I'm working with robust design data consisting of six primary sessions each of 5-14 days (secondary sessions) in length over two summer seasons (three trapping sessions per season). I've entered my time intervals as months so that survival (S) and temporary emigration (Gamma) will be estimated as monthly rates. Between the primary sessions there are: 1.87, 0.53, 8.8, 1.6, 0.53 months, thus two short intervals followed by one long interval and then two more short intervals.
As the species is a gecko, capture probabilities are highly dependent on temperature. I have been doing some analyses to determine the best model for Gamma and capture/recapture probabilities (p/c) before modelling survival and abundance (N). So far I've found that temporary emigration is random and constant (Gamma"=Gamma'(dot)) and there is no behaviour effect on capture probability. The top model for capture probability is p=c(minimum temperature + session + group). Group has two levels based on time since fire in the habitat: recently burnt and unburnt.
For the models I just mentioned, I have been keeping S and N dependent on group. Occasionally there were times when Gamma wasn't estimated properly, but when I re-ran the model using the simulated annealing option, this problem disappeared. However, I'm now starting to look at more detailed models for survival (keeping p and Gamma as described above). I want to model S as a function of two time variables (i) the default "time" in MARK, and (ii) another variable I've called "time_season", a factor with two levels (1,2) which separates within season survival (1= short time intervals) from between season survival (2= the long time interval).
My problem is that as soon as I introduce any time dependence on S, the Gamma" parameter is not estimated properly. I do not have time dependence on Gamma", therefore all of the parameters should be individually identifiable. In the model table below the top four models have all the parameters counted except Gamma. The bottom two have all parameters counted, including Gamma. It looks as though there is strong support for time dependent survival models, but I'm sure the estimates can't be accurate when Gamma has not been estimated properly.
I have tried fixing this by using the simulated annealing option. I tried setting "retry=1" (I'm using RMark) to rerun the models with the initial value for the singular beta set to zero. I have tried manually setting the initial beta to -1.4283997, which is the beta value for Gamma in the S.group model where all parameters were counted. And finally I've tried all combinations of the above, but still, no Gamma.
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model npar AICc DeltaAICc weight Deviance
6 S.time_season+group 14 896.4824 0.000000 7.382443e-01 812.8970
5 S.time+group 17 898.6955 2.213072 2.441392e-01 808.7318
4 S.time_season 13 904.0432 7.560774 1.684221e-02 822.5674
3 S.time 16 910.2057 13.723282 7.730852e-04 822.3763
2 S.group 13 924.1806 27.698154 7.138755e-07 842.7048
1 S.dot 12 925.0823 28.599837 4.548042e-07 845.7080
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Real Function Parameters of { S.time_season+group }
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
------------------------- -------------- -------------- -------------- --------------
1:S gEarly c1 a0 t1 0.4856040 0.0679880 0.3563899 0.6167717
2:S gEarly c1 a2.4 t3. 0.9741229 0.0321746 0.7551844 0.9978279
3:S gMedium c1 a0 t1 0.7316598 0.0469022 0.6306101 0.8132526
4:S gMedium c1 a2.4 t3 0.9908865 0.0117343 0.8949186 0.9992801
5:Gamma'' gEarly c1 a0 0.6825531E-010 0.5898824E-007 -.1155487E-006 0.1156852E-006
As you can see from the top model results there is a large difference bewteen within and between season survival (which kind of makes sense because they are probably completely inactive and underground over winter), as well as a difference in S between groups. I feel that if I ignore these differences and just go with S(group) (the best model with a Gamma estimate), then my estimates won't be accurate. But the time dependent estimates can't be accurate without an estimate of Gamma.
I understand that the huge number of parameters in RD models combined with a limited amount of data means that I can't do everything I might want to do. However, I was wondering if perhaps there is something else going on here, rather than just sparseness of data?
S~time_season has the same number of parameters as S~group, so why does S~time_season cause Gamma to be not estimated properly?
Any advice will be much appreciated as always,
Annabel.