For various reasons, I have a crappy experimental design that I am forced to deal with. I present subjects with 20 yes/no trials, out of which 5 are no-signal catch trials and 15 are signal trials. Each signal presentation has a unique level with unequal spacing (as small as one in arbitrary units and as great as 5). All subjects run the trials in the same order (i.e., the 5th trial is always at signal level, and the 11th and the 6th trial are always catch trials).
To make things more complicated, at the population level, and likely at the individual level, the criterion is not stable such that the probability of saying yes on a no-signal trial depends on the catch trial number (the 1st catch trial has a probability of yes of about 0.05 and the 5th catch trial has a probability of yes of about 0.2). From this mess, I need a single number representing threshold. I don't really care what the threshold corresponds to (e.g., the signal level leading to 50% yes, 50% correct, or a d' of 1 would all be fine). Ideally, the method would provide a confidence interval on the estimate at the individual subject level.
I was thinking of using a Spearman-Karber type approach, but the unequal step sizes seems problematic (along with the variable criteria). I also don't understand the history of Spearman-Karber (i.e., where was it first published and how it relates to Spearman 1908) and kind of hope that maybe in the past 100 years we have improved on things.