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A Mixed Model for Binary Data: The Modified X 2 ‐test
Author(s) -
Hecker Hartmut,
Wiese Birgitt
Publication year - 1994
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710360703
Subject(s) - bernoulli trial , mathematics , bernoulli's principle , statistics , test statistic , binary data , statistic , binary number , bernoulli distribution , constant (computer programming) , statistical hypothesis testing , random variable , computer science , thermodynamics , physics , arithmetic , programming language
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.