  The small $x$ behavior of the flavor non-singlet $g_{1}$ structure function
is analysed numerically by taking into account the all-order resummation of
$\alpha_{s} \ln^{2}x $ terms. We include a part of the next-to-leading
logarithmic corrections coming from the resummed ``coefficient function'' which
are not considered in the literatures to respect the factorization scheme
independence. The resummed coefficient function turns out to give unexpectedly
large suppression effects over the experimentally accessible range of $x$ and
$Q^{2}$. This fact implies that the higher order logarithmic corrections are
very important for $g_{1}$ in the small $x$ region.
