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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.

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  • $\begingroup$ Welcome to Psychology.SE. Although the subject of the equations may fit the scope of this site, I am wondering if this may be more suited to math.stackexchange.com Whether it is better suited here or at math.stackexchange.com, what have you done to try and solve this and what are the results of your attempts? Where are your specific difficulties in solving in relation to "untangling the inverse CDFs"? $\endgroup$ – Chris Rogers Nov 4 '18 at 1:11
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    $\begingroup$ Although I'm in psychophysics, there's basically just one person that can help out here - StrongBad. They are not too active here, but they do seem to keep track of questions marked with the 'Psychophysics' tag, which is now included. They are knowledgeable on the topic and let's wait a bit until they show up. Otherwise, Math is surely another option. Please do not cross-post, but let the mod team migrate the question instead. $\endgroup$ – AliceD♦ Nov 4 '18 at 20:42
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    $\begingroup$ @AliceD I am not sure if there is an analytical expression unless you want to approximate the Gaussian CDF. If I am the best hope, then it is probably best to ask over at math.se, but if the OP can phrase the question in terms of the underlying integrals, that is probably better. $\endgroup$ – StrongBad Nov 6 '18 at 18:24
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    $\begingroup$ @AliceD I suggest keeping this question here, asking the underlying mathematical question there, and then posting an answer here from the gained knowledge. The current formulation of the question is really nice for cogsci.se. $\endgroup$ – StrongBad Nov 6 '18 at 18:25
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    $\begingroup$ @AliceD If it gets an answer over at math.se, than we/I/someone can translate from math speak to psych speak. $\endgroup$ – StrongBad Nov 6 '18 at 20:59

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