closed capture p(t)=c(t) in design matrix

[scoster]

Hi,

I'm trying to figure out how to construct p(t)=c(t) in the design matrix for closed population mark-recapture

I started with a DM using a common intercept for p(t),c(t) (6 trap sessions, huggins with full heterogeneity)

1 0 0 0 0 0 0 0 0 0 0 0 0 pi
0 1 1 1 0 0 0 0 0 1 0 0 0 p
0 1 1 0 1 0 0 0 0 0 1 0 0 p
0 1 1 0 0 1 0 0 0 0 0 1 0 p
0 1 1 0 0 0 1 0 0 0 0 0 1 p
0 1 1 0 0 0 0 1 0 0 0 0 0 p
0 1 1 0 0 0 0 0 0 0 0 0 0 p
0 1 0 0 1 0 0 0 0 0 0 0 0 c
0 1 0 0 0 1 0 0 0 0 0 0 0 c
0 1 0 0 0 0 1 0 0 0 0 0 0 c
0 1 0 0 0 0 0 1 0 0 0 0 0 c
0 1 0 0 0 0 0 0 0 0 0 0 0 c


But I'd like to constrain p so that the parameters are estimable, and would therefore like to construct p(t)=c(t) using the design matrix, but I'm having some problems.

My first thought is to follow a p(.)=c(.) model, so make all p=1 and all c=1 and delete the intercept because it is redundant, but keep in the time variation. The model then looks like this:


1 0 0 0 0 0 0 0 0 0 0 0 pi
0 1 1 0 0 0 0 0 1 0 0 0 p
0 1 0 1 0 0 0 0 0 1 0 0 p
0 1 0 0 1 0 0 0 0 0 1 0 p
0 1 0 0 0 1 0 0 0 0 0 1 p
0 1 0 0 0 0 1 0 0 0 0 0 p
0 1 0 0 0 0 0 0 0 0 0 0 p
0 1 0 1 0 0 0 0 0 0 0 0 c
0 1 0 0 1 0 0 0 0 0 0 0 c
0 1 0 0 0 1 0 0 0 0 0 0 c
0 1 0 0 0 0 1 0 0 0 0 0 c
0 1 0 0 0 0 0 0 0 0 0 0 c

But this model has a different deviance than the p(t)=c(t) model derived from the PIMs (reference page 14.14).

Can anyone help point me in the right direction with regards to this?

Much appreciated,
Stephanie

[tpuettker]

Hello Stephanie,

I think you have to take out the interaction terms (beta-columns 8-12), because there is only one variable (time) and therefore there cannot be any interaction.

Hope it solves your problem,

thomas

[scoster]

Thomas,

Thanks for the tip. I'm still having problems with the deviances not matching, but I'm not sure what else to try.

Stephanie

[cooch]

Hi,

I'm trying to figure out how to construct p(t)=c(t) in the design matrix for closed population mark-recapture

I started with a DM using a common intercept for p(t),c(t) (6 trap sessions, huggins with full heterogeneity)

<snip>

But this model has a different deviance than the p(t)=c(t) model derived from the PIMs (reference page 14.14).

Can anyone help point me in the right direction with regards to this?

Much appreciated,
Stephanie

The page you refer to concerns simple closed population estimation without heterogeneity. To deal with the heterogeneity models, you should look at section 14.6 (and subsections therein). In particular, pay attention to the particular nuance in heterogeneity models concerning interaction terms, and the effect that pi is assumed to be constant over time. These models are a bit twitchy to construct, because of various non-identifiable bits.

[scoster]

Hi Cooch,

Thanks for your response. I did go back and look at the heterogeneity part of the chapter. I think in trying to simplify my question I used an inappropriate matrix. I have many questions, but I'll try to get better at asking them. Here's one: Is it even possible to create a DM (closed capture, no heterogeneity) with both a behavioral effect (enc group) and time effect? For the behavioral effect I know you would use p(.),c(.), and for the time effect p(t)=c(t) which seem incompatible. I know you could make a DM with p(t),c(t) but those parameters are inestimable. Does that mean that all models with behavior and time in them are inestimable?

Thanks again,
Stephanie

[cooch]

Hi Cooch,

Thanks for your response. I did go back and look at the heterogeneity part of the chapter. I think in trying to simplify my question I used an inappropriate matrix. I have many questions, but I'll try to get better at asking them. Here's one: Is it even possible to create a DM (closed capture, no heterogeneity) with both a behavioral effect (enc group) and time effect? For the behavioral effect I know you would use p(.),c(.), and for the time effect p(t)=c(t) which seem incompatible. I know you could make a DM with p(t),c(t) but those parameters are inestimable. Does that mean that all models with behavior and time in them are inestimable?

Thanks again,
Stephanie

Of course - that's one of the standard models (model M(tb))- but you need to make sure you set the p=c constraint for the model to work. The DM shown at the top of p 22. in Chapter 14 is an example of just such a DM.