convergence problems, age model with fixed parameters

[abreton]

Perhaps someone can help me see the obvious? After several hours and lots of 'guesses' my problem remains elusive. I'm working with a data set that consists of male and female herring gulls marked as adults (breeders; high site fidelity) with four band types (bt) from 1978-1984. In order to assess GOF with MARK (c-hat, bootstrap) and U-CARE (many tests) I specified encounter histories (7 occasions) with eight groups: 2 sexes * 4 band types = 8 groups and ran model Phi (g+t) p (g+t) where g = group and t = time (design matrix, logit link, no covariates fitted or included in the EHs). The model converges in MARK and all unconfounded parameters are estimable (18; 2 confounded parameters); 'interactions' led to convergence failure; hence the additive model.

One important detail is that two of the band types enter the study after 1978: first release for band type (bt) 3 is 1979; and bt 4 is 1981. To accomodate this staggered release, I fixed appropriate parameters (e.g., survival from 1978-79 (phi1) for bt 3) to 0 from the "Numerical Estimation Run" window.

If the g+t model was appropriate for the data, then I'd be home by now with my feet up. But as we suspected, this was not the case. The primary objective of the analysis is to determine if 'marking' affected survival immediately after bands were applied - an acute (one interval) effect. Both MARK and U-CARE detected 'heterogeneity' in model Phi (g+t) p (g+t); and U-CARE suggested 'transience' was the cause, which is exactly what we'd expect if marking had an immediate affect on survival.

I went ahead and fit a simple age model in MARK which allows survival immediately after banding to differ from subsequent intervals: Phi (s+a2-c/c+bt) p (s+bt+t) where s = sex, a2-c/c = two age model no time dependence (constant), bt = band type, and t = time. As before, I fixed some paramters to zero to account for the fact that birds with band types 3 and 4 were not released until 1979 (one year after the study began) and 1981, respectively; however, this time I specified a 1 in the PIMs anywhere that I wanted to fix a paramter to zero; subsequently, I fixed Parm Phi:1 to zero prior to running the model. Note that this model has 16 parameters all of which should be estimable (no confounding) - and that this is fewer parameters than model Phi (g+t) p (g+t) described above, which converegd appropriately

When I run this model in MARK, two out of three band type effects are not estimable as either effects fitted to survival or resighting probabilities. And the bogus estimates for all four of these parameters are identical:
Parm, B, SE, UCI, LCI
0.1000000,35.316547,69.320434,69.120434

Why is it that a more complex model (Phi (g+t) p (g+t)) converges but this simpler age model does not? Out of desperation, I tried fixing PIM Phi:1 to one rather than zero and an interesting thing happened - one of the band effects was now 'estimated'. Perhaps this clue is helpful? I've taken a close look (over and over again) at the EHs, m-arrays (each group), PIMs, and design matrix and can find no flaws...which ultimately leads me to suspect something associated with fixing paramters?

[cooch]

Out of desperation, I tried fixing PIM Phi:1 to one rather than zero and an interesting thing happened - one of the band effects was now 'estimated'. Perhaps this clue is helpful? I've taken a close look (over and over again) at the EHs, m-arrays (each group), PIMs, and design matrix and can find no flaws...which ultimately leads me to suspect something associated with fixing paramters?

Haven't got time for more than a quick read (and may be missing something), but it might be as simple as this: if you fix any survival parameter to zero which you tried, you're 'telling' MARK that no individual can be encountered after that interval (if survival is zero, then no individual survives past that point). If the first diagonal is fized at zero, then there are no possible encounters after that. I'm pretty sure this isn't what you want, and would easily explain why nothing was estimable when you did this. It would also explain why when, instead, you set it to 1, things became estimable (although the estimates may not make any sense).

[abreton]

Thanks for you thoughts evan. The following may be more helpful than what I wrote last night...

The parameters that are not 'estimable' are those that require data from the two band types (bt) that were released after 1978 (start of my EHs). Clearly, this suggests, in my brain anyway, that somehow my 'zeros' (fixed parms) are getting into the data*ln(pr) terms that make up the likelihood function specified by model Phi (s+bt+a2-c/c) p (s+bt+t). But what I can't figure out is how parameters before the first release for these groups can get into the EH probability statements. A simple example might help expose my flawed thinking; I'll use the standard open pop/live encounter CJS model parameterization:

Imagine a scenario with two groups of adults, and the second group is released on the second occasion of the study. Therefore, there are no individuals available from group two to estimate survival from occasion 1 to 2 or resighting on occasion 2. For ease of model building, a "1" in the PIMs will be reserved for these 'non-existent' probabilities and fixed to zero at model runtime. An example EH from this group is 0101 0 1;. As in my study, we'll assume that an acute marking effect is suspected and build age into phi and only time dependence into p; we'll assume group dependence; for simplicity, we'll also assume that survival is constant in both age classes (i.e., not include time in phi):

PIM Group One, Phi, 2 age classes
2 3 3
2 3
2
PIM Group Two, Phi, 2 age classes
1 4 5
1 4
1
PIM Group One, p, time
6 7 8
7 8
8
PIM Group Two, p, time
1 9 10
9 10
10

The probability statement for EH 0101 0 1 is phi2*1-p3*phi3*p4. Phi1 and p2, which are fixed to zero for this group (parm 1 in the PIMs) at model runtime, do not enter into the probability statement or "(pr)" component of the data*ln(pr) term. A simple model to construct in the design matrix is Phi (group+age) p (group+time):
B1 B2 B3 B4 Parm
1 0 0 0 Phi:1
0 1 0 1 Phi:2
0 1 0 0 Phi:3
0 1 1 1 Phi:4
0 1 1 0 Phi:5
Where B1 is fixed = 0, B2 = intercept, B3 = group 2, and B4 = the survival interval immediately after first release (age1).
B5 B6 B7 B8 Parm
1 0 0 0 p:6
1 0 1 0 p:7
1 0 0 1 p:8
1 1 1 0 p:9
1 1 0 1 p:10
Where B5 is the intercept, B6 = group 2, B7 = t2, B8 = t3.

So, when I run this model, B3 and B6 are not estimable. If phi1 or p2 get into group two's probability statements, then this would result result in a lot of zeros in teh likelihood. More specifically, all the terms in the likelihod funciotn that included B3 and B6 would be zero - so we couldn't begin to estimate these 'group' betas. This is what appears to be happening in my case, which is identical to this example but includes two more groups. But I can see no way of getting phi1 and p2 into group two's probability statements.

Note that (last night) I reduced my groups to the two most data rich and tried the same age model described earlier. These two groups offer the same scanario as the one I just described (group two released on the 2nd occasion) just with more occasions. The result was again that the phi and p group effects were not estimable. In my mind, this result and others that I tried rules out 'data sparsity' as the underlying problem. Thanks to anyone who patiently read through my posts.

[abreton]

The PIM structure changed after I posted my reply. To interpret these correctly, make each column a diagonal: the first column = first diagonal, second the second, etc. Once you do this, you'll see that the Phi and p PIMs are specified with an age and time structure, respectively.