  The double logarithmic terms $\alpha_{s} \ln^{2}x $ are important to predict
precisely the small $x$ behavior of the spin structure function $g_{1}$. We
numerically analyze the evolution of the flavor non-singlet $g_{1}$ including
the all-order resummed effect of these terms. It is pointed out that the
next-to-leading logarithmic corrections produce an unexpectedly large
suppression factor over the experimentally accessible range of $x$ and $Q^{2}$.
This implies that the next-to-leading logarithmic contributions are very
important in order to obtain a definite prediction.
