adding covariates in a multiple-season analysis

[Diego.Pavon]

I have a question about how to enter covariates in Presence in a multiple-season model. I have data on predators and I would like to model psi as a function of their main prey. I also have an index of prey abundance. So, I would like to enter in PRESENCE the prey index for each year. The prey index will be the same for each site, but, of course, it changes from year to year. In this sense, for a a year t (season 1), all the sites will have the same value of prey index. For year t+1 (season 2) all the sites will have the same value (which is different to year t (season 1) and year t+2 (season 3), t+3 (season 4), and so on). As far as I understood and given how the Input Data Form window looks, I only can enter values for each site but not across time. As an example, I would like my Input Data Form (covariates tab) to look like:

Site Season 1 2 3 4 5 6....
1 1.3 0.32 -1.0 3.2 -0.02 -2.1
2 1.3 0.32 -1.0 3.2 -0.02 -2.1
3 1.3 0.32 -1.0 3.2 -0.02 -2.1
4 1.3 0.32 -1.0 3.2 -0.02 -2.1
5 1.3 0.32 -1.0 3.2 -0.02 -2.1
6 1.3 0.32 -1.0 3.2 -0.02 -2.1

where the first column is the site ID, the first row is the season (i.e. year, from 1989
to 2010) and the values are the standardized prey abundance Index.

I was wondering if adding 22 columns (one per year) and entering the prey index (the
value of prey abundance index for each year in each column) is one solution?
Any idea about how can I do it otherwise?

Thank you very much.

Diego

[jhines]

If your covariate is the same for all sites, you don't need to enter it as a site or survey covariate. You can enter the values directly into the design matrix. So, if your seasonal prey index values are 1.3 0.32 -1.0 3.2 -0.02 -2.1, your design matrix for occupancy would be:

a1 a2
psi1 1 1.3
psi2 1 0.32
psi3 1 -1.0
psi4 1 3.2
psi5 1 -0.02
psi6 1 -2.1

This assumes we're doing one of the alternate parameterizations (psi(t),gam(t),p(), or psi(t),eps(t),p()).

In this design matrix, logit(occupancy) is modeled as a linear function of prey index. So, logit(psi1)=a1+a2*1.3, logit(psi2)=a1+a2*0.32,... etc. So, if the estimate of a2 is positive, there is a positive relationship between occupancy and prey (more prey -> higher occupancy).

You could also model colonization or extinction in a similar fashion.

Jim

[Diego.Pavon]

Hi Jim,

Thank you very much for your reply! I am running the models as you suggested although it is taking ages to run them! If I would like to constrain also "p", could I do it the same way? Let's say:
a1
1-1 1.3
1-2 1.3
1-3 1.3
2-1 0.32
2-2 0.32
2-3 0.32
3-1 -1
3-2 -1
3-3 -1


Thank you,

Diego

[jhines]

Hi Diego,

Yes, you can model p as a function of the covariate in the design matrix, but don't forget that you probably need an intercept term. For example:

Code: Select all

    a1 a2
1-1 1 1.3
1-2 1 1.3
1-3 1 1.3
2-1 1 0.32
2-2 1 0.32
2-3 1 0.32
3-1 1 -1
3-2 1 -1
3-3 1 -1


Without the intercept term, you're forcing the detection probability to be 0.5 (logit of zero) when the covariate is zero. Even though you might not have a covariate value of zero, you'd be forcing the regression line to go through that point.

Jim
Thank you,

[Diego.Pavon]

Hi,

Thanks again for your useful comment!

Yes, I forgot to write in the example the first column full of ones. However, if I want to have a season effect, then I don't need the first column (intercept), do I?
In that case, if I want to have season effect+covariate, I would not need to have the first column, am I right? Or... does it make sense to do it like this? Perhaps it is not correct to write it like this in the DM... I am not sure about this approach. Do you think it makes "biological" sense?
a1 a2 a3
1-1 1.3 0 0
1-2 1.3 0 0
1-3 1.3 0 0
2-1 0 0.32 0
2-2 0 0.32 0
2-3 0 0.32 0
3-1 0 0 -1
3-2 0 0 -1
3-3 0 0 -1

Or would it be like:

a1 a2 a3 a4
1-1 1.3 1
1-2 1.3 1
1-3 1.3 1
2-1 0.32 0 1
2-2 0.32 0 1
2-3 0.32 0 1
3-1 -1 0 1
3-2 -1 0 1
3-3 -1 0 1

Thank you very much

Diego

[jhines]

Yes, you still need an intercept column. For a seasonal effect on p, you need a different intercept for each season. Then an additional column for the prey covariate.

[code]
a1 a2 a3 a4
1-1 1 0 0 1.3
1-2 1 0 0 1.3
1-3 1 0 0 1.3
2-1 0 1 0 0.32
2-2 0 1 0 0.32
2-3 0 1 0 0.32
3-1 0 0 1 -1
3-2 0 0 1 -1
3-3 0 0 1 -1
[code]

This design matrix would yield a model where (logit of) detection probability is a linear function of the prey covariate for each season. Each season would have the same effect of the prey covariate, but a different intercept. So, if you plotted logit(p) vs prey, you would get 3 parallel lines.

Jim