Definition of hits anf fale alarms using likelihood criterion

The probability of hits and false alarm should be defined as

\[p_{H}=\int_{x \in R} f_s(x) \,dx \]

\[p_{FA}=\int_{x \in R} f_n(x) \,dx\]

where \(R=\{x: \Lambda(x) > \beta\}\).

Hit and false alarm for unequal variance using the likelihood criterion (for \(\sigma >1\))

\[p_{H}= 1 - \left(\Phi\left(\frac{c_2-d'}{\sigma}\right) - \Phi \left(\frac{c_1-d'}{\sigma}\right) \right)\]

\[p_{FA}= 1 - \left(\Phi(c_2)- \Phi(c_1)\right)\]

Hit and false alarm for unequal variance using a sensory criterion

But this are not the expression used to deri The zROC curve is a straight line only for the decision rule using the sensory criterion.

\[p_{FA} = 1 - \Phi(c) = \Phi(-c)\]

\[p_{H} = 1 - \Phi(\frac{c-d'}{\sigma}) = \Phi(\frac{d'-c}{\sigma})\]

zROC for unequal variance

For what I understood, the typical derivation of a linear relationship between the z for hits and false alarms is as follows

\(\Phi^{-1}(p_{FA}) = -c\)

\(\Phi^{-1}(p_{H}) = \frac{d'-c}{\sigma}\)

\(\Phi^{-1}(p_{H}) = \frac{1}{\sigma}d' + \frac{1}{\sigma}\Phi^{-1}(p_{FA})\)

which is using the definition of hits and false alarms using the sensory criterion rather than the likelihood criterion.

When one uses the optimal decision rule (the likelihood ratio), the zROC curves are not linear (see last part of this http://www.dlinares.org/sdtyesnofreqnormalunequal.html).