species interaction model

[darryl]

If X is your covariate, setting up your design matrix like this:
a1 a2 a3 a4
psiA 1 X 0 0
psiBA 0 0 1 X
psiBa 0 0 1 X

says the effect of the covariate is different on the different species, but because the psiBA and psiBa rows are identical, the probabilities for spB are the same regardless of whether species A is present or not, with a4 being the effect of covariate X on occupancy of species B.

In this next design matrix, the effect of covariate X on spB is the same regardless of whether spA is also present, but a5 allows the overall probability of occupancy for spB to be different if species A was present.
a1 a2 a3 a4 a5
psiA 1 X 0 0 0
psiBA 0 0 1 X 1
psiBa 0 0 1 X 0

This final design matrix allows the effect of the covariate X on SpB to be different if spA is also present, with a6 telling us how different (once estimated).
a1 a2 a3 a4 a5 a6
psiA 1 X 0 0 0 0
psiBA 0 0 1 X 1 X
psiBa 0 0 1 X 0 0

[jhines]

I think you'd want to model both with the covariate, but perhaps in an additive model (different intercept but common covariate column(s)) like the 2nd one Darryl in his post.

Jim