Lets say I have an equal variance SDT framework with an equal number of target present and target absent trials:
$d'=Z^{-1}(HR)-Z^{-1}(FAR)$
$C=\frac{Z^{-1}(HR)+Z^{-1}(FAR)}{-2}$
$ACC=\frac{HR+(1-FAR)}{2}$
where $HR=$ hit rate, $FAR=$ false alarm rate, $d'=$ sensitivity, $C=$ bias, and $ACC=$ accuracy.
If I know the value of $C$ and $ACC$, how do I solve for $HR$ and $FAR$?
To put it another way. In ROC space, I know my iso-bias curve and I know my iso-accuracy curve, and I want to know the coordinates where they intersect in terms of $HR$ and $FAR$.
It seems that I should take my equations for $C$ and $ACC$ in terms of $HR$ and $FAR$, and work them to give me $HR$ and $FAR$ in terms of $C$ and $ACC$, but I haven't managed to successfully untangle the inverse CDFs in order to do that.
Edit:
I can rearrange the equation for $ACC$, to express $FAR$ in terms of $HR$ and $ACC$:
$FAR=HR+1-2ACC$
And then I can plug that into the equation for $C$:
$C=\frac{Z^{-1}(HR)+Z^{-1}(HR+1-2ACC)}{-2}$
And rearrange that a bit:
$Z^{-1}(HR)=-2C-Z^{-1}(HR+1-2ACC)$
And then I could do this:
$HR=Z(-2C-Z^{-1}(HR+1-2ACC))$
And then I get stuck, because I'm not sure how to "free" the $HR$ buried on the right-hand side.
