Model structure strongly alters covariate beta's

[qrobinson]

Hi everyone,

I am running single-season, multi-species models of two predator species and exploring the influence of prey density covariates on occupancy of each predator. I'm getting some puzzling results from a subset of my models and am hoping someone might have a suggestion as to what's going on.

The results in question have to do with my dominant predator species. I have estimated the density of two different guilds of prey that I expect to be important to occupancy of this species. If I run a model containing either single prey covariate in isolation, the betas for the prey covariate are large positive numbers. Some example beta's output is below:

estimate std.error
A2 psiA.Prey1 : 65.774868 17.218891

And

estimate std.error
A2 psiA.Prey2 : 165.237071 49.130936

However, when both prey covariates are included in the model, the beta coefficient for prey type 1 becomes a large negative number, and the beta coefficient for prey type 2 increases dramatically. Example:

estimate std.error
A2 psiA.Prey1 : -379.002298 151.823845
A3 psiA.Prey2 : 1121.332696 387.611769

Considering the biology of my situation, the negative beta for prey type 1 is counter-intuitive at best -- my dominant predator species was detected much more frequently where this prey is abundant. Also, models containing an interaction term between the two prey types received relatively little support. I have a few ideas, but I'd welcome any suggestions regarding what might be going on here.

Thanks,

Quinn

[jhines]

My first suspicion is that the covariates might need to be standardized or scaled. Have you tried that? What do the design matrices look like for the models? What did you get for the intercept beta's?

Jim

[qrobinson]

Hi Jim,

Thanks for the reply. The values of the covariates in the input file have been scaled to between 0 and 1. I did this in a spreadsheet prior to importing the values into PRESENCE; is it a better idea to import the raw data and scale it within PRESENCE?

A simple example of the psi component of the design matrices for the two models in question:

Prey Type 1:
a1 a2 a3 a4
psiA 1 Prey.1 0 0
psiB 0 0 1 Prey.1
psiBa 0 0 1 Prey.1

(The structure for the Prey 2-only model is equivalent.)

Both prey species:
a1 a2 a3 a4 a5 a6
psiA 1 Prey.1 Prey.2 0 0 0
psiB 0 0 0 1 Prey.1 Prey.2
psiBa 0 0 0 1 Prey.1 Prey.2


The intercept betas are as follows (from the simplest, psiA psiB pA pB parameterization):

Prey type 1 only:

estimate std.error
A1 psiA : -1.340027 0.456643
A2 psiA.Prey.1 : 65.774868 17.218891

Prey type 2 only:

estimate std.error
A1 psiA : -1.497743 0.501076
A2 psiA.Prey.2 : 165.237071 49.130936

Both prey species:

estimate std.error
A1 psiA : -2.507339 0.694193
A2 psiA.Prey.2 : 1121.332696 387.611769
A3 psiA.Prey.1 : -379.002298 151.823845


As you can see, the inclusion of both covariates dramatically changes the resulting beta values. I hope this information will help clarify the situation, and most sincere thanks for your assistance!

Quinn

[jhines]

Quinn,

It doesn't matter if you scale the covariates in Excel or PRESENCE and the design matrices look OK.

One thing to try is an interactive model, where you aren't forcing occupancy when one prey species to be present to be a fixed amount greater/less than occupancy when both prey species are present. To do this, you'll need to add a covariate which is the product of prey.1 and prey.2 (named prey12 below). You would have to multiply the scaled covariates. Here is what the design matrix would look like:

Code: Select all

      a1     a2       a3     a4   a5   a6     a7    a8
psiA   1   Prey.1   Prey.2 Prey12  0    0      0     0
psiB   0      0        0     0     1  Prey.1 Prey.2 Prey12
psiBa  0      0        0     0     1  Prey.1 Prey.2 Prey12


I don't have an answer as to why you get a negative effect of prey.1 when it is included with prey.2. My guess is that there is a funny distribution of covariate values (ie. most values bunched up near a single value). I'd be happy to look at it if you'd like to send it to me.

Jim

[darryl]

Hi Quinn
Have you checked how much these two covariates are correlated? It looks a bit like you may have multi-colinearity to me. What happens to the -2log-like value for the model when you put both covariates in? If it doesn't change a whole lot compared to the single-covariate values then that would really suggest to me it's an issue with correlated covariates. If there's a big reducation in the -2log-like value, and your estimated probabilities start being near 0 and 1, then you may also be over fitting the data.

You might also want to scale your covariates back up too because of the magnitude of the coefficients and their SE, eg multiply by 100. Usually this isn't advised, but there's always exceptions.

Cheers
Darryl

[qrobinson]

Hi everyone,

Thanks for the further comments.

Multi-colinearity strikes me as a likely scenario; we've demonstrated that (generally speaking) abundance patterns are strongly correlated between these two prey types at our site. The deviance value changes from ~953 (when either prey type covariate is in the model alone) to ~950 with both prey covariates included.

Regarding scaling of the covariates, is it still appropriate to scale or normalize if there is a huge range of covariate values between sampling units or if the distribution of their values isn't normal? I have a large proportion of sites, for example, where biomass (in g) of prey type 1 was estimated to be essentially 0.0, and a significant proportion of sites where this estimate is over 4000. I know that normalization is the preferred technique if covariates do not share units (which mine do not), but I'm wondering about its applicability here.

Best, and thanks again.

-Quinn