Covariate design matrix in multi-species models

[qrobinson]

Hi all,

I have what I believe is a simple question and I apologize if it has been addressed elsewhere.

I am running a multi-species model of two predators, and am interested in the effect of a covariate (density of a shared prey species) on their occupancy. I hypothesize that occupancy of predator species A will respond differently to changes in prey density than will occupancy of predator B. Given this, my question is:

Do I need a shared column in my design matrix representing prey density, followed by two columns representing the prey's effect on each predator (i.e. prey x species interaction columns), or can I omit the shared column altogether? Simplified design matrices of these two possibilities would look like:

prey.dens prey.dens*A prey.dens*B
psiA 1 1 0
psiB 1 0 1


Or:

prey.dens*A prey.dens*B
psiA 1 0
psiB 0 1


I would appreciate any thoughts or guidance, and again I apologize if a similar issue has been raised and addressed previously.


Cheers,

Quinn

[jhines]

The design matrix should be:

Code: Select all

-     a1    a2       a3       a4
psiA   1  prey.dens   0        0
psiBA  0      0       1    prey.dens
psiBa  0      0       1    prey.dens


The first column (a1) is the intercept for occupancy for species A, or said another way, (the logit of) occupancy for species A when there are no prey around. The 2nd column is the effect of prey density on occupancy of species A. The third column is the intercept for occupancy for species B, and the last column is the effect of prey density on occupancy for species B. This model assumes occupancy is independent for the two species, or occupancy for species B does not depend on occupancy of species A (psiBA=psiBa).

If species A is dominant and gets most of the prey when it's around, then a model where occupancy of species B depends on occupancy of species A might make sense. The design for that model would be:

Code: Select all

-     a1    a2       a3       a4       a5       a6
psiA   1  prey.dens   0        0        0         0
psiBA  0      0       1    prey.dens    0         0
psiBa  0      0       0        0        1    prey.dens


Jim

[qrobinson]

Jim,

Thanks for the rapid response and helpful clarification. Apologies for not including intercept terms in my example design matrix -- I was attempting to simplify things but I realize now that this may have created some confusion.

Best,

Quinn