Sample size determination for clinical trials with co-primary outcomes: exponential event times

Abstract

Clinical trials with event-time outcomes as co-primary contrasts are common in many areas such as infectious disease, oncology, and cardiovascular disease. We discuss methods for calculating the sample size for randomized superiority clinical trials with two correlated time-to-event outcomes as co-primary contrasts when the time-to-event outcomes are exponentially distributed. The approach is simple and easily applied in practice.

Figure 1

Relationship between correlations for Clayton, positive stable, and Frank copulas with limited recruitment and censoring, where T₀ = 2, λ_T1/λ_C1 = λ_T2/λ_C2 and S_T1(T) = S_T2(T).

Figure 2

Behavior of the sample size with common correlation ρ_T = ρ_C = ρ: sample size (equally sized groups: r 0.5) was calculated to detect the joint reduction in both of the time-to-event outcomes with the overall power of 1 − β = 0.80 at the significance level of α = 0.025, where T₀ = 2 and T = 5, λ_T1/λ_C1 = 0.667 and S_T1(5) = S_T2(5) = 0.5 (C: Clayton copula, PS: positive stable copula, F: Frank copula).

Figure 3

Behavior of the empirical overall power for the log-rank test with common correlation ρ_T = ρ_C = ρ: sample size (equally sized groups: r = 0.5) was calculated to detect the joint reduction in both of the time-to-event outcomes with the overall power of 1 − β = 0.80 at the significance level of α = 0.025, where T₀ = 2 and T = 5, λ_T1/λ_C1 = 0.667 and S_T1(5) = S_T2(5) = 0.5 A: Data were generated from the Clayton copula. B: Data were generated from the positive stable copula. C: Data were generated from the Clayton copula.