An Approach to Determine Sample Size and to Allocate Sample Size for a Specific Region in a Multiregional Trial for Survival (Time-to-Event) Data under Accelerated Failure Time Model

Abstract

For the time-to-event outcome, current methods for sample determination are based on the proportional hazard model. However, if the proportionality assumption fails to capture the relationship between the hazard time and covariates, the proportional hazard model is not suitable to analyze survival data. The accelerated failure time (AFT) model is an alternative method to deal with survival data. In this paper, we address the issue that the relationship between the hazard time and the treatment effect is satisfied with the AFT model to design a multiregional trial. The log-rank test is employed to deal with the heterogeneous effect size among regions. The test statistic for the overall treatment effect is used to determine the total sample size for a multiregional trial, and the proposed criteria are used to rationalize partition sample size to each region.

Keywords:

• Heterogeneity
• Multiregional study
• Regional sample size

Mathematics Subject Classification:

• Primary 62K99
• Secondary 62-07

Appendix

In Appendix, we show the mathematical expressions of assurance probability (AP) for criteria (i), (ii), and (iii).

The AP of criterion (i) can be expressed as where (a, b) is the solution of u = c₁v + c₂ and c₃u + c₄v = c₅. , , , , c₅ = − z_1 − ι.

Similarly, the APs of criteria (ii) and (iii) can be represented by where (a, b) is the solution of u = c₁v + c₂ and c₃u + c₄v = c₅. c₁ = ρ, , , , c₅ = − z_1 − ι. where Z_{1 : K − 1} is the Z-statistic for min (D₁, …, D_i, …, D_K − 1) and p_{1 : K − 1} is the proportion of the min (D₁, …, D_i, …, D_K − 1) region. Φ( · ) is the c.d.f of the standard normal distribution and φ( · ) is the p.d.f of the standard normal distribution.

Supplement

In Supplement, we show the complete derivation of assurance probability for criteria (i), (ii), and (iii). where (a, b) is the solution of u = c₁v + c₂ and c₃u + c₄v = c₅. , , , , c₅ = − z_1 − ι.

Thus, can be obtained. where (a, b) is the solution of u = c₁v + c₂ and c₃u + c₄v = c₅. c₁ = ρ, , , , c₅ = − z_1 − ι.

Thus, can be obtained. where Z_{1 : K − 1} is the Z-statistic for min (D₁, …, D_i, …, D_K − 1) and p_{1 : K − 1} is the proportion of the min (D₁, …, D_i, …, D_K − 1) region. Φ( · ) is the c.d.f of the standard normal distribution and φ( · ) is the p.d.f of the standard normal distribution.