  We compare, with data from the quasars, the Hubble parameter measurements,
and the Pantheon+ type Ia supernova, three different relations between X-ray
luminosity ($L_X$) and ultraviolet luminosity ($L_{UV}$) of quasars. These
three relations consist of the standard and two redshift-evolutionary
$L_X$-$L_{UV}$ relations which are constructed respectively by considering a
redshift dependent correction to the luminosities of quasars and using the
statistical tool called copula. By employing the PAge approximation for a
cosmological-model-independent description of the cosmic background evolution
and dividing the quasar data into the low-redshift and high-redshift parts, we
find that the constraints on the PAge parameters from the low-redshift and
high-redshift data, which are obtained with the redshift-evolutionary
relations, are consistent with each other, while they are not when the standard
relation is considered. If the data are used to constrain the coefficients of
the relations and the PAge parameters simultaneously, then the observations
support the redshift-evolutionary relations at more than $3\sigma$. The Akaike
and Bayes information criteria indicate that there is strong evidence against
the standard relation and mild evidence against the redshift-evolutionary
relation constructed by considering a redshift dependent correction to the
luminosities of quasars. This suggests that the redshift-evolutionary
$L_X$-$L_{UV}$ relation of quasars constructed from copula is favored by the
observations.
