Transcript SSC2002 - Department of Mathematics

Another look at Adjusting Radio-Telemetry Data for Tag-Failure
L. Cowen and C.J. Schwarz
Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC
Motivation
Mark-recapture theory studies a cohort of marked individuals that are
recaptured at a later time and/or space. Animals are marked with unique tags
allowing the estimation of both survival and capture rates.
Historically, mark-recapture studies involving fish have used PIT-tags. These
tags have a low recapture rate, thus large sample sizes are needed to get robust
survival estimates. For animals that are listed as endangered, large sample
sizes are not ideal. With the advent of radio-telemetry, this problem is
somewhat alleviated.
Table 1. Capture histories and associated expected probabilities of the two
release groups.
Release
Group
Capture
Expected Probability
History
R1
11
DP S (t ) p2
n11,t 2 
1  n11,t1
1 p2 

 ˆ

ˆ
N  S (t1 )
S (t 2 ) 

DP S (t )1  p2   DP 1  S (t )1  p2   D 1  P   1  D 
10
R2
First we make the assumption that the support for this integral consists of the
travel times that we have observed for the fish (i.e. the ti’s). Thus our integral
becomes a summation as we discretise the problem. Our estimator of 1p2
becomes (5).
01
P S (t ) p2
00
P S (t )1  p2   P 1  S (t )1  p2   1  P 
This estimate can intuitively be thought of as inflating each observed history
by the survival probability of the radio-tags.
In radio-telemetry studies, small radio-transmitters are attached to the animal.
Associated with each radio-tag is a unique radio frequency.
The major problem with the use of radio tags is their reliance on battery
power. Each radio-tag requires a battery and failure of the battery before the
end of the study can negatively bias survival estimates. If information is
available on the life of the radio-tags used in the study, a tag-failure curve can
be developed. Given the tag-failure curve, adjustments can be made to the
known detections to account for the proportion of tags that could not be
detected because a proportion of the tags were no longer active. Thus our
objective is to develop a method to adjust survival estimates given radio-tag
failure-time data.
To illustrate the method, consider the simple study of having only 2 dams.
Fish are released above the first dam (release group 1) and below the first dam
(release group 2). They are later recaptured using radio antennas before the
second dam. Since we are interested in the time of failure of the radio-tag, we
must keep track of the time it takes each fish to travel from dam 1 to dam 2.
To simplify the problem even more we only allow for 2 possible travel times
for the fish, from dam 1 to dam2 (although this is not the case in the study).
In order to obtain the radio-tag failure curve, tags are placed in a tank. The
time of radio failure for each tag is recorded and a radio-failure curve can then
be constructed.
Notation
n11,Ri= the number of fish released at dam 1, recaptured at dam 2 from release
group Ri.
n10,Ri= the number of fish released at dam 1, not recaptured at dam 2 from
release group Ri.
We can now estimate dam survival by taking the ratio of the parameters for
each release group (1) and then substitution tin the maximum likelihood
estimator (2).
R1 = release group 1.
R2 = release group 2.
DR p2
D 
 R p2
(1)


n11, R1


 n11, R  n10, R 

1
1 
D  


n11, R2


 n11, R  n10, R 
2
2 

(2)
L(1 , p2 )   1 p2 g (ti ) S ti n11,ti 1  1 p2 g (ti ) S ti n10
n10 = the number of fish released at dam 1, not recaptured at dam 2.
For illustration, we only use two time periods for traveling from dam 1 to dam
2: t1 and t2. Due to identifiability problems we cannot get an estimate for 1
on its own. However we can get an estimate for 1p2 as shown in (4).
1 p2 
n11,t1  n11,t 2
(4)
N  g (u ) S (u ) du
g(ti) = travel time distribution for the fish from dam 1 to dam 2.
The difficulty here is in the estimation of
(6)
Case Study
Radios
Fish Released
(R1/R2)
Before
Adjustment 1
Adjustment
Adjustment 2
19
360/324
0.68210
1.05763
1.33382
When making an adjustment for radio-failure, travel time of the fish must be
taken into consideration. This is more evident in the 3 dam case (which I am
still working on).
As we now have a survival estimate, we have to get a variance estimate. This
will be done using bootstrapping.
References
Lebreton, J-D., Burnham, K.P., Clobert, J. and Anderson, D.R. (1992).
Modeling Survival and testing biological hypotheses using marked animals: a
unified approach with case studies. Ecological Monographs, 62(1): 67-118.
1 = Pr(survival from dam 1 to dam 2).
p2 = Pr(recapture at dam 2).
(3)
i


Future Work
ti = the travel time of the fish from dam 1 to dam 2, i=1,2,…; 0ti.
n11,ti = the number of fish released at dam 1, traveled to dam 2 in ti time and
were recaptured at dam 2, i=1,2.

N r  nti  1
ˆ
S (ti ) 
N r  1
Using data for both fish travel times and radio-tag failure times for the 2001
season we obtained estimates for 1) unadjusted dam survival, 2) adjusted dam
survival, and 3) adjusted dam survival adjusting radio survival first.
We model this type of data assuming a multinomial distribution. The likelihood
is given in equation (3).
1 = Pr(survival from dam 1 to dam 2).
p2 = Pr(recapture at dam 2).
Problems arise when the travel time of a fish is large so that the survival
probability of the radio is very small and the adjusted counts explode. To
account for this we add 1 to the numerator and the denominator of the radiotag survival estimate. Motivation for this comes from the adjusted Petersen
estimate where adding 1 is found to reduce bias. (Note that we are using the
Product Limit survival estimate.)
If we have 2 release groups, 1 before dam1 and 1 after dam1 we can estimate
dam survival. 1 for group1, released above the dam would be made up of 2
components: survival through the dam, D and survival between dam 1 and
dam 2, R. In the second release group, released after the dam, survival is
only that between dam 1 and dam 2, R.
Capture histories are represented by a series of 1’s and 0’s. A ‘1’ signifies a
recapture and a ‘0’ signifies a non-recapture. Table 1 describes the possible
outcomes and capture histories for a fish in the radio-telemetry study. Note
that table 1 does not take into consideration the travel time of the fish.
(5)
 g (u ) S (u )du.
English, K.K., Skalski, J.R., Lady, J., Koski, W.R., Nass, B.L., and
Sliwinski, C. An assessment of project, pool and dam survival for run-of-river
steelhead smolts at Wanapum and Priest Rapids projects using radio-telemetry
techniques, 2000. Public Utility District No. 2 of Grant County, Washington,
draft report.
Cowen and Schwarz. Adjusting survival for premature radio-tag failure.
SSC, Burnaby BC, 2001.
S(ti) = survival distribution of the radio-tag.
Nr= the number of initial radio-tags.
nr,ti= the number of radio tags that have failed by time ti.
N   n11,t n10
t i
i
= the total number of fish released.
Acknowledgements
We would like to thank Karl English of LGL Limited for providing the data
for the radio-tag failure curve.