Testing a primary and a secondary endpoint in a group sequential design

Abstract

We consider a clinical trial with a primary and a secondary endpoint where the secondary endpoint is tested only if the primary endpoint is significant. The trial uses a group sequential procedure with two stages. The familywise error rate (FWER) of falsely concluding significance on either endpoint is to be controlled at a nominal level α. The type I error rate for the primary endpoint is controlled by choosing any α-level stopping boundary, e.g., the standard O'Brien-Fleming or the Pocock boundary. Given any particular α-level boundary for the primary endpoint, we study the problem of determining the boundary for the secondary endpoint to control the FWER. We study this FWER analytically and numerically and find that it is maximized when the correlation coefficient ρ between the two endpoints equals 1. For the four combinations consisting of O'Brien-Fleming and Pocock boundaries for the primary and secondary endpoints, the critical constants required to control the FWER are computed for different values of ρ. An ad hoc boundary is proposed for the secondary endpoint to address a practical concern that may be at issue in some applications. Numerical studies indicate that the O'Brien-Fleming boundary for the primary endpoint and the Pocock boundary for the secondary endpoint generally gives the best primary as well as secondary power performance. The Pocock boundary may be replaced by the ad hoc boundary for the secondary endpoint with a very little loss of secondary power if the practical concern is at issue. A clinical trial example is given to illustrate the methods.

Figure 1

FWER as a function of ρ and Δ₁ when $(c_1,c_2)=(1.678\sqrt{2},1.678)$, n₁ = n₂ and d₁ = d₂ = z_.05 = 1.645

Figure 2

FWER as a function of ρ and Δ₁ when $(c_1,c_2)=(1.678\sqrt{2},1.678)$, n₁ = n₂ and d₁ = d₂ = z_.0303 = 1.876

Figure 3

FWER as a function of ρ and Δ₁ when $(c_1,c_2)=(d_1,d_2)=(1.678\sqrt{2},1.678)$ and n₁ = n₂

Figure 4

Secondary powers of the six boundary combinations (OF = O’Brien-Fleming, PO = Pocock, AH = Ad-Hoc, 1 = Primary, 2 = Secondary) for the primary and secondary endpoints as functions of Δ₁ for ρ = 0.4 and Δ₂ = 1 when α = 0.05 and n₁ = n₂

Figure 5

Secondary powers of the six boundary combinations (OF = O’Brien-Fleming, PO = Pocock, AH = Ad-Hoc, 1 = Primary, 2 = Secondary) for the primary and secondary endpoints as functions of Δ₁ for ρ = 0.4 and Δ₂ = 2 when α = 0.05 and n₁ = n₂

Figure 6

Secondary powers of the six boundary combinations (OF = O’Brien-Fleming, PO = Pocock, AH = Ad-Hoc, 1 = Primary, 2 = Secondary) for the primary and secondary endpoints as functions of Δ₁ for ρ = 0.4 and Δ₂ = 3 when α = 0.05 and n₁ = n₂