Phi models differ (RD Pradel Recruit vs Lambda)

[ctlamb]

Hi All,

I am attempting to tease apart the influence of human harvest rates and annual food productivity on grizzly bear demographics using 8 years of DNA-based mark-recapture data.

I have been using the Robust Design Huggins Closed Population models but have not yet settled on the Recruitment (RDPdfHuggins) or Lambda (RDPdLHuggins) version.

I began with the Lambda version, where I used a step approach where I tested the fit of a few models to p, while keeping Phi and Lambda constant (~1). Following this I retained the best model for p, and tested models for Phi, while leaving Lambda constant, and finally retained the top Phi model and then modelled Lambda.

When I used the exact same approach outlined above for the Recruitment version, my top model for p was identical, as expected, but my top model for Phi was different when using (RDPdfHuggins vs RDPdLHuggins).

My top Phi model for RDPdfHuggins, while leaving f constant (~1) was also a constant survival (phi~1), but when I did the same thing RDPdLHuggins, human harvest came out as my top survival parameter.

Am I just not understanding how the model derivation is conducted, as the Phi, Lambda and Recruitment parameters are obviously not mutually exclusive? Once I get this small issue figured, out, I'm also wondering whether the Phi and f model is a better approach than the Phi and Lambda model given that Phi and f are different processes, while Lambda is simply Phi+f, which makes modelling somewhat difficult as we are trying to fit a model to multiple processes.

Any input on this issue?

Cheers,
Clayton

[cooch]

Partial answer to some of your question...

Hi All,

I
I began with the Lambda version, where I used a step approach where I tested the fit of a few models to p, while keeping Phi and Lambda constant (~1). Following this I retained the best model for p, and tested models for Phi, while leaving Lambda constant, and finally retained the top Phi model and then modelled Lambda.

When I used the exact same approach outlined above for the Recruitment version, my top model for p was identical, as expected, but my top model for Phi was different when using (RDPdfHuggins vs RDPdLHuggins).

My top Phi model for RDPdfHuggins, while leaving f constant (~1) was also a constant survival (phi~1), but when I did the same thing RDPdLHuggins, human harvest came out as my top survival parameter.

Am I just not understanding how the model derivation is conducted, as the Phi, Lambda and Recruitment parameters are obviously not mutually exclusive? Once I get this small issue figured, out, I'm also wondering whether the Phi and f model is a better approach than the Phi and Lambda model given that Phi and f are different processes, while Lambda is simply Phi+f, which makes modelling somewhat difficult as we are trying to fit a model to multiple processes.

Cheers,
Clayton

People seem to like the parameterization of Pradel models with lambda in the likelihood (as opposed to a derived paramter), because this allows them to fit linear constraints to lambda. However, this is (often) problematic, since putting a constraint on lambda imposes a covariance structure on survivla and recuritment which doesn't always make good sense. From the 'Pradel models' chapter,

Since , if a model is fit with time invariant (i.e., constant) λ, or where λ is constrained to follow a linear trend, but with time varying f, then this implies a direct inverse relationship between survival and recruitment (e.g., if λ is held constant, then if ϕ(i) goes up, then f(i) must go down). While this may be true in a general sense, it is doubtful that the link between the two operates on small time scales typically used in mark-recapture studies. As noted by Franklin (2001), models where ϕ is time invariant while λ is allowed to vary over time are (probably) reasonable, as variations in recruitment are the extra source of ‘variation’ in λ. More complex models involving covariates have the same difficulty.

Moreover, suppose you fin covariation of λ with some covariate. What do you do next? Generally, you want to decompose variation in λ due to variation in the lower-level processes, f and ϕ. For most applications, the parameterization using f and ϕ is recommended.

[ctlamb]

Fantastic, thanks so much 'cooch'.

This is exactly what I was conceptually struggling with. I began with the Lambda formulation, then switched to Recruitment given that I couldn't rationalize the linear fit of the time-varying covariates to lambda given that lambda is composed of two processes that may (and often will) have varying and sometimes opposite relationships with my variables of interest.

In regards to the Phi models differing between the Phi-Lambda and Phi-Recruitment models, is this expected given the formulation of the model? I am somewhat confused about this, as I would expect the top survival model to be the same between the two model-types, given the same data?

[cooch]

Fantastic, thanks so much 'cooch'.

This is exactly what I was conceptually struggling with. I began with the Lambda formulation, then switched to Recruitment given that I couldn't rationalize the linear fit of the time-varying covariates to lambda given that lambda is composed of two processes that may (and often will) have varying and sometimes opposite relationships with my variables of interest.

Good choice. I assume you've read chapter 13 of the MARK book (I would be the 'Cooch' of 'Cooch & White'):

http://www.phidot.org/software/mark/docs/book/

In regards to the Phi models differing between the Phi-Lambda and Phi-Recruitment models, is this expected given the formulation of the model? I am somewhat confused about this, as I would expect the top survival model to be the same between the two model-types, given the same data?

For 'straight Pradel models', which don't allow for temporary emigration, everything is (or should be) identical. For the RD formulation, which allows for temporary emigration, it s possible (and I can imagine scenarios where it might even be likely) such that things don't line up exactly. The RD Pradel models aren't documented (meaning, I haven't gotten around to it yet), but if you stick with the survival and recrutiment parameterization, and focus inference on models built with that as the common likelihood structure, your inference across models will be robust.

[ctlamb]

Yes I certainly knew the name, I just wasn't sure of the protocol for addressing other users on the site. Straight Cooch with no quotes seemed informal

I've read the book nearly front to back (many things didn't stick the first time, but as I bump into issues they are starting to reverberate), and I have gone back to the chapters often. Very valuable resource.

Thanks again, I really appreciate the advice here,
Clayton

[cooch]

Yes I certainly knew the name, I just wasn't sure of the protocol for addressing other users on the site. Straight Cooch with no quotes seemed informal

Not to worry. Just pulling your chain.