Concurrent borrowing methods

Concurrent borrowing is applied in the master protocol. It is usually used in the study design with multiple parallel arms, such as basket trial design and platform trial design, to borrowing information within multiple sub studies. These arms are of equal importance and are analyzed simultaneously without a chronologic order. Bayesian hierarchical model (BHM) and its extensions and multisource exchangeability models (MEMs) are commonly used in concurrent borrowing. The I-SPY2 trial was designed as a platform trial that used a BHM to adaptively borrow information between running arms [79].

Bayesian hierarchical model (BHM) was first proposed by Thall et al. [80], using variance in a hierarchical model to control the extent of borrowing (Table 2). BHM assumes the interested parameters in different trials comes form same normal distribution in which the variance τ reflects the heterogeneity. Specifically, let θ _j denote the interested parameters from arm j, j=1, …, J.j follows the normal distribution with mean μ. A standard hierarchical model is:

HN(․) Denotes half-normal distribution. τ is the heterogeneity measurement parameter of different arms which controls the amount of borrowing from all J arms. Larger τ represents larger heterogeneity, so that the less information would be borrowed. The posterior distribution of τ is updated through the Bayesian approach.

The extensions of BHM focus on the specification of τ and which arms should be included into borrowing. Calibrated Bayesian hierarchical model (CBHM), Bayesian hierarchical classification and information sharing (BaCIS), Bayesian cluster hierarchical model (BCHM), and clustered Bayesian hierarchical model (CLBHM) are the four commonly used extensions of BHM (Table 2).

Calibrated Bayesian hierarchical model (CBHM) was first proposed by Chu et al. [81]. CBHM uses the heterogeneity measurement function to determine the extent of information borrowing, instead of specifying distribution of variance τ. Therefore, CBHM has the advantage of controlling type I error inflation, especially in the study with a small amount of arms.

Bayesian hierarchical classification and information sharing (BaCIS) was first proposed by Chen et al. [82]. Compared to CBHM, BaCIS clusters the arms adaptively with bipartition instead of calculating the heterogeneity across all arms directly. BaCIS uses a latent variable γ to divide arms into two categories: effective cluster and ineffective cluster. Then BHM is used to borrow information within each cluster.

Bayesian cluster hierarchical model (BCHM) was first proposed by Chen et al. [83]. Compared to BaCIS, BCHM uses non-parametric Dirichlet process G to adaptively and dynamically determine the numbers of clusters instead of fixed two clusters. Therefore, BCHM could bring information more flexibility.

Clustered Bayesian hierarchical model (CLBHM) was first proposed by Jiang et al. [84]. This approach first clusters treatment arms into active and inactive clusters based on the posterior probability of the treatment effect, and then apply BHM to each cluster for information borrowing. CLBHM is simple to implement, and often yields better and more robust performance than more complicated BaCIS and BCHM methods [84].

Multisource exchangeability models (MEMs) were first proposed by Hobbs et al. in the scenario of multiple parallel arms existing [85], which specify all possible pairwise exchangeability models among arms by a symmetric matrix and weight above models (Table 2). The posterior estimate is the average of all models in which the weights are adaptively determined according to the similarity between arms.