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

Observations favor the redshift-evolutionary $L_X$-$L_{UV}$ relation of quasars from copula