POPAN - fixed values double in derived estimates

questions concerning analysis/theory using program MARK

POPAN - fixed values double in derived estimates

Postby wijw » Wed Mar 17, 2021 3:46 pm

I have a dataset with 258 individuals, 6 occasions (years), and 6 groups. The population has no births or immigration/emigration, but deaths occur. I am trying to estimate the population size using POPAN formulation. I know the N and pent for some of the occasions. I was able to fix the known N using the ‘fix parameters’ button.

The model I am running is phi(.)p(.)pent(t)

My problem is that when I look at the derived estimates notepad - the Group 1 occasion 1 N-hat value is twice as large as the fixed value. But correctly identifies the remaining “fixed N” values for other occasions.

I cannot figure out why the first value is almost doubled although it was fixed.

Any help is appreciated.
wijw
 
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Re: POPAN - fixed values double in derived estimates

Postby keevilm » Thu Mar 18, 2021 7:51 am

Barring a really strange coincidence, I suspect that wijw are working on the exact same problem, which I have continued to pursue without being able to get to the bottom of it. Having prepared the following description of the issue, I am going to go ahead an add it here in hopes that some further details will help:

I am studying recapture observations of released cohorts in a reintroduction program. I am not considering any natural recruitment. I am using the POPAN JS model and I am trying to fix the known number released for each cohort by fixing b[t] (i.e. pent) parameters and N (the superpopulation). However I am getting unexpected results: if the known number of releases just before occasion t is called R[t-1] then I expect Nhat[t] would be exactly R[t-1], which would be exactly B[t-1] which would be exactly N. Instead these quantities are B[t-1] = Nhat[t] = N + the number of encounter histories.
I have simulated some encounter histories, which are pasted at the end, for a simpler 1 cohort (= 1 release occasion) example. I simulated a set of 5-occasion encounter histories for 30 individuals released between the 1st and the 2nd occasion with constant p = 0.5 and constant phi = 0.7. After removing all-zero encounter histories there were 24 remaining. I set up a the PIM matrices for 5 parameters: (parm1): p(.), (parm2): phi(.), (parm3):b[1] = 1 (fixed), (parm4):b[2] = b[3] = b[4] = 0 (fixed), and (parm5): N = 30 (fixed). This gives incorrect results and two warnings:
** WARNING ** Error number 0 from VA09AD optimization routine.
** WARNING ** Invalid calculation occurred during processing derived parameters of this model.


The output shows that N = 30 but Nhat[2] =Bhat[1] = 54 (i.e. 30 + 24 encounter histories). 
Code: Select all
Parameter                  Estimate       Standard Error      Lower           Upper
 --------------------------  --------------  --------------  --------------  --------------
     1:Phi                    0.6832247       0.0950899       0.4768883       0.8361389                          
     2:p                      0.2571128       0.0509876       0.1701962       0.3686945                          
     3:pent                   1.0000000       0.0000000       1.0000000       1.0000000      Fixed              
     4:pent                   0.0000000       0.0000000       0.0000000       0.0000000      Fixed              
     5:N                      30.000000       0.0000000       30.000000       30.000000      Fixed               
 Grp. Occ.   B-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    54.000000       0.0000000       54.000000       54.000000    
   1     2    0.0000000       0.0000000       0.0000000       0.0000000    
   1     3    0.0000000       0.0000000       0.0000000       0.0000000    
   1     4    0.0000000       0.0000000       0.0000000       0.0000000
  Grp. Occ.   N-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    0.0000000       0.0000000       0.0000000       0.0000000    
   1     2    54.000000       0.0000000       54.000000       54.000000    
   1     3    36.894136       5.1348557       28.122526       48.401673    
   1     4    25.206987       7.0165202       14.756587       43.058206    
   1     5    17.222037       7.1907901       7.8496743       37.784825
 
   

If I re-run the model with N fixed to 6 (i.e. the number of released but never-encountered individuals) the Nhat and Bhat values are correct and phi and p are close to the simulation values. However, this kludge does not make sense to me based on my understanding of what N represents. I understand that there is a b[0] value that is not shown and that is obtained as 1 - sum(b[1] ... b[n.occ-1]) but I don't see how that explains the result because the fixed values should ensure that sum(b[1] ... b[n.occ-1]) = 1 and therefore b[0] should be 0.

Possibly relevant: when I only fix parm 4 (i.e. b[2 ... 4]) to 0 the resulting estimate of b[1] is 1, which matches my understanding of the multinomial logit link BUT if I switch it around and fix b[1] = 1 and let b[2 ...4] be estimated I get all pent approximately = 1 (how is that possible if they share a mlogit(1) link?) negative abundances at early occasions, and 30 individuals "being born" every occasion.

Please help; what am I missing?

encounter histories to reproduce the analysis:

Code: Select all
00001 1;
00010 1;
00011 1;
00100 5;
00101 3;
00111 1;
01000 9;
01010 1;
01100 1;
01101 1;
keevilm
 
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Re: POPAN - fixed values double in derived estimates

Postby keevilm » Thu Mar 18, 2021 1:36 pm

I have done a bit more, and I believe I have narrowed down the problem, which stems from fixing N not from fixing the pents because the problem remains even when the pents are not fixed but N is. I constrained the last three pent parameters using the design matrix:
Code: Select all
            
1   0   0   0   0      1:Phi
0   1   0   0   0      2:p
0   0   1   0   0      3:pent
0   0   0   1   0      4:pent
0   0   0   1   0      5:pent
0   0   0   1   0      6:pent
0   0   0   0   1      7:N


I ran the model without fixing any parameters and recovered decent estimates with b_2=b_3=b_4close to 0 and b_1close to 1, N ~ 38 (a bit high but CI included true value of 30):
Code: Select all
  Parameter                  Estimate       Standard Error      Lower           Upper
 --------------------------  --------------  --------------  --------------  --------------
     1:Phi                    0.7183856       0.0958561       0.5019484       0.8658951                         
     2:p                      0.3433264       0.0873211       0.1966066       0.5276290                         
     3:pent                   1.0000000       0.7061540E-04  0.3700331E-300   1.0000000                         
     4:pent                   0.5009396E-08   0.2366513E-04  -0.4637865E-04   0.4638867E-04                     
     5:pent                   0.5009396E-08   0.2366513E-04  -0.4637865E-04   0.4638867E-04                     
     6:pent                   0.5009396E-08   0.2366513E-04  -0.4637865E-04   0.4638867E-04                     
     7:N                      38.347435       8.2424428       29.037134       64.866273   

 Grp. Occ.   B-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    38.347435       8.2424533       25.283452       58.161590   
   1     2    0.1920975E-06   0.9075458E-03   0.6053803E-10   0.6095580E-03
   1     3    0.1920975E-06   0.9075458E-03   0.6053803E-10   0.6095580E-03
   1     4    0.1920975E-06   0.9075458E-03   0.6053803E-10   0.6095580E-03
            Population Estimates of {phi(.)p(.)pent  Dmatrix)}
                                                95% Confidence Interval
 Grp. Occ.   N-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    0.1821665E-09   0.1278167E-05   0.4765015E-13   0.6964225E-06
   1     2    38.347435       8.2424533       25.283452       58.161590   
   1     3    27.548246       6.0420492       18.013094       42.130788   
   1     4    19.790264       5.7906917       11.284442       34.707479   
   1     5    14.217041       5.6643384       6.7000828       30.167425   



I then fixed N to 30 but no other parameters were fixed and I got:
Code: Select all
  Parameter                  Estimate       Standard Error      Lower           Upper
 --------------------------  --------------  --------------  --------------  --------------
     1:Phi                    0.6832247       0.0950900       0.4768882       0.8361389                         
     2:p                      0.2571128       0.0509876       0.1701962       0.3686946                         
     3:pent                   1.0000000       0.8665327E-04  0.3052136E-300   1.0000000                         
     4:pent                   0.6065413E-08   0.2876711E-04  -0.5637747E-04   0.5638960E-04                     
     5:pent                   0.6065413E-08   0.2876711E-04  -0.5637747E-04   0.5638960E-04                     
     6:pent                   0.6065413E-08   0.2876711E-04  -0.5637747E-04   0.5638960E-04                     
     7:N                      30.000000       0.0000000       30.000000       30.000000      Fixed 

 Grp. Occ.   B-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    53.999997       0.0073673       53.985559       54.014439   
   1     2    0.9111062E-06   0.0024503       0.3772928E-09   0.0022002   
   1     3    0.9111062E-06   0.0024503       0.3772928E-09   0.0022002   
   1     4    0.9111062E-06   0.0024503       0.3772928E-09   0.0022002   
            Population Estimates of {phi(.)p(.)pent  Dmatrix)}
                                                95% Confidence Interval
 Grp. Occ.   N-hat           Standard Error      Lower           Upper
 ---- ----   --------------  --------------  --------------  --------------
   1     1    0.6366955E-08   0.2574365E-04   0.2161732E-11   0.1875261E-04
   1     2    53.999997       0.0073560       53.985581       54.014417   
   1     3    36.894139       5.1348584       28.122524       48.401682   
   1     4    25.206992       7.0165225       14.756589       43.058218   
   1     5    17.222043       7.1907929       7.8496766       37.784839   



so once again the number of individuals on the occasion after release (\hat{N}_2) is not 30 as intended but rather (approximately) 30 + the 24 individuals with encounter histories.

As far as I can tell, things are working as expected IF fixing N is actually fixing the number of undetected individuals, f_0 (using closed capture chapter notation), and not the superpopulation.
keevilm
 
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Re: POPAN - fixed values double in derived estimates

Postby gwhite » Thu Mar 18, 2021 2:26 pm

Although you think you are fixing N, you are really fixing N - u. = Number of animals never captured, but available for capture. The likelihood is maximizing u. and N is then obtained as u. + the known captures.
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Re: POPAN - fixed values double in derived estimates

Postby keevilm » Thu Mar 18, 2021 5:18 pm

Thank you for confirming that. Is there a reference that we could cite for this?
keevilm
 
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Re: POPAN - fixed values double in derived estimates

Postby gwhite » Thu Mar 18, 2021 7:16 pm

Not that I'm aware of. I programmed the model like that to force the estimate of the unmarked animals >0. Otherwise, the likelihood is not defined. So using a log link on the estimated unmarked animals simplifies the optimization process and makes the code more efficient.
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Re: POPAN - fixed values double in derived estimates

Postby cooch » Thu Mar 18, 2021 7:24 pm

So, you simply say 'As implemeneted in MARK,...', then describe the constraint as Gary has described.
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Re: POPAN - fixed values double in derived estimates

Postby keevilm » Thu Mar 18, 2021 7:54 pm

I will cite Schwarz and Arnason 1996 who state "By conditioning upon the total number of unmarked animals observed (u.) ... " and then write something like: 'we specified U_T, as the known number of released individuals that remained undetected after the final occasion. Fixing U_T, constrains the superpopulation, as implemented in MARK, to N = u. + U_T.' I hope that's coherent. Thank you again.
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