Survival Endpoints

In many clinical trials, the primary endpoint is time‑to‑event, most commonly overall survival (OS) or progression‑free survival (PFS).
In such studies, each patient contributes a survival time and an event indicator, and comparisons between treatment groups rely on survival curves and hazard functions rather than simple proportions.

When two groups are compared—typically a control arm receiving standard of care and an experimental arm receiving a new treatment—improvement can be expressed in several equivalent ways:

• Difference in survival probability at a specific time, e.g.
  \(S_e(\tau) = S_c(\tau) + 0.10\): at time \(\tau\), the treatment increases survival by 10 percentage points.

• Difference in median survival, e.g.
  \(S_c(t_c)=0.5\) and \(S_e(t_c+1)=0.5\): the treatment extends the median survival time by one year.

• Hazard ratio (HR) under the proportional hazards assumption:
  \(S_e(t) = S_c(t)^{\text{HR}}\).
  Here, the HR summarizes the relative risk of the event at any time.

with \(S_c(t)\) and \(S_e(t)\) denoting the survival functions of the control and experimental groups, respectively.

The proportional hazards condition states that

\[ \exists r \in \mathbb{R}^+_*, \forall t, \quad r = \frac{\log S_e(t)}{\log S_c(t)} , \]

so that the treatment effect is constant on the log‑hazard scale.

This chapter focuses on methods for designing clinical trials with survival endpoints, with an emphasis on how different statistical software and R packages compute sample sizes under various scenarios.

More specifically:

• In Sample size computation methods for log-rank test, we introduce the classical formulas for calculating the number of required events and sample size in survival trials, and we discuss their assumptions and limitations.

• In Two-Arm Fixed Design with Survival endpoints, we compare results from different software and R packages for trials without interim analyses.

• In Two-Arm Group-Sequential Design with Survival endpoints, we extend the comparison to trials with interim analyses, also known as group‑sequential designs.

• Finally, in One-Arm Design with Survival endpoints, we examine sample size calculations for single‑arm designs, frequently used in rare diseases or in early‑phase oncology trials where randomized comparisons are not feasible.