Interpreting beta parameter estimates

questions concerning analysis/theory using program MARK

Interpreting beta parameter estimates

Postby natprice » Mon Oct 22, 2018 12:42 pm

I am confused about the relationship between the beta parameter estimates and the real parameter estimates for the abundance parameters in POPAN and Jolly-Seber Lambda -- Burnham models. For a model with only intercept term and log link function I thought the relationship should be N_real = exp(Beta_N). What is the relationship between the model parameter estimates and the values on the real scale?

For example, running a POPAN model on dipper data with log link function the coefficient is 2.72 and the real estimate is 309.

PARM-SPECIFIC Link Function Parameters of { Phi(~1)p(~1)pent(~1)N(~1) }
95% Confidence Interval
Parameter Beta Standard Error Lower Upper
------------------------- -------------- -------------- -------------- --------------
1:Phi:(Intercept) 0.2382585 0.1016105 0.0391019 0.4374150
2:p:(Intercept) 2.2914677 0.3323917 1.6399799 2.9429554
3:pent:(Intercept) 0.6683266 0.2233367 0.2305866 1.1060667
4:N:(Intercept) 2.7195525 0.4296113 1.8775143 3.5615907


Real Function Parameters of { Phi(~1)p(~1)pent(~1)N(~1) }
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:Phi g1 a0 t1 0.5592844 0.0250455 0.5097742 0.6076429
2:p g1 a0 t1 0.9081679 0.0277211 0.8375322 0.9499295
3:N g1 a0 t1 309.17353 6.5187202 300.77275 327.99446
4:pent g1 a1 t2 0.1535493 0.0026990 0.1483335 0.1589143
5:pent g1 a2 t3 0.1535493 0.0026990 0.1483335 0.1589143
6:pent g1 a3 t4 0.1535493 0.0026990 0.1483335 0.1589143
7:pent g1 a4 t5 0.1535493 0.0026990 0.1483335 0.1589143
8:pent g1 a5 t6 0.1535493 0.0026990 0.1483335 0.1589143
9:pent g1 a6 t7 0.1535493 0.0026990 0.1483335 0.1589143


EDIT:

I ran the JS model on the dipper data using marked package in R and coefficient of N is 2.72 and the estimated real value of N is 15.17. This agrees with my expectations that N_real = exp(Beta_N). Is there a bug in MARK or am I doing something wrong?
natprice
 
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Joined: Wed Nov 01, 2017 10:50 am

Re: Interpreting beta parameter estimates

Postby cooch » Mon Oct 22, 2018 4:06 pm

Simple - because what is being reported as a beta N should in fact be beta f(0). What is being reported is the beta value for the estimate (f0) that is added to M(t+1), which yields the value of N you're after (and which is reported in the reals).

For example, take just the male dippers. M(t+1) = 141. If you fit model phi(t)p(.)pent(t)N (using the MLogit link for pent), you get the following beta estimates (note, I used the default sin link for phi and p, mlogit for pent, and the log link for 'N'):

Code: Select all
                                                             95% Confidence Interval
 Parameter                    Beta         Standard Error      Lower           Upper
 ------------  --------------  --------------  --------------  --------------
    1:Phi       0.2203577       0.3047551      -0.3769623       0.8176778
    2:Phi      -0.0843960       0.1995335      -0.4754816       0.3066896
    3:Phi      -0.0079520       0.1695325      -0.3402358       0.3243318
    4:Phi       0.2264415       0.1657479      -0.0984244       0.5513075
    5:Phi       0.1476554       0.1533389      -0.1528889       0.4481997
    6:Phi       0.2585709       0.1623499      -0.0596349       0.5767768
    7:p         1.0308557       0.1416162       0.7532880       1.3084235
    8:pent      0.4834153       0.3752593      -0.2520929       1.2189236
    9:pent      0.7068022       0.3546445       0.0116990       1.4019054
   10:pent      0.5652694       0.3650444      -0.1502176       1.2807564
   11:pent      0.5613525       0.3656803      -0.1553808       1.2780858
   12:pent      0.6109237       0.3618835      -0.0983679       1.3202153
   13:pent      0.2815496       0.3892568      -0.4813938       1.0444930
   14:N         1.6039344       0.7658604       0.1028481       3.1050208       


So, reported beta for N (which probably should be f(0)) is 1.6039344. Here are the reconstituted real parameter estimates:

Code: Select all
                                                               95% Confidence Interval
  Parameter                  Estimate       Standard Error      Lower           Upper
 -----------------  --------------  --------------  --------------  --------------
     1:Phi           0.6092894       0.1486930       0.3143378       0.8413850
     2:Phi           0.4578521       0.0994116       0.2780896       0.6493020
     3:Phi           0.4960240       0.0847636       0.3361513       0.6567139
     4:Phi           0.6122557       0.0807583       0.4477046       0.7546479
     5:Phi           0.5735597       0.0758352       0.4227947       0.7117895
     6:Phi           0.6278496       0.0784764       0.4662282       0.7651812
     7:p             0.9288696       0.0364014       0.8160186       0.9746501
     8:pent          0.1431326       0.0305990       0.0929181       0.2140794
     9:pent          0.1789593       0.0335625       0.1222688       0.2543186
    10:pent          0.1553414       0.0320156       0.1023346       0.2288063
    11:pent          0.1547342       0.0320202       0.1017630       0.2282714
    12:pent          0.1625978       0.0326115       0.1082770       0.2369298
    13:pent          0.1169686       0.0285535       0.0715384       0.1854854
    14:N             145.97256       3.8082853       142.31305       159.83127   


So, where does 145.97256 come from?

M(t+1)+f(0) = 141 + exp(1.6039344) = 141+4.972558 = 145.97256

And there you have it...the confusion is because the beta is labeled N, when it should probably be f(0). This is something Gary would need to change.
cooch
 
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Re: Interpreting beta parameter estimates

Postby natprice » Mon Oct 22, 2018 4:37 pm

Thanks, it is not so simple for me because I am just learning. M(i) is the number of marked individuals at time (i). M(t+1) is the number of marked individuals at the end of the study. I thought f was the per capita recruitment rate. The super population size is equal to the number of individuals marked over the study plus f(0)?
natprice
 
Posts: 5
Joined: Wed Nov 01, 2017 10:50 am

Re: Interpreting beta parameter estimates

Postby cooch » Mon Oct 22, 2018 5:53 pm

natprice wrote:Thanks, it is not so simple for me because I am just learning.

If you're 'just learning', the POPAN datatype isn't the best place to start. A fair amount of the POPAN parameterizations takes some time to wrap your brain around (took me many attempts before I finally figured out what the gross derived parameters mean...I digress).

M(i) is the number of marked individuals at time (i). M(t+1) is the number of marked individuals at the end of the study. I thought f was the per capita recruitment rate. The super population size is equal to the number of individuals marked over the study plus f(0)?


f(0) is the estimate of the number of individuals that are in the population, but not observed (see Chapter 14). Logically, the estimate of N is 'how many you did observe' (at least once), M(t+1), plus the number that you estimate were there but not observed (f(0)).

So $\hat{N}=\hat{f}_0+M_{t+1}$.

And, I'm guessing you don't fully understand the concept of a 'super-population', which is a sort of abstract thing that the fish squeezers (who predominate amongst POPAN users) use. Have a read of the literature of the definition of a 'super-population' in a POPAN model. It is likely different than what you might think it is.
cooch
 
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