We compare the constraints from two (2019 and 2021) compilations of HII starburst galaxy (HIIG) data and test the model-independence of quasar angular size (QSO) data using six spatially flat and non-flat cosmological models. We find that the new 2021 compilation of HIIG data generally provides tighter constraints and prefers lower values of cosmological parameters than those from the 2019 HIIG data. QSO data by themselves give relatively model-independent constraints on the characteristic linear size, $l_{rm m}$, of the QSOs within the sample. We also use Hubble parameter ($H(z)$), baryon acoustic oscillation (BAO), Pantheon Type Ia supernova (SN Ia) apparent magnitude (SN-Pantheon), and DES-3yr binned SN Ia apparent magnitude (SN-DES) measurements to perform joint analyses with HIIG and QSO angular size data, since their constraints are not mutually inconsistent within the six cosmological models we study. A joint analysis of $H(z)$, BAO, SN-Pantheon, SN-DES, QSO, and the newest compilation of HIIG data provides almost model-independent summary estimates of the Hubble constant, $H_0=69.7pm1.2 rm{km s^{-1} Mpc^{-1}}$, the non-relativistic matter density parameter, $Omega_{rm m_0}=0.293pm0.021$, and $l_{rm m}=10.93pm0.25$ pc.
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