I'm performing a psychophysical experiment where an observer has to detect the presence of an auditory stimulus (yes/no paradigm). Sometimes the auditory stimulus is paired with an irrelevant visual stimulus. In all, I'm presenting four types of trials: catch trials (no stimulus whatsoever), auditory (target) trials, visual (nontarget) trials, and audiovisual (target nontarget) trials.

I use signal detection theory to convert hits (H; p(Y|auditory)) into a bias-free "probability correct" measure by convolving them with false alarms (FA):

p(c) = z^-1((z(H)-z(FA))/2)

When I do this for auditory alone detection, my False Alarm is the proportion of "yes" responses during catch trials (p(Y|catch)). This is easy given that the assumption of SDT is that there is a noise distribution and a signal plus noise distribution. However, when I present a visual stimulus, observers might make an errant response to a visual stimulus, just as they might make an errant response during a catch trial. I would like to account for these potential responses if possible.

I'm not sure what the best "False Alarm" measure would be when a visual stimulus is present. I've considered using releases during just catch trials, just visual trials, and some combination ([release during catch trial releases during visual trials]/[total catch trials total visual trials]; or p(Y|visual)*p(Y|catch)).

What is your opinion about this stats people? I am unsatisfied with using just visual release rate. However, I can make a case for using just catch trials OR some combination of catch and visual trials. On one hand, observers are trained to release for auditory ONLY, which would argue for using catch trials. On the other hand, observers could possibly release for visual trials, in which case, I would want to account for those releases when calculating p(c) for audiovisual trials.

Stated differently, should the internal representation during visual trials be represented in the "noise" distribution. And if so, how?