Within-subjects proportional data

I may be missing something very obvious, but I am stuck on the following
problem and hope someone can advise:

I want to do a psychology experiment where participants will be tested on
their ability to detect targets under two conditions. One each trial they
see one target paired with one distractor. They can either pick the target
(correct), pick the distractor (false alarm) or pick nothing (miss).

I am predicting that one condition will produce more correct hits, but am
also interested in whether condition affects the proportion of each type of
error (false alarm or miss). There are 12 trails per participant. I can't
use signal detection because there are 3 choices on each trail, and I can't
use chi-square because there are multiple trails per participant.

Testing the number of correct hits in each condition is easy enough by
calculating number of correct identifications out of 12 for each
participant. I could also separately compare the number of false alarms and
the number of misses between conditions. But obviously these measures are
not independent, and apart from the problem of multiple testing, what I
really want to know is whether the proportion of each type of error differs
between conditions.

I could take the errors for each participant and calculate the proportion of
false hits to misses in each condition and use these proportions for
analysis. However, the overall number of errors will be different for each
participant, and may be very small, meaning that proportions will tend to
take extreme values and the range of variation in proportions is restricted.
Also one condition is predicted to produce a smaller number of errors than
the other, meaning it will have more extreme proportions. I presume I should
do an arcsin transformation on the proportions, but I am concerned that the
test would still be unsound.

Does this sound like a serious violation of t-test assumptions, and should I
abandon this design and go back to the drawing board? Or is there some other
way of treating the data that I have missed? I would rather make sure that
the data will be analysable before collecting it.

Any thoughts appreciated...

Janet


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