multi-season: sampling covariates effect on psi,eps, gam

[giulia.santulli]

Hello,
I apologize if this issue has been addressed elsewhere, but I didn't manage to find it.
I'm using the two-species multi-season model with two small carnivores.
I have 204 sites sampled, 2 sampling occasions per year during 12 years (of course many missing data). I want to explore the effect of a sampling covariate (i.e. number of cells occupied by each of the two species in a X km radius in each sampling occasion) on psi, gam and eps. In other words I want to test if the number of sites occupied by species A around a target site occupied by species B has an influence on the occupancy, colonization and extinction of B.

First question is: can I do that? Can I test the effect of a sampling covariate on occupancy, colonization and extinction? Or psi, gam and eps can be related only to site covariates and time? Are sampling covariates only relevant for detection?

In my case each sampling covariate is a table of 24 columns (2 sampling occasions during 12 years).
My problem is that presence calculates gam and eps using the first 12 columns of the sampling covariates...which is bad, because they represent only the first 12 sampling session, 6 years of 12. And in the case of psi, it is related only to the first column of my sampling covariate (first sampling occasion of the first year).

Second question: is there something wrong in my design matrix?
My design matrix for psi looks like (EM_t is one of my sampling covariate)

psiA 1 0 0 EM_t 0 0
psiBA 0 1 0 0 EM_t 0
psiBa 0 0 1 0 0 EM_t

My design matrix for gam looks like

gamAB[1] 1 0 0 0 EM_t 0 0 0
gamAB[2] 1 0 0 0 EM_t 0 0 0
gamAB[3] 1 0 0 0 EM_t 0 0 0
gamAB[4] 1 0 0 0 EM_t 0 0 0
gamAB[5] 1 0 0 0 EM_t 0 0 0
gamAB[6] 1 0 0 0 EM_t 0 0 0
gamAB[7] 1 0 0 0 EM_t 0 0 0
gamAB[8] 1 0 0 0 EM_t 0 0 0
gamAB[9] 1 0 0 0 EM_t 0 0 0
gamAB[10] 1 0 0 0 EM_t 0 0 0
gamAB[11] 1 0 0 0 EM_t 0 0 0
gamAb[1] 0 1 0 0 0 EM_t 0 0
gamAb[2] 0 1 0 0 0 EM_t 0 0
gamAb[3] 0 1 0 0 0 EM_t 0 0
gamAb[4] 0 1 0 0 0 EM_t 0 0
gamAb[5] 0 1 0 0 0 EM_t 0 0
gamAb[6] 0 1 0 0 0 EM_t 0 0
gamAb[7] 0 1 0 0 0 EM_t 0 0
gamAb[8] 0 1 0 0 0 EM_t 0 0
gamAb[9] 0 1 0 0 0 EM_t 0 0
gamAb[10] 0 1 0 0 0 EM_t 0 0
gamAb[11] 0 1 0 0 0 EM_t 0 0
gamBAA[1] 0 0 1 0 0 0 EM_t 0
gamBAA[2] 0 0 1 0 0 0 EM_t 0
gamBAA[3] 0 0 1 0 0 0 EM_t 0
gamBAA[4] 0 0 1 0 0 0 EM_t 0
gamBAA[5] 0 0 1 0 0 0 EM_t 0
gamBAA[6] 0 0 1 0 0 0 EM_t 0
gamBAA[7] 0 0 1 0 0 0 EM_t 0
gamBAA[8] 0 0 1 0 0 0 EM_t 0
gamBAA[9] 0 0 1 0 0 0 EM_t 0
gamBAA[10] 0 0 1 0 0 0 EM_t 0
gamBAA[11] 0 0 1 0 0 0 EM_t 0
gamBAa[1] 0 0 1 0 0 0 EM_t 0
gamBAa[2] 0 0 1 0 0 0 EM_t 0
gamBAa[3] 0 0 1 0 0 0 EM_t 0
gamBAa[4] 0 0 1 0 0 0 EM_t 0
gamBAa[5] 0 0 1 0 0 0 EM_t 0
gamBAa[6] 0 0 1 0 0 0 EM_t 0
gamBAa[7] 0 0 1 0 0 0 EM_t 0
gamBAa[8] 0 0 1 0 0 0 EM_t 0
gamBAa[9] 0 0 1 0 0 0 EM_t 0
gamBAa[10] 0 0 1 0 0 0 EM_t 0
gamBAa[11] 0 0 1 0 0 0 EM_t 0
gamBaA[1] 0 0 0 1 0 0 0 EM_t
gamBaA[2] 0 0 0 1 0 0 0 EM_t
gamBaA[3] 0 0 0 1 0 0 0 EM_t
gamBaA[4] 0 0 0 1 0 0 0 EM_t
gamBaA[5] 0 0 0 1 0 0 0 EM_t
gamBaA[6] 0 0 0 1 0 0 0 EM_t
gamBaA[7] 0 0 0 1 0 0 0 EM_t
gamBaA[8] 0 0 0 1 0 0 0 EM_t
gamBaA[9] 0 0 0 1 0 0 0 EM_t
gamBaA[10] 0 0 0 1 0 0 0 EM_t
gamBaA[11] 0 0 0 1 0 0 0 EM_t
gamBaa[1] 0 0 0 1 0 0 0 EM_t
gamBaa[2] 0 0 0 1 0 0 0 EM_t
gamBaa[3] 0 0 0 1 0 0 0 EM_t
gamBaa[4] 0 0 0 1 0 0 0 EM_t
gamBaa[5] 0 0 0 1 0 0 0 EM_t
gamBaa[6] 0 0 0 1 0 0 0 EM_t
gamBaa[7] 0 0 0 1 0 0 0 EM_t
gamBaa[8] 0 0 0 1 0 0 0 EM_t
gamBaa[9] 0 0 0 1 0 0 0 EM_t
gamBaa[10] 0 0 0 1 0 0 0 EM_t
gamBaa[11] 0 0 0 1 0 0 0 EM_t

Third question: in the two-species multi-season model, how I can calculate seasonal occupancy over the studied period, as in the one-species multi-season models? which is the design matrix definition for that?

Many thanks in advance

Giulia

[darryl]

1. No, you shouldn't use a sampling covariate on those parameters. What you could do though is define a series of site-specific covariates (1 for each year) indicating number of occupied cells around the focal cell. You'd then use the series of covariates in the design matrix, eg viewtopic.php?f=11&t=1996&p=6006&hilit=site+specific+covariate#p6006. However, how good is your detection rate? Is there the potential for some nearby cells to be occupied, but undetected? Do you have 100% coverage of all cells? These could create some issues for you in terms of interpretation of the results. Note that you're trying to fit an autologisitc model to your data. There's been some progress on this for the single-species multi-season model, but not sure if Jim Hines has extended it to 2-species models.

2. Whether it's 'wrong' depends on what model you're trying to fit to the data. I note that the way you've set up the design matrix for colonization you're saying that colonization probability for species B depends on the presence/absence of species A at time t, but not time t+1.

3. You can derive it using matrix multiplication. If v_psi[t] is a 1x4 row vector of the probability of a cell being in each of the 4 possible states (both present, only A, only B, or both absent) at time t, and m_phi[t] is the transition probability matrix for between times t and t+1, then:
v_psi[t+1] = v_psi[t] %*% m_phi[t] (where %*% indicates matrix multiplication)

Hope this helps
Darryl

[giulia.santulli]

Hello Darryl,
thank you very much for your help.
Now I'm trying to derive the seasonal occupancy using the transition probability matrix. In chapter 8 the dynamic parameters of the transition matrix are defined as:

εtAB = prob. both species A and B go locally extinct between season t and t+1
εtA= prob. species A go locally extinct between season t and t+1, given both species present in season t
εtB= prob. species B go locally extinct between season t and t+1, given both species present in season t
υtA prob. species A go locally extinct between season t and t+1, given species B absent in season t and t+1
υtB prob. species B go locally extinct between season t and t+1, given species A absent in season t and t+1
γtAB prob. both species A and B colonize a site between season t and t+1
γtA prob. species A colonizes a site between season t and t+1, given species B absent in season t
γtB prob. species B colonizes a site between season t and t+1, given species B absent in season t
ηtA prob. species A colonizes a site between season t and t+1, given species B present in season t and t+1
ηtB prob. species B colonizes a site between season t and t+1, given species A present in season t and t+1
ωtAB species A is replaced by B between seasons t and t+1
ωtBA species B is replaced by A between seasons t and t+1

I'm not sure on how to calculate the parameters. Shall I calculate them directly from my data season by season, and average them? Can I derive ε and γ from eps and gam of my model? How do I derive υ, η, ω? Or is there some model to estimate them?

εtAB = εtA x εtB
εtA = epsAB?
εtB = epsBAA and epsBAa?

Any help would be much appreciated!

Giulia

[darryl]

The parameterisation in the book is an old one and been replaced by the one you're using. If the 4 states are AB, Ab, aB and aa (upper case indicating that species is present and lower case indicating absence), then transition matrix will be (rows indicating state at t, columns state a t+1):
AB Ab aB ab
AB (1-epsAB)*(1-epsBAA) (1-epsAB)*epsBAA epsAB*(1-epsBAa) epsAB*epsBAa
Ab (1-epsAb)*gamBAA (1-epsAb)*(1-gamBAA) epsAb*gamBAa epsAb*(1-gamBAa)
aB gamAB*(1-epsBaA) gamAB*epsBaA (1-gamAB)*(1-epsBaa) (1-gamAB)*epsBaa
ab gamAb*gamBaA gamAb*(1-gamBaA) (1-gamAb)*gamBaa (1-gamAb)*(1-gamBaa)

Sorry, column alignment might be slightly screwed.

Cheers
Darryl