  We give sufficient conditions for a positive stochastic matrix to be similar
and strong shift equivalent over $\mathbb{R}_+$ to a positive doubly stochastic
matrix through matrices of the same size. We also prove that every positive
stochastic matrix is strong shift equivalent over $\mathbb{R}_+$ to a positive
doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive
stochastic matrices over $\mathbb{R}$ with positive trace and the set of
nonzero spectra of positive doubly stochastic matrices over $\mathbb{R}$ are
identical. We exhibit a class of $2\times 2$ matrices, pairwise strong shift
equivalent over $\mathbb R_+$ through $2\times 2$ matrices, for which there is
no uniform upper bound on the minimum lag of a strong shift equivalence through
matrices of bounded size. In contrast, we show for any $n\times n$ primitive
matrix of positive trace that the set of positive $n\times n$ matrices similar
to it contains only finitely many SSE-$\mathbb R_+$ classes.
