Table \[tabella1\]. At first, we fix $\gamma$ to the SS expectation values (also listed in Table \[tabella1\]) and we derive $C$ and $\beta$ at each [*independent*]{} redshift (Sections 5.1 and 5.2). Later, to study the evolution of the normalization, we let $\gamma$ free to vary (Section 5.3).

We employ the routines provided by [@2007ApJ...665.1489K] to investigate the parameter space and to perform the linear regression in logarithmic space. The `linmix_err.pro` routine is applied to the single linear regression with $C$ and $\beta$ as parameters. The `mlinmix_err.pro` is adopted when we include $\gamma$ as free parameter. The independent variables are modeled as Gaussian distribution from which we extract the mean and standard deviation that we refer as the parameter best-fit value with its uncertainty. The Bayesian approach allows us to treat the intrinsic scatter, $\sigma$, estimated via the method of Monte Carlo Markov Chains (MCMC), as a free parameter. The MCMC is carried out by using Metropolis-Hastings (in `linmix_err.pro`) and Gibb Samplers (in `mlinmix_err.pro`) algorithms.

Comparison between simulated and observed scaling relations
===========================================================

Observational data sets
-----------------------

In this Section, we compare our theoretical predictions for the scaling relations with some observational results. For the mass-proxy relations, due to uncertainties associated 