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We consider a system of dependent Poisson variables, where each variable isthe sum of an independent variate and a common variate. It is the commonvariate that creates the dependence. Within this system, a test of independencemay be constructed where the null hypothesis is that the common variate isidentically zero. In the present paper, we consider the maximum log likelihoodratio test. For this test, it is well-known that the asymptotic distribution ofthe test statistic is an equal mixture of zero and a chi square distributionwith one degree of freedom. We examine a Bartlett correction of this test, inthe hope that we will get better approximation of the nominal size formoderately large sample sizes. This correction is explicitly derived, and itsusefulness is explored in a simulation study. For practical purposes, thecorrection is found to be useful in dimension two, but not in higherdimensions.

Bartlett correction of an independence test in a multivariate Poisson model