  Searching for departures from general relativity (GR) in more than one
post-Newtonian (PN) phasing coefficients, called a \emph{multi-parameter test},
is known to be ineffective given the sensitivity of the present generation of
gravitational-wave (GW) detectors. Strong degeneracies in the parameter space
make the outcome of the test uninformative. We argue that Principal Component
Analysis (PCA) can remedy this problem by constructing certain linear
combinations of the original PN parameters that are better constrained by
gravitational-wave observations. By analyzing binary black hole events detected
during the first and second observing runs (O1 and O2) of LIGO/Virgo, we show
that the two dominant principal components can capture the essence of a
multi-parameter test. Combining five binary black hole mergers during O1/O2, we
find that the dominant linear combination of the PN coefficients obtained from
PCA is consistent with GR within the 0.38 standard deviation of the posterior
distribution. Furthermore, using a set of simulated \emph{non-GR} signals in
the three-detector LIGO-Virgo network with designed sensitivities, we find that
the method is capable of excluding GR with high confidence as well as
recovering the injected values of the non-GR parameters with good precision.
