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

For the case of two categorical variables, each with two levels ( 2× 2 table with present
and absent of the species), to evaluate if an association exists between the variables the
following model can be used:
A B AB
ln( F ) = µ + γ + γ + γ . (5.35)
ij i j ij
ln( F ij
) is the log of the expected cell frequency of the cases for cell i,
j in the
contingency table.
µ is the overall mean of the natural log of the expected frequencies
γ represent the ‘effects’, which the variables have on the cell frequencies
A and B are two categorical variables
i and j refer to the categories within the variables
Therefore:
A
γ
i
= the main effect for variable A
B
γ
j
= the main effect for variable B
AB
γ
ij
= the interaction effect for variables A and B.
The model presented by equation (5.35) is called the saturated model. It includes all
possible one-way and two-way effects. Given that the saturated model has the same
number of effects as there are cells in the contingency table, the expected cell
frequencies will always exactly match the observed frequencies, with no degrees of
freedom remaining (Agresti, 2002). To find a more parsimonious model that will isolate
the effects best explaining the data, a non-saturated model must be discovered. This
model could be achieved by setting some of the effect parameters to zero. For instance,
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