  The discovery of gravitational waves (GWs) provides an unprecedented arena to
test general relativity, including the gravitational Lorentz invariance
violation (gLIV). In the propagation of GWs, a generic gLIV leads to
anisotropy, dispersion, and birefringence. GW events constrain the anisotropic
birefringence particularly well. Kosteleck\'y and Mewes (2016) performed a
preliminary analysis for GW150914. We improve their method and extend the
analysis systematically to the whole GW transient catalog, GWTC-1. This is the
first global analysis of the spacetime anisotropic Lorentzian structure with a
catalog of GWs, where multiple events are crucial in breaking the degeneracy
among gLIV parameters. With the absence of abnormal propagation, we obtain new
limits on 34 coefficients for gLIV in the nonminimal gravity that surpass
previous limits by $\sim 10^2$-$10^5$.
