  The Laser Interferometer Space Antenna (LISA) will provide us with a unique
opportunity to observe the early inspiral phase of supermassive binary black
holes (SMBBHs) in the mass range of $10^5-10^6\,M_{\odot}$, that lasts for
several years. It will also detect the merger and ringdown phases of these
sources. Therefore, such sources are extremely useful for multiparameter tests
of general relativity (GR), where parametrized deviations from GR at multiple
post-Newtonian orders are simultaneously measured, thus allowing for a rigorous
test of GR. However, the correlations of the deviation parameters with the
intrinsic parameters of the system make multiparameter tests extremely
challenging to perform. We demonstrate the use of principal component analysis
(PCA) to obtain a new set of deviation parameters, which are best-measured
orthogonal linear combinations of the original deviation parameters. With the
observation of an SMBBH of total redshifted mass,
$\sim\mathrm{7\times10^5\,M_{\odot}}$ at a luminosity distance of 3 Gpc, we can
estimate the five most dominant PCA parameters, with 1-$\sigma$ statistical
uncertainty of $\lesssim 0.2$. The two most dominant PCA parameters can be
bounded to $\sim \mathcal{O}(10^{-4})$, while the third and fourth-dominant
ones to $\sim \mathcal{O}(10^{-3})$. Measurement of the PCA parameters with
such unprecedented precision with LISA makes them an excellent probe to test
the overall PN structure of the GW phase evolution.
