Link-Barker and Pradel unequal time interval discrepancy

questions concerning analysis/theory using program MARK

Link-Barker and Pradel unequal time interval discrepancy

Postby stshroye » Fri Nov 19, 2021 11:45 am

I am interested in estimating Jolly-Seber parameters for a Muskellunge population that was sampled each spring in 2011-2013 and 2015-2017. Effective Sample Size = 543. I initially coded capture histories with six occasions and set time intervals to 1,1,2,1,1. Capture histories were grouped by sex (1 = female, 2 = male). I ran Release GOF with the following results:

Code: Select all
 Goodness of Fit Results (TEST 2 + TEST 3) by Group
 
 Group      Chi-square   df   P-level
 -----         ----------     ----  -------
 1              2.5084         9    0.9807
 2             13.9081       10    0.1772
 Total       16.4165       19    0.6293


I built a set of candidate Link-Barker models using PIM coding and sine links. Fletcher c-hat for my global model Phi(g*t) p(g*t) f(g*t) was 1.11.

The best-supported model was Phi(g) p(t) f(g). Since there was no time variation in Phi or f, I expected a single estimate of Phi, f, and Lambda for each group. Real parameter estimates looked as expected:

Code: Select all
Parameter          Estimate       Standard Error      Lower           Upper     
 ----------------- -----------     --------------     --------------  --------------
    1:Phi             0.8063951       0.0298723       0.7411067       0.8583655     
    2:Phi             0.8885083       0.0211434       0.8398695       0.9237152     
    3:p                0.7539629       0.1089191       0.4922595       0.9064200     
    4:p                0.2471774       0.0414449       0.1750481       0.3368905     
    5:p                0.3905809       0.0456618       0.3055799       0.4827877     
    6:p                0.1943251       0.0341582       0.1358997       0.2700195     
    7:p                0.7183033       0.0513023       0.6080497       0.8073684     
    8:p                0.4941700       0.0525771       0.3927942       0.5960274     
    9:f                 0.3146688       0.0552829       0.2174087       0.4314515     
   10:f                0.2351896       0.0495188       0.1520240       0.3453238   


However, occasion 3 (the two-year interval) had substantially different derived estimates for Lambda:

Code: Select all
 Grp. Occ. Lambda-hat        Standard Error      Lower           Upper
 ---- ----    --------------          --------------     --------------     --------------
   1     1    1.1210639            0.0474039       1.0319363       1.2178893   
   1     2    1.1210639            0.0474039       1.0319363       1.2178893   
   1     3    1.2567842            0.1062857       1.0651281       1.4829263   
   1     4    1.1210639            0.0474039       1.0319363       1.2178893   
   1     5    1.1210639            0.0474039       1.0319363       1.2178893   
  2     1    1.1236979            0.0446491       1.0395386       1.2146706   
   2     2    1.1236979            0.0446491       1.0395386       1.2146706   
   2     3    1.2626970            0.1003442       1.0808388       1.4751539   
   2     4    1.1236979            0.0446491       1.0395386       1.2146706   
   2     5    1.1236979            0.0446491       1.0395386       1.2146706


When I fitted the equivalent model Phi(g) p(t) f(g) using the Pradel recruitment formulation, results were similar for Phi and p, but much different for f and Lambda:

Code: Select all
                                                             
 Parameter                   Estimate       Standard Error      Lower           Upper     
 -------------------------  --------------     --------------     --------------  --------------
    1:Phi                      0.8032024       0.0300957       0.7375461       0.8556486     
    2:Phi                      0.8866474       0.0213586       0.8375912       0.9222613     
    3:p                        0.7023161       0.0822840       0.5217225       0.8361365     
    4:p                        0.2535039       0.0400336       0.1832251       0.3395334     
    5:p                        0.4428963       0.0495322       0.3491266       0.5409205     
    6:p                        0.1692410       0.0296090       0.1188069       0.2353661     
    7:p                        0.7001491       0.0513179       0.5911943       0.7903624     
    8:p                        0.5158839       0.0532063       0.4124123       0.6180114     
    9:f                        0.2026007       0.0362209       0.1406916       0.2827868     
   10:f                       0.1454143       0.0286644       0.0976923       0.2109968

 Grp. Occ. Lambda-hat   Standard Error      Lower           Upper
 ---- ----   --------------      --------------     --------------     --------------
   1     1    1.0058031       0.0326810       0.9437614       1.0719235   
   1     2    1.0058031       0.0326810       0.9437614       1.0719235   
   1     3    1.0116399       0.0657412       0.8907750       1.1489044   
   1     4    1.0058031       0.0326810       0.9437614       1.0719235   
   1     5    1.0058031       0.0326810       0.9437614       1.0719235   
   2     1    1.0320617       0.0277792       0.9790351       1.0879604   
   2     2    1.0320617       0.0277792       0.9790351       1.0879604   
   2     3    1.0651514       0.0573397       0.9585645       1.1835901   
   2     4    1.0320617       0.0277792       0.9790351       1.0879604   
   2     5    1.0320617       0.0277792       0.9790351       1.0879604


Since the unequal intervals seemed to be a problem, I edited the capture histories to have seven occasions with occasion 4 (2014) coded as zeroes. I then used the default one-year time intervals and fixed p = 0 for occasion 4. The estimates for both the Link-Barker and Pradel formulation were then exactly the same and there was no time variation in Lambda-hat:

Code: Select all
   Parameter                  Estimate     Standard Error      Lower           Upper
 ------------------------  --------------     --------------      --------------    --------------
     1:Phi                    0.8032024       0.0286032       0.7411127       0.8533476                         
     2:Phi                    0.8866474       0.0202994       0.8403817       0.9207673                         
     3:p                      0.7023161       0.0782033       0.5312593       0.8308273                         
     4:p                      0.2535039       0.0380482       0.1863225       0.3349375                         
     5:p                      0.4428963       0.0470758       0.3535736       0.5360713                         
     6:p                      0.0000000       0.0000000       0.0000000       0.0000000      Fixed               
     7:p                      0.1692410       0.0281406       0.1209667       0.2317023                         
     8:p                      0.7001491       0.0487729       0.5969241       0.7863984                         
     9:p                      0.5158839       0.0505677       0.4174393       0.6131111                         
    10:f                      0.2026007       0.0344246       0.1455579       0.2819981                         
    11:f                      0.1454143       0.0272429       0.1010434       0.2092697                         

 Grp. Occ. Lambda-hat   Standard Error      Lower           Upper
 ---- ----   --------------      --------------     --------------  --------------
   1     1    1.0058031       0.0326810       0.9437613       1.0719234   
   1     2    1.0058031       0.0326810       0.9437613       1.0719234   
   1     3    1.0058031       0.0326810       0.9437613       1.0719234   
   1     4    1.0058031       0.0326810       0.9437613       1.0719234   
   1     5    1.0058031       0.0326810       0.9437613       1.0719234   
   1     6    1.0058031       0.0326810       0.9437613       1.0719234   
   2     1    1.0320617       0.0277792       0.9790351       1.0879604   
   2     2    1.0320617       0.0277792       0.9790351       1.0879604   
   2     3    1.0320617       0.0277792       0.9790351       1.0879604   
   2     4    1.0320617       0.0277792       0.9790351       1.0879604   
   2     5    1.0320617       0.0277792       0.9790351       1.0879604   
   2     6    1.0320617       0.0277792       0.9790351       1.0879604


I assume these estimates are correct, but I don't understand why the unequal intervals made such a difference for the Link-Barker formulation. My understanding from the MARK book was that either method of coding the capture histories should have given the same results.
stshroye
 
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby stshroye » Wed Dec 08, 2021 5:09 pm

Does anyone know the reason for the discrepancy?
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby cooch » Mon Mar 14, 2022 9:19 pm

stshroye wrote:Does anyone know the reason for the discrepancy?


As you note phi the same, but the two methods do seem to handle unequal intervals differently. In fact, the general recommendation is to be highly suspicious of unequal intervals for TSM-like models (see below).

Take the canonical capsid data. Make one of the intervals '2' (I picked interval 5 out of 12). Fit Pradel 'recruitment and survival' - look at phi (for example)

Code: Select all
    Parameter                Estimate     
--------------------------  --------------
    1:Phi                    0.6411063     
    2:Phi                    1.0000000     
    3:Phi                    0.8950196     
    4:Phi                    0.6665493     
    5:Phi                    0.8310037     
    6:Phi                    0.7608369     
    7:Phi                    0.6394752     
    8:Phi                    0.9916983     
    9:Phi                    0.7294168     
   10:Phi                    0.8528419     
   11:Phi                    0.7876797     
   12:Phi                    0.3001666     



Change data type to Link-Barker Jolly-Seber (which is identical to Pradel, except for how it handles losses on capture (Pradel doesn't -- in fact, the general recommendation is to just routinely use Link-Barker). Get the *exact same phi estimates* (below), even interval 5.

Code: Select all
     Parameter                Estimate       
 --------------------------  -------------- 
     1:Phi                    0.6411100     
     2:Phi                    1.0000000     
     3:Phi                    0.8950195     
     4:Phi                    0.6665490     
     5:Phi                    0.8310029     
     6:Phi                    0.7608401     
     7:Phi                    0.6394765     
     8:Phi                    0.9916924     
     9:Phi                    0.7294188     
    10:Phi                    0.8528447     
    11:Phi                    0.7876728     
    12:Phi                    0.3160629     


What about lambda (a derived parameter)? For Pradel

Code: Select all
 Grp. Occ.   Lambda-hat     
 ---- ----   -------------- 
   1     1    1.5605253       
   1     2    1.4390146       
   1     3    1.3397122       
   1     4    0.9862104       
   1     5    0.8871995 
   1     6    0.8914071       
   1     7    0.8162201       
   1     8    0.9916992       
   1     9    0.7980453       
   1    10    1.0111625       
   1    11    0.9527089       
   1    12    0.3130484       


and for Link-Barker

Code: Select all
 Grp. Occ.   Lambda-hat     
 ---- ----   -------------- 
   1     1    9.2319100       
   1     2    1.4331465       
   1     3    1.3391482       
   1     4    0.9844238       
   1     5    1.6206453       
   1     6    0.8905043       
   1     7    0.8153864       
   1     8    0.9916929       
   1     9    0.7976359       
   1    10    1.0107722       
   1    11    0.9519993       
   1    12    0.3295952       


So, estimates largely the same, except for the unequal interval. Short answer is, I wouldn't trust any estimate from the unequal interval anyway. What MARK typically does is simply take the nth root of the estimate over that interval, which means...what? Not much. You can't really constrain it in any sort of useful way to be a function of covariates so, I often just toss them as garbage placeholders, and move on to the intervals I have faith in. There are lots of data types in MARK where unequal intervals basically make things uninterpretable (robust design, multi-state models, some of the dynamic occupancy models...). So much so, that MARK in restricts if you can even tweak the intervals.

Longer answer, I'll check with Gary. Superficially, not obvious why the two data types should treat the funky interval differently, but they seem to.
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby stshroye » Tue Mar 15, 2022 9:41 am

Thank you! I have learned to be wary of unequal time intervals. I will be curious to see if Gary has anything to add.
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby gwhite » Tue Mar 15, 2022 10:34 am

You’ve opened this can of worms, and found out that they don’t all wiggle in the same direction. I’ll try to explain why.

First, let’s start with the CJS data type = 1, where phi is corrected for unequal time intervals (I’ll use T as time interval length for simplicity) as phi^T. This makes complete sense for T = 2, as phi*phi = phi^T. All is good.

Now consider how Pradel conceptualized recruitment in his models. He considered what phi is for the trailing zeros in the encounter history, that is, that phi is the probability that an animal is still alive and not seen. So for the first trailing zero, the probability of the animal being alive and not seen is phi*(1-p). Then Pradel conceptualized what is phi if the encounter history is reversed and run through a CJS model, so that trailing zeros become leading zeros. Then phi becomes the probability that the animal is alive but not yet observed. He called this reversed phi parameter gamma and modeled recruitment as gamma in his likelihood. This is data type 15. Because phi^T makes complete sense, so does gamma^T for recruitment. All is good.

I reparameterized the Pradel model of data type 15 to include the relationship that gamma = phi/lambda, providing data type 16. The lambda parameter is a natural parameter for biologists: lambda = N(t+1)/N(t). So making gamma in the likelihood now phi/lambda is a natural, and gamma^T = (phi/lambda)^T = phi^T/lambda^T looks great. All is good.

Then I went one step further and incorporated the relationship lambda = phi + f to create data type 17. Then gamma = phi/lambda = phi/(phi + f). Logically, gamma^T = [phi/(phi + f)]^T = phi^T/(phi + f)^T, but NOT phi^T/(phi^T + f^T). All is NOT good.

Now enter data type 31, the Link-Barker version. They parameterized the likelihood in terms of a probability of entry, much like the POPAN data type. The then converted the probability of entry into f. I included f with the usual f^T for unequal intervals, but the relationship between data type 17 and data type 31 for f^T breaks down, as you found. All is NOT good.

Another example in MARK of the failure of 2 equivalent models is the Kendall robust design and the multi-state robust design. Kendall had the gamma’ and gamma’’ parameters to model the probability of the animal remaining on the study area or remaining away from the study area. I used the ^T trick on these parameters because it seemed reasonable that the longer the interval, the length of T should affect them just like phi. Note that a critical assumption often forgotten with the robust design data types is that S is the same for animals on or off the study area. Sometimes this is a really bad assumption.

Now consider the equivalent robust design multi-state model where psi parameters are used in place of the 2 gammas. Suppose state A is on the study area and state B is off the study area. If S(A) = S(B), the model is identifiable and equivalent to the robust design above. But multi-state models make a different assumption about the transitions -- they can ONLY occur at the end of the interval. Otherwise, survival across the interval is a function of 2 survival rates. When S(A) = S(B), no problem, but that is not the usual multi-state model. So if you run the robust design data type and parameterize the robust design multi-state data type with all time intervals equal 1, you get the identical -2log likelihood value and you can get identical psi and gamma estimates if interpreted correctly. But when you make time intervals unequal, i.e., T=2, you still get identical -2log likelihood values, but your estimates of the psi and gamma parameters no longer mesh because the gamma parameters are corrected for T and the psi parameters are not.

To further complicate the issue, a standard trick is to use a log link for phi in the CJS model, and enter the time interval length in the design matrix instead of the usual 1’s. For T=2, you put a 2 in the design matrix. This works because exp(beta*2) = exp(log(phi)*2) = phi^2. All is good. So try using this trick with the Pradel and Link-Barker data types and see what happens. Normally you use a link function to constrain phi and gamma to the interval 0-1. But usually the log link that constrains the parameter to >0 works fine for phi and gamma, and the log link is automatically used for lambda and f because they are only required to be >0 with no upper bound. Not sure what you’ll get, but it might make more sense than what you’ve seen so far.
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby cooch » Tue Mar 15, 2022 10:48 am

gwhite wrote:To further complicate the issue, a standard trick is to use a log link for phi in the CJS model, and enter the time interval length in the design matrix instead of the usual 1’s. For T=2, you put a 2 in the design matrix. This works because exp(beta*2) = exp(log(phi)*2) = phi^2. All is good. So try using this trick with the Pradel and Link-Barker data types and see what happens. Normally you use a link function to constrain phi and gamma to the interval 0-1. But usually the log link that constrains the parameter to >0 works fine for phi and gamma, and the log link is automatically used for lambda and f because they are only required to be >0 with no upper bound. Not sure what you’ll get, but it might make more sense than what you’ve seen so far.


Said log link (in the context of uneven intervals) is demonstrated in Chapter 4 -- about half-way into the very (too) long -sidebar- that begins on p. 24. If your browser supports a link to a specific page in a PDF file: http://www.phidot.org/software/mark/doc ... df#page=30 (or simply search the PDF for 'log link').
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby egc » Tue Mar 15, 2022 1:54 pm

gwhite wrote:
<snip>

...I reparameterized the Pradel model of data type 15 to include the relationship that gamma = phi/lambda, providing data type 16. The lambda parameter is a natural parameter for biologists: lambda = N(t+1)/N(t). So making gamma in the likelihood now phi/lambda is a natural, and gamma^T = (phi/lambda)^T = phi^T/lambda^T looks great. All is good.

</snip>


One of the main motivations for putting a parameter 'in the likelihood' is that it then becomes 'available' to you to model via the design matrix.
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby gwhite » Thu Mar 17, 2022 6:03 pm

I have posted a new version of MARK that addresses part of this issue. I found that the derived lambda values being computed with the Pradel f and Pradel gamma data types was getting corrected for the time interval twice, so fixed that.

I also attempted to get the Link-Barker data type to agree with the Pradel lambda estimates. Not sure whether I just don't understand the recruitment model in this data type well enough to make it agree with the Pradel estimate, or whether they will never agree with time intervals not equal to 1. What I posted comes close, but not exactly the same.

Gary
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby murray.efford » Thu Mar 17, 2022 6:11 pm

For what it's worth, I struggled with scaling of recruitment in 'openCR' and applied the ad hoc solution below. I think it connects with the difficulty of using an instantaneous rate for recruitment, as Torbjorn Ergon has advocated for mortality.
Murray

[from openCR-vignette]

## Sampling intervals

We have seen the role of the intervals attribute in defining primary and secondary sessions. Between-session intervals need to be specified only if they vary, or if you would like rates (phi, gamma, lambda, f) to be reported in time units other than the (implicitly constant) sampling interval. Scaling from the standardised parameter $\theta_j$ to the interval-specific value $\theta^\prime_j$ uses $\theta^\prime_j = \theta_j^{T_j}$ where $\theta_j$ is one of $\phi_j$ or $\lambda_j$, and $T_j$ is the duration of interval $j$.

Scaling $\gamma$ follows the same pattern except that the relevant duration for $\gamma_j$ is $T_{j-1}$. Scaling per capita recruitment $f_j$ is more tricky. We use $f^\prime_j = (\phi_j + f_j)^{T_j} - \phi_j^{T_j}$.
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Re: Link-Barker and Pradel unequal time interval discrepancy

Postby ehileman » Fri Mar 18, 2022 6:33 pm

I have posted a new version of MARK that addresses part of this issue. I found that the derived lambda values being computed with the Pradel f and Pradel gamma data types was getting corrected for the time interval twice, so fixed that.

I also attempted to get the Link-Barker data type to agree with the Pradel lambda estimates. Not sure whether I just don't understand the recruitment model in this data type well enough to make it agree with the Pradel estimate, or whether they will never agree with time intervals not equal to 1. What I posted comes close, but not exactly the same.

Gary


Acknowledging the additional complications that arise when using unequal intervals in a robust design framework, has this recent Pradel f fix also been applied to the robust design Pradel f?

Thanks,

Eric
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