A Mixed Model for Binary Data: The Modified X 2 ‐test

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions P g in treatment group g(g = 1, 2, …, G ). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H 0 : E ( P 1 ) = E ( P 2 ) = … = E ( P G ). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects ) follows the X 2 ‐distribution.