time intervals in CJS simulator - bad design or bug?

[bp]

Hi,
I'm using MARK to assess different sampling designs to estimate survival under a CJS framework. Our sampling intervals are not equal so I am specifying the time intervals to make the survival rates for each of the intervals comparable. In our case we may sample about 4 consecutive months, then not sample for 9 months, then 4 additional months. So I am inputting time intervals of 0.08 between the consecutive monthly samples then a time interval of 0.75 for the 9 month interval. No big deal. I understand that the time intervals are used as the exponent for the survival estimate for the time interval. However, when I run these types of simulations, the phi estimates for the small time intervals are about 0.24 (true value is 0.85 annual) and the larger time interval is about 0.82 (true time interval, pretty good estimate). When I try and alternative sampling design of equal time intervals quarterly, and specify a time interval of 0.25, the phi estimates are about 0.6. These are all with a capture probability of 0.15 (low yes, but I work with fish, always low) and inputs of 500 animals each interval run for about 100 simulations over 24 intervals. I've also tried this with higher capture probabilities and it doesn't improve very much. I'm curious if the time interval correction is working for these small time intervals....so I set up a little test below with higher p-hat...

So here are results from a simple example with a phat=0.5 and Phi =0.85, 4 events, 500 animals released each time. True model is Phi(t) p(t) estimated model is the same. The first result keeps time intervals =1. The second has time intervals 1, 1, 0.5. The third one has time intervals 1, 0.5, 1. The first simulation isn't that interesting, the third phi isn't a good estimate because it is the last event. The second simulation is not very interesting either. The third one is (1, 0.5, 1) because real estimate 2 (phi events 2 to 3) is much lower than it was for the second simulation. Is this a bug in the time interval correction in the Phi or is the effect of changing the sampling interval just that great? Thanks very much for your time



###################################################

Time interval 1, 1, 1
True Phi = 0.85
True phat = 0.5
500 released per time
Phi(t),p(t) model (real and estimated)


Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.911 0.0723 0.0229 0.866 0.956
REAL2 0.811 0.1146 0.0363 0.740 0.882
REAL3 0.661 0.0550 0.0174 0.627 0.695
REAL4 0.493 0.0917 0.0290 0.436 0.550
REAL5 0.506 0.0688 0.0218 0.464 0.549
REAL6 0.661 0.0550 0.0174 0.627 0.695



###################################################




Time interval 1, 1, 0.5
True Phi = 0.85
True phat= 0.5
500 released per time
Phi(t),p(t) model (real and estimated)

Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.855 0.0828 0.0262 0.804 0.907
REAL2 0.868 0.0674 0.0213 0.826 0.910
REAL3 0.726 0.1230 0.0389 0.649 0.802
REAL4 0.505 0.0788 0.0249 0.456 0.554
REAL5 0.513 0.1175 0.0372 0.440 0.586
REAL6 0.512 0.0845 0.0267 0.459 0.564


###################################################


Time interval 1, 0.5, 1
True Phi = 0.85
True phat = 0.5
500 released per time
Phi(t),p(t) model (real and estimated)

Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.875 0.0695 0.0220 0.832 0.918
REAL2 0.660 0.2272 0.0718 0.519 0.801
REAL3 0.669 0.0667 0.0211 0.627 0.710
REAL4 0.495 0.0465 0.0147 0.466 0.524
REAL5 0.536 0.1214 0.0384 0.460 0.611
REAL6 0.669 0.0667 0.0211 0.627 0.710

[cooch]

Hi,
I'm using MARK to assess different sampling designs to estimate survival under a CJS framework. Our sampling intervals are not equal so I am specifying the time intervals to make the survival rates for each of the intervals comparable. In our case we may sample about 4 consecutive months, then not sample for 9 months, then 4 additional months. So I am inputting time intervals of 0.08 between the consecutive monthly samples then a time interval of 0.75 for the 9 month interval. No big deal. I understand that the time intervals are used as the exponent for the survival estimate for the time interval. However, when I run these types of simulations, the phi estimates for the small time intervals are about 0.24 (true value is 0.85 annual) and the larger time interval is about 0.82 (true time interval, pretty good estimate). When I try and alternative sampling design of equal time intervals quarterly, and specify a time interval of 0.25, the phi estimates are about 0.6. These are all with a capture probability of 0.15 (low yes, but I work with fish, always low) and inputs of 500 animals each interval run for about 100 simulations over 24 intervals. I've also tried this with higher capture probabilities and it doesn't improve very much. I'm curious if the time interval correction is working for these small time intervals....so I set up a little test below with higher p-hat...

So here are results from a simple example with a phat=0.5 and Phi =0.85, 4 events, 500 animals released each time. True model is Phi(t) p(t) estimated model is the same. The first result keeps time intervals =1. The second has time intervals 1, 1, 0.5. The third one has time intervals 1, 0.5, 1. The first simulation isn't that interesting, the third phi isn't a good estimate because it is the last event. The second simulation is not very interesting either. The third one is (1, 0.5, 1) because real estimate 2 (phi events 2 to 3) is much lower than it was for the second simulation. Is this a bug in the time interval correction in the Phi or is the effect of changing the sampling interval just that great? Thanks very much for your time



###################################################

Time interval 1, 1, 1
True Phi = 0.85
True phat = 0.5
500 released per time
Phi(t),p(t) model (real and estimated)


Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.911 0.0723 0.0229 0.866 0.956
REAL2 0.811 0.1146 0.0363 0.740 0.882
REAL3 0.661 0.0550 0.0174 0.627 0.695
REAL4 0.493 0.0917 0.0290 0.436 0.550
REAL5 0.506 0.0688 0.0218 0.464 0.549
REAL6 0.661 0.0550 0.0174 0.627 0.695



###################################################




Time interval 1, 1, 0.5
True Phi = 0.85
True phat= 0.5
500 released per time
Phi(t),p(t) model (real and estimated)

Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.855 0.0828 0.0262 0.804 0.907
REAL2 0.868 0.0674 0.0213 0.826 0.910
REAL3 0.726 0.1230 0.0389 0.649 0.802
REAL4 0.505 0.0788 0.0249 0.456 0.554
REAL5 0.513 0.1175 0.0372 0.440 0.586
REAL6 0.512 0.0845 0.0267 0.459 0.564


###################################################


Time interval 1, 0.5, 1
True Phi = 0.85
True phat = 0.5
500 released per time
Phi(t),p(t) model (real and estimated)

Statistical Summary of Numerical Variables
(Number of Observations = 10)

95% Confidence Interval
Variable Mean Standard Dev. Standard Error Lower Upper
---------- --------------- -------------- -------------- --------------- ---------------
REAL1 0.875 0.0695 0.0220 0.832 0.918
REAL2 0.660 0.2272 0.0718 0.519 0.801
REAL3 0.669 0.0667 0.0211 0.627 0.710
REAL4 0.495 0.0465 0.0147 0.466 0.524
REAL5 0.536 0.1214 0.0384 0.460 0.611
REAL6 0.669 0.0667 0.0211 0.627 0.710

I suspect you don't understand (and thus made a mistake) simulating intervals. (Quick aside -- doing simulations with only 3 occasions is not a good idea, unless this really matches your sampling situation. Final phi and p are confounded, meaning not many free parameters to explore. So, in the following, I use 5 occasions (4 intervals)). True model phi(t)p(.).I'll let my sampling intervals are[1,1,0.5,1]. If true *annual* survival is 0.85 (let p=0.5), then this is what will happen, depending on how I specify the simulation:

1\ if I enter [0.85,0.85,0.85,0.85,0.5] for the true betas (identity link), with intervals [1,1,0.5,1], then MARK will (correctly) generate estimates ~[0.85,0.85,0.7225,0.85]. Where does the 0.7225 come from? Easy, its 0.85^2, which is exactly what it should be. By specifying the interval of 0.5, you are telling MARK you want the estimate for 'half the interval'. You've told mark that the true value for this half-interval is 0.85, so MARK reports the square of this as 0.7225.

2\ If the true estimate of annual survival over that interval is in fact 0.85, then the half-interval true value should be sqrt(0.85)=0.922. So, if you use [0.85,0.85,0.922,0.85,0.5] for the true betas, with intervals [1,1,0.5,1], then MARK will (correctly) generate estimates ~[0.85,0.85,0.85,0.85,0.5].

3\ where the unequal interval falls won't affect what happens (as you OP implies). For example, if I use [1,1,1,0.5], an a beta vector of [0.85,0.85,0.85,0.922,0.5], MARK reports real ~[0.85,0.85,0.85,0.85,0.5], as expected.

Remember, MARK will try to report estimates as if the intervals were even. As such, you need to be careful about what you enter as the true beta parameters when simulating with uneven intervals. MARK is doing things correctly -- what is probably incorrect is what you've entered for the betas (incorrect, given where your interest may lie). As a test, what do you think will happen if instead of [1,1,0,5,1] I had used [1,1,2,1]? I'll leave that to you.

[bp]

Thank you very much for the reply. I did not specify my Beta values correctly for the time intervals. Have a good day.