  An estimation of the higher twist contribution to the polarized nonsinglet
structure function $g_1$ is given. Its first moment is generally assumed to be
small but this could be due to a cancellation of its contribution in different
Bjorken-$x$ ranges. We estimate the magnitude of $g_1^{tw4}$ as a function of
$x$ in the framework of the renormalon method. The calculated
higher-twist-correction to $g_1$ turn out to be in fact small for all $x$
values. The correction to the determination of the nonsinglet matrix element
$d_{NS}^{(2)}$ from data for transversely nucleon polarization due to the
calculated higher twist effects is of order $10\%$.
