Phi vs Lambda in two species occupancy estimation

[jluscie]

Regarding estimating co-occurrence patterns of two species with Program PRESENCE, both Donovan and Hines (2007; http://www.uvm.edu/envnr/vtcfwru/spread ... upancy.htm) and Mackenzie et al. (2004; Journal of Animal Ecology) refer to the species interaction factor (SIF) as Gamma = PsiAB/(PsiA*PsiB), however MacKenzie et al. 2006 (the book) refers to this as Phi. Program PRESENCE refers to this parameter as “Lambda,” right?? I just want to be sure this is indeed the same parameter with 3 different names/codes.

Also, the “two-species” analysis engine in PRESENCE also estimates “Phi.” This “Phi” represents the probability that a site will be occupied by both species, occupied by one or the other species but not both, or not occupied by either species?? So, this should essentially be 1.0??

Thanks in advance for any clarification.[/img]

[jlaake]

One of the authors or developers will have to help with your question, but I thought I'd just mention the parallel I saw between the model you describe here and double observer distance sampling models. If I understand it correctly the measure

Gamma = PsiAB/(PsiA*PsiB)

is the ratio of probability that a site is occupied by both species A and B divided by the product of the probability that the site is occupied by A and the probability that the site is occupied by B. If the species are occupying sites independently then you would expect PsiAB=PsiA*PsiB because under independence the product of an intersection (ie both A and B) is the product of the probabilities. If there is positive dependence (attraction) then PsiAB > PsiA*PsiB and the opposite is true if there is negative dependence (avoidance).

In double observer distance sampling, gamma is the same as delta (Borchers et al 2006; Laake et al 2008), which measures the dependence in the detection probabilities by 2 observers. If there is heterogeneity in detection probabiliity (some critters are easier to see than others) that is not modeled, this will create a positive dependence (delta >1), which means both observers will see those that are easy to see and they will both miss those that are harder to see.

So while this may not be relevant to the question I thought it wouldn't hurt to point out the similarity. Also, for those of you that monitor all of the forum components, this is relevant to the discussion to the question posed by db regarding removal models for double observer point transect data. Some of the issues raised by Murray Efford in the discussion and his recent paper (Efford and Dawson 2009) could be relevant to occupancy modelling.

BORCHERS, D. L., J. L. LAAKE, C. SOUTHWELL and C. G. M. PAXTON. 2006. Accommodating unmodeled heterogeneity in double-observer distance sampling surveys. Biometrics 62: 372-378.

EFFORD, M. and D. K. DAWSON. 2009. Effect of distance-related hetoreneity on population size estimates from point counts. Auk 126: 100-111.

LAAKE, J., M. J. DAWSON and J. HONE. 2008. Visibility bias in aerial survey: mark-recapture, line-transect or both? Wildlife Research 35: 299-309.

[jhines]

In the two-species model, the (occupancy) species interaction factor is labeled 'Phi' in PRESENCE, and you've got the formula right. There is a similar parameter for detection probability which is labeled 'Lam'. This is the species interaction factor for detection and is computed as:

Lam = rAB/(rA*rB)

where rAB is the probability of detection of both species, given both are present, rA is the prob. of detection of only species A, given both are present and rB is the prob. of detection of only species B, given both are present.

So, Gamma = Phi = SIF(occupancy), and Lam=SIF(detection)

Sorry for the confusion.