multivariate poisson hidden markov models for analysis of spatial ...

The loglinear model is one special case of Generalized Linear Models (GLM) for
Poisson distributed data (Agresti, 2002 and Brijs, 2002). Further, loglinear analysis can
be considered as an extension of the two-way contingency table to where the
conditional relationship between two or more discrete categorical variables is analyzed
by taking the natural logarithm of the cell frequencies within the contingency table.
Loglinear models are generally used to summarize multi-way contingency tables that
involve three or more variables. Therefore, loglinear models are very useful to evaluate
the association between variables. PROC CATMOD procedure in SAS software
(SAS/STAT, 2003) can be used to fit the models.
The fundamental strategy in loglinear analysis involves fitting models to the observed
frequencies in the cross-tabulation of categorical variables. The models can then be
represented by a set of expected frequencies that may or may not look like the observed
frequencies. Different models can be described in terms of marginal models that they fit
and in terms of the constraints they impose on the associations that are present in the
data. Using expected frequencies, different models can be fitted and compared that are
hierarchical to one another. The idea of modelling is then to choose a preferred model,
which is the most suitable model that fits the data. The choice of the preferred model is
based on a formal comparison of goodness-of-fit statistics (likelihood ratio test)
associated with models that are related hierarchically (i.e. models containing higher
order terms also implicitly include all lower order terms).
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