My textbook simply use the turn ratio as the mechanical advantage, but I'm not sure how that works.
My thinking: If the worm has a radius of $r$, and a turn ratio $n$ (turns/meter), the input distance is $2\pi r$, and the output distance is $1/n$ for each turn of the worm.
Therefore, mechanical advantage is the former divided by the latter which is $2\pi r \times n$. I'm not sure how this relates to the turn ratio. Is their definition of mechanical advantage different?