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

AB
if the effects parameter γ
ij
is set to zero (i.e. assume that variable A has no effect on
variable B, or vice versa), the unsaturated model is obtained:
A B
ln( F ) = µ + γ + γ . (5.36)
ij i j
Furthermore, it can be said that the models presented above are hierarchically related to
each other, i.e. they are nested. In other words, the unsaturated model is nested within
the saturated model.
From the collection of unsaturated models that have been fitted, it is required to decide
2
which of the unsaturated models provides the best fit. The likelihood ratio test ( G ) can
be carried out to find out the best-fitted model, since the models are nested within each
other. If F ij
represents the fitted frequency and f ij
the observed frequency, then the
likelihood ratio test statistic (Agresti, 2002) is denoted by:
G
2
= 2
∑∑
⎛ f
log⎜
⎝ F
ij
f
ij
i j ij
⎟ ⎞
. (5.37)
⎠
The
2
G test is distributed chi-square with degrees of freedom (df) equal to the number
of cells minus the number of non-redundant parameters (number of model parameters)
in the model. In other words, the df equals the number of γ parameters set equal to
zero. When the models get more complex, the df value decreases, with the df=0 for the
saturated model. As a result, the
2
G tests the residual frequency not accounted for by
the effects in the model. (i.e. the γ parameters set equal to zero). Therefore, larger
values indicate that the model does not fit the data well, and thus the model should be
2
G
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