Slightly OT-AIC Differences and Parameter Estimates

[bacollier]

All,

This is slightly off-topic but I have been looking for an answer for a few days with no luck. I am probably looking in the wrong spot in the model selection literature, so if someone has a paper/book recommendation I would appreciate it.

I am estimating breeding season daily survival for turkeys over 117 days using ragged telemetry (nest survival) approach (~300 hens). The best AIC model indicates that DSR differs daily based on the number of days (at that day) hen has been on nest (following Dinsmore et al. 2002 example for nest age). DSR estimate for all 117 days is 0.998x (x=4th significant digit--is all that varies). Basically a flat line.

I also looked at a model which assumes constant DSR over time and got an DSR estimate for the 117 days also of 0.998x; fair enough, also basically a flat line. But, this model is >8 delta AIC units from the best model?

I was surprised that these two models, differing by 1 parameter, while having basically the same parameter estimates (at least out to the 4th significant digit) would have such difference in AIC? Any thoughts as to why? I assume it has something to do with how the likelihood function is optimized, but I cannot find enough evidence to satisfy myself that is the case.

Thanks
Bret

[bacollier]

All,
Ok, after an email exchange with Jay Rotella, we have determine my mistake. I was interpreting how to use of the beta estimates incorrectly.

With his permission, I am posting his comments regarding my earlier post here (without some of our earlier discussion) just in case someone else is dealing with the same issue.

(Quick note: The parameter of interest (Days on a Nest) is set up like Dinsmore et al. 2002. As nesting is not constant (e.g., renests occurred so there could be breaks between nesting attempts), I could not use 'add' function in MARK, so I had to set up each day as its own individual covariate)

From Jay:

Ok - if I look at the beta's for the 'days on nest' model, I see
LOGIT Link Function Parameters of {DSR (DN)--Days on
Nest-Unstandardized Covariate}
95%
Confidence Interval
Parameter Beta Standard Error Lower
Upper
------------------------- -------------- --------------
-------------- --------------
1: 6.6205449 0.2557944 6.1191880
7.1219019
2: -0.0589883 0.0146658 -0.0877333
-0.0302432

So, the effect of spending time on a nest is negative and you see that
the CI doesn't include 0 so things are estimated with pretty good
precision.

So, for that model you would calculate the DSR for a female as a
function of days on nest using:

DSR(dn) = exp(6.6205449-0.0589883*dn)/(1+ exp(6.6205449-0.0589883*dn))
Where 'dn' is the number of days on the nest.

So you'd get the following dsr estimates and monthly survival rates
(DSR30) for birds that had spent 0 to 25 days on nests.

dn DSR DSR30
[1,] 0 0.9986691 0.9608331
[2,] 1 0.9985883 0.9585049
[3,] 2 0.9985027 0.9560417
[4,] 3 0.9984118 0.9534358
[5,] 4 0.9983155 0.9506797
[6,] 5 0.9982133 0.9477651
[7,] 6 0.9981050 0.9446835
[8,] 7 0.9979900 0.9414260
[9,] 8 0.9978682 0.9379832
[10,] 9 0.9977389 0.9343454
[11,] 10 0.9976019 0.9305025
[12,] 11 0.9974565 0.9264439
[13,] 12 0.9973024 0.9221588
[14,] 13 0.9971389 0.9176356
[15,] 14 0.9969656 0.9128627
[16,] 15 0.9967818 0.9078277
[17,] 16 0.9965870 0.9025182
[18,] 17 0.9963803 0.8969211
[19,] 18 0.9961612 0.8910231
[20,] 19 0.9959289 0.8848105
[21,] 20 0.9956826 0.8782694
[22,] 21 0.9954215 0.8713853
[23,] 22 0.9951446 0.8641439
[24,] 23 0.9948511 0.8565304
[25,] 24 0.9945400 0.8485301
[26,] 25 0.9942101 0.8401279

So, you do get different estimates of DSR depending on your nesting
effort. Those differences aren't all that large on a daily basis but
such differences amount to important differences as DSRs are multiplied
together over time.

MARK outputs estimates of DSR for each day based on the AVERAGE value of DN1, DN2, ..., DN118. If you look at those in your output you'll see that the averages were all quite low in your dataset and often 0. Thus,
in the MARK output the DSR values aren't going to vary much (i.e.,
they're all using similar values of 'dn'). But, those aren't what you
really care about. You care about the actual range of values for
individuals, i.e., how many days did you observe birds spending, which
is 0, 1, 2, ..., max.

So, MARK is finding a beta for 'dn' that's negative and so that model is
much better supported than a model that ignores 'dn'.
To get the DSR's for various values of 'dn' you need to apply the beta's
to various 'dn' values as I did above.

Sorry for flooding folks inbox's,

Bret