Figure 2
Visual representation of one-shot distributed algorithm for hurdle regression (ODAH). In the initialization round, coefficient (\(\hat{\beta }_{i} , \hat{\gamma }_{i}\)) and variance (\(\hat{\sigma }_{i}^{2} , \hat{\tau }_{i}^{2}\)) estimates from fitting separate hurdle models at each collaborating site are sent to the lead site; these estimates are then used together with lead site estimates in a meta-analysis to produce initial estimates (\(\overline{\beta }, \overline{\gamma }\)) for ODAH, which are sent to each collaborating site. In the surrogate likelihood estimation round, first-order (\(\nabla L_{1i} , \nabla L_{2i}\)) and second-order (\(\nabla^{2} L_{1i} , \nabla^{2} L_{2i}\)) gradients are computed at each site, evaluated at the received initial estimates and sent to the lead site. These gradients are used in conjunction with data from the lead site to construct surrogate likelihood functions \(\tilde{L}_{1} \left( \beta \right)\) and \(\tilde{L}_{2} \left( \gamma \right)\), which are then maximized to produce surrogate maximum likelihood estimates \(\tilde{\beta }\) and \(\tilde{\gamma }\).
