How do you convert a log odds ratio into a marginal effect?

Question

Basically, how do you convert a one unit change in $x_1$ to a $Z\%$ change in $Y$?

Answer 1

You can't (not without more information). The point here is that the logit / logistic is not a linear transformation. Therefore, you cannot get a constant correspondence between a starting percentage and a subsequent percentage even though you use the same log odds ratio to move from the one to the other each time. Here are a few demonstrative numbers for $1$-unit changes in $X$ with a log odds ratio of $1$ (thus, the log odds will simply increase by $1$):
\begin{array}{c} \text{starting %} &\text{starting lo} & &\text{subsequent lo} &\text{subsequent %} & &\text{% difference} \\ \hline 0.20 &-1.37\quad &\Rightarrow &-0.37\quad &0.40 & &0.20 \\ 0.50 &0\quad\ &\Rightarrow &1.0\ \ &0.73 & &0.23 \\ 0.90 &2.20 &\Rightarrow &3.20 &0.96 & &0.06 \end{array}

Alternatively, if you simply want to compare two groups (which could be coded $0$ and $1$, for 'control' and 'treatment'), you would need to know the base rate in the control group.