Additive model: GOF and number of parameters

[simone77]

Just two short questions about GOF testing of an additive model.

I have a capture-resighting dataset, 10 years, 6 sessions per year (two months resightings for each session), individuals marked as chicks (just during the breeding season) and I want to test as most general model this: Phi(age+session+year) P(age+session+year).
I eliminated the "ones" of the chicks and let the others (in case they were still alive and seen again), in this manner I had two classes of age (two groups in *.inp file).
I constructed the design matrix for this model and ran the model, now I'm trying to test his GOF (by median c-hat).

Question: Is there something wrong by testing the GOF of an additive model like this?

I'm looking for some help on how I should count the number of parameters of this model (MARK tell me there are 22 estimable parameters). I believe I read something somewhere on how to calculate the number of parameters of an additive model but I cannot try it.

Thanks in advance

[simone77]

Something to correct: I realize now I spoke about 10 years-study. Really I have 7 years (42 occasions). I remark it because of the sentence on the 22 estimable parameters MARK calculated.

I also would like to better explain this question: "...Is there something wrong by testing GOF of an additive model like this?..."

Of course, I don't refer to the biological features supporting or not this model, simply I' m not completely sure (I'm a new MARK user as you can easily realize) it is possible to test the GOF of an additive model. I have not found any analogous example but I suppose there is no trouble by a mathematical point of view, is there?

Simone

[dhewitt]

I'll say up front that I am rather unclear about the motivation for this model. Perhaps you can explain which particular model you are using, and why you want the additive structure involving all three "factors", from a biological standpoint.

In any case, with 42 occasions and that much structure on both Phi and p it seems like you would have many more parameters. How many of the estimates from this model are sensible? When data is sparse you simply cannot rely on MARK to count them. And counting parameters in additive models is a bit of a pain, but you appear to have other problems.

[simone77]

Hi,

I believe all the three factor may affect the probability one individual surviving and being resighted. In fact, it is plausible that:
1) AGE - the first year survival rate (juveniles) is lower than adults one (I used two age classes), age could affect resighting probability because of different "circumspection" habits.
2) SESSION - the session can affect the survival because of different environmental conditions depending on stage of the year as well as it could affect the resighting rate because of different catch effort and visibility itself (i.e. we go more frequently to the field in certain season).
3) YEAR - the year factor is a very typical factor could affect survival and resighting by indicating different environmental and demographical conditions varying through time.

Perhaps I'm wrong but, from a biological standpoint, I don't locate interactions among these factors.

Apart of being this model more or less appropriate, I'm very interested in finding the way to calculate the number of parameters of an additive model: do you know how can I do it (or any reference about this issue)?

Thanks again

Simone

[dhewitt]

Additivity is more than just knowing that the three factors should be in the model. The structure you are using implies a particular design matrix, so it is just a caution about knowing the details of the model you are fitting. One thing that stands out in your descriptions is that session is nested in time.

For me, the best way to understand the model is to sketch out what you expect the estimates to look like according to the structure. Additive models can be tricky, at least to me. The sketching of the estimates according to your factors is also handy for parameter counting. There are established formula based on groups and time for standard CJS models. For example, with 2 groups and 10 occasions, the number of theoretically estimable parameters is: 2*2[the no. of groups]+(10[the no. of occasions]-2) = 12. Without a clearer understanding of your model structure, I can't tell how to count the parameters in your model.

[simone77]

Hi,

I append the design matrix of this model in another, more explicit, post whose subject is "how many parameters" in this forum. You can see it there to better understand the model structure.
Anyway I would like to take the opportunity to discuss something inherent this subject by means of the example you did. I don't understand the model you are considering for your example. If it would be a CJS (fully dependent) mode with two groups and 10 occasions I calculate 34 estimable parameters.
In fact, considering the saturated encounter history for each group (1111111111) as suggested in 4-61 of the last version of Gentle Mark Introduction for CJS model {Phi(t) P(t)}, it would have this probability:

Phi(1)P(2) Phi(2)P(3) Phi(3)P(4) Phi(4)P(5) Phi(5)P(6) Phi(6)P(7) Phi(7)P(8) Phi(8)P(9) Beta(10)
(Unfortunately the 8th survival and recapture rate are black glass smiles!)

Where:

Beta(10)=Phi(9)P10 that are each one an unestimable parameter.

The same for the other group (suppose colony A and colony B). So, resuming for each group, there would be:

8 Phi
8 P
1 Beta

That is: 17 parameters for each group. And 34 parameters for the whole model. It is very different from 12 and I suppose you were not talking about CJS because in that case I'm very confused on the way to calculate how many parameters one model has!

Thanks

Simone

[dhewitt]

Simone,

Sorry, I should have been more explicit. You were asking about additive models, so the calculation I gave was for:

Phi(group), p(group+time) = 12 parms

You are correct on the 34 parameters for the full model:

Phi(group*time), p(group*time)

For a model with additive effects on both parameters:

Phi(group+time), p(group+time)

it would be 19 parameters.