Limit to unevenness of unequal time intervals?

[khone]

Hi all,

I'm wondering if there is a statistical or theoretical limit to the unevenness of unequal time intervals in the CJS model in MARK?

My motivation for this question follows:
I am working with a CMR data set with 6 capture sessions separated by unequal time intervals. These data are for an amphibian. My goal is to estimate apparent survival (from here on just "survival" or "phi") using the CJS model.

The capture sessions took place over two consecutive summers with 3 sessions in each summer. Therefore, the intervals between sessions within summers are considerably different in length from the interval between summers. The intervals (specified in years) are 0.06, 0.05, 0.88, 0.06, 0.05 ,
where 0.88 is the interval between the 3rd session in summer 1 and the 1st session in summer 2.

Below I show results accounting for unequal time intervals ("Model.1"), and then, not accounting for unequal time intervals ("Model.2"):

"Model.1"
From a phi(t)p(t(with p4 = p5)) model accounting for the unequal time intervals,
where I entered the time intervals in the "set time intervals" box.
(I used the profile likelihood method for computation of the CIs becasue so close to 0)

Parameter, Estimate, Standard Err, Lower, Upper
----------------------- ----------- ----------------------
1:Phi 0.0129691, 0.0231837, 0.000341, 0.4032208
2:Phi 0.0099307, 0.0172054, 0.000332, 0.3128192
3:Phi 0.6231185, 0.0642324, 0.5080025, 0.7620658
4:Phi 0.0000434, 0.0000735, 0.0000016, 0.0012062
5:Phi 0.0000613, 0.0001187, 0.0000014, 0.0028809
6:p 0.2690342, 0.0504200, 0.1821201, 0.3782442
7:p 0.2883755, 0.0335182, 0.2273355, 0.3582067
8:p 0.3377836, 0.0344027, 0.2739579, 0.4081206
9:p 0.3812565, 0.0375178, 0.3108949, 0.4569822

The phi estimates for the intervals during the summer are MUCH lower than the phi estimated for the winter period.

"Model.2"
To test the math and explore what is going on I ran the same model without accounting for unequal time intervals
(I falsely assume the time intervals are equal by entering 1 in each "set time intervals" box)

Parameter Estimate Standard Err Lower Upper
---------- ------------ ------------ ---------- --------------
1:Phi 0.7585744, 0.0862316, 0.5552753, 0.8877290
2:Phi 0.7930590, 0.0690705, 0.6268140, 0.8973716
3:Phi 0.6585982, 0.0599425, 0.5335824, 0.7648714
4:Phi 0.5756607, 0.0530745, 0.4698192, 0.6749896
5:Phi 0.6320271, 0.0579111, 0.5132146, 0.7367172
6:p 0.2690331, 0.0504196, 0.1821196, 0.3782424
7:p 0.2883763, 0.0335179, 0.2273369, 0.3582068
8:p 0.3377819, 0.0344026, 0.2739564, 0.4081188
9:p 0.3812555, 0.0375168, 0.3108957, 0.4569790

In model.2, estimates of phi are relatively similar across the intervals.

A check of the math shows things are working as they should:
model.2.phi.1 ^ (1/interval.1) = model.1.phi.1 or ( 0.758^(1/0.0635) = 0.012969)

If phi were biologically similar between seasons, in model.2(when assuming intervals are equal) I would expect that reported phi estimates for intervals during summer would be higher (because of shorter true time periods)than phi estimates for the winter period.

In model.1, if phi were biologically similar between seasons, we would expect all calculated phi's to be similar.

Surely it is not unreasonable there is a seasonal effect here.
Test 3 in U-CARE does indicate this study "population" includes many transients,
which likely drives some of this result. There could also be other biological factors driving phi lower in summer, such as predation.

But the seasonal differences in model.1 seem extreme.

So, is it possible that the severe difference in length of time intervals is causing a problem for the model,
or is it possbible the seasonal effect is really very strong?

Thanks in advance!

[gstauffer]

In model.1, did you enter the intervals as 0.06, 0.05, ..., or as 1/0.06, 1/0.05, ...? If you did the latter, I suspect it might explain the unexpected results you got.

[khone]

In model.1, did you enter the intervals as 0.06, 0.05, ..., or as 1/0.06, 1/0.05, ...? If you did the latter, I suspect it might explain the unexpected results you got.

Thanks for your reply. In model.1, I entered the intervals into MARK as 0.06, 0.05, ect...

[Eurycea]

I don't know what's going on, but I would suggest coding your time intervals on a different scale, for example, where your shortest interval is equal to 1. Try that and see if you get a different result.

[khone]

I don't know what's going on, but I would suggest coding your time intervals on a different scale, for example, where your shortest interval is equal to 1. Try that and see if you get a different result.

Thanks for your suggestion Eurycea.

I took your suggestion and reran the models using the shortest time period (18.25 days) as the time period of interest. I will call this "model.3" I provided MARK the following time intervals: 1.2, 1.0, 17.6, 1.2, 1.0. The results from model.3 are shown below:

Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:Phi 0.7943233 0.0752457 0.6102731 0.9049873
2:Phi 0.7930616 0.0690703 0.6268165 0.8973736
3:Phi 0.9765495 0.0050500 0.9643228 0.9846528
4:Phi 0.6311584 0.0484934 0.5321822 0.7202036
5:Phi 0.6320260 0.0579124 0.5132108 0.7367182

These results do appear "nicer" than results from model.1. However, if I extrapolate the estimates to annual survival rates we see the results have the same patterns as in model.1 from the OP.
365/18.25 = 20 time periods in each year
> model.3.estimates = c(0.7943233 ,0.7930616 ,0.9765495 ,0.6311584 ,0.6320260 )
> model.3.estimates^20
[1] 0.0099987576, 0.0096858656, 0.6221358672, 0.0001006392, 0.0001034425

One option might be to rerun with month as the time length of interest (instead of 18 days) and report these "nicer" looking results.

However, I am still wondering if for some reason the disparity among the time intervals is causing problems in the analysis? Is there some limit to how different the time intervals can be before exponentiation of the estimate causes problems?

Thanks in advance for further insight.

[Eurycea]

So it looks like things are working as they should be, so I doubt there is any numerical funniness going on here. I could guess few possibilities for your low summer survival rates, without knowing anything about your study design or organism:

1) there is a lot of migration away from your study site
2) predation or other factors that influence survival during the summer, but are not as prevalent during the winter (as you mentioned)
3) If you captured the individuals just before and after dormancy, and they do well during dormancy (e.g., low exposure to predators, they aren't moving), it might just be way better than summer months.

I'm sure there are other explanations.

How does your model fit?

[khone]

So it looks like things are working as they should be, so I doubt there is any numerical funniness going on here. I could guess few possibilities for your low summer survival rates, without knowing anything about your study design or organism:

1) there is a lot of migration away from your study site
2) predation or other factors that influence survival during the summer, but are not as prevalent during the winter (as you mentioned)
3) If you captured the individuals just before and after dormancy, and they do well during dormancy (e.g., low exposure to predators, they aren't moving), it might just be way better than summer months.

I'm sure there are other explanations.

How does your model fit?

Thanks!
I think all 3 of those could explain a portion of the differences between summer and winter survival rates, but the differences still seem extreme.

As far as fit, the data are definitely overdispersed (from results in Release); note I did not adjust c-hat for the results I provided in models 1 through 3. Results from U-CARE do indicate a high level of transience in the "population" which I assume is a major cause of the overdispersion. The U-CARE results also validate migration from the study area as a partial cause of low summer survival.

I have a follow-up question for this post:
Considering how different the survival rates between summer and winter are, would it be invalid/erroneous to describe the system using a model with constant survival over time?

[Eurycea]

As far as fit, the data are definitely overdispersed (from results in Release); note I did not adjust c-hat for the results I provided in models 1 through 3. Results from U-CARE do indicate a high level of transience in the "population" which I assume is a major cause of the overdispersion. The U-CARE results also validate migration from the study area as a partial cause of low summer survival.

That makes sense. What is the animal/life stage?

I have a follow-up question for this post:
Considering how different the survival rates between summer and winter are, would it be invalid/erroneous to describe the system using a model with constant survival over time?

Well the constant phi model would probably not perform very well, in which case you probably would not be able to make as good an inference from it as the time-specific model. You'd probably end up with a very wide confidence interval on phi, making the estimate not very reliable anyway. Because you have different seasons, you could also run a phi(season) model and see if that performs better than a phi(time) model. What is your goal? If it is just to estimate survival, you should model-average. But what your data are telling you with regard to seasonal variation is probably the more interesting story here.

[khone]

Thanks for your continued help here. This analysis is for juveniles and adults of a stream frog. The animals are thought to live for 5+ years or so after metamorphosing. My original goal was to report an annual estimate of survival. Good point about the effect of the seasonal variation on confidence intervals in a time constant model. I agree the seasonal variation is interesting; I'll work to include that information in the results.