Distributed Analysis for Vertically Sample Clauses

Distributed Analysis for Vertically. Partitioned Data When data are vertically partitioned, to simplify our problem, we assume the data are from two sites/institutions. Suppose institution 1 has a data set A for n subjects and institution 2 has another data set B for the same subjects, thus X = (A, B). We further make an assumption that both institutions know the outcome variable Y . Considering a Figure 1.5: Distributed logistic regression on data that are horizontally distributed across K = 3 sites linear regression on Y with X, the goal is to obtain βˆ of Equation 1.2 when the data are vertically partitioned. Note that, , and XT X = AT A AT B   BT A BT B AT Y  XT Y =  BT Y  Du et al. (2004) develop secure multi-party computation protocols for privacy-preserving calculations of matrix product and matrix inverse. By using their secure technique, we can obtain βˆ without passing A to institute 1 and passing B to institute 2. An alternative approach of distributed linear regression is proposed by Xxxxx et al. (2004). Details will be discussed in Chapter 4. As for logistic regression, Slavkovic et al. (2007) propose an algorithm to aggregate information among institutions through secure multi-party computation protocols. How- ever, the algorithm induces very high computational cost and is not scalable as K is large. Li et al. (2015) publish a distributed algorithm to solve the maximum likelihood problem by dual optimization. Their method is shown to be more efficient from the simulation study.