2-species Model, binary covariate for competition

questions concerning analysis/theory using program PRESENCE

2-species Model, binary covariate for competition

Postby jhubbell072786 » Mon Apr 17, 2017 5:57 pm

To my fellow occupancy analysts,

I'm trying to aproximate the co-occurrence of two species using MacKenzie's 2-species model, and am interested in assessing to what extent (if any) the species are in competition with one another. I'm specifically interested in looking at the influence of biotic interactions on co-occurrence. Therefore, I included covariates in my models to assess the importance of difference biotic interactions (abundance of competitors, abundance of predators, etc). However, I'd also like to be able to directly assess whether there is competition is occurring between the two species for whom I'm modeling co-occurrence.

I'm well aware that the parameter, φ , provides you with some of idea of whether the occurrence of both species is purely random or not. However, this parameter is obviously not a direct measure of competition. To directly infer that competition was actually occurring, I've included a binary covariate indicating sites where both species were present and then observed whether any of my best models contained this covariate. My concern, is that there is a limited number (roughly 20%) of sites where the species do co-occur. Therefore, my expectation is that this covariate will falsely describe much more of the variance than it normally would assuming there were more sites where the species co-occurred. Any helpful suggestions on how I might get around this problem?

Thanks,

Josh
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Re: 2-species Model, binary covariate for competition

Postby darryl » Mon Apr 17, 2017 9:01 pm

Hi Josh,
The problem with your approach is that the binary variable you've defined may not be where both species are present due to imperfect detection. If you use the 2nd or 3rd parametrizations for the 2 species models you can get something similar to what you are attempting to do. Using the 2nd parametrization with the following design matrix (note, check the ordering of the rows is correct; I've done it from memory) then a5 indicates how the probability of sp B occupancy changes in the presence of sp A. This equivalent to using the presence/absence of sp A as a covariate for occupancy on sp B (with a5 the estimated effect size on the logit-scale), but accounts for imperfect detection of both species. This is similar to what you are trying to do.

\begin{tabular}{l|rrrrr}
	& a1 &	a2	& a3	& a4	& a5 \\
psiA	& 1	& X	& 0 & 0 &	0\\
psiBA &	0	& 0	& 1	& X	& 1\\
psiBa &	0	& 0	& 1	& X	& 0\\
\end{tabular}

The other option is using the 3rd parametrization where the following design matrix is equivalent to the above. Interpretation of a5 is the same where nu is an odds ratio (so 1 = independence; <1 avoidance; >1 co-occurrence).
\begin{tabular}{l|rrrrr}
	& a1 &	a2	& a3	& a4	& a5 \\
psiA	& 1	& X	& 0 & 0 &	0\\
psiBa &	0	& 0	& 1	& X	& 0\\
nu &	0	& 0	& 0	& 0	& 1\\
\end{tabular}

Hope that helps.

Darryl
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Re: 2-species Model, binary covariate for competition

Postby jhubbell072786 » Tue Apr 18, 2017 10:09 am

Darryl,

I sincerely appreciate your rapid reply. I do have one further question about the design matrix setup for the 2-species model. If I were interested in only examining the influence of habitat covariate/s on co-occurrence (psiBA) using the psiBa/rBa parameterization, my understanding is that the approach is the same as the single species model. For example:

For one habitat covariate:


a1 a2 a3 a4 a5 a6
psiA 1 0 0 1 0 0
psiBA 0 1 0 0 Hab1 0
psiBa 0 0 1 0 0 1


For two habitat covariates:

a1 a2 a3 a4 a5 a6 a7 a8 a9
psiA 1 0 0 1 0 0 1 0 0
psiBA 0 1 0 0 Hab1 0 0 Hab2 0
psiBa 0 0 1 0 0 1 0 0 1

Fyi- I sincerely apologize for the formatting of these design matrices. Hopefully its clear what I'm trying to do though.

Regards,


Josh
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Re: 2-species Model, binary covariate for competition

Postby jhines » Tue Apr 18, 2017 5:28 pm

Hi Josh,

Since Darryl's probably asleep now, I'll step in. There are a couple of problems with the 1st design matrix. First, columns a4 and a6 are exactly the same, meaning that you have redundant parameters. Second, by putting Hab1 only on psiBA, you're building a model where occupancy of species B depends on that habitat variable when species A is present, but occupancy of species B doesn't depend on habitat when species A is not present.
Code: Select all
      a1 a2 a3 a4 a5 a6
psiA  1  0  0  1   0  0
psiBA 0  1  0  0 Hab1 0
psiBa 0  0  1  0   0  1

I think you want Hab1 on both psiBA and psiBa. The fact that you have different intercepts for psiBA (a2) and psiBa (a3) will give you different occupancy estimates for species B when A is present versus when species A is absent.

You could use the following design matrix and get a single parameter (a3) representing the difference in occupancy for species B when species A is absent versus when A is present.
Code: Select all
      a1 a2 a3 a4
psiA  1  0  0  0
psiBA 0  1  0  Hab1
psiBa 0  1  1  Hab1

In this design matrix, a1 is the intercept parameter for psiA, a2 is the intercept parameter for psiBA and psiBa. A3 is the difference in occupancy for species B when species A is absent versus when A is present. A4 is the effect of Hab1 on occupancy of species B (equally affecting species B occupancy whether A is present or absent). If parameter a3 is estimated equal to zero, then there is no difference in occupancy of species B when species A is present or absent (occupancy of species B independent of presence/absence of species A). If a3>0 then occupancy of species B is higher when species A is absent (species B "avoids" species A).

For two habitat covariates:
Code: Select all
      a1 a2 a3 a4 a5
psiA  1  0  0   0    0
psiBA 0  1  0  Hab1 Hab2
psiBa 0  1  1  Hab1 Hab2


Hope this helps,

Jim
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Re: 2-species Model, binary covariate for competition

Postby darryl » Tue Apr 18, 2017 9:17 pm

Hi Josh,
Following on from Jim's post.

You have to be clear about your interpretation of psiBA (and other parameters), it's not coocurrence of the species per se, it's probability of occupancy of Sp B when A is also present. Cooccurrence is defined by how the design matrix for psiBA and psiBa differ (ie what factors are effecting psiBA but aren't effecting psiBa). Aside from the other issues, if you only have an extra covariate on psiBA like you've tried that means the level of cooccurrence completely depends upon only the value of that covariate; there's no offset or intercept term for the cooccurrence (ie it's like fitting a 'no intercept' model). In fact as you've done it you're saying that when the covariate =0, the species occur independently, but for other values they either avoid each other, or cooccur depending on the sign of the covariate and sign of the estimated effect size. Generally, you probably want to do something as Jim suggests. If you what the level of cooccurrence to depend on a covariate, then you could something like the following:

covariate also has general effect on occupancy of sp B:
Code: Select all
          a1 a2 a3    a4 a5
    psiA  1  0  0     0  0
    psiBA 0  1  Hab1  1  Hab1
    psiBa 0  1  Hab1  0  0


also general effect on sp A
Code: Select all
          a1 a2    a3 a4    a5 a6
    psiA  1  Hab1  0  0     0  0
    psiBA 0  0     1  Hab1  1  Hab1
    psiBa 0  0     1  Hab1  0  0


only influences cooccurrence
Code: Select all
          a1 a2 a3 a4
    psiA  1  0  0  0
    psiBA 0  1  1  Hab1
    psiBa 0  1  0  0


Good luck!
Darryl
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