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Consider the following type of data:

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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?

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You could model this data using BUGS (or JAGS or Stan), which would allow you to model the data with any distribution you want (binomial model for observations and beta for the parameters (p) of a binomial distribution) and in any way you want = basically you build model for p taking into account dependence between days + treatment. See the wealth of examples at:

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