  We consider a system of dependent Poisson variables, where each variable is
the sum of an independent variate and a common variate. It is the common
variate that creates the dependence. Within this system, a test of independence
may be constructed where the null hypothesis is that the common variate is
identically zero. In the present paper, we consider the maximum log likelihood
ratio test. For this test, it is well-known that the asymptotic distribution of
the test statistic is an equal mixture of zero and a chi square distribution
with one degree of freedom. We examine a Bartlett correction of this test, in
the hope that we will get better approximation of the nominal size for
moderately large sample sizes. This correction is explicitly derived, and its
usefulness is explored in a simulation study. For practical purposes, the
correction is found to be useful in dimension two, but not in higher
dimensions.
