# Probability distribution for a binomial proportion 'derived' from serially dependent data

Consider the following type of data:

This is data from a single-case experiment: an experiment in which one entity (i.e. one person) is observed repeatedly over time (cf. measurement times 1 to 20). From a certain measurement point on, an experimental manipulation (i.e. a treatment) is introduced which divides the whole experiment into a baseline phase (A) and a treatment phase (B).

I was wondering if it would be possible to derive a binomial proportion ,reflecting the treatment effect, by making ordinal comparisons between the observed data points of both phases in such a way that a bèta-binomial model would be appropriate for significance testing and constructing confidence intervals for this proportion.

To give a concrete example:

One could calculate the median of phase A and count the number of B observations that exceed the A phase median. This value divided by the number of comparisons (i.e. trials) could then be considered as a binomial proportion that reflects the treatment effect of the experiment (with 0.50 indicating no treatment effect and 1 indicating a maximal effect).

Given the serial dependence of the data, the individual trials would not be independent, making the binomial distribution inadequate. However, I was wondering if a bèta-binomial model would be adequate to model the dependencies between subsequent trials.

If so, how can I estimate or model the correlations between the individual trials using the bèta distribution?