  In multicenter research, individual-level data are often protected against
sharing across sites. To overcome the barrier of data sharing, many distributed
algorithms, which only require sharing aggregated information, have been
developed. The existing distributed algorithms usually assume the data are
homogeneously distributed across sites. This assumption ignores the important
fact that the data collected at different sites may come from various
sub-populations and environments, which can lead to heterogeneity in the
distribution of the data. Ignoring the heterogeneity may lead to erroneous
statistical inference. In this paper, we propose distributed algorithms which
account for the heterogeneous distributions by allowing site-specific nuisance
parameters. The proposed methods extend the surrogate likelihood approach to
the heterogeneous setting by applying a novel density ratio tilting method to
the efficient score function. The proposed algorithms maintain the same
communication cost as the existing communication-efficient algorithms. We
establish a non-asymptotic risk bound for the proposed distributed estimator
and its limiting distribution in the two-index asymptotic setting which allows
both sample size per site and the number of sites to go to infinity. In
addition, we show that the asymptotic variance of the estimator attains the
Cram\'er-Rao lower bound when the number of sites is in rate smaller than the
sample size at each site. Finally, we use simulation studies and a real data
application to demonstrate the validity and feasibility of the proposed
methods.
