Interaction Effect Betas

questions concerning analysis/theory using program PRESENCE

Interaction Effect Betas

Postby saldanh » Wed Apr 07, 2021 3:19 pm

Hi everyone,

For my study design, I have pre- and post- disturbance data collected in and outside the effected area. Whether a site falls in or outside the effected area is encoded in the site covariate "disturbed" (1 is in, 0 is out). I added pre- and post- disturbance betas (a3 and a4 respectively). The interaction betas are a5 and a6, and are supposed to be used to estimate psi for sites effected by the disturbance and those which were not.

I am wondering whether this is the right way to input my betas for my question.
............a1.....a2............a3....a4............a5............a6
psi(1).....1.....disturbed.....1.....0.............disturbed.....0
psi(2).....1.....disturbed.....1.....0.............disturbed.....0
psi(3).....1.....disturbed.....0.....1.............0...............disturbed
psi(4).....1.....disturbed.....0.....1.............0...............disturbed

Also, when would you use an additive model (like the one above), versus a model where only the interaction effect (a2 and a3) is used to estimate psi (shown below)?
..........a1.....a2............a3
psi(1).....1.....disturbed...0
psi(2).....1.....disturbed...0
psi(3).....1.....0.............disturbed
psi(4).....1.....0.............disturbed

Thanks!!
saldanh
 
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Joined: Tue Mar 30, 2021 9:51 pm

Re: Interaction Effect Betas

Postby jhines » Wed Apr 07, 2021 4:19 pm

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:
Code: Select all
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:
Code: Select all
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:
Code: Select all
          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?
Code: Select all
          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.
jhines
 
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Location: Laurel, MD, USA

Re: Interaction Effect Betas

Postby darryl » Wed Apr 07, 2021 4:24 pm

I see Jim has beaten me to it, but here goes...

Hi There,
Your design matrix is overparameterised for an interaction model, although there are multiple ways it could be set up and what changes is how to interpret the beta parameters (you should get identical psi estimates). Some of your options are to use either:
a1, a2, a4, a6, or
a1, a2, a3, a5, or
a3, a4, a5, a6

In any case, you should only need to use 4 columns to define the interaction model you've outlined. Which to use depends on how you want to interpret the coefficients.

Your second design matrix only allows for a pre- and post- difference at disturbed units, while undisturbed units have the same value right through. That's different to the interaction model.

Cheers
Darryl
darryl
 
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Location: Dunedin, New Zealand

Re: Interaction Effect Betas

Postby saldanh » Wed Apr 14, 2021 9:50 pm

Thank you both very much!
saldanh
 
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Joined: Tue Mar 30, 2021 9:51 pm


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