Marginal Log-Linear Models

Using the marginal log-linear parameterization of joint distributions developed in Chap. 7 , and their smoothness property established in Sect. 9.3 , this chapter defines marginal log-linear models as the assumption that some of the marginal log-linear parameters are zero, and proves many of their attractive properties. A detailed discussion of the algorithms available to determine maximum likelihood estimates for marginal log-linear models is given. These algorithms are developed based on a discussion of the Newton-Raphson and the Fisher scoring algorithms. The maximum likelihood estimates are shown to be identical, whether the available data were generated by a Multinomial or by a multivariate Poisson sampling procedure. As the main areas of application, Markov models associated with DAGs and with chain graphs are considered. Path models for categorical data as marginal log-linear models are also defined and smoothness results are obtained.