Bayesian credible intervals for binomial proportions in a single patient trial

Practitioners are often asking if the treatment successfully improved performance. Many times this question is directed towards the outcome of a single individual. In this article, we develop a method to assess the improvement of a single individual who is administered a test of percent correct at pre-treatment and post-treatment. A Bayesian approach is taken where the number correct is modelled as a binomial random variable and the percent correct is set to a beta prior distribution. The first model assumes percent correct at pre-test is equal to the percent correct at post-test and the posterior predictive distribution is used to evaluate the change in the number correct. We subsequently model the proportions correct at pre-test and post-test as unequal. The second model then assumes independent proportions and the third assumes correlated beta distributions for the two proportions. 95% credible intervals are calculated for the various methods for number of correct at post-test given a particular level at pre-test. An example using data from a cochlear implant clinical trial is presented where clinicians recorded percent correct in a consonant-nucleus-consonant test.