  In recent years information on the transversity distribution $h_1$ has been
obtained combining the Collins asymmetry results from semi-inclusive deep
inelastic scattering (SIDIS) data on transversely polarized nucleon targets
with the information on the fragmentation function of a transversely polarized
quark from the asymmetries measured in $e^+e^-$ annihilation into hadrons. An
alternative method was proposed long time ago, which does not require the
$e^+e^-$ data, but allows one to get ratios of the $u$ and $d$ quark
transversity distributions from the SIDIS data alone. The method utilizes the
ratio of the difference of the Collins asymmetries of positively and negatively
charged hadrons produced on transversely polarized proton and deuteron targets.
We have applied this method to the COMPASS proton and deuteron data, and
extracted the ratio $h_1^d/h_1^u$. The results are compared to those obtained
in a previous point--by--point extraction based both on SIDIS and $e^+e^-$
data.
