\begin{table}%t1
\caption{\label{tab:results}Correlations of flux ratios $\mathcal{R}$ with SN~Ia absolute magnitudes
and standard deviations of Hubble diagram fits.}
%\centerline
 {\small \begin{tabular}{ccccrcccc}
\hline \hline
Correction & \multicolumn{3}{c}{Correlation with absolute magnitude} & & \multicolumn{3}{c}{Hubble diagram residual scatter} \\
parameter(s)    & Training & Validation & Combined & $\gamma$~~~~~ & Training & Validation        & Combined & $\sigma_{\rm core}$ \\
\hline
$\mathcal{R}_{642/443}$  & 0.94 & 0.96 & 0.95 &  $3.5 \pm 0.2$  & $0.130 \pm 0.018$ & $0.134 \pm 0.018$ & $0.128 \pm 0.012$ & 0.108 \\
$\mathcal{R}_{642/417}$  & 0.95 & 0.91 & 0.91 &  $4.9 \pm 0.2$  & $0.114 \pm 0.016$ & $0.185 \pm 0.025$ & $0.166 \pm 0.016$ & 0.162 \\
$\mathcal{R}_{772/437}$  & 0.92 & 0.94 & 0.93 &  $7.3 \pm 0.3$  & $0.142 \pm 0.020$ & $0.160 \pm 0.021$ & $0.152 \pm 0.014$ & 0.125 \\
$\mathcal{R}_{642/512}$  & 0.90 & 0.95 & 0.93 &  $4.7 \pm 0.3$  & $0.162 \pm 0.022$ & $0.146 \pm 0.020$ & $0.154 \pm 0.015$ & 0.152 \\
$\mathcal{R}_{728/398}$  & 0.90 & 0.93 & 0.91 &  $7.9 \pm 0.3$  & $0.162 \pm 0.022$ & $0.168 \pm 0.022$ & $0.172 \pm 0.016$ & 0.138 \\
\hline
$c, \mathcal{R}^c_{642/519}$  & 0.96 & 0.96 & 0.96 &  $3.5 \pm 0.3$  & $0.106 \pm 0.015$ & $0.129 \pm 0.018$ & $0.119 \pm 0.011$ & 0.128 \\
$c, \mathcal{R}^c_{577/642}$  & 0.95 & 0.95 & 0.95 & $-1.4 \pm 0.1$  & $0.115 \pm 0.016$ & $0.150 \pm 0.020$ & $0.135 \pm 0.013$ & 0.126 \\
$c, \mathcal{R}^c_{642/536}$  & 0.95 & 0.96 & 0.95 &  $2.3 \pm 0.2$  & $0.116 \pm 0.016$ & $0.134 \pm 0.018$ & $0.125 \pm 0.012$ & 0.126 \\
$c, \mathcal{R}^c_{676/642}$  & 0.94 & 0.93 & 0.93 & $-4.2 \pm 0.5$  & $0.131 \pm 0.019$ & $0.178 \pm 0.024$ & $0.157 \pm 0.015$ & 0.163 \\
$c, \mathcal{R}^c_{642/443}$  & 0.95 & 0.96 & 0.96 &  $3.2 \pm 0.3$  & $0.121 \pm 0.017$ & $0.125 \pm 0.017$ & $0.119 \pm 0.011$ & 0.104 \\
\hline
$c$, $x_1$         & 0.91  & 0.93  & 0.92 & ...~~~~~  & $0.154 \pm 0.022$ & $0.171 \pm 0.023$ & $0.161 \pm 0.015$ & 0.156 \\
\hline
\end{tabular}}
\medskip
Notes: $\gamma$ is a fit parameter in the distance modulus $\mu_B = (m_B - M^\prime) + \gamma \mathcal{R}$, and $\sigma_{\rm core} = 1.4826\times {\rm median}(|\Delta\mu_B - {\rm median}(\Delta\mu_B)|)$.
\end{table}