Semiparametric clustered overdispersed multinomial goodness-of-fit of   log-linear models
  Traditionally, the Dirichlet-multinomial distribution has been recognized as a key model for contingency tables generated by cluster sampling schemes. There are, however, other possible distributions appropriate for these contingency tables. This paper introduces new test-statistics capable to test log-linear modeling hypotheses with no distributional specification, when the individuals of the clusters are possibly homogeneously correlated. The estimator for the intracluster correlation coefficient proposed in Alonso-Revenga et al. (2016), valid for different cluster sizes, plays a crucial role in the construction of the goodness-of-fit test-statistic. 
Juana M. Alonso-Revenga, Nirian Martin, Leandro Pardo
stat.ME