  Aims. We focus on how to homogenize the comparison between galaxy samples
having different characteristics. We check the projections of the fundamental
metallicity relation (FMR) and the evolution of these projections between a
sample selected at $z\sim0$ (SDSS) and $z\sim0.7$ (VIPERS). We check, in
particular, whether and to what extent selection criteria can affect the
results. Methods. We checked the influence of different biases introduced
either by physical constraints (evolution of the luminosity function and
differences in the fraction of blue galaxies) or data selection (the
signal-to-noise ratio and quality of the spectra) on the FMR and its
projections. To separate the differences occurring due to the physical
evolution of galaxies with redshift from the false evolution mimed by these
biases, we first analyzed the effects of these biases individually on the SDSS
sample, and next, starting from the SDSS data, we built a VIPERS-equivalent $z
\sim 0$ sample, replicating the main characteristics of VIPERS sample at
$z\sim0.7$ for a fair comparison. Results. We found that the FMR projections
are all sensitive to biases introduced by the selection on S/N and the quality
flags of the emission line measurements in the spectra, especially the
$\left[\text{O{\,\sc{iii}}}\right]\lambda 4959$ line. The exception is the
metallicity versus the sSFR plane which is insensitive to these biases. The
evolution of the luminosity function introduces a bias only in the plane
metallicity versus the star formation rate (SFR) while the fraction of blue
galaxies has no impact on results.
