplete event reporting, substantial departures from planned allocation, and too few events for the normal approximations to be reliable. Investigate these issues before interpreting a boundary crossing; changing a boundary after observing results compromises its error-rate guarantee.

Common mistake: Treating an event target as if it were a participant target, or assuming that 50% of the target events always means 50% of the final statistical information. The event count and information are related but not identical. Plan participant numbers using an event-fraction assumption, then assess the realized logrank variance and information fraction at each analysis.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For a balanced design with \(HR=0.80\), does the approximate event requirement increase or decrease relative to \(HR=0.70\), all else equal? Why?
A design targets 360 events at 40%, 70%, and 100% information. What are the cumulative event targets?
At a look, \(S=18\) and \(V=81\). Calculate \(Z\) and interpret its direction.
Why is a planned event fraction not itself a measure of realized logrank information?

Solutions. (1) It increases: \(|\log(0.80)|<|\log(0.70)|\), so the squared effect in the denominator of the event formula is smaller. (2) The targets are \(0.40(360)=144\), \(0.70(360)=252\), and 360 events. (3) \(Z=18/\sqrt{81}=18/9=2.00\), a positive statistic favoring treatment under the stated score convention; whether it crosses a boundary depends on the prespecified threshold. (4) Logrank variance also depends on the allocation within risk sets and the event contributions, so the variance fraction can differ from the event-count fraction.