Lambda or recruitment in Pradel model?

questions concerning analysis/theory using program MARK

Lambda or recruitment in Pradel model?

Postby sixtystrat » Mon May 08, 2017 4:41 pm

I'm running 9 years of CMR data on bears and am investigating the effect of a flooding event on recruitment (f). I compared a model with constant phi and constant f except that I varied 1 year of f due to the effects of the flood event. This is a robust design Pradel full heterogeneity model with pi, c, p, and f0. When I run the model based on the recruitment model, I get low recruitment the year of the flood (0.02, normally about 0.22) but the beta for the effect includes zero. However, if I parameterize the model in terms of lambda, I get low lambda during the flood (0.87, normally 1.08) but the beta does NOT include zero.
Why the difference? The deviance is the same for both models and the derived estimate of lambda in the recruitment model is correct. I read in the manual that there are questions about constraining 2 parameters (phi and f, or phi and lambda), which I do not completely understand, so that may be part of it. My gut tells me that Mark is having a harder time estimating recruitment because it is a low number close to zero but I do not have any real evidence for that. I tried the same model with phi as time-varying and the flood effect was not significant (95% CI did not include zero) in either case. Can anyone offer any insight?
Thanks!!
Joe
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Re: Lambda or recruitment in Pradel model?

Postby cooch » Mon May 08, 2017 7:17 pm

sixtystrat wrote:I'm running 9 years of CMR data on bears and am investigating the effect of a flooding event on recruitment (f). I compared a model with constant phi and constant f except that I varied 1 year of f due to the effects of the flood event. This is a robust design Pradel full heterogeneity model with pi, c, p, and f0. When I run the model based on the recruitment model, I get low recruitment the year of the flood (0.02, normally about 0.22) but the beta for the effect includes zero. However, if I parameterize the model in terms of lambda, I get low lambda during the flood (0.87, normally 1.08) but the beta does NOT include zero.
Why the difference? The deviance is the same for both models and the derived estimate of lambda in the recruitment model is correct. I read in the manual that there are questions about constraining 2 parameters (phi and f, or phi and lambda), which I do not completely understand, so that may be part of it. My gut tells me that Mark is having a harder time estimating recruitment because it is a low number close to zero but I do not have any real evidence for that. I tried the same model with phi as time-varying and the flood effect was not significant (95% CI did not include zero) in either case. Can anyone offer any insight?
Thanks!!
Joe


Simple: lambda is the sum of survival + recruitment. Meaning, the 'beta' for lambda is a function of two processes, and there is no reason to expect that the beta for the sum of two processes will be the same (or even 'onsistent') with the beta for one of the processes alone.
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Re: Lambda or recruitment in Pradel model?

Postby sixtystrat » Tue May 09, 2017 7:47 am

Understood; I wouldn't expect the betas to be the same either. However, if I model phi as constant and lambda changing only during the flood year, wouldn't the change in lambda (slope) necessarily have to be due to the change in recruitment? So if the beta for lambda is "significant" (does not include zero), and phi does not change, wouldn't that change have to be due to a change in recruitment and would not that effect also be considered "significant"? I think that the model may be having trouble estimating recruitment when parameterized that way. When parameterized to estimate lambda however, recruitment as a derived parameter is not presented so I do not have the standard errors to check to see if the estimates are the same. Thanks for the help!
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Re: Lambda or recruitment in Pradel model?

Postby cooch » Tue May 09, 2017 8:54 am

sixtystrat wrote:Understood; I wouldn't expect the betas to be the same either. However, if I model phi as constant and lambda changing only during the flood year, wouldn't the change in lambda (slope) necessarily have to be due to the change in recruitment?


Correct.

So if the beta for lambda is "significant" (does not include zero), and phi does not change, wouldn't that change have to be due to a change in recruitment and would not that effect also be considered "significant"?


Not quite. If \varphi = \mbox{constant}, then \Delta\lambda\propto\Delta\varphi. There is a \beta for \varphi, even though it is a constant. In other words, the \beta for recruitment is not simply the \beta for \lambda. The \beta for recruitment is a function of whatever the \beta is for \varphi.

I think that the model may be having trouble estimating recruitment when parameterized that way. When parameterized to estimate lambda however, recruitment as a derived parameter is not presented so I do not have the standard errors to check to see if the estimates are the same. Thanks for the help!


You can, in fact, get MARK to 'derive' recruitement from the Pradel 'survival and lambda' data type, using any number of approaches. The one I would likey use is to make use of the chains from MCMC, and derive the parameter that way. See the addendum to the Delta method appendix (appendix B) for an example of same. Its actually quite easy (although it does assume you're famiuliar with how to use MCMC in MARK -- Appendix G).
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Re: Lambda or recruitment in Pradel model?

Postby cooch » Tue May 09, 2017 9:23 am

For example, using the capsid data from Chapter 13 (Pradel models), fitting model phi(.)p(.)f(t) to the data yields the following estimates for recruitment (showing just the first 3 intervals to make the point):

Code: Select all
     3:f                     1.8996889       0.4002188       1.1152601       2.6841177
     4:f                     0.3431272       0.1084128       0.1690820       0.5728208
     5:f                     0.4501956       0.1040704       0.2642550       0.6511751   


Now, fitting model phi(.)p(.)lambda(t) to the same data, and using MCMC to create a derived parameter for recruitment (f), I get the following estimates:

Code: Select all
 basic summary stats of recruitment values
       f1              f2                f3       
 Min.   :0.699   Min.   :0.02645   Min.   :0.1060 
 1st Qu.:1.640   1st Qu.:0.27420   1st Qu.:0.3829 
 Median :1.888   Median :0.34514   Median :0.4522 
 Mean   :1.922   Mean   :0.35114   Mean   :0.4562 
 3rd Qu.:2.164   3rd Qu.:0.42036   3rd Qu.:0.5223 
 Max.   :4.349   Max.   :0.88662   Max.   :0.9237 


Since the posterior distributions are somewhat skewed (which you'd see if you plotted the posterior densities for the parameters), the median is arguably the more approrpiate moment to use -- and as you can see, the median values are very close to the estimates of recruitment coming out of model phi(.)p(.)f(t).

For the confidence bounds, and again taking into account the skew in the posterior density, I'd recommend using HPD (highest posterior density), which you can generate using the coda package in R. For the first 3 recruitment parameters

Code: Select all
       lower    upper
var1 1.190457 2.733546
attr(,"Probability")
[1] 0.95

         lower     upper
var1 0.1461542 0.5667102
attr(,"Probability")
[1] 0.95

         lower     upper
var1 0.2548271 0.6573581
attr(,"Probability")
[1] 0.95


which are quite close to (and, depending on who you ask, "even better than") the 95% CI shown above for estimates from model phi(.)p(.)f(t).

So, yes, you can get an estimate of recruitment from the Pradel "survival and lambda" model. You could do it using the Delta method (which for this problem, is easy, and could be 'programmed' into an Excel spreadsheet), or you could 'run with the cool kids', and use MCMC, as shown here (although showing the 'math' for the Delta method in your paper is just a different flavor of 'cool').
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Re: Lambda or recruitment in Pradel model?

Postby sixtystrat » Wed May 10, 2017 8:44 am

Thank you for that detailed response Evan. Your help is always appreciated!
Joe
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