Using Pradel models when p=0 for a capture occasion

[jbauder]

I am running some Pradel models with the intention of estimate lambda for three different populations. I have 17 capture occasions for each population. I am running the analyses separately for each population and I would like to report a "mean" lambda for each population. Since the model with phi(.)p(time)lambda(time) was the best supported model (w > 0.80), my plan was to use a variance component approach to fit a means model on the real parameter estimates for lambda to estimate a mean lambda and sigma-hat. However, I have noticed some "wonky" estimates of lambda. I know that in a fully time-dependent model, the first and last lambda are confounded and that the lambda for the second time period may also be biased. However, I have what appear to be abnormal estimates of lambda for other time periods that should otherwise be valid. These "wonky" estimates are all related to capture occasions where p was estimated to be zero (because we had no recaptures on those occasions). Each den has at least one interval where this was the case. The corresponding lambda for that interval and the interval immediately following it appear suspect.
For example (this output is for my phi(.)p(time)lambda(time) model):
In 1997, we had no recaptures. The lambda for 1996 and 1997 are clearly suspect and I do not trust the lambda for 1998. Why would p=0 cause lambda estimates like these?
Similarly, p in 2007 is 0.01 and lambda for 2007 is very high with a large SE (6.58, 3.71) and lambda for 2008 is also suspect (0.32, 0.18).
How should I treat these estimates when using variance-components to calculate an overall mean for lambda?

Thanks
Javan

1995:p 0.6252162 5.1421408 0.3486598E-018 1.0000000
1996:p 0.0433507 0.0301100 0.0108021 0.1582812
1997:p 0.8018377E-007 0.0000000 0.8018377E-007 0.8018377E-007
1998:p 0.0148198 0.0147589 0.0020699 0.0983632
1999:p 0.2122208 0.0483711 0.1325408 0.3220201
2000:p 0.2885619 0.0374596 0.2209711 0.3670853
2001:p 0.1983199 0.0267794 0.1509766 0.2560321
2002:p 0.0564701 0.0140461 0.0344690 0.0911878
2003:p 0.1368769 0.0234025 0.0971123 0.1895063
2004:p 0.2224641 0.0309522 0.1676899 0.2889192
2005:p 0.0537839 0.0136666 0.0324883 0.0877720
2006:p 0.1328863 0.0247773 0.0913605 0.1893532
2007:p 0.0134931 0.0078637 0.0042784 0.0417230
2008:p 0.1194101 0.0260611 0.0770016 0.1806049
2009:p 0.0377810 0.0129431 0.0191661 0.0731278
2010:p 0.0182898 0.0092945 0.0067093 0.0488755
2011:p 0.0672391 0.0199292 0.0372300 0.1184608
1995:Lambda 10.877728 89.778753 0.1925909 614.38508
1996:Lambda 226395.46 0.0000000 226395.46 226395.46
1997:Lambda 0.1083449E-004 0.0000000 0.1083449E-004 0.1083449E-004
1998:Lambda 0.2866825 0.2865032 0.0561095 1.4647589
1999:Lambda 0.9629792 0.2235598 0.6145975 1.5088397
2000:Lambda 1.1369747 0.1709328 0.8481901 1.5240823
2001:Lambda 1.1854443 0.2701869 0.7626450 1.8426372
2002:Lambda 0.8089045 0.1930962 0.5099283 1.2831734
2003:Lambda 1.0411904 0.1673752 0.7613173 1.4239497
2004:Lambda 0.6346401 0.1132976 0.4484936 0.8980464
2005:Lambda 0.9310476 0.1755247 0.6454999 1.3429122
2006:Lambda 6.5775686 3.7145685 2.3453746 18.446694
2007:Lambda 0.3156881 0.1820284 0.1104611 0.9022090
2008:Lambda 1.2144282 0.4093744 0.6384370 2.3100728
2009:Lambda 1.2499296 0.6781218 0.4618726 3.3825861
2010:Lambda 0.5226802 0.2676529 0.2029846 1.3458881

[cooch]

I am running some Pradel models with the intention of estimate lambda for three different populations. I have 17 capture occasions for each population. I am running the analyses separately for each population and I would like to report a "mean" lambda for each population. Since the model with phi(.)p(time)lambda(time) was the best supported model (w > 0.80), my plan was to use a variance component approach to fit a means model on the real parameter estimates for lambda to estimate a mean lambda and sigma-hat. However, I have noticed some "wonky" estimates of lambda. I know that in a fully time-dependent model, the first and last lambda are confounded and that the lambda for the second time period may also be biased. However, I have what appear to be abnormal estimates of lambda for other time periods that should otherwise be valid. These "wonky" estimates are all related to capture occasions where p was estimated to be zero (because we had no recaptures on those occasions). Each den has at least one interval where this was the case. The corresponding lambda for that interval and the interval immediately following it appear suspect.
For example (this output is for my phi(.)p(time)lambda(time) model):
In 1997, we had no recaptures. The lambda for 1996 and 1997 are clearly suspect and I do not trust the lambda for 1998. Why would p=0 cause lambda estimates like these?
Similarly, p in 2007 is 0.01 and lambda for 2007 is very high with a large SE (6.58, 3.71) and lambda for 2008 is also suspect (0.32, 0.18).
How should I treat these estimates when using variance-components to calculate an overall mean for lambda?

Thanks
Javan

1995:p 0.6252162 5.1421408 0.3486598E-018 1.0000000
1996:p 0.0433507 0.0301100 0.0108021 0.1582812
1997:p 0.8018377E-007 0.0000000 0.8018377E-007 0.8018377E-007
1998:p 0.0148198 0.0147589 0.0020699 0.0983632
1999:p 0.2122208 0.0483711 0.1325408 0.3220201
2000:p 0.2885619 0.0374596 0.2209711 0.3670853
2001:p 0.1983199 0.0267794 0.1509766 0.2560321
2002:p 0.0564701 0.0140461 0.0344690 0.0911878
2003:p 0.1368769 0.0234025 0.0971123 0.1895063
2004:p 0.2224641 0.0309522 0.1676899 0.2889192
2005:p 0.0537839 0.0136666 0.0324883 0.0877720
2006:p 0.1328863 0.0247773 0.0913605 0.1893532
2007:p 0.0134931 0.0078637 0.0042784 0.0417230
2008:p 0.1194101 0.0260611 0.0770016 0.1806049
2009:p 0.0377810 0.0129431 0.0191661 0.0731278
2010:p 0.0182898 0.0092945 0.0067093 0.0488755
2011:p 0.0672391 0.0199292 0.0372300 0.1184608
1995:Lambda 10.877728 89.778753 0.1925909 614.38508
1996:Lambda 226395.46 0.0000000 226395.46 226395.46
1997:Lambda 0.1083449E-004 0.0000000 0.1083449E-004 0.1083449E-004
1998:Lambda 0.2866825 0.2865032 0.0561095 1.4647589
1999:Lambda 0.9629792 0.2235598 0.6145975 1.5088397
2000:Lambda 1.1369747 0.1709328 0.8481901 1.5240823
2001:Lambda 1.1854443 0.2701869 0.7626450 1.8426372
2002:Lambda 0.8089045 0.1930962 0.5099283 1.2831734
2003:Lambda 1.0411904 0.1673752 0.7613173 1.4239497
2004:Lambda 0.6346401 0.1132976 0.4484936 0.8980464
2005:Lambda 0.9310476 0.1755247 0.6454999 1.3429122
2006:Lambda 6.5775686 3.7145685 2.3453746 18.446694
2007:Lambda 0.3156881 0.1820284 0.1104611 0.9022090
2008:Lambda 1.2144282 0.4093744 0.6384370 2.3100728
2009:Lambda 1.2499296 0.6781218 0.4618726 3.3825861
2010:Lambda 0.5226802 0.2676529 0.2029846 1.3458881

Simple. If p=0 for a particularly sampling occasion, then you are, in effect, estimating a parameter (S, lambda) over an interval of 2 years. You're problem is strictly analogous to manually changing the interval between occasions (MARK defaults to interval = 1). So, if true lambda for successive years is 1.05 and 1.02, then if you set things up correctly, MARK will report square-root of product of (1.05 * 1.02) as its estimate of lambda for each of those two years. With a bit of work, you might be able to figure out how to set things up correctly -- the key is to think about the appropriate interval.

However, if you have unequal intervals (as implicit from p -> 0), you shouldn't use random effects. See point (2), page 44 of Appendix D.