# Statistics for a probability parameter

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)?

1. The way I would go about it is to use a Bayesian multilevel model, to fit psychometric functions for all participants in one go. In this case you could set $$\text{Beta} (a,b)$$ as a prior for your lapse parameter. Say you estimate $$a$$ and $$b$$ via MCMC sampling, then you can easily calculated the expected lapse rate as $$\frac{a}{a+b}$$, and use any methods (percentile, HPDI) to calculate a confidence interval. In practice, you'd need also an hyperprior for $$a$$ and $$b$$, which could be a Gamma distribution (since they both can take only positive values). You can find an example of how to run this approach using Stan here (use left/right arros keys to move through the slides; see also here for a repository containing all the example code).