Event-Driven Designs with a Prespecified Treatment Lag

Why a Treatment Lag Changes an Event-Driven Design

An event-driven design stops or conducts an analysis after a target number of events, not after a fixed calendar duration. That target alone does not determine power when treatment benefit is delayed. Events occurring before the benefit begins can add information and variance to a weighted logrank statistic while contributing no expected treatment-benefit signal. Events after the lag contribute signal according to the hazard contrast and the prespecified weight at their event times.

The key design question is therefore not simply how many events are expected, but when those events occur relative to the lag and how the test weights them. The calculations below describe this relationship for a standard event-driven approximation. In a group sequential design, the same lag-sensitive mean and information calculations feed into the joint distribution of the interim statistics; the sequential boundaries still determine the stopping and rejection rules.

Definition: Let $t$ denote time since randomization, let $\tau$ be a prespecified treatment-effect lag, and let $w(t)$ be a fixed, prespecified weight. A simple delayed-effect model has control hazard $\lambda_0(t)$ and treatment hazard $$ \lambda_1(t)= \begin{cases} \lambda_0(t), & t<\tau,\\ \theta\lambda_0(t), & t\geq \tau, \end{cases} \qquad 0<\theta<1. $$ Thus, the treatment and control hazards are equal before $\tau$, and the treatment hazard is lower by a constant hazard ratio $\theta$ afterward. The parameter $\tau$ is measured on the stated analysis-time scale; it is not automatically a calendar-time point shared by participants enrolled on different dates.

For clarity, suppose treatment assignment is randomized, censoring is independent of the event process conditional on treatment and any design variables used in the calculation, and the weight is chosen in advance rather than adapted to observed treatment differences. We also use a large-sample approximation in which the fraction of individuals at risk in the treatment arm is approximately $\pi$ over the relevant event times. For equal allocation and comparable follow-up, $\pi$ is often approximated by $0.5$. If risk-set balance changes substantially over time, use time-varying risk-set proportions in the corresponding calculation or simulate the design.

The score convention here is that a positive score points toward treatment benefit. This is the negative of the usual treatment-minus-expected weighted logrank score. Its sign does not change the test when the direction is handled consistently, but it makes the expected drift under a beneficial treatment positive.

Expected Drift and Information per Event

At an event time, write $Y_1$ and $Y_0$ for the numbers at risk in the treatment and control groups, and $Y=Y_1+Y_0$. The benefit-oriented weighted score increment is

dB_w(t)=w(t)\left\{\frac{Y_1}{Y}\,dN_0(t)-\frac{Y_0}{Y}\,dN_1(t)\right\},

where $dN_1(t)$ and $dN_0(t)$ indicate treatment and control events. Given the risk sets, their expected event increments over a short interval are $Y_1\lambda_1(t)\,dt$ and $Y_0\lambda_0(t)\,dt$. Substituting these rates gives the expected score drift:

E\{dB_w(t)\mid Y_1,Y_0\} =w(t)\frac{Y_1Y_0}{Y}\{\lambda_0(t)-\lambda_1(t)\}\,dt.

Before the lag, $\lambda_0(t)-\lambda_1(t)=0$, so the expected drift is zero. After the lag, it is positive under the assumed benefit model. To express this as a contribution per event, let $\pi=Y_1/Y$ and define the combined event rate per person at risk as $\bar{\lambda}(t)=\pi\lambda_1(t)+(1-\pi)\lambda_0(t)$. Dividing the drift rate by the total event rate yields

\delta(t) =w(t)\pi(1-\pi) \frac{\lambda_0(t)-\lambda_1(t)} {\pi\lambda_1(t)+(1-\pi)\lambda_0(t)}.

This is the approximate expected score contribution per event at time $t$. It is zero before $\tau$, even though such events still enter the test statistic. Under the null, the variance contribution per event is approximately $w(t)^2\pi(1-\pi)$, the usual weighted logrank information increment. The null information is a variance quantity; it should not be mistaken for expected evidence of benefit.

Key result: Suppose a design targets $D$ events and let $g(t)$ denote the event-time distribution across those events under the design model, normalized so that its integral is one. With approximately stable risk-set fraction $\pi$, the expected score and null information are $$ E(B_w)\ \approx\ D\int g(t)\delta(t)\,dt, \qquad I_0\ \approx\ D\pi(1-\pi)\int g(t)w(t)^2\,dt. $$ The approximate standardized mean is $E(B_w)/\sqrt{I_0}$. These expressions emphasize that the fraction of events before the lag affects both the accumulated score and, through the weights, the information.

The event-time distribution $g(t)$ is a design quantity, not necessarily uniform in time. It reflects when events arise under the assumed event and enrollment patterns. For a simple calculation, it can be represented by event fractions in intervals before and after $\tau$. In a more detailed design evaluation, the expected score and information can be accumulated across event times or estimated by simulating the specified design.

Worked Calculations

Worked Example: Half the Events Occur Before the Lag

Hypothetical design. Suppose there are $D=200$ target events, with half expected before $\tau$ and half after it. Allocation and risk sets are balanced, so $\pi=0.5$. Take $w(t)=1$, and suppose the post-lag hazard ratio is $\theta=0.70$, with control hazard scaled to $\lambda_0=1$.

Before the lag the hazards are equal, so the mean score contribution is zero. After the lag, the combined event rate is $\bar{\lambda}=0.5(0.70)+0.5(1)=0.85$. The expected benefit score per post-lag event is

\delta_{\mathrm{post}} =1(0.5)(0.5)\frac{1-0.70}{0.85} =0.08824.

There are 100 expected post-lag events. Thus, the expected total score is $100(0.08824)=8.824$. Because the weight is one throughout, the approximate null information is $200(0.5)(0.5)=50$, and the standardized mean is $8.824/\sqrt{50}=1.248$. The pre-lag events add to the information but not to the expected benefit drift. If all 200 events were instead post-lag under the same assumptions, the expected score would be $200(0.08824)=17.65$, with information 50 and standardized mean 2.496.

Worked Example: A Late-Emphasis Weight

Hypothetical design. Suppose $D=240$, with 40% of events before the lag and 60% after it. Use $\pi=0.5$, a post-lag hazard ratio of $0.60$, and a prespecified weight of $0.5$ before the lag and $1.5$ afterward. Let $\lambda_0=1$ for the post-lag rate calculation.

Before the lag, the hazards are equal and the expected score contribution remains zero. After the lag, $\bar{\lambda}=0.5(0.60)+0.5(1)=0.80$. The post-lag expected score contribution per event is

\delta_{\mathrm{post}} =1.5(0.5)(0.5)\frac{1-0.60}{0.80} =0.1875.

The expected post-lag event count is $240(0.60)=144$, so the expected score is $144(0.1875)=27.00$. For the null information, the mean squared weight across events is $0.40(0.5^2)+0.60(1.5^2)=1.45$. Therefore,

I_0=240(0.25)(1.45)=87.00, \qquad \frac{E(B_w)}{\sqrt{I_0}}=\frac{27.00}{\sqrt{87.00}}=2.894.

For comparison, an ordinary logrank score with weight one has expected score $144(0.25)(0.40/0.80)=18.00$, information $240(0.25)=60.00$, and standardized mean $18/\sqrt{60}=2.324$. In this hypothetical pattern, late emphasis increases the standardized mean because it assigns greater weight to the events with treatment benefit. This is not a general guarantee: the result depends on the lag, event-time distribution, and chosen weight.

Using the Lag Model to Approximate an Event Target

A simple event-target calculation can show how much larger a design may need to be when some target events occur before the treatment effect begins. It is an approximation for a single final analysis, not a replacement for calibrating group sequential boundaries. For a one-sided normal test with significance level $\alpha$ and target power $1-\beta$, a common planning condition is that the alternative standardized mean be approximately $z_{1-\alpha}+z_{1-\beta}$, where $z_q$ is the $q$-quantile of the standard normal distribution.

Worked Example: Approximate Event Target With a Prespecified Lag

Hypothetical planning calculation. Suppose allocation is balanced, the weight is one, 35% of events are expected before the lag and 65% afterward, and the post-lag hazard ratio is $0.75$. Plan for one-sided $\alpha=0.025$ and power $0.80$. Assume the event fractions remain approximately as stated as the target is increased.

For a post-lag event, the combined event rate relative to the control hazard is $0.5(0.75)+0.5(1)=0.875$. Thus its expected score contribution is

\delta_{\mathrm{post}} =(0.5)(0.5)\frac{1-0.75}{0.875} =0.07143.

Averaged across all target events, the expected score per event is $0.65(0.07143)=0.04643$. The null information per event is $0.25$. Hence the approximate standardized mean at target $D$ is

\frac{D(0.04643)}{\sqrt{D(0.25)}} =0.09286\sqrt{D}.

The required normal mean is approximately $z_{0.975}+z_{0.80}=1.960+0.8416=2.802$. Solving $0.09286\sqrt{D}=2.802$ gives $D=(2.802/0.09286)^2=910.3$, or about 911 events after rounding up. If one incorrectly assumed that all events occurred after the lag, the coefficient would be $0.07143/\sqrt{0.25}=0.1429$, leading to $D=(2.802/0.1429)^2\approx385$. The difference illustrates why the event mix relative to the lag matters.

The 911-event figure is not a group sequential sample-size result. For a sequential design, the planned information times, score means and covariances, and prespecified boundaries must be evaluated together. The single-look calculation is a transparent scale check and a useful diagnostic for whether ignoring the lag would be consequential.

Assumptions, Diagnostics, and Alternatives

These calculations are useful when the lag and weight are specified on a meaningful time scale, event timing can be represented adequately, and risk-set balance is reasonably stable or explicitly modeled. Before using the result for planning, check that the assumed post-lag hazard ratio and the fraction of events before the lag are plausible for the intended population and follow-up. Also check that the weight has the intended sign and relative emphasis: a large weight before the lag can increase information without creating benefit drift under this model.

Warning signs include a projected event distribution with a large pre-lag fraction but a power calculation that treats every event as post-lag; a weight that changes after looking at unblinded treatment differences; substantial arm imbalance in risk sets that is ignored by a fixed $\pi$; or a lag defined on calendar time when the treatment effect is actually expected to depend on time since randomization. Such discrepancies can make the approximate mean, information, or both misleading. They should be addressed by refining the design model and recalculating the planned score process, not by interpreting a null-information count as if it were accumulated treatment benefit.

The ordinary logrank test is a natural comparison when a constant hazard ratio is credible and no time-specific emphasis is desired. A prespecified weighted logrank test can be more efficient for the particular delayed-effect pattern it targets, but its performance depends on the chosen weight and event timing. An analysis based on restricted mean survival time (RMST) answers a different estimand-focused question and does not reduce to this weighted-score calculation. For group sequential monitoring, the prespecified weighted statistic and its boundary calibration must be maintained; a final event target alone does not specify interim operating characteristics.

A further limitation is that this derivation treats the lag model as prespecified and summarizes event timing through expected proportions or a distribution. If the assumed lag or event pattern is uncertain, the computed event target is conditional on those assumptions; it is not a guarantee of power across other patterns. The next design step is to make that uncertainty explicit when evaluating the operating characteristics, while preserving the planned testing procedure.

Common pitfall: “Every event contributes one unit of treatment-effect information, so a fixed event target has the same power regardless of the lag.” This is wrong because an event before the onset has zero expected score drift under the stated model, even though it contributes variance. The correction is to calculate expected drift and information separately, using the event-time distribution and the prespecified weight for each.

Check Your Understanding

Use the assumptions and formulas in this tutorial. Answers are given below each question.

With equal allocation, weight one, and equal hazards before the lag, what is the expected score contribution of a pre-lag event? What does that event still contribute?
For a post-lag hazard ratio of $0.80$, equal risk sets, and control hazard scaled to one, calculate the expected score contribution per event when $w(t)=1$.
Suppose half of 100 events occur before the lag and half after it, with the post-lag per-event score contribution from Question 2. Find the expected total score.
Why can increasing the post-lag weight increase the expected score but also increase the null information?

Solutions. (1) Its expected score contribution is zero because the hazards are equal; it still contributes to the variance or null information according to its squared weight. (2) The combined event rate is $0.5(0.80)+0.5(1)=0.90$, so the contribution is $(0.5)(0.5)(1-0.80)/0.90=0.05556$ per event. (3) There are 50 post-lag events, so the expected total score is $50(0.05556)=2.778$; the 50 pre-lag events have zero expected drift. (4) The score drift is proportional to $w(t)$, while the null information is proportional to $w(t)^2$. Thus, emphasizing post-lag events can raise both the mean and the variance, and the effect on the standardized mean depends on their relative changes and on the event-time pattern.