Parameter Counting Question

[Evan]

Hello all,

I am hoping to get some help regarding counting identifiable parameters with some CJS models I am creating. I am modeling survival of turtles in five different ponds that have been sampled over a four year period. My collaborators and I anticipate that survival (and recapture) rates may differ among ponds, between the sexes, and/or over time. Thus, the global model we use is:

Phi (Pond and Sex * t), p (Pond and Sex * t)

indicating that both survivorship and recapture rate depend on the pond where a turtle is located, the sex of the turtle, and the effect of time. The model is built with 10 groups (to code for each sex at each of the five ponds). I understand how to count parameters for this model (20 unique survival parameters + 20 unique recapture parameters + 10 confounded beta parameters = 50 identifiable parameters total), but I'm having trouble with some other, "simpler" models that also include time dependent effects on both survivorship and recapture rate. For example, how many confounded parameters exist in a model where survivorship and recapture rate are both group and time dependent, but with different numbers of groups? I'm specifically thinking of the following models we're interested:

Phi (Pond * t), p (Pond and Sex * t); meaning Phi is modeled with 5 groups, p with 10
Phi (Sex * t), p (Pond and Sex * t); meaning Phi is modeled with 2 groups, p with 10
Phi (Sex * t), p (Pond * t); meaning Phi is modeled with 2 groups, p with 5
also, Phi (Pond and Sex * t), p (Pond * t) which MARK indicates has a different number of identifiable parameters from Phi (Pond * t), p (Pond and Sex * t)

It's not immediately clear to me which parameters would be confounded in the above models, although I assume some are because of the time dependence in both survival and recapture rate. I ask all of this because I know that MARK is having problems correctly counting parameters with our more parameterized models. For example, it reports 49 identifiable parameters for Phi (Pond and Sex * t), p (Pond and Sex * t) rather than 50.

If anyone can provide some general guidelines for counting parameters in these situations or help me with the parameter counts of the four models in question it would be greatly appreciated.

Thanks,
Evan

[aswea]

Hi Evan,

I asked a question on parameter counting in January that seems similar to yours and I received a good explanation. My first post was on January 08 2010 and the topic was Parameter Counting. You might want to scroll to the reply from Carl Schwarz on January 20th 2010.

~Aswea

[tpuettker]

Hello Evan,

Let´s see if I got it right:

model Phi (Pond * t), p (Pond and Sex * t):

pond 1 sex 1 : phi1 (pond1) p2 (sex1, pond1) phi2 (pond1) p3 (sex1,pond1) phi3 (pond1) p4 (pond1, sex1)

The red parameter are not uniquely identifyable ("beta"-term).
5 unique parameters: phi1, phi2, p2, p3, beta

pond 1 sex 2 : phi1 (pond1) p2 (sex2, pond1) phi2 (pond1) p3 (sex2,pond1) phi3 (pond1) p4 (pond1, sex2)

this means you get additional 3 uniquely identifyable parameters, because the "beta"-term is different from the first one: p2, p3, beta
This sums up to 8 uniquely identifyable parameters per pond, hence 40 parameters, 10 of which are "beta"-terms (2 per pond)

If I got it right, you will have 10 "beta"-terms in each of your models, because as long as you include time-dependence in the model, phi and p will not be uniquely identifyable in the last capture occasion.

Hope it helps,

thomas