You have 4 categories of occupancy estimates: 1) occupancy for sites in/out of disturbed areas, and 2) occupancy before/after the disturbance. So, you should have at most, 4 beta columns.

Here are some models to test for the effect of disturbance. Starting with the simplest assumption... both areas have the same occupancy before and after the disturbance:

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`psi(1).....1`

psi(2).....1

psi(3).....1

psi(4).....1

Next, both areas have the same occupancy before the disturbance, then the area where the disturbance occurs, psi's differ:

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`psi(1) 1 0`

psi(2) 1 0

psi(3) 1 disturb

psi(4) 1 disturb

That gives 2 occupancy estimates: 1 for undisturbed sites (anytime on undisturbed sites, or before the disturbance in disturbed sites), and 1 for disturbed sites, after the disturbance.

The next model assumes constant occupancy for sites in undisturbed areas, and different psi’s before and after the disturbance in disturbed areas:

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` a1 a2 a3`

psi(1) 1 disturb 0

psi(2) 1 disturb 0

psi(3) 1 disturb disturb

psi(4) 1 disturb disturb

Here, the 2nd column produces a different occupancy for sites in disturbed areas, and the 3rd column produces another estimate for occupancy in disturbed areas after the disturbance. For undisturbed areas, logit(psi) = a1, for disturbed areas, before the disturbance (psi(1),psi(2)), logit(psi) = a1+a2, and for disturbed areas after the disturbance, logit(psi)= a1+a2+a3.

The last, most complicated model would be where occupancy differs before/after the disturbance on both sites in/out of disturbed areas. This model can easily be created, but doesn’t make much sense… why would sites out of disturbed areas have different occupancy estimates before/after the disturbance?

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` a1 a2 a3 a4`

psi(1) 1 disturb 0 0

psi(2) 1 disturb 0 0

psi(3) 1 disturb disturb 1

psi(4) 1 disturb disturb 1

So, your first matrix has too many columns. It appears you’re using a3,a4 as your before/after disturbance effect columns, but only one of those is needed since you have an intercept column (a1). That means you only need one interaction column instead of two (a5,a6). Deleting columns a3 and a5 would yield a matrix similar to the one above.

The 3-column matrix you listed is another way of representing the 3-column matrix I listed… one psi for sites out of the disturbed areas (logit(psi)=a1), and two psi’s for before/after disturbance in disturbed areas.

After running these models, you can use AIC to determine if disturbance affects occupancy.