  We use measurements of 59/58 quasars (QSOs), over a redshift range
$0.0041\leq z \leq 1.686$, to do a comparative study of the radius--luminosity
($R-L$) and X-ray$-$UV luminosity ($L_X-L_{UV}$) relations and the implication
of these relations for cosmological parameter estimation. By simultaneously
determining $R-L$ or $L_X-L_{UV}$ relation parameters and cosmological
parameters in six different cosmological models, we find that both $R-L$ and
$L_X-L_{UV}$ relations are standardizable but provide only weak cosmological
parameter constraints, with $L_X-L_{UV}$ relation data favoring larger current
non-relativistic matter density parameter $\Omega_{m0}$ values than $R-L$
relation data and most other available data. We derive $L_X-L_{UV}$ and $R-L$
luminosity distances for each of the sources in the six cosmological models and
find that $L_X-L_{UV}$ relation luminosity distances are shorter than $R-L$
relation luminosity distances as well as standard flat $\Lambda$CDM model
luminosity distances. This explains why $L_X-L_{UV}$ relation QSO data favor
larger $\Omega_{m0}$ values than do $R-L$ relation QSO data or most other
cosmological measurements. While our sample size is small and only spans a
small $z$ range, these results indicate that more work is needed to determine
whether the $L_X-L_{UV}$ relation can be used as a cosmological probe.
