  In the near future, space-borne gravitational wave (GW) detector LISA can
open the window of low-frequency band of GW and provide new tools to test
gravity theories. In this work, we consider multi-parameter tests of GW
generation and propagation where the deformation coefficients are varied
simultaneously in parameter estimation and the principal component analysis
(PCA) method are used to transform posterior samples into new bases for
extracting the most informative components. The dominant components can be more
sensitive to potential departures from general relativity (GR). We extend
previous works by employing Bayesian parameter estimation and performing both
tests with injections of GR and injections of subtle GR-violated signals. We
also apply multi-parameter tests with PCA in the phenomenological test of GW
propagation. This work complements previous works and further demonstrates the
enhancement provided by the PCA method. Considering a supermassive black hole
binary system as the GW source, we show that subtle departures will be more
obvious in posteriors of PCA parameters. The departures less than $1\sigma$ in
original parameters can yield significant departures in first 5 dominant PCA
parameters.
