I am measuring how humans perform a perceptual test. The graph shows the proportion of correct responses as a function of an independent variable x for one person.
The model includes a lapse parameter that corresponds to the asymptotic proportion for x large; for this person the estimated lapse parameter is around 0.94.
Several participants performed the perceptual test and I have an estimation of the lapse parameter for each participant. For many participants the estimated lapse parameter is 1.
How should I perform statistics across participants for the lapses given that they are not normally distributed? Also they are probabilities, but I don't have a size and number of successes for each one. For example,
how can I calculate confidence intervals for the mean lapse rate across participants?
how can I perform a t-test or ANOVA type of analysis for the lapses (the participants performed the perceptual test in different conditions)?