$\beta$ and $\mathcal{M}$ with the redshift, splitting the samples in redshift bins and performing the fit separately for each one. In this situation, we found evidence of evolution in the parameter $\beta$ and also in $\sigma_{\text{int}}$.

While this paper was in preparation two articles addressing the issue of using $\chi^2$ with unknown variances appeared in the arXiv. The first one [@Kim2011] focused on the question of the determination of $\sigma_\text{int}$, using simulated data for the distance modulus. The second one [@March2011] studied the SALT2 case and proposed a sophisticated Bayesian analysis. We did not find in our work with the SDSS first year compilation any kind of catastrophic biases of parameters when adopting the likelihood approach, a possibility suggested in [@March2011].

In summary, we used current data to compare the traditional $\chi^2$ and the likelihood approaches to determine best fit and confidence regions from SN Ia. We argued that when the variance is not completely known, minimizing the traditional $\chi^2$ is not formally equivalent to maximizing the likelihood function, since the normalization of the likelihood, assumed Gaussian, is also a function of parameters to be determined. We conclude suggesting the adoption of the likelihood framework instead of the traditional $\chi^2$ 